Mathematics Applied to Engineering, Modelling, and Social Issues [1st ed.] 978-3-030-12231-7, 978-3-030-12232-4

This book presents several aspects of research on mathematics that have significant applications in engineering, modelli

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Mathematics Applied to Engineering, Modelling, and Social Issues [1st ed.]
 978-3-030-12231-7, 978-3-030-12232-4

Table of contents :
Front Matter ....Pages i-xv
On the Reynolds Equation and the Load Problem in Lubrication: Literature Review and Mathematical Modelling (Hassán Lombera Rodríguez, J. Ignacio Tello)....Pages 1-43
On Dynamic Interactions Between Body Motion and Fluid Motion (Frank T. Smith, Samire Balta, Kevin Liu, Edward R. Johnson)....Pages 45-89
Certain Aspects of Problems with Non Homogeneous Reactions (Alejandro Omón Arancibia)....Pages 91-119
A Survey on the Melnikov Theory for Implicit Ordinary Differential Equations with Applications to RLC Circuits (Michal Fečkan)....Pages 121-160
Numerical Solution of Space-Time-Fractional Reaction-Diffusion Equations via the Caputo and Riesz Derivatives (Kolade M. Owolabi, Hemen Dutta)....Pages 161-188
An Extended Langhaar’s Solution for Two-Dimensional Entry Microchannel Flows with High-Order Slip (R. Rasooli, B. Çetin)....Pages 189-212
Dynamics of Solitons in High-Order Nonlinear Schrödinger Equations in Fiber Optics (Gholam-Ali Zakeri)....Pages 213-243
MHD Mass Transfer Flow Past an Impulsively Started Semi-Infinite Vertical Plate with Soret Effect and Ramped Wall Temperature (N. Ahmed)....Pages 245-279
Secure Communication Systems Based on the Synchronization of Chaotic Systems (Samir Bendoukha, Salem Abdelmalek, Adel Ouannas)....Pages 281-311
Numerical Techniques for Fractional Competition Dynamics with Power-, Exponential- and Mittag-Leffler Laws (Kolade M. Owolabi, Hemen Dutta)....Pages 313-332
Existence of Periodic Solutions for First Order Differential Equations with Applications (Smita Pati)....Pages 333-361
Dynamic Programming Viscosity Solution Approach and Its Applications to Optimal Control Problems (Bing Sun, Zhen-Zhen Tao, Yang-Yang Wang)....Pages 363-420
A Simple Model of Periodic Reproduction: Selection of Prime Periods (Raul Abreu de Assis, Mazílio Coronel Malavazi)....Pages 421-438
Transport on Networks—A Playground of Continuous and Discrete Mathematics in Population Dynamics (Jacek Banasiak, Aleksandra Puchalska)....Pages 439-487
Augmenting and Decreasing Systems (J. M. Bilbao, A. Jiménez-Losada, M. Ordóñez)....Pages 489-528
Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases (Khalid Hattaf, Noura Yousfi)....Pages 529-549
Approximation of Short-Run Equilibrium of the N-Region Core-Periphery Model in an Urban Setting (Minoru Tabata, Nobuoki Eshima)....Pages 551-567
New Phase-Field Models with Applications to Materials Genome Initiative (Peicheng Zhu, Yangxin Tang, Yeping Li)....Pages 569-598
Optimal Control Measures for Tuberculosis in a Population Affected with Insurgency (A. O. Egonmwan, D. Okuonghae)....Pages 599-627
Insurance Model to Estimate the Financial Risk Due to Direct Medical Cost on Dengue Outbreaks (S. S. N. Perera)....Pages 629-663
Dynamics of Zika Virus Epidemic in Random Environment (Yusuke Asai, Xiaoying Han, Peter E. Kloeden)....Pages 665-684
Incidence Graph Models for the Analysis of Active Illegal Immigration Routes and Human Loss (Sunil Mathew, John N. Mordeson)....Pages 685-699

Citation preview

Studies in Systems, Decision and Control 200 2 0 0 t h Vo l u m e o f S S D C · 2 0 0 t h Vo l u m e o f S S D C · 2 0 0 t h Vo l u m e o f S S D C

Frank T. Smith Hemen Dutta John N. Mordeson Editors

Mathematics Applied to Engineering, Modelling, and Social Issues

Studies in Systems, Decision and Control Volume 200

Series editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. ** Indexing: The books of this series are submitted to ISI, SCOPUS, DBLP, Ulrichs, MathSciNet, Current Mathematical Publications, Mathematical Reviews, Zentralblatt Math: MetaPress and Springerlink.

More information about this series at http://www.springer.com/series/13304

Frank T. Smith Hemen Dutta John N. Mordeson •



Editors

Mathematics Applied to Engineering, Modelling, and Social Issues

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Editors Frank T. Smith Department of Mathematics University College London London, UK

Hemen Dutta Department of Mathematics Gauhati University Guwahati, Assam, India

John N. Mordeson Department of Mathematics Center for Mathematics of Uncertainty Creighton University Omaha, NE, USA

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-12231-7 ISBN 978-3-030-12232-4 (eBook) https://doi.org/10.1007/978-3-030-12232-4 Library of Congress Control Number: 2018968112 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book contains several aspects of current researches in mathematics having significant applications in engineering, modelling and social issues. This book is primarily designed for researchers, graduate students and educators interested in mathematics having diverse uses and applications. The book should also be useful and interesting for several other readers interested in current developments of mathematics having uses and applications in engineering science and modelling of natural phenomena including certain social issues having current research significance. There are 22 chapters in this book, and they are organized as follows: The Chap. “On the Reynolds Equation and the Load Problem in Lubrication: Literature Review and Mathematical Modelling” reviews the theory of hydrodynamic lubrication. It presents some mathematical models of lubrication with an emphasis on the load or equilibrium problems. It also presents the derivation of the most used equation in lubrication, i.e. the Reynolds equation as well as the equations of equilibrium in certain cases. Different models of cavitation and the generalized Reynolds equation are also presented. The Chap. “On Dynamic Interactions Between Body Motion and Fluid Motion” deals with dynamic fluid-body interactions, concentrating on applying mathematical/ analytical ideas to complement direct numerical studies. It also presents a review of ideas developed over the last decade for cases of high flow rates. The chapter first addresses inviscid approaches to one or more bodies free to move within a channel flow, a skimming sharp-edged body on a free surface, the sinking of a body in water and the rocking or rolling of a body on a solid surface, before moving on to more recent viscous-inviscid approaches for channel flows and boundary layers. The chapter also outlines the beginnings of certain current research projects. The Chap. “Certain Aspects of Problems with Non Homogeneous Reactions” reviews certain aspects of systems with reaction terms which are nonhomogeneous reactions. These types of reactions are frequent in problems arise in various models of physical processes, in particular, in those where the temperature is the main variable. Also, it discusses both stationary and evolution problems.

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The Chap. “A Survey on the Melnikov Theory for Implicit Ordinary Differential Equations with Applications to RLC Circuits” deals with the development of the Melnikov theory in studying implicit ordinary differential equations with small amplitude perturbations, and in particular, the persistence of orbits connecting singularities in finite time provided that certain Melnikov like conditions hold. Further, it considered the achievements on reversible implicit ordinary differential equations. Some applications to nonlinear systems of RLC circuits are also presented. The Chap. “Numerical Solution of Space-Time-Fractional Reaction-Diffusion Equations via the Caputo and Riesz Derivatives” considered the numerical solution of space-time-fractional reaction-diffusion problems used to model complex phenomena that are governed by dynamic of anomalous diffusion. The time- and space-fractional reaction-diffusion equation is modelled by replacing the first-order derivative in time and the second-order derivative in space, respectively, with the Caputo and Riesz operators. It proposed some numerical approximation schemes such as the matrix method, average central difference operator and L2 method. It applied the Laplace transform technique in time and the Fourier transform method in space to give a general two-dimensional representation of the analytical solution in terms of the Mittag-Leffler function. The proposed methods are tested for applicability on a range of practical problems. In the Chap. “An Extended Langhaar’s Solution for Two-Dimensional Entry Microchannel Flows with High-Order Slip”, Langhaar’s assumptions for the entrance region of two-dimensional micro-channels (micro-tube, slit channel and concentric annular micro-channel) have been implemented using high-order slip models. Different slip models have been used, and velocity profile, entrance length and apparent friction factor have been obtained in an integral form. The advances in micro-fabrication technology have brought numerous applications to the field of micro-scale science, and engineering and micro-channels are inseparable part of microfluidic technology. The Chap. “Dynamics of Solitons in High-Order Nonlinear Schrödinger Equations in Fiber Optics” aims to construct kink, bright and dark solitons of a generalized higher-order nonlinear Schrödinger equation in a cubic-quintic non-Kerr medium by applying a modified extended mapping method. It also presented the formation conditions on solitary wave parameters in which kink, dark and bright solitons can exist, and graphically illustrated the collision of the constructed soliton solutions that help realizing the physical phenomena of nonlinear Schrödinger equation. Further, it outlined descriptions of various issues on integrability. The stability of the model in normal dispersion and anomalous regime are discussed by using the modulation instability analysis. The Chap. “MHD Mass Transfer Flow Past an Impulsively Started Semi-Infinite Vertical Plate with Soret Effect and Ramped Wall Temperature” presents an exact solution to the problem of a hydromagnetic natural convective mass transfer flow of an incompressible viscous electrically conducting non-Gray optically thin fluid past an impulsively started semi-infinite vertical plate with ramped wall temperature in

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presence of appreciable radiation, thermal diffusion and uniform transverse magnetic field. It also studied graphically the influences of thermal radiation, ramped parameter, magnetic field, thermal diffusion and time on the flow and transport characteristics. The Chap. “Secure Communication Systems Based on the Synchronization of Chaotic Systems” aims at giving an overview of secure communications and chaos. It summarizes the latest advancements made in the field of chaos-based communications. A case study is also considered assuming antipodal chaos shift keying modulation and described the complete communication system. Simulation results are further incorporated highlighting the performance of chaotic modulation systems. The Chap. “Numerical Techniques for Fractional Competition Dynamics with Power-, Exponential- and Mittag-Leffler Laws” deals with modelling and analysing fractional competition system with power law, exponential law and the Mittag-Leffler law in which the standard derivative in time is replaced with the Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. Further, it formulates a fractional version of the Adams-Bashforth scheme for the approximation of these derivatives and justifies the usability of these derivatives by drawing comparison through an application of them to solve certain problems. It also considers a number of fractional competition dynamics arising in applied science and engineering in the simulation framework. The Chap. “Existence of Periodic Solutions for First Order Differential Equations with Applications” used a fixed point theorem in cones in a Banach space to present different sufficient conditions for the existence of at least two positive periodic solutions of first-order functional differential equations. The results obtained are also applied to the Nicholson’s Blowflies model and the generalized MichaelisMenton-type single-species growth model. The Chap. “Dynamic Programming Viscosity Solution Approach and Its Applications to Optimal Control Problems” is concerned with optimal control problems of dynamical systems described by partial differential equations. First, by the Dubovitskii-Milyutin functional analytical approach, it studies the Pontryagin maximum principle of an age-structured population dynamics for spread of universally fatal diseases. Then an optimal control problem of a McKendrick-type age-structured population dynamics is solved by the dynamic programming viscosity solution, and its corresponding numerical solutions of optimal feedback control are constructed. Finally, it proves the convergence of a well-adapted upwind finite-difference numerical scheme for the HJB equation solution. The Chap. “A Simple Model of Periodic Reproduction: Selection of Prime Periods” studies a discrete-time model of periodic reproduction with inter- and intra-specific competition as a tool to investigate the selection of prime reproduction cycle lengths observed in certain species of cicadas. It also proposed an approximation for the average populations and analysed for the case of 2 and 13 populations. It observed that prime periods have an advantage when compared with composite ones suggesting that the prime periods displayed by cicada species in nature might arise by the process of natural selection of adaptive values and not as a random result of evolutionary constraints.

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The Chap. “Transport on Networks—A Playground of Continuous and Discrete Mathematics in Population Dynamics” studied structured population models in which the population is subdivided into states according to certain feature of the individuals. It considered various rules allowing individuals to move between the states. It observed that depending on the type of the migration rule, the models can vary from a system of coupled McKendrick equations to a system of transport equations on a graph. The chapter aims also to address the well-posedness of such problems. The Chap. “Augmenting and Decreasing Systems” deals with cooperative games in which there exists a feasible coalition structure. Augmenting and decreasing systems are set systems specially introduced for analysing certain situations of partial cooperation, and they are dual structures. The chapter studied the core and the Weber set for games on augmenting systems. Two known classical solutions for games are also defined on augmenting systems: the Shapley value and the Banzhaf one. It also uses the duality relationship to analyse the values for decreasing systems. The Chap. “Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases” proposed and analysed a class of three spatiotemporal models describing infectious diseases caused by viruses such as the human immunodeficiency virus and the hepatitis B virus. The qualitative analysis of the models, such as positive invariance, boundedness and global stabilities of steady states, has also been studied. Biological findings of the analytical results are incorporated, and further extended and generalized some mathematical virus models and previous results. The Chap. “Approximation of Short-Run Equilibrium of the N-region CorePeriphery Model in an Urban Setting” aims to give an approximation of short-run equilibrium of the N-region core-periphery model in an urban setting. The approximation is claimed to be sufficiently accurate, which is expressed explicitly in terms of the distribution of workers that is contained as known function in the model. It further argued that the approximation can be used to analyse the behaviour of short-run equilibrium. The Chap. “New Phase-Field Models with Applications to Materials Genome Initiative” reviews some types of phase-field models formulated by Alber and Zhu. These models may be used to describe important phenomena, such as solid-solid phase transitions occurring in, e.g. smart materials like shape memory alloys and interface motion by interface diffusion. The first chapter presents the background of these models. Then, mathematical and numerical investigations of these models are presented as well as some open problems related to the models are also listed. Further, it introduces phase-field crystal method which can be regarded as an extension of phase-field approach. The Chap. “Optimal Control Measures for Tuberculosis in a Population Affected with Insurgency” applied optimal control theory to a mathematical model describing the population dynamics of tuberculosis with variability in susceptibility due to difference in awareness level. Aiming at minimize the number of high-risk

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susceptible individuals with low level of tuberculosis awareness and maximizing the number of isolated actively-infected individuals placed under Directly Observed Treatment Short-Course, it incorporated time-dependent control functions that represent educational campaign programmes in the midst of insurgency, and case finding techniques for chronic tuberculosis cases. It also characterized the optimal controls in terms of the optimality systems and solved numerically. Further, numerical simulations are performed to illustrate the effect of the controls on the population dynamics of the disease in a population. The Chap. “Insurance Model to Estimate the Financial Risk Due to Direct Medical Cost on Dengue Outbreaks” attempts to build a bridge between epidemiological and insurance modelling and set up an actuarial based tool that provides financial arrangements to cover the future medical expenses resulting from the medical treatments of dengue disease. It converted classical SIR model into probability model and then developed the insurance plan to cover the future financial burden due to direct medical expenses. The premium, the present financial burden due to future expenses is defined by means of the equivalence principle and discussed the sensitivity with respect to model parameters and external variables. It introduced several control measures and discussed the variability of the present financial burden with respect to such measures. Further, it analysed the efficiency of the controls and discussed necessary and sufficient criterion for the existence of insurance plans. The Chap. “Dynamics of Zika Virus Epidemic in Random Environment” developed a mathematical model for Zika virus dynamics under randomly varying environmental conditions, in which the birth and loss rates for mosquitoes and environmental influence are modelled as random processes. It studied the system of random ordinary differential equations by the theory of random dynamical systems and dynamical analysis. It first discussed the existence, uniqueness, positiveness and boundedness of solutions, and then investigated the long-term dynamics in terms of existence and geometric structures of random attractors and forward omega limit sets. Finally, it proved some conditions ensuring that the prevalence of Zika virus among human beings decreases monotonically to zero, and also established some conditions under which an epidemic may happen. The Chap. “Incidence Graph Models for the Analysis of Active Illegal Immigration Routes and Human Loss” studied connectivity in fuzzy incidence graphs. Different connectivity aspects of fuzzy incidence graphs are discussed. It introduced the notions like incidence connectivity and incidence connectivity of pairs and presented results similar to Whitney’s Theorem. It also investigated the notion of t-connected fuzzy incidence graphs as well as obtained some characterizations. An application in connection with illegal migration is presented and evaluated certain risks focusing on some vulnerable routes. The editors wish to thank the contributors for their timely contribution and cooperation while the chapters were being reviewed and processed. The reviewers also deserve sincere thanks for their great efforts and time voluntarily offered

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towards the successful completion of this book project. The editors are indebted to several well-wishers, colleagues, editors and supporting staff at Springer for timely and efficient cooperation which helped in executing this project smoothly. London, UK Guwahati, India Omaha, USA February, 2019

Frank T. Smith Hemen Dutta John N. Mordeson

Contents

On the Reynolds Equation and the Load Problem in Lubrication: Literature Review and Mathematical Modelling . . . . . . . . . . . . . . . . . . Hassán Lombera Rodríguez and J. Ignacio Tello

1

On Dynamic Interactions Between Body Motion and Fluid Motion . . . . Frank T. Smith, Samire Balta, Kevin Liu and Edward R. Johnson

45

Certain Aspects of Problems with Non Homogeneous Reactions . . . . . . Alejandro Omón Arancibia

91

A Survey on the Melnikov Theory for Implicit Ordinary Differential Equations with Applications to RLC Circuits . . . . . . . . . . . . . . . . . . . . 121 Michal Fečkan Numerical Solution of Space-Time-Fractional Reaction-Diffusion Equations via the Caputo and Riesz Derivatives . . . . . . . . . . . . . . . . . . 161 Kolade M. Owolabi and Hemen Dutta An Extended Langhaar’s Solution for Two-Dimensional Entry Microchannel Flows with High-Order Slip . . . . . . . . . . . . . . . . . . . . . . . 189 R. Rasooli and B. Çetin Dynamics of Solitons in High-Order Nonlinear Schrödinger Equations in Fiber Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Gholam-Ali Zakeri MHD Mass Transfer Flow Past an Impulsively Started Semi-Infinite Vertical Plate with Soret Effect and Ramped Wall Temperature . . . . . . 245 N. Ahmed Secure Communication Systems Based on the Synchronization of Chaotic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Samir Bendoukha, Salem Abdelmalek and Adel Ouannas

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Numerical Techniques for Fractional Competition Dynamics with Power-, Exponential- and Mittag-Leffler Laws . . . . . . . . . . . . . . . . 313 Kolade M. Owolabi and Hemen Dutta Existence of Periodic Solutions for First Order Differential Equations with Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Smita Pati Dynamic Programming Viscosity Solution Approach and Its Applications to Optimal Control Problems . . . . . . . . . . . . . . . . 363 Bing Sun, Zhen-Zhen Tao and Yang-Yang Wang A Simple Model of Periodic Reproduction: Selection of Prime Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Raul Abreu de Assis and Mazílio Coronel Malavazi Transport on Networks—A Playground of Continuous and Discrete Mathematics in Population Dynamics . . . . . . . . . . . . . . . . 439 Jacek Banasiak and Aleksandra Puchalska Augmenting and Decreasing Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 J. M. Bilbao, A. Jiménez-Losada and M. Ordóñez Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 Khalid Hattaf and Noura Yousfi Approximation of Short-Run Equilibrium of the N-Region Core-Periphery Model in an Urban Setting . . . . . . . . . . . . . . . . . . . . . . 551 Minoru Tabata and Nobuoki Eshima New Phase-Field Models with Applications to Materials Genome Initiative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 Peicheng Zhu, Yangxin Tang and Yeping Li Optimal Control Measures for Tuberculosis in a Population Affected with Insurgency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 A. O. Egonmwan and D. Okuonghae Insurance Model to Estimate the Financial Risk Due to Direct Medical Cost on Dengue Outbreaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 S. S. N. Perera Dynamics of Zika Virus Epidemic in Random Environment . . . . . . . . . 665 Yusuke Asai, Xiaoying Han and Peter E. Kloeden Incidence Graph Models for the Analysis of Active Illegal Immigration Routes and Human Loss . . . . . . . . . . . . . . . . . . . . . . . . . . 685 Sunil Mathew and John N. Mordeson

Contributors

Salem Abdelmalek Department of Mathematics, University of Tebessa, Tebessa, Algeria N. Ahmed Department of Mathematics, Gauhati University, Guwahati, India Yusuke Asai Department of Hygiene, Graduate School of Medicine, Hokkaido University, Sapporo, Japan Samire Balta University College London, London, UK Jacek Banasiak Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa; Institute of Mathematics, Łódź University of Technology, Łódź, Poland Samir Bendoukha Department of Electrical Engineering, Taibah University, Yanbu, Saudi Arabia J. M. Bilbao Department of Applied Mathematics II, University of Seville, Escuela Superior de Ingenieros, Camino de los Descubrimientos, Sevilla, Spain B. Çetin Microfluidics and Lab-on-a-chip Research Group, Mechanical Engineering Department, İ.D. Bilkent University, Ankara, Turkey Raul Abreu de Assis Departamento de Matemática, UNEMAT, Sinop, MT, Brazil Hemen Dutta Department of Mathematics, Gauhati University, Guwahati, India A. O. Egonmwan Department of Mathematics, University of Benin, Benin City, Nigeria Nobuoki Eshima Center for Educational Outreach and Admissions, Kyoto University, Kyoto, Japan

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Contributors

Michal Fečkan Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Bratislava, Slovakia; Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovakia Xiaoying Han Department of Mathematics and Statistics, 221 Parker Hall Auburn University, Auburn, AL, USA Khalid Hattaf Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Casablanca, Morocco; Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, Sidi Othman, Casablanca, Morocco A. Jiménez-Losada Department of Applied Mathematics II, University of Seville, Escuela Superior de Ingenieros, Camino de los Descubrimientos, Sevilla, Spain Edward R. Johnson University College London, London, UK Peter E. Kloeden Department of Mathematics and Statistics, 221 Parker Hall Auburn University, Auburn, AL, USA Yeping Li Department of Mathematics, East China University of Science and Technology, Shanghai, People’s Republic of China Kevin Liu University College London, London, UK Hassán Lombera Rodríguez Centro de Informática Industrial, Universidad de las Ciencias Informáticas, La Habana, Cuba Mazílio Coronel Malavazi Instituto de Ciências Naturais, Humanas e Sociais (ICNHS)—UFMT, Sinop, MT, Brazil Sunil Mathew Department of Mathematics, National Institute of Technology Calicut, Calicut, India John N. Mordeson Department of Mathematics, Center for Mathematics of Uncertainty, Creighton University, Omaha, NE, USA D. Okuonghae Department of Mathematics, University of Benin, Benin City, Nigeria Alejandro Omón Arancibia Departamento Universidad de La Frontera, Temuco, Chile

de

Ingeniería

Matemática,

M. Ordóñez Department of Applied Mathematics II, University of Seville, Escuela Superior de Ingenieros, Camino de los Descubrimientos, Sevilla, Spain Adel Ouannas Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Larbi Tebessi, Tebessa, Algeria

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Kolade M. Owolabi Faculty of Natural and Agricultural Sciences, Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa; Department of Mathematical Sciences, Federal University of Technology, Akure, Ondo State, Nigeria Smita Pati Department of Mathematics, Amity School of Engineering and Technology, Amity University Jharkhand, Ranchi, Jharkhand, India S. S. N. Perera Research and Development Centre for Mathematical Modeling, Faculty of Science, University of Colombo, Colombo, Sri Lanka Aleksandra Puchalska Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland R. Rasooli Microfluidics and Lab-on-a-chip Research Group, Mechanical Engineering Department, İ.D. Bilkent University, Ankara, Turkey Frank T. Smith University College London, London, UK Bing Sun School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China; Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, China Minoru Tabata Department of Mathematical Sciences, Osaka Prefecture University, Osaka, Japan Yangxin Tang Department of Mathematics, Shanghai University, Shanghai, People’s Republic of China; Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, Anhui Province, People’s Republic of China Zhen-Zhen Tao School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China J. Ignacio Tello Depto Matemática Aplicada a las TIC. ETSIS Sistemas Informáticos, UPM, Madrid, Spain Yang-Yang Wang School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China Noura Yousfi Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, Sidi Othman, Casablanca, Morocco Gholam-Ali Zakeri Department of Mathematics, and Interdisciplinary Research Institute for the Sciences (IRIS), California State University - Northridge, Northridge, CA, USA Peicheng Zhu Materials Genome Institute and Department of Mathematics, Shanghai University, Shanghai, People’s Republic of China

On the Reynolds Equation and the Load Problem in Lubrication: Literature Review and Mathematical Modelling Hassán Lombera Rodríguez and J. Ignacio Tello

Abstract In this chapter, we provide a literature review concerning the theory of hydrodynamic lubrication, especially applied to journal bearings. The device consists of an external cylinder surrounding a rotating shaft, both separated by a lubricant to prevent contact. In particular, we derive the fluid film thickness model for journal bearings, considering both the parallel and the misaligned case. The hydrodynamic Reynolds equation with cavitation phenomenon, through both Reynolds and ElrodAdams models are fully derived in this chapter. Subsequently, we pose two suitable variational formulations for the hydrodynamic problem considering both cavitation models. In addition, we present the admissible range of misalignment angle projections for prescribed values of the shaft eccentricity and angular coordinate. Finally, we properly state the problem of a loaded misaligned journal bearing for stationary regime, considering the balance of force and torque components involved. Keywords Reynolds equation · Hydrodynamic lubrication · Journal bearing · Misalignment · Cavitation · Inverse problem

1 On the Beginning of the Theory of Hydrodynamic Lubrication Hydrodynamic lubrication is a phenomenon characterized by a lubricant flowing in the narrow gap between two closely spaced surfaces in relative motion. Important and well-known scientists, engineers and tribologists investigated in the past H. Lombera Rodríguez Centro de Informática Industrial, Universidad de las Ciencias Informáticas, 19370 La Habana, Cuba e-mail: [email protected] J. I. Tello (B) Depto Matemática Aplicada a las TIC. ETSIS Sistemas Informáticos, UPM, 28037 Madrid, Spain e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_1

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H. Lombera Rodríguez and J. I. Tello

the relationship between friction, wear and lubrication, especially applied to journal bearings. Gustave Adolph Hirn (1815–1890), accomplished in 1847 the first experiments on hydrodynamic lubrication and rediscovered the laws of Amontons and of Coulomb. In 1879, Robert Henry Thurston (1839–1903), published the results of his study on friction and lubrication. He showed that, with increasing speed, the friction coefficient of a lubricated bearing diminishes below its static value, passes through a minimum and then increases. He also specified that the speed corresponding to the minimum of friction depends on the load applied to the bearing [48]. In 1883, Nikolai Pavlovich Petrov (1836–1920) introduced the results of his studies and tests on lubricated bearings. He proved that, among the physical characteristics of an oil, the viscosity has a preponderant role in bearing friction. He stipulated that a fluid film totally separates the surfaces of both shaft and bearing, and that a constant pressure should be produced in this film. Petrov, also looked through the work of Hirn and reused the term of mediate friction to characterize hydrodynamic lubrication [48]. In 1885, a remarkable discovery was the existence of hydrodynamic pressure in the lubricant film of a bearing by Beauchamp Tower (1845–1904), which served as a basis for accomplishing the theory of lubrication. Fortunately, Tower’s discovery results provided experimental confirmation to Reynolds, who was working on a hydrodynamic theory of lubrication at that time. The result of this was a theory of hydrodynamic lubrication published in the Proceedings of the Royal Society by Reynolds; see [82]. In that early work, Reynolds proposed the equation that at present is named after him and provided the first analytical proof that a viscous liquid can physically separate two sliding surfaces by hydrodynamic pressure, resulting in low friction and theoretically zero wear [90]. That work represents the seminal paper on Lubrication Theory and in fact, most of mathematical models of hydrodynamic lubrication processes between solid surfaces have the Reynolds Equation (RE) as their key point. A rigorous approach for the deduction of the classical linear RE from Navier–Stokes may be found in [4]. It was not until the beginning of the 20th century that Reynolds theory on hydrodynamic lubrication was used for calculating thrust and journal bearings. In 1902, Richard Stribeck confirmed the hydrodynamic effects and performed the original research into the limits of hydrodynamic lubrication. He proposed the relationship between friction, load, speed and viscosity that is still used today to present the various types of lubrication. In most cases, the friction and lubrication relationship is (oil viscosity × sliding velocity/normal load) factor, characterized with basis on μv F in a diagram called Stribeck curve. This diagram summarizes the limits of hydrodynamic lubrication; see Fig. 1. Three zones can be identified, each one corresponding to a type of lubrication depending on the level of pressure established in the contact. For low pressure (0.1 to 50 MPa), zone 1 corresponds to boundary lubrication; surface separation is ensured by lubricant molecules attached to the surfaces; see Fig. 2a. This type of lubrication is related to the physico-chemistry of surfaces and of lubricants, for low and moderate speeds and for relatively low loads. In zone 2, the hydrodynamic effect described by RE takes some importance and tends to separate the areas still in contact over a part of their asperities; this type of lubrication is the mixed lubrication; see Fig. 2b. Zone

On the Reynolds Equation and the Load Problem in Lubrication …

3

2

friction

1

3

v F

0

Fig. 1 Schematic diagram for the limits of hydrodynamic lubrication

(a) Boundary lubrication regime

(b) Mixed lubrication regime

(c) Hydrodynamic lubrication regime Fig. 2 Types of lubrication regimes present on the Stribeck curve

3 corresponds to hydrodynamic lubrication and is described by RE; see Fig. 2c. In this region a full film separates the surfaces and friction is proportional to the speed if the lubricant viscosity is constant with temperature [48]. Notice that depending on lubrication regime, different surface interaction mechanisms occur, leading to distinct wear and friction responses. Friction behaviour in Stribeck curve is used to explain rubbing phenomena occurring in lubricated con, friction tacts. Transition between zones is explained as follows. For high values of μv F

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H. Lombera Rodríguez and J. I. Tello

coefficient is linearly ascending due to fluid film lubrication. When load increases or factor falls. It means that the sliding oil viscosity and/or velocity decreases, the μv F velocity v and the oil viscosity μ are unable to generate sufficient oil-film pressure p to support the entire load F [58]. Then, the fluid film becomes thinner and, therefore, friction coefficient decreases up to a minimum value. Note that for even smaller val, fluid film thickness is further reduced, and contact appears. Then, friction ues of μv F factor decreases [69]. Such rise in friction coefficient coefficient increases as the μv F is also related to oil viscosity increase in some regions at contact area under high contact pressure. These phenomena characterize the mixed lubrication regime. Addifactor makes contact stronger. Film thickness becomes smaller tional reduction in μv F than the height of surface asperities and then boundary lubrication regime will occur. Due to this behaviour, the Stribeck curve is also represented with the film thickness along the horizontal axis [48]. In 1904, Arnold Johannes Wilhelm Sommerfeld used a change of variables and succeeded in obtaining an analytical solution of the RE for infinitely long journal bearings ( ∂∂xp = 0), where p stands for the pressure and x, the geometrical coordinate, is in the direction of the journal axis [48]. Nevertheless, the used boundary conditions did not correspond to the physical reality, as they did not take into account the film rupture in the bearing. In addition, the pressure distribution obtained was negative in the divergent zone of the film. In 1914, Ludwig Karl Friedrich Gümbel suggested that only the positive part of the pressure distribution should be included for the calculation of the bearing load, omitting the negative part. The load calculated was not exact either. Herbert Walker Swift in 1932, and later W. Stieber in 1933 independently presented boundary conditions for film exit, representing the reality in an improved way [91, 95]. With their proposals, the fluid film breaks in its divergent zone along the boundary, where the pressure has the value of the saturated pressure and its gradient is zero. Such conditions agree with the continuity of flow at the film exit and are named the boundary conditions of Reynolds. They are universally used for calculating bearings with constant loads [48]. This model considers as interface conditions: ∂p = 0, (1) pc = ∂n where pc stands for the cavitation pressure and n stands for the unitary normal vector to the free boundary. In 1941, a numerical method for solving the RE with such boundary conditions was proposed in Christopherson [25]. In 1953, Fred William Ocvirk proposed to neglect circumferential pressure gradient compared to the axial one in the RE. This way he was presenting his approximate method for short bearing [73]. The solution is analytical and uses Gümbel’s boundary conditions. The results are almost exact for journal bearings having an L/D ratio (bearing length over diameter) smaller than 0.25. Calculations are in consequence considerably simplified [48]. A type of slider, with steps, consisting of two parallel parts, but shifted, was described by John William Strutt, Lord Rayleigh. He showed that this type of slider exhibits a load carrying capacity for a given minimum film thickness greater than for

On the Reynolds Equation and the Load Problem in Lubrication …

5

any known slider type [48]. In 1917, Lord Rayleigh was the first one in calculating the load and the friction torque of a hydrostatic thrust bearing. The review of all existing papers since the beginning of last century represents a considerable effort that goes beyond the aims of this chapter. We presented here only a brief review of major achieved progresses in the beginning of hydrodynamic lubrication, that will allow us to succeed “standing on the shoulders of giants”.

2 On Misalignment and Cavitation in Journal Bearings Misalignment In the case where misalignment is not allowed, the mathematical model assumes ¯ only depends on the circumferential that the clearance normalized film thickness, h, coordinate and is expressed as: h¯ = 1 + ρ¯ cos(θ − α),

(2)

where ρ¯ ∈ [0, 1) stands for the normalized shaft eccentricity, α is the shaft angular coordinate and θ represents a point on the external circumference. The assumption posed by Eq. (2) is equivalent to the supposition that both shaft and bearing axes are perfectly parallel to each other, and that the eccentricity ρ¯ does not depend on the axial coordinate. Thus, this expression restricts the physical simulating capacity of the study since in reality it becomes impossible to fully avoid radial and angular misalignments [51]. The most common causes of misalignment are elastic and thermal deflections of the shaft and bearing misalignment, as a result of assembly errors. Large misalignment can decrease the bearing clearance and its load capacity. It can increase the temperature and has the potential to reduce the operating velocity threshold [51]. In addition, misalignment and residual unbalance are the typical causes for rotor vibration. Both excitations are responsible for most common machine dynamic problems happening in the field. One of the first documented researches on journal bearing misalignment is reported by McKee and McKee [70], who experimentally observed that measured peak pressures move from the bearing mid-plane towards the bearing ends when the journal is subjected to misalignment. Same result was found by Bouyer and Fillon [18] in an experimental analysis of misalignment effects on hydrodynamic plain journal bearing performance. They experimentally studied the hydrodynamic plain journal bearing submitted to a misalignment torque. The misalignment caused more significant changes in bearing performance when the rotational speed or load was low [18]. Piggott showed that a 40% reduction in bearing load capacity was induced by a 0.0002 rad misalignment. These observations clearly revealed the importance of misalignment in bearing performance [78]. Subsequently, Dubois et al. [41] showed that the pressure distribution of a misaligned bearing was not symmetric, and reported that the maximum pressure was located as well at the bearing ends. They

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H. Lombera Rodríguez and J. I. Tello

observed that when a bearing is subjected to severe misalignment, the maximum pressure increases and the bearing performance deteriorates due to the permanent deformation at bearing ends [41]. In presence of cavitation it has been shown that the maximum pressure is shifted to the bearing ends as well. The location of the maximum pressure is influenced by the orientation of the misalignment. Besides, the maximum pressure is greater than that for the aligned bearing and an increase in the degree of misalignment could yield two peak values in the pressure, axially near both ends [56]. Representative numerical studies about loaded misaligned journal bearings by References [3, 52, 80, 98], suggest that misaligned bearings have a finite load capacity as the end-plane film thickness goes to zero. Moreover, perfectly aligned journal bearings have a theoretically infinite load capacity, see [29, 81] for instance. Conversely, Boedo and Booker [16] suggest (but no prove) that misaligned bearings have infinite load and moment capacity as the end-plane minimum film thickness approaches zero under transient journal squeeze motion and under steady load and speed conditions. These results differ markedly from finite capacity trends reported in previously mentioned numerical and experimental studies. Nikolakopoulos and Papadopoulos [71] presented an analysis of misaligned journal bearing operating, considering both the linear and non-linear plain journal bearing characteristics. The Finite Element Method (FEM) was used to solve the RE. They calculated the linear and non-linear dynamic properties for misaligned bearings depending on the developed forces and moments as functions of the displacements and misalignment angles [71]. An analytical solution for misaligned journal bearing axes (short bearing) at its steady-state was obtained in [51]. The same approach for modelling misalignment was used in [52]. The solution is expanded in series over a small parameter a, which characterizes the non parallelism of journal bearing axes. In our model we characterize misalignment using an equivalent derivation procedure, but we do not make that power series expansion, in order to propose a general characterization for misaligned journal bearings; see Sect. 5.2 for details. Thus, as a journal bearing almost always operates with some misalignment between its shaft and bearing, it is important to include this issue in the analysis. Cavitation models. Their numerical resolution Mathematical models that we consider in Lubrication Theory, assume that the unknown pressure p is constant through the thickness of the fluid film, which allows one to approximate the three dimensional Navier–Stokes equations by the bidimensional RE; see [4] for details. In presence of cavitation, the RE is no longer valid and this condition makes the use of cavitation models mandatory. A review on the mathematical and physical analysis for different cavitation models is presented in [5]. Also, Álvarez [2] studied two different models for describing the fluid pressure distribution in journal bearings: a stationary model and a transient model. He considered cavitation and demonstrated uniqueness of the solution. The common feature of the models lies in the domain decomposition into two parts: a lubricated region and a cavitated region. In the former the RE is verified

On the Reynolds Equation and the Load Problem in Lubrication …

7

while in the latter the pressure is taken to be a constant [12]. The main difference between models comes from the way to obtain the free boundary that separates lubricated and cavitated areas. In Sects. 7 and 10 we give details on the derivation of both cavitation models. Several papers have used the theory of variational inequalities taking advantage that the pressure in the full filled area is greater than the saturation pressure. In fact, the idea was reinforced when Cryer justified the work of Christopherson [25], associating that study to an obstacle problem [33]. This is known as the Reynolds cavitation model [12]. In 1975, Rohde and McAllister presented a variational formulation for hydrodynamic lubrication, from which the associated free boundary problem arose naturally. The Finite Difference Method (FDM) and the FEM were discussed as strategies for obtaining approximate solutions [84]. In fact, due to the nature of the Reynolds cavitation model and easy computational approach, it has been used in a large list of mathematical works; see [22, 26, 27, 34, 66, 83] for instance. In general, for the numerical resolution of this model, techniques based on the FEM have been widely used. The discrete problem has been solved by the classical Gauss-Seidel method or a point-overrelaxation method, including both a projection technique to consider cavitation; see [21, 34] for instance. In this work, we also use the Reynolds cavitation model including a FEM discretization. Nevertheless, we propose to solve the system of linear equations by minimizing a convex functional, using a Preconditioned Conjugate Gradient Method (PCG) with both projection and restarting strategies. The choice obeys two major reasons: the fact that we solve a convex functional and that matrices resulting from the discretization of Partial Differential Equations (PDE) [e.g. FEM or Finite Volume Method (FVM)], in addition to be sparse are usually ill-conditioned, for which preconditioning is widely recommended. Another model to describe cavitation is the Elrod–Adams model [47]. In that work, the authors introduce the hypothesis that the cavitation region is a fluid-air mixture and an additional unknown ϑ appears (the saturation of fluid in the mixture1 ). This model, which still relies on the RE has been widely used in Tribology [68]. Unlike some other models, such as the Reynolds cavitation model, it does allow the starvation phenomena to take place. Its interest also relies on the evidence that it is a mass-preserving model. In [5, 45] comparisons for journal bearings are made, between their operating parameters computed by the Reynolds and the Elrod–Adams cavitation models. Vijayaraghavan and Keith [97] analysed the effect of cavitation on the performance of a line-grooved misaligned bearing for both flooded and starved inlet conditions. They used the mass-conserving cavitation algorithm in their analysis. They took into account the lubricant rupture and the reformation phenomena. One year later, they showed that at the higher degrees of misalignment, the performance characteristics of the bearing are significantly different from those for an aligned journal bearing [98].

1 It

represents the lubricant concentration.

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H. Lombera Rodríguez and J. I. Tello

Numerical methods for solving the Elrod–Adams model for cavitation in different devices and conditions were presented in [6, 13, 45, 46], among others. Similarly, numerical experimentations of various schemes based both on stationary upwind methods and pseudo-stationary techniques were conducted in [21]. These methods are mainly based on the characteristics discretization for the non-linear convection term and a duality method for the multivalued non-linear saturation-pressure relation, posed by the Heaviside operator. Namely, they use an approach based on the Method of Characteristics (MC) to discretize a total derivative in the final formulation. This technique was also used in [42–44, 67] among others, and it is the strategy proposed to solve our problem as well. Additionally, the first three used a Yosida regularization for the Heaviside operator as in [14]. In contrast, in [67] it is used a regularization of this function by a cubic interpolating Hermite polynomial that allowed to express the solution of the direct problem as a minimum of a convex functional.

3 The Inverse Problem. Its Numerical Resolution Most of the papers previously mentioned deal with imposed geometry in the associated RE, i.e. the gap function h for the journal bearing is a given datum and the unknown is the pressure p. In real engineering applications the position of the shaft, that defines the gap function h, is unknown. So, Newton’s second law is introduced to obtain that position. The problem consists in finding the pressure of the lubricant, its concentration ϑ in the cavitation area and the shaft position. If misalignment is considered, two more variables need to be found, which stand for the angular misalignment projections. The problem is considered as an inverse problem where the coefficient h depends on the unknown p. Díaz and Tello [36] addressed such a problem, considering the simple case in which the surfaces are two parallel planes, and assuming prescribed the total force applied upon one of the surfaces. They provided some sufficient conditions on the total force in order to solve the inverse problem. Ciuperca et al. [28] also studied analytically the inverse problem for a more general geometry. Specifically, they studied the asymptotic behaviour of the position in the evolution problem. Furthermore, Ciuperca et al. [29] studied the inverse problem for journal bearings using the Reynolds cavitation model. In that work the inner cylinder is parallel to the exterior one and misalignment is not allowed. They proved the existence of shaft equilibrium positions when the hydrodynamic force created by the pressure film balances an external radial force. The authors proved the non-existence of contact for any force, even for the case where the shape of the external surface presents some rugosity. Additionally, Ciuperca and Tello considered the problem for both cases, a rigid surface moving over a flat plane and the elastohydrodynamic problem; see [31, 32] for instance. Similarly, Ciuperca, Jai and Tello studied the existence of equilibrium positions for the load problem in Lubrication Theory. In their work, considering the Elrod–Adams model, the balance of forces allows to obtain the unknown position of the surfaces, defined with one degree of freedom [30].

On the Reynolds Equation and the Load Problem in Lubrication …

9

As for the numerical resolution of the inverse problem which entails the balance between an imposed load on the device and the hydrodynamic load we can mention the work of [42]. They developed a numerical scheme which combines fixed point algorithms, the MC, duality techniques and finite element approximations. In [44] the authors used an implicit Euler method to deal with the dynamical shaft problem coupled with the fluid hydrodynamic problem. At each time step the resulting nonlinear system is solved by the Broyden method combined with the Armijo–Goldstein criterion to choose a proper step length in the descent direction. Conversely, in [67] the authors proposed a different approach to deal with the shaft model. It was based on first solving the Elrod–Adams equation for a known position by minimizing a convex and lower semi continuous (l.s.c) functional and then using an iterative method to reach the equilibrium, namely a trust-region strategy. In general, there are a lot of gradient based algorithms for continuous optimization that can be used for solving problems like the one addressed in this work. They allow to find a local minimum, but the optimized function needs to be continuous and differentiable. Thus, their usefulness is limited due to such prerequisites. Line search and trustregion approaches are two of the fundamental strategies in optimization algorithms that must be mentioned; see [72] for a wide explanation on these approaches. On the other hand, metaheuristics are a family of optimization techniques, which have seen increasingly rapid development and application to numerous problems in computer science and other related fields. Normally, they require the problem to be partitioned into a set of components to look for the solution in an optimal combination or permutation of them. One of the more recent, prominent and actively developed metaheuristic is Ant Colony Optimization (ACO) which was inspired by the ants’ foraging behaviour. It was originally introduced by Dorigo [38], to solve discrete optimization problems where each decision variable is characterized by a set of components; see [39, 92] for instance. Many successful implementations of the ACO metaheuristic have been applied to a number of different discrete optimization problems [63]. These applications mainly concern NP-hard combinatorial optimization problems including problems in routing [49], assembly sequence planning [37], bioinformatics [15] and many other areas. ACO was initially designed to solve the Traveling Salesman Problem (TSP), where a salesman must visit a list of cities exactly once, using the shortest possible route. The cities and paths between them can be represented as a connected graph, and the ants move from one city to another following the pheromone trails on the edges. Let Ti j (t) be the trail intensity on edge (i, j) at time t. Then, each ant chooses the next city to visit depending on the intensity of the associated trail. When the ants have completed their city tours, the trail intensity is updated according to: Ti j (t + 1) = Ti j (t) + Ti j ,

 ∈ [0, 1],

(3)

where  is a coefficient such that (1 − ) represents the evaporation of trail and

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H. Lombera Rodríguez and J. I. Tello

Ti j =

m 

Ti kj ,

(4)

k=1

where Ti kj is the pheromone quantity laid by the kth ant on edge (i, j), defined as:  Ti kj

=

1 , Wk

0,

if edge (i, j) is in the trajectory of the kth ant, otherwise,

(5)

with Wk the tour length of the kth ant [38]. The transition probability Pikj from city i to city j for the kth ant is defined as: [Ti j ]a [ηi j ]b , a b l∈allowedk [Til ] [ηil ]

Pikj = 

(6)

where ηi j = 1/di j is called visibility and di j is the associated cost to travel from city i to city j; a and b are parameters that control the relative importance of trail versus cost, and allowedk is the set of allowed cities the kth ant can move to from city i [50]. Genetic Algorithm, Simulated Annealing, Tabu Search and Particle Swarm Optimization are other approaches we find in the literature to deal with combinatorial optimization problems; see for instance [75, 79, 89, 100] for a detailed explanation of them. Since the emergence of these approaches as combinatorial optimization tools, attempts have been made to use them for addressing continuous problems [87]. Now, these metaheuristics that were originally developed for combinatorial optimization are adapted to the continuous case. Examples include the Continuous Genetic Algorithm [24], Enhanced Simulated Annealing [86], or Enhanced Continuous Tabu Search [23]. There are also included some ant related methods. In this sense, Socha and Dorigo [88], proposed one of the most popular and easy to implement ACO algorithms for continuous domains, called Ant Colony Optimization for continuous domain (ACOR ). It uses a solution archive as a form of pheromone model for the derivation of a probability distribution over the search space. However, its use in problems with many decision variables have some limitations, reported in Leguizamón and Coello [60]. Thus, Leguizamón and Coello [61] proposed an Alternative Ant Colony Optimization for continuous domain (DACOR )2 which could be more appropriate for large scale unconstrained continuous optimization problems. Later on, Liao et al. [62] proposed an Incremental Ant Colony Optimization with Local Search for continuous domain (IACOR -LS). This algorithm uses a growing solution archive as an extra search diversification mechanism and a local search to intensify the search. Subsequently, Liao et al. [64] proposed an ACO algorithm for continuous optimization that combines algorithmic components from ACOR , DACOR and IACOR -LS. They called it Unified Ant Colony Optimization for continuous domain (UACOR ). It is unified, because from UACOR , we can instantiate the original ACOR , 2 “D”

stands for diversity.

On the Reynolds Equation and the Load Problem in Lubrication …

11

DACOR and IACOR -LS algorithms by using specific combinations of the available algorithmic components and parameter settings. Since in our inverse problem, we only deal with four decision variables and considering that ACOR has proven to be an efficient, versatile and easy to implement tool for continuous optimization, we propose its use in our work. However, we do deal with a large scale direct problem and motivated by the inherent parallelism of the ACOR and possible computation speed up we suggest an implementation of the algorithm with parallel regions for time-consuming tasks, using Open Multi-Processing (OpenMP).

4 Other Topics Moreover, we mention other topics which have also received attention on the subject of misaligned journal bearings. Literature concerning the topics of thermohydrodynamic and elastohydrodynamic lubrication can be found in [1, 19, 54, 59, 76, 77, 93, 96, 99]. Besides, lubrication is not the only way to decrease the effect of friction; the materials used and the quality of polished surfaces are also of major concern. However, if surfaces are extremely polished, it is probable a contrary trend to decrease load capacity. It has often been observed in engineering practice that there is a risk of sudden seizure if the surface is too smooth. In this sense, it is commonly believed that small asperities play a useful role as a reservoir for the lubricant between asperities [90]. The effect of surface roughness on the performance characteristics of bearings can be found in [53, 85]. In general, roughness is one of the challenges of the field; see [7–11, 68] and references there in, to study its effects in journal bearings for different scenarios. More recent results on this topic can be found in [93, 94]. The current research on journal bearings also includes the applications of nonNewtonian fluids to improve performance of modern machines [56]. Such nonNewtonian fluids have shown that the stress is not directly proportional to the shear strain, and the formulation of the governing equations needs to be changed. Literature on non-Newtonian fluids includes the works of [1, 17, 35, 57, 74]. The general characterization of the behaviour of a misaligned journal bearing considering all mentioned factors and including the prediction of its final position is very complex. For that reason most researches focus on specific topics.

5 Fluid Film Thickness In this section we depict the formulations of the fluid film thickness of a journal bearing, for the parallel and the misaligned case. Actually, there are similar results that can be found in the literature; see [48, 51] for instance. However, for making this chapter self-contained we present their derivations in this section.

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H. Lombera Rodríguez and J. I. Tello

5.1 Parallel Case Figure 3 depicts the cross section of a journal bearing. The inner cylinder, the shaft of radius R, rotates in counter-clockwise direction at a constant velocity ω, about the X axis. The film pressure generated by the moving surfaces, forces the lubricant through a wedge shaped zone of thickness h, which varies according to the angle α. It is assumed a coordinate system in which “y” represents the circumferential coordinate, “z” is the coordinate across the fluid film and “x” depicts the journal bearing axial dimension, orthogonal to the zy-plane. Let Ob and O j be the centres of the bearing and shaft, respectively. The origin of coordinate “y” is located over the line segment Ob O j , to place the minimum gap of the device at an angle  = π. Moreover, the reference z = 0 is taken on the bearing surface. The model characterizes the parallel misalignment, where the shaft is allowed to move with two degrees of freedom. Let M be a point over the bearing surface, represented by the angular coordinate  = (Ob A, Ob M), where Ob A and Ob M are line segments. The fluid film thickness is given by: (7) h = Ob M − Ob M  = Rb − Ob M  . 

Applying the sine rule for the triangle O j M Ob we have: O j M R R R ρ = = = = , sin(O j Ob M  ) sin(O j Ob M  ) sin(π − ) sin() sin(α) where sin(α) =

ρ sin() R

and

Ob M  =

R  sin(Ob O j M ). sin()

(8)

(9)

Y

Fig. 3 Cross section of a journal bearing

A

Φ

R O M’ M

h

˜

R O

Z

On the Reynolds Equation and the Load Problem in Lubrication …

Notice that:



(Ob O j M ) =  − α˜ =  − arcsin

ρ R

 sin() ,

13

(10)

and therefore: Ob M  =

 ρ  R sin  − arcsin sin() . sin() R

(11)

Taking into account that: 

ρ 2 1/2 sin() = arccos 1 − sin() arcsin , R R ρ



(12)

we can calculate the sine of the sum indicated in Eq. (11), from which we obtain:

   ρ 2 1/2 R sin() Ob M = sin() cos arccos 1 − sin() R   ρ sin() , − cos() sin arcsin R

  2 1/2 ρ R ρ  sin() Ob M = − cos() sin() , sin() 1 − sin() R R  2 1/2 ρ sin() Ob M  = R 1 − − ρ cos(). (13) R 

Substituting Eq. (13) in (7) we have:  

2 1/2 ρ sin() h = Rb − R 1 − − ρ cos() . R

(14)

Let C = Rb − R be the radial clearance. It must be noticed the relation: C ρ <  1. R Rb

(15)

2 Thus, the term Rρ sin() can be neglected compared to unit [48, p. 116]. The fluid film thickness becomes: h ≈ Rb − R + ρ cos(), h ≈ C + ρ cos().

(16)

In Fig. 3 we can notice the relation  = θ − α. Thus, Eq. (16) can be rewritten as:

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H. Lombera Rodríguez and J. I. Tello

h ≈ C + ρ cos(θ − α).

(17)

where ρ and α depict the shaft position in polar coordinates and θ represents a point on the external circumference at the height Y = y. The non-dimensional expression For the non-dimensional expression, we introduce the following non-dimensional variables: h h¯ = , C

ρ¯ =

ρ . C

(18)

Then, Eq. (17) can be rewritten as: h¯ = 1 + ρ¯ cos(θ − α),

ρ¯ ∈ [0, 1).

(19)

¯ Notice that it occurs when cos(θ − α) = −1, Let h¯ min be the minimum value of h. so: ¯ h¯ min = 1 − ρ. A contact exists for ρ¯ = 1.

5.2 Misaligned Case In Fig. 4 we show the journal bearing axes along with their projections on the x y-plane and x z-plane. The origin O is located at the centre of the bearing “left” end-plane Ob , and the shaft rotates at a constant velocity ω about the X axis. The coordinate system has been rotated, in favour of the graphic comprehension. Notice that the axis is rotated an angle ψ, characterized by its projections ϕ and β on the x y-plane and x z-plane, respectively. We create a cross section of the inner cylinder along the plane X = x. The cross section is approximated by a circumference. We compute the position of the inner cylinder centre, on that plane, considering the eccentricity λ of its axis. Taking into account the auxiliary coordinate system (x  , y  , z  ) located in the yzplane, at the position (ρ sin α, ρ cos α), the coordinates of the shaft, on the plane X = x, will be: S y = ν y + ρ sin α = x tan ϕ + ρ sin α, Sz = νz + ρ cos α = x tan β + ρ cos α.

On the Reynolds Equation and the Load Problem in Lubrication …

15

Fig. 4 Axes and projections in a misaligned journal bearing

Therefore, the eccentricity λ is: 1/2 , λ = S y 2 + Sz 2

1/2 λ = (x tan ϕ + ρ sin α)2 + (x tan β + ρ cos α)2 , 2 2 1/2 2 2 λ = x tan ϕ + 2xρ tan ϕ sin α + x tan β + 2xρ tan β cos α + ρ2 . We introduce the angle γ, computed as a function of the eccentricity components: Sy , Sz x tan ϕ + ρ sin α . γ = arctan x tan β + ρ cos α γ = arctan

By analogy to the parallel case, see Sect. 5.1, we approximate the fluid film thickness as follows: h(ρ, α, ϕ, β, θ, x) = C + λ(ρ, α, ϕ, β, θ, x) cos(θ − γ(ρ, α, ϕ, β, x)),

(20)

where C represents the radial clearance and θ represents a point on the external circumference at the height Y = y. To simplify the notation, we drop arguments of function h, λ and γ from now on.

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The non-dimensional expression For the non-dimensional expression, we introduce the following non-dimensional variables in addition to the ones defined in Eq. (18): x¯ =

x , L

λ λ¯ = . C

(21)

Then, Eq. (20) can be rewritten as:

where

h¯ = 1 + λ¯ cos(θ − γ), ¯

(22)

1/2 1 (x¯ tan ϕ + C ρ¯ sin α)2 + (x¯ tan β + C ρ¯ cos α)2 λ¯ = C

(23)

and γ¯ = arctan

x¯ tan ϕ + C ρ¯ sin α . x¯ tan β + C ρ¯ cos α

(24)

Admissible range for the misalignment angle projections given ρ¯ and α In this section we present the admissible range of the misalignment angle projections ϕ and β to ensure we have no contact for given values of ρ¯ and α. In the normalized case, we have one point contact at x¯ = 0 and x¯ = 1. The solution of the former is trivial (ρ¯ < 1) and we will focus on the latter. We have the condition λ¯ 2 < 1 which expands to: (x¯ tan ϕ + C ρ¯ sin α)2 + (x¯ tan β + C ρ¯ cos α)2 < C 2 .

(25)

Working on the first term we get (x¯ tan ϕ + C ρ¯ sin α)2 < C 2 . Considering the negative solution and the contact: − tan ϕ − C ρ¯ sin α < C, −C(1 + ρ¯ sin α) < tan ϕ.

(26)

Considering the positive solution and the contact: tan ϕ + C ρ¯ sin α < C, tan ϕ < C(1 − ρ¯ sin α).

(27)

On the Reynolds Equation and the Load Problem in Lubrication …

17

Putting Eqs. (26) and (27) together we can set the range for tan ϕ: − C(1 + ρ¯ sin α) < tan ϕ < C(1 − ρ¯ sin α).

(28)

Following the same procedure we set the range for tan β as a function of tan ϕ: 1/2

− C 2 − (tan ϕ + C ρ¯ sin α)2 1/2

− C ρ¯ cos α < tan β < C 2 − (tan ϕ + C ρ¯ sin α)2 − C ρ¯ cos α.

(29)

Note that constraint (29) was obtained in the same way as the solution of inequality x 2 + y 2 < 1, which represents a circle, i.e. − 1 < x < 1, −(1 − x 2 )1/2 < y < (1 − x 2 )1/2 .

(30)

6 Derivation of the Reynolds Equation The derivation of the RE can be found in different references; see [65] for instance. To make this chapter self-contained we provide its derivation in the current section. Thus, we obtain the governing equation for an incompressible fluid, confined at small distance h between a surface and an object that slips on the first one; see Fig. 5. For a better understanding of the phenomenon, we will place the coordinate system on the object, as in Liñan [65], in such a way that we get a moving surface with respect to the object, that will remain fixed. The following assumptions are used in the analysis. The distance between the surface and the object in Fig. 5 is assumed to be defined by the function h(x, t), where 0 < h(x, t) < H˜ , with H˜ the largest distance that can exist between them. Similarly, 0 < h(x, t)  L and 0 < h(x, t)  B. The origin of the analysis are the conservation principles of NavierStokes for incompressible fluids, where u = (u, v, w) and v = (U, V, W ) stand for the fluid and object velocity vector respectively. Let , μ, g, p and t be the fluid density, the fluid viscosity, the gravity vector, the fluid pressure and the time, respectively. L B

h

Fig. 5 A fluid example, confined between a surface and a moving object

18

H. Lombera Rodríguez and J. I. Tello

Mass conservation for incompressible fluids ∇ · (u) = 0.

(31)

Momentum conservation for incompressible fluids 

∂u μ + (u∇u) = −g − ∇ p + ∇ · (∇u + (∇u)T ). ∂t 2

(32)

Let w = u − v be the relative velocity between the fluid and the moving object. We rewrite the Eq. (32) in terms of w:   ∂ (u − v) +  (u − v)∇(u − v) = −g − ∇ p ∂t μ + ∇ · (∇(u − v) + (∇(u − v))T ), 2   ∂  (u − v) +  (u − v)∇(u − v) = −g ∂t μ − ∇ p + ∇ · (∇u − ∇v + (∇u)T − (∇v)T ), 2 

Since v only depends on time, ∇v vanishes and we obtain: 

∂v μ ∂u − + (u − v)∇u = −g − ∇ p + ∇ · (∇u + (∇u)T ), ∂t ∂t 2 ∂v μ ∂u + u∇u =  + v∇u − g − ∇ p + ∇ · (∇u + (∇u)T ).  ∂t ∂t 2 (33)

We perform a dimensional analysis of Eqs. (31) and (33), for computing the order of magnitude of each term. We assume that the dimensions of the object are close, e.g. BL  1. The distance h  H˜ between the object and the surface is negligible ˜

˜

compared to the dimensions of the surface, HL  1 and HB  1. We assume that the velocity components have variations Uc , Vc , Wc in a characteristic time Tc . Thus, we have: Uc ∼

L B H˜ , Vc ∼ , Wc ∼ . Tc Tc Tc

Then: Uc L ∼ , Wc H˜

Wc H˜ ∼ 0, since H˜  L and therefore Wc  Uc . ∼ Uc L

(34)

On the Reynolds Equation and the Load Problem in Lubrication …

Similarly:

19

H˜ Wc ∼ 0, and Wc  Vc . ∼ Vc B

Performing the dimensional analysis in Eq. (31) we have: ∂u ∂x

+

∂v ∂y

∂w ∂z

+

Uc L

Vc B

Wc H˜

1 Tc

1 Tc

1 Tc

=

0,

using Eq. (34).

Therefore, all terms are comparable and equally important. Next, we analyse the x component of Eq. (33). Notice that in this direction there is no gravity component. ∂p μ ∂u ∂u ∂u ∂U ∂u ∂u ∂u T  ∂u ∂t + u ∂x + v ∂ y + w ∂z =  ∂t + U ∂x + V ∂ y + W ∂z − ∂x + 2 ∇ · (∇u + (∇u) ).

c U T c

 ULc

2

 VcBUc

 Wc˜Uc H

c U T c

 ULc

2

 VcBUc

 Wc˜Uc H

p L

(μ U2c μ U2c μ U˜ c2 ) L B H

The dominant term is μ HU˜ c2 . When comparing it with the remaining orders, the other terms vanish, except Lp and  UTcc . The former, because it involves the unknown, the latter because since Tc is large enough, the term HU˜ c2 UTcc . Finally, we get: −

∂2u ∂p + μ 2 = 0. ∂x ∂z

(35)

After applying the same analysis for the y component we have: −

∂2v ∂p + μ 2 = 0. ∂y ∂z

(36)

In the case of the z component we firstly analyse the orders of magnitude of Eqs. (35–36). p Uc =μ , L H˜ 2

p=μ

L2 . H˜ 2 Tc

20

H. Lombera Rodríguez and J. I. Tello

Similarly, p Vc , =μ B H˜ 2

p=μ

B2 . H˜ 2 Tc

Then, analysing the z component of Eq. (33) we obtain: + u ∂w + v ∂w + w ∂w  ∂w ∂t ∂x ∂y ∂z  WTcc

 UcLWc

 VcBWc

 WH˜c

2

and  ∂W + U ∂w + V ∂w + W ∂w − gˆ − ∂t ∂x ∂y ∂z  WTcc

 UcLWc

 VcBWc

 WH˜c

2

∂p ∂z

gˆ

p H˜

+

μ ∇ 2

· (∇w + (∇w)T ).

c c c (μ W μW μW ) L2 B2 H˜ 2

Applying the orders of magnitude of Eqs. (35–36) and the order order of magnitude of term

∂p : ∂z

p H˜

we get the real

∂p ∂z

L2 H˜ 2 Tc H˜

L2 , H˜ 3 Tc

Therefore, it is the dominant term. We can conclude that: −

∂p = 0. ∂z

(37)

Notice the physical meaning of Eq. (37) under the assumptions used in the derivation: the pressure variations in the height, are negligible with respect to the pressure variations along the other directions. Finally, we obtain the following system of equations:

On the Reynolds Equation and the Load Problem in Lubrication …

21

∂2u ∂p + μ 2 = 0, ∂x ∂z ∂2v ∂p + μ 2 = 0, − ∂y ∂z ∂p − = 0. ∂z with the following boundary conditions: ∂h , at z = h. ∂t u − U = v − V = w = W = 0, at z = 0. u = v = 0, w =

(38) (39)

We then integrate Eq. (35): −

∂2u ∂p +μ 2 ∂x ∂z ∂2u μ 2 ∂z  z ∂2u μ 2 dz ∂z 0 ∂u ∂z  z ∂u dz 0 ∂z

= 0,

∂p , ∂x  z ∂p dz, = 0 ∂x 1 ∂p = z + C1 , μ ∂x  z 1 ∂p = (z + C1 )dz, μ ∂x 0 1 ∂ p z2 + C1 z + C2 , u= μ ∂x 2 z2 ∂ p + C1 z + C2 . u= 2μ ∂x =

(40)

Substituting and evaluating the initial condition u(0) = U we obtain: U = C2 .

(41)

Substituting and evaluating the initial condition u(h) = 0, together with the result in Eq. (41) we have: h2 ∂ p + C1 h + U, 2μ ∂x h ∂p U . C1 = − − h 2μ ∂x 0=

(42)

22

H. Lombera Rodríguez and J. I. Tello

Substituting results from Eqs. (42) and (41) in Eq. (40) we obtain: z2 ∂ p  U h ∂p z + U, + − − 2μ ∂x h 2μ ∂x h ∂p z2 ∂ p U − z− z + U, u= 2μ ∂x h 2μ ∂x z 1 ∂p z(z − h) + U (1 − ). u= 2μ ∂x h

u=

(43)

Similarly, we obtain the expression for the v component: v=

1 ∂p z z(z − h) + V (1 − ). 2μ ∂ y h

(44)

This way, we obtain the expressions for the velocity components in terms of the pressure gradient and the object velocity. Then, we must find p which is independent of z. Therefore, we define the flow rate qi as the amount of fluid flowing in the direction i, from 0 to h. The flow rate qx is: 

h

qx =



 1 ∂p z  z(z − h) + U (1 − ) dz, 2μ ∂x h 0 3 3 h ∂p Uh h ∂p − + Uh − , = 6μ ∂x 4μ ∂x 2 h

udz =

0

2h 3 ∂∂xp − 3h 3 ∂∂xp Uh + , 2 12μ h3 ∂ p Uh − . = 2 12μ ∂x =

(45)

Similarly, we found the expression for the flow rate q y :  qy =

h

vdz =

0

h3 ∂ p Vh − . 2 12μ ∂ y

(46)

We now substitute the expressions for u, v, w in the mass conservation Eq. (31), and we integrate along the z direction:  0

h

∂u dz + ∂x



h 0

∂v dz + ∂y

 0

h

∂w dz = 0. ∂z

(47)

We then apply the Leibniz’s integral rule for the first two terms: ∂ ∂x



h 0

udz − u(h)

∂0 ∂ ∂h + u(0) + ∂x ∂x ∂y



h 0

vdz − v(h)

h ∂0 ∂h  + v(0) + w  = 0. 0 ∂y ∂y

On the Reynolds Equation and the Load Problem in Lubrication …

Since u(h) = v(h) =

∂0 ∂x

=

∂0 ∂y



h

∂ ∂x

0

= 0 and w(h) = ∂ udz + ∂y



h

∂h ∂t

vdz +

0

23

we have: ∂h = 0. ∂t

(48)

Substituting the flow rates of Eqs. (45) and (46) in Eq. (48) we obtain: ∂ Uh h3 ∂ p  ∂ Vh h 3 ∂ p  ∂h − + − + = 0, ∂x 2 12μ ∂x ∂y 2 12μ ∂ y ∂t ∂ Uh  ∂  h3 ∂ p  ∂ Vh ∂  h 3 ∂ p  ∂h − + − + = 0, ∂x 2 ∂x 12μ ∂x ∂y 2 ∂ y 12μ ∂ y ∂t ∂ ∂  3∂p ∂ ∂  3∂p ∂h 6μ Uh − h + 6μ Vh − h + 12μ = 0, ∂x ∂x ∂x ∂y ∂y ∂y ∂t from which we obtain the well-known RE for incompressible fluids: − ∇ · (h 3 ∇ p) = −6μ∇ · (hv) − 12μ

∂h , ∂t

(49)

where h stands for the fluid film thickness, μ stands for the fluid viscosity, and v stands for the velocity of the moving object.



7 The Reynolds Cavitation Model In this section we present the governing equations in a stationary regime for the hydrostatic pressure in a journal bearing, considering a small fluid film thickness and the cavitation phenomenon. Namely, we introduce the Reynolds cavitation model, which includes boundary conditions for film exit that were previously proposed in the works of Swift and Stieber; see [91, 95]. In fact, this model is known as Reynolds cavitation model or Swift-Stieber model. Let L, R be the length and the cross section radius respectively of the shaft. We consider the circumferential coordinate unfolded y ∈ (0, 2π R) and the axial dimension x ∈ [0, L]. We define the region  : [0, 2π R] × [0, L]. The unknowns of the problem are: p(y, x) :  → R+ , ρ ∈ [0, C), α ∈ [0, 2π], β ∈ [−2C/L , 2C/L], ϕ ∈ [−2C/L , 2C/L],

fluid pressure, left end-plane eccentricity, left end-plane angular position, misalignment angle projection on x z-plane, misalignment angle projection on x y-plane,

and they satisfy the constraint C 2 > λ2 ,

(50)

24

H. Lombera Rodríguez and J. I. Tello

i.e. C 2 > x 2 tan2 β + 2xρ tan ϕ sin α + x 2 tan2 β + 2xρ tan β cos α + ρ2 . Under certain operating conditions of a journal bearing, the fluid pressure can reach a minimum value, related to the lubricant vapour pressure. Below this value, cavitation occurs, and the cavitated area is filled by a vapour at a constant pressure pc . We will consider pc = 0 as an approximation of the pressure value at which the phenomenon occurs. In this region the RE is no longer valid and a cavitation model is needed to describe the phenomenon. We look for a function p ≥ 0 in  satisfying the RE, where p > 0 (the lubricated region). We assume there is not mass exchange through the free boundary which separates both regions (the lubricated region and the cavitated region) and we consider ∂ p/∂n = 0, with n the unitary normal vector to the free boundary. We then use the Reynolds cavitation model to describe this phenomenon, whose weak formulation is given by the following inequality: 

 

h 3 ∇ p∇(φ − p)d ≥



6μhv∇(φ − p)d ∀φ, p ∈ K ,

(51)

with    1/2  1/2 K = φ ≥ 0, (∇φ)2 d + φ2 d < ∞, φ(y, 0) = φ(y, L) = pa , 



(52) pa = atmospheric pressure.

In Eq. (51), μ stands for the fluid viscosity, h stands for the fluid film thickness and the velocity v = (U, V, W ) makes reference to the shaft velocity vector. For steadystate bearing operation the shaft presents only one non-zero velocity component, W = ω R, with ω the angular velocity. Then Eq. (51) becomes: 

 h ∇ p∇(φ − p)d ≥ 3



6μhW 

∂(φ − p) d, ∀φ ∈ K , ∂y

(53)

defined in Eq. (52). We define the bilinear form a and the function f by:  a(u, v) :=



h 3 ∇u∇vd,

f := −6μ∇ · (hv) ∈ H −1 (), where H −1 () is the dual space of H01 (), a Sobolev’s space. We then reformulate Eq. (53) as follows: a(u, v − u) ≥ < f, v − u > .

On the Reynolds Equation and the Load Problem in Lubrication …

25

We express the problem as a minimization problem of the following convex functional, J (v) =

1 a(v, v)− < f, v > on K . 2

(54)

Taking the parameters of Eq. (53) and substituting them in Eq. (54) we obtain the dimensional functional to minimize:   ∂φ 1 3 T d, (55) h (∇φ) ∇φd − 6μhW J (φ) = 2  ∂y  with h defined in Eq. (20).

7.1 The Non-dimensional Reynolds Cavitation Model To make our solution to suffice for a variety of different problems, we introduce the following non-dimensional variables in addition to the ones defined in Eqs. (18) and (21): θ=

y , R

dy = Rdθ,

dx = Ld x, ¯

W =

W , ωR

(56) φ¯ =

C2 φ, μ0 ω R 2

p¯ =

C2 μ0 ω R 2

p,

λ¯ =

z¯ = hz ,

λ , C

μ¯ =

μ , μ0

where μ0 stands for the reference viscosity. Thus, we transform our domain into the ¯ = [0, 2π] × [0, 1] for the (θ, x) dimensionless domain  ¯ coordinates. Then, the dimensionless equation for the functional in Eq. (55) is: ¯ = J (φ)

1 2

 ¯ 

¯ T ∇ φd ¯  ¯ − L R h¯ 3 (∇ φ)

 ¯ 

¯ 6μ¯ hW

μ0 ωL R 2 ∂ φ¯ ¯ d, C2 ∂θ

(57)

with h¯ defined in Eq. (22):

8 Derivation of the Generalized Reynolds Equation The derivation of the Generalized Reynolds Equation (GRE) is a well-known result that can be found in the literature; see Dowson [40] for instance. However, we provide its derivation in the current section to make this chapter self-contained. The geometry and coordinate system are shown in the cross section of a journal bearing shown in

26

H. Lombera Rodríguez and J. I. Tello

Fig. 3. Suffixes 1 and 2 will be used to denote conditions on surfaces z = 0 and z = h, respectively. The following assumptions are used in the analysis as in Dowson [40]: 1. The radius of curvature of the bearing components is large compared with the film thickness. 2. The lubricant is a Newtonian and isoviscous fluid. 3. Inertia and body force terms are small compared with the viscous and pressure terms in the equations of motion. 4. Owing to the geometry of the fluid film the derivatives of u and v with respect to z are large compared with all other velocity gradients. 5. There is no slip between the fluid and boundary solids at common boundaries. After applying an order of magnitude analysis to the Navier-Stokes equations for Newtonian fluids we obtain the following system of equations. See Sect. 6 for details: ∂ ∂p = ∂x ∂z ∂p ∂ = ∂y ∂z ∂p = 0. ∂z

  ∂u μ , ∂z  μ

∂v ∂z

(58)

 ,

(59) (60)

The gradient of the velocity component u across the film can be found by integrating Eq. (58).  0

z

∂ ∂z

   z ∂u ∂p μ dz = dz, ∂z 0 ∂x ∂u ∂p z B(x, y) = + . ∂z ∂x μ μ

(61)

Integrating again, introducing the following boundary conditions z = 0,

u = U1 ,

v = V1 ,

(62)

z = h,

u = U2 ,

v = V2 ,

(63)

we have: 

h 0

 h  h ∂u ∂p z B(x, y) dz = dz + dz, ∂z μ 0 ∂x μ 0  h  h 1 ∂p h z  u = dz + B(x, y) dz, 0 ∂x 0 μ μ 0

On the Reynolds Equation and the Load Problem in Lubrication …

27

h z U2 − U1 ∂ p 0 μ dz B(x, y) =  h − .  1 ∂x h 1 dz dz 0 μ 0 μ Defining



h

F0 = 0



1 dz, μ

we have:

h

F1 = 0

(64)

z dz, μ

(65)

U2 − U1 ∂ p F1 − . F0 ∂x F0

B(x, y) =

(66)

The equation for the u component would be: ∂p u(z) = u(0) + ∂x



z

0



s ds + B(x, y) μ

z 0

1 ds. μ

(67)

Substituting the proper boundary condition from Eqs. (63) and (66) in Eq. (67) we have:  z   U2 − U1 1 ∂p z s ∂ p F1 u(z) = U1 + ds + ds. (68) − ∂x 0 μ F0 ∂x F0 μ 0 Similarly, the equation for the v component after applying the proper boundary conditions is:  z   V2 − V1 1 ∂p z s ∂ p F1 ds + ds. (69) v(z) = V1 + − ∂y 0 μ F0 ∂ y F0 0 μ We then return to the continuity equation and perform an integration with respect to z between the limits 0 and h: 

h

0

∂ dz + ∂t



h

0

∂(u) dz + ∂x



h

0

∂(v) dz + ∂y



h

0

∂(w) dz = 0. ∂z

(70)

We then apply the Leibniz’s integral rule: 

h2 h1

∂ ∂ f (x, y, z) dz = ∂x ∂x



h2

f (x, y, z)dz − f (x, y, h 2 )

h1

∂h 2 ∂h 1 + f (x, y, h 1 ) , ∂x ∂x

which gives:  0

h

∂ ∂ dz + ∂t ∂x

 0

h

(u)dz +

∂ ∂y

 0

h

(v)dz − (U )2

∂h ∂h − (V )2 + [w]0h = 0. ∂x ∂y

(71)

28

H. Lombera Rodríguez and J. I. Tello

The integrals of (u) and (v) are evaluated by parts to give:  0

h

  h  h  ∂ ∂ ∂ ∂u  z dz + (uz) − u+  dz 0 ∂t ∂x ∂z ∂z 0   h  h  ∂ ∂v ∂  (vz) − v+  dz + z 0 ∂y ∂z ∂z 0 ∂h ∂h − (V )2 + [w]0h = 0. − (U )2 ∂x ∂y

(72)

We then expand and simplify to obtain: 

h 0

   h ∂ ∂ ∂(U )2 ∂(V )2 ∂u ∂ zu dz + h + − + z dz ∂t ∂x ∂y ∂x 0 ∂z ∂z   h ∂ ∂ ∂v zv dz + [w]0h = 0. − + z ∂y 0 ∂z ∂z

∂ Substituting Eqs. (61) and (68) in term − ∂x

h 0

(73)

 zu ∂ dz of Eq. (73): + z ∂u ∂z ∂z

 z     h  U2 − U1 ∂ 1 ∂ ∂p z s ∂ p F1 U1 + ds + ds dz z − ∂x 0 ∂z ∂x 0  F0 ∂x F0 0 μ    h  1 U2 − U1 ∂ ∂p z ∂ p F1 dz. + − z − ∂x 0 ∂x μ μ F0 ∂x F0 −

We then expand to get:  h  z ∂ ∂ ∂ ∂ p U1 dz − dz z ∂z ∂x 0 ∂z ∂x 0 0  h  z  h ∂ ∂ p F1 1 ∂ ∂ + ds − z dz ∂x 0 ∂z ∂x 0 0 ∂x F0 μ  h z ∂ p F1 ∂ + dz. ∂x 0 μ ∂x F0



∂ ∂x



h

z

Grouping terms with

and

U2 −U1 F0



h

z

respectively in Eq. (74) we have:

 z  h  z  h 2 ∂ ∂ ∂ ∂ p ∂ s ∂ p F1 1 z  ∂p dz ds + ds − dz z z dz ∂z ∂x  ∂x ∂z ∂x F μ ∂x μ ∂x 0 0 0 0 0 0   z   h  h  h z ∂ p F1 1 ∂ ∂ ∂ ∂ ∂ U2 − U1 −+ dz ds dz z U1 dz − z ∂x 0 μ ∂x F0 ∂x 0 ∂z ∂x 0 ∂z F0 μ 0  h z U2 − U1 ∂ dz. − ∂x 0 μ F0

∂ − ∂x



∂p ∂x

   z 1 ∂ U2 − U1 ds dz ∂z F0 0 0 μ  h z2 ∂ p z U2 − U1 ∂ dz − dz (74) μ ∂x ∂x 0 μ F0

s ∂ ds −  ∂x

h

On the Reynolds Equation and the Load Problem in Lubrication …

29

We then apply a factorization:   h   z  h z s ∂ ∂ F1 − dz + z z− ds − ∂x F0 ∂z 0 μ 0 0 μ   h    h  ∂ z U2 − U1 ∂ ∂ ∂ z − z U1 dz − ∂x 0 ∂z ∂x F0 ∂z 0 0 Defining F2 = F3 =



h

0 h 0

   ∂p F1 z ds dz F0 0 μ ∂x   h ds z dz + dz . μ 0 μ (75)

   z   h  ∂ F1 F1 z ds z s dz, G 1 = z z− ds − dz, μ F0 F0 0 μ 0 h  ∂z  z0 μ ∂ z ds z dz, G2 = dz, μ 0 h ∂z 0 μ ∂ G3 = z dz, ∂z 0

we have:    ∂ U2 − U1 ∂ ∂p − − (F3 + G 2 ) + U1 G 3 . (F2 + G 1 ) ∂x ∂x ∂x F0 Similarly, we derive the expressions for ∂v and ∂z   h  ∂ ∂ ∂v − ∂ y 0 zv ∂z + z ∂z dz of Eq. (73) we yield: −

∂ . ∂z

(76)

Substituting them in term

   ∂ ∂ V2 − V1 ∂p − (F3 + G 2 ) + V1 G 3 . (F2 + G 1 ) ∂y ∂y ∂y F0

(77)

Substituting Eqs. (76) and (77) in Eq. (73) we have:   h ∂(U )2 ∂(V )2 ∂ dz + h + ∂x ∂y 0 ∂t    ∂ U2 − U1 ∂p ∂ − − (F2 + G 1 ) (F3 + G 2 ) + U1 G 3 ∂x ∂x ∂x F0    ∂ V2 − V1 ∂p ∂ − − (F2 + G 1 ) (F3 + G 2 ) + V1 G 3 + (W )2 − (W )1 = 0. ∂y ∂y ∂y F0

(78) Balancing both sides of Eq. (78) we found the GRE:    ∂ ∂(U )2 ∂(V )2 ∂ ∂p ∂p + =h + (F2 + G 1 ) (F2 + G 1 ) ∂x ∂x ∂y ∂y ∂x ∂y

30

H. Lombera Rodríguez and J. I. Tello



∂ ∂x



U2 − U1 F0



  V2 − V1 ∂ (F3 + G 2 ) + U1 G 3 − (F3 + G 2 ) + V1 G 3 ∂y F0

 h ∂ dz + (W )2 − (W )1 , + 0 ∂t

where F0 = F1 = F2 = F3 =



h

0 h 0 h 0 h 0

(79)

 z   h  ∂ F1 z ds 1 s z dz, G1 = ds − dz, μ F0 0 μ 0 h  ∂z  z0 μ ∂ z ds z dz, dz, G2 = μ  ∂z 0 μ  0 h F1 z ∂ dz, G 3 = z− z dz. μ F0 ∂z 0 z dz, μ

9 Simplifications to the GRE for Journal Bearings The following assumptions are used in the simplification. We consider in the lefthand side of Eq. (79) the components of the pressure-driven flow (Poiseuille flow) along both circumferential “y” and axial “x” directions. Similarly, we consider in the right-hand side of Eq. (79) the component of the drag-driven flow (Couette flow) which is only present along the circumferential direction. We also consider an incompressible fluid ( = const) and a stationary bearing (V1 = 0). In addition, there is no slip between the shaft and the fluid, so U2 = ω R. Under these assumptions Eq. (79) reduces to:    ∂ ∂V2 ∂ ∂p ∂p F3 ∂ + (W2 − W1 ). F2 + F2 = h − V2 ∂y ∂y ∂x ∂x ∂y ∂y F0

(80)

Then, the fluid density vanishes and in consequence F3 = F1 . Thus, Eq. (80) becomes:    ∂ ∂(hV2 ) ∂ ∂p ∂p F1 ∂h ∂ − V2 F2 + F2 = − V2 + (W2 − W1 ). ∂y ∂y ∂x ∂x ∂y ∂y F0 ∂y (81) For journal bearings, generally W2 − W1 = V2 ∂∂hy , see Jang and Khonsari [55] for details. Then, we obtain the simplified GRE for journal bearings:      F1 ∂ ∂ ∂p ∂p ∂ . F2 + F2 = V2 h − ∂y ∂y ∂x ∂x ∂y F0

(82)

On the Reynolds Equation and the Load Problem in Lubrication …

31

9.1 The Non-dimensional GRE for Journal Bearings To make Eq. (82) to suffice for a variety of different problems, we follow the nondimensional variables defined in Eqs. (18), (21) and (55). As in Sect. 7.1, we transform ¯ = [0, 2π] × [0, 1] for the (θ, x) our domain into the dimensionless domain  ¯ coordinates. Since we consider an isoviscous lubricant the non-dimensional viscosity integrals are defined as follows, see [20] for details.  F0 =

1

0

 F2 =

1

0

with F0 =

¯ F0 hC , μ0

 1 1 1 1 z¯ d¯z = , d¯z = , F1 = μ¯ μ¯ ¯ 2μ¯ 0 μ

 z¯ F1 2F0 − 3F1 , d¯z = z¯ − μ¯ F0 6μF ¯ 0

F1 =

h¯ 2 C 2 F1 , μ0

F2 =

h¯ 3 C 3 F2 . μ0

(83)

Then, by applying the chain rule, Eq. (82) is posed into the normalized reference domain:     ∂ h¯ 3 C 3 F2 μ0 ω R 2 ∂ p¯ 1 ∂ h¯ 3 C 3 F2 μ0 ω R 2 ∂ p¯ 1 R + ∂θ μ0 C 2 ∂θ R ∂ x¯ μ0 C 2 ∂ x¯ L L ⎞⎤ ⎡ ⎛ ¯ 2 C 2 F1 ∂ ⎣ h ¯ − ⎠⎦ (84) = ω R ⎝hC ¯ F0 μ0 hC ∂θ μ0

After simplifying we obtain the non-dimensional RE for journal bearings:

     R 2 ∂ ¯ 3 ∂ p¯ ∂ ¯ ∂ ¯ 3 ∂ p¯ F1 + 2 = . h F2 h F2 h 1− ∂θ ∂θ L ∂ x¯ ∂ x¯ ∂θ F0

(85)

˜ = [0, 2π] × [0, L ] by introducing the new variWe transform the domain into  R L able x˜ = x¯ R . Equation (85) becomes:

     ∂ ˜ 3 ∂ p¯ ∂ ˜ ∂ ˜ 3 ∂ p¯ F1 + = . h F2 h F2 h 1− ∂θ ∂θ ∂ x˜ ∂ x˜ ∂θ F0

(86)

where: h˜ = 1 + λ˜ cos(θ − γ), ˜

(87)

32

with

H. Lombera Rodríguez and J. I. Tello

1/2 1 (x˜ tan ϕ + C ρ¯ sin α)2 + (x˜ tan β + C ρ¯ cos α)2 λ˜ = C

and γ˜ = arctan

x˜ tan ϕ + C ρ¯ sin α . x˜ tan β + C ρ¯ cos α

(88)

(89)

¯ to , ˜ the It is important to remark that, due to the domain transformation from  admissible range of the misalignment angle projections provided in Eqs. (28) and (29) for prescribed values of ρ¯ and α changes to: −

CR CR (1 + ρ¯ sin α) < tan ϕ < (1 − ρ¯ sin α) . L L

⎞ ⎛ 2 1/2  L R⎝ 2 tan ϕ + C ρ¯ sin α − C ρ¯ cos α⎠ < tan β − C − L R ⎞ ⎛ 2 1/2  R⎝ 2 L tan β < tan ϕ + C ρ¯ sin α − C ρ¯ cos α⎠ . C − L R

(90)

(91)

10 The Elrod–Adams Cavitation Model An additional unknown is introduced to address the cavitation phenomenon in the Elrod–Adams model, the saturation ϑ that represents the lubricant concentration. As it was explained in Sect. 2, when cavitation occurs, the domain presents two regions. ˜ into two parts: the active zone + where positive Thus, we split the domain  pressure values are present with the GRE governing the lubricant behaviour, and the 0 where the pressure is zero and the conservation of mass equation needs to ˜ including its be changed. See the whole configuration of the normalized domain  ˜ in Fig. 6, i.e. boundary ∂  ˜ = + ∪ 0 ,  ˜ = 0 ∪  1 ∪  2 . ∂

(92) (93)

The saturation ϑ takes the value 1 in the active zone + and takes any other value in the range [0, 1] within the cavitated region 0 . Notice that 0 is the zero width boundary where the lubricant is axially supplied through, with known concentration (ϑ = ϑ0 );  depicts the free boundary between the active zone and the cavitated one. The hydrodynamic problem is stated as follows. To find ( p, ¯ ϑ) such as the following conditions are verified:

On the Reynolds Equation and the Load Problem in Lubrication …

33

˜ Fig. 6 Configuration for the normalized hydrodynamic domain 

     ∂ ˜ 3 ∂ p¯ ∂ ˜ ∂ ˜ 3 ∂ p¯ F1 + = , h F2 h F2 h 1− ∂θ ∂θ ∂ x˜ ∂ x˜ ∂θ F0 



∂ F1 ϑh˜ 1 − ∂θ F0

p¯ > 0 and ϑ = 1 in + ,

(94)

 = 0,

p¯ = 0, 0 ≤ ϑ ≤ 1 in 0 ,

 ˜h 3 F2 ∂ p¯ = (1 − ϑ)h˜ 1 − ∂n

F1 F0

(95)

cos(n, i),

p¯ = 0 in , (96)

ϑ = ϑ0 in 0 , p¯ = p¯ s in 0 ,

(97) (98)

˜ − 0 , p¯ = 0 in ∂  p¯ s = normalized supply pressure,

(99) (100)

where n is the normal vector to  and i is the unitary vector in the θ direction. Equation (94) states that the pressure, in the active zone is governed by the GRE. Equation (95) states that in the cavitated zone, the mass conservation law must be satisfied in the θ direction. Equation (96) is the flow continuity condition through the free boundary between the active and the cavitated zone. Equations (97–100) correspond to the boundary conditions related to the concentration and pressure at the supply groove, and the pressure at the front and back boundaries of the device, respectively.

10.1 Deduction of the Flow Continuity Condition Through the Free Boundary The amount of flux in a fluid element is the difference between the amount of incoming and outcoming fluxes. Mathematically it is:  ˜ 

˜ = ∇ · (h˜ 3 F2 ∇ p)d ¯ 



∂ p¯ ds. h˜ 3 F2 ∂n ˜ ∂

(101)

34

H. Lombera Rodríguez and J. I. Tello

It is important to remark that the flux has two components: the pressure-driven flow (Poiseuille flow) along both circumferential “y” and axial “x” directions and the dragdriven flow (Couette flow) which is only present along the circumferential direction. The flux in + is: 

˜h 3 F2 ∇ p¯ − h˜ 1 − F1 i, (102) F0 where i is the standard basis vector i = (1, 0), pointing to the θ direction in this case. At the boundary of + the flux comprises the normal and the tangential components. Since a mass interchange occurs along the normal direction, we have: 

˜h 3 F2 ∂ p¯ − h˜ 1 − F1 i · n. ∂n F0

(103)

In the same way, the amount of outcoming mass from 0 through the boundary  is: 

F1 ˜ ϑh 1 − i · n. (104) F0 Then, there must be a balance, so:



 ˜h 3 F2 ∂ p¯ = h˜ 1 − F1 i · n − ϑh˜ 1 − F1 i · n, ∂n F0 F0

(105)

i.e. 

∂ p¯ F 1 = (1 − ϑ)h˜ 1 − h F2 cos(i, n). ∂n F0 ˜3

(106)

11 Weak Formulation of the Elrod–Adams Cavitation Model The starting point of this derivation is the strong form of the Elrod-Adams model provided by Eqs. (94–100). We consider the space of test function: # " ˜ : φ¯ |∂ − = 0 . K = φ¯ ∈ H 1 () ˜ 0 We first multiply by φ¯ such as φ¯ |1 ∪2 = 0,

(107)

On the Reynolds Equation and the Load Problem in Lubrication …

 

 F1 3 ¯ ¯ ˜ ˜ ¯ = φ∇ · h 1 − φ∇ · (h F2 ∇ p) ,0 F0

p¯ > 0, ϑ = 1 in + .

35

(108)

The boundary of + denoted by ∂+ is split into two parts 2 and , i.e. ∂+ = 2 ∪ .

(109)

So, we apply Green’s formula on the left-hand side and we have: 

  ∂ p¯ ¯ ∂ p¯ ¯ ¯ + (h˜ 3 F2 ∇ p) ¯ ∇ φd h˜ 3 F2 h˜ 3 F2 φ d2 + φ d − ∂n ∂n 2  +  

  F 1 ¯ = ∇ · h˜ 1 − , 0 φ. F0 +

(110)

Similarly, we apply Green’s formula on the right-hand side of Eq. (110) to have: 

  ∂ p¯ ¯ ∂ p¯ ¯ ¯ + (h˜ 3 F2 ∇ p) ¯ ∇ φd h˜ 3 F2 h˜ 3 F2 φ d2 + φ d − + ∂n ∂n 2  

    F 1 = h˜ 1 − , 0 n φ¯ d2 F0 2



       F F 1 1 ¯ +. + h˜ 1 − , 0 n φ¯ d − h˜ 1 − , 0 ∇ φd F0 F0  +

(111)

Since the function φ¯ is chosen such as φ¯ |2 = 0 we have: 

∂ p¯ ¯ h˜ 3 F2 φ d − ∂n 

    F 1 ¯ += (h˜ 3 F2 ∇ p) ¯ ∇ φd h˜ 1 − , 0 n φ¯ d F0 + 

    F 1 ¯ +. − h˜ 1 − , 0 ∇ φd F0 + (112)



We now multiply by the function φ¯ the Eq. (95). Since ∂0 = 0 ∪ 1 ∪  and ϑ = ϑ0 in 0 we have after applying Green’s formula:

36

H. Lombera Rodríguez and J. I. Tello



   



  F F 1 1 ¯ ϑ0 h˜ 1 − ϑ h˜ 1 − , 0 n φ d0 + , 0 n φ¯ d1 F0 F0 0 1    



   F F 1 1 ¯ 0 = 0. + ϑ h˜ 1 − ϑ h˜ 1 − , 0 n φ¯ d − , 0 ∇ φd 0 F0 F0   (113)

Since the function φ¯ is chosen such as φ¯ |1 = 0 we have: 

 

 F 1 ϑ0 h˜ 1 − , 0 n φ¯ d0 F0 0    



   F F 1 1 ¯ 0 = 0, + ϑ h˜ 1 − ϑ h˜ 1 − , 0 n φ¯ d − , 0 ∇ φd F0 F0  0 (114)

i.e.  



 F1 F1 ¯ ˜ ˜ ϑ0 h 1 − ϑh 1 − cos(n, i) φ d0 + cos(n, i) φ¯ d F0 F0 0  

 ∂ φ¯ 0 F 1 d = 0. ϑh˜ 1 − (115) − F0 ∂θ 0



We then substitute Eq. (96) in Eq. (112) and obtain: 

 

F1

(1 − ϑ)h˜ 1 −

=



 

h˜ 1 −

F1

cos(n, i) φ¯ d −

F0

F0

cos(n, i) φ¯ d −





+

+

¯ + (h˜ 3 F2 ∇ p) ¯ ∇ φd 

h˜ 1 −

F1 F0



∂ φ¯ + d . ∂θ

(116)

Reducing like terms we have:  −



 ϑh˜ 1 −

=−



 +

F1



F0

h˜ 1 −

cos(n, i) φ¯ d − F1 F0



 +

¯ + (h˜ 3 F2 ∇ p) ¯ ∇ φd

∂ φ¯ + d . ∂θ

Then, adding up Eqs. (115) and (117) and reducing like terms we obtain:

(117)

On the Reynolds Equation and the Load Problem in Lubrication …

 −

+

37

 



 ¯ F1 ∂ φ¯ ˜ ˜h 1 − F1 ∂ φ d+ + d0 ϑh 1 − F0 ∂θ F0 ∂θ + 0 

 F1 − ϑ0 h˜ 1 − (118) cos(n, i) φ¯ d0 . F0 0

¯ + =− (h˜ 3 F2 ∇ p) ¯ ∇ φd



˜ = + ∪ 0 , ϑ = 1 in + and p¯ = 0 in 0 we have: Taking into account that   −

˜ 

¯  ˜ =− (h˜ 3 F2 ∇ p) ¯ ∇ φd 





0

ϑ0 h˜ 1 −

F1 F0







F1 ∂ φ¯ ˜ ˜ d ϑh 1 − ˜ F0 ∂θ 

cos(n, i) φ¯ d0 ,

⎧ if p¯ > 0, ⎨ 1, ϑ ∈ H ( p) ¯ = [0, 1], if p¯ = 0, ⎩ 0, if p¯ < 0,

(119)

(120)

where the space of test functions is given by Eq. (107). By using integration by parts we rewrite Eq. (119) as follows:  −

˜ 

¯  ˜ = (h˜ 3 F2 ∇ p) ¯ ∇ φd



∂ ˜ ∂θ 



 F1 ˜ ˜ ∀φ¯ ∈ K, ϑ = ϑ0 in 0 . φ¯ d, ϑh 1 − F0



(121)

12 Shaft Stationary Model The shaft stationary model is based on the balance of force and torque components involved in the device. For that, the hydrodynamic RE is coupled to Newton’s second law. It is well-known that the pressure p is a physical magnitude that measures the force projection in a perpendicular direction to the surface per unit area. Then, the resultant dimensional fluid film force components, acting on the bearing and accordingly to Fig. 4 are:   

p(θ, x) sin θdθdx = Fy ,

(122)

p(θ, x) cos θdθdx = Fz ,

(123)

where sin θ and cos θ stand for the unitary normal vector components to the bearing surface.

38

H. Lombera Rodríguez and J. I. Tello

Moreover, to determine the resultant torque generated by the fluid force on the shaft, we start finding the torque in an arbitrary position vector q located on the bearing surface with coordinates q = (x, Rb sin θ, Rb cos θ) with Rb the bearing radius. The coordinate system corresponds to the one in Fig. 4. Let p be, the pressure at the position vector q. The force Fq exerted by the fluid pressure at that point will be: Fq = (0, p sin θ, p cos θ), Mathematically the torque τ is defined as a cross product of two vectors, which produces rotation: τ = r × Fq , where r represents the displacement vector from the rotation axis to the point where Fq is applied. Thus, the torque generated at the position vector q is: τ = q × Fq , = (x, Rb sin θ, Rb cos θ) × (0, p sin θ, p cos θ), = (0, −x p cos θ, x p sin θ). Then, the resultant dimensional torque components acting on the bearing are:  −

 

x p(θ, x) cos θdθdx = τy ,

(124)

x p(θ, x) sin θdθdx = τz .

(125)

Notice that the equilibrium of the external torque τ and the one exerted by the fluid film is taken also with respect to the origin O. Accordingly to Eq. (55), we obtain the following dimensionless expressions for the equilibrium of force components:  μ0 ωL R 3 1 p¯ sin θdθd x¯ = F¯y , |F| ¯ C2  1 μ0 ωL R 3 p¯ cos θdθd x¯ = F¯z , |F| ¯ C2

(126) (127)

where |F| denotes the modulus of the external load F, used for scaling. The righthand side terms F¯y and F¯z stand for the normalized components of F. Similarly, the dimensionless expressions for the torque are:  μ0 ωL 2 R 3 1 x¯ p¯ cos θdθd x¯ = τ¯y , − |τ | ¯ C2  1 μ0 ωL 2 R 3 x¯ p¯ sin θdθd x¯ = τ¯z , |τ | ¯ C2

(128) (129)

On the Reynolds Equation and the Load Problem in Lubrication …

39

where |τ | denotes the modulus of the external torque τ , used for scaling. The righthand side terms τ¯y and τ¯z stand for the normalized components of τ .

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On Dynamic Interactions Between Body Motion and Fluid Motion Frank T. Smith, Samire Balta, Kevin Liu and Edward R. Johnson

Abstract This contribution on dynamic fluid-body interactions concentrates on applying mathematical/analytical ideas to complement direct numerical studies. The typical body may be of given shape or flexible depending on the context. In the background there are numerous real-world motivations in industry, biomedical and environmental applications, many of which involve high flow rates. A review of ideas developed over the last decade for cases of high flow rates first addresses inviscid approaches to one or more bodies free to move within a channel flow, a skimming sharp-edged body on a free surface, the sinking of a body in water and the rocking or rolling of a body on a solid surface, before moving on to more recent viscous-inviscid approaches for channel flows and boundary layers. The beginnings of certain current research projects are also outlined. These concern models of liftoff of a body from a solid surface, the impact of a smooth body during skimming and viscous-inviscid effects in the presence of more than one freely moving body. Linear and nonlinear mathematical properties as appropriate are described. Keywords Fluid-solid interactions · Channel flow · Skimming · Lift-off

1 Introduction Dynamic fluid-body interactions where the body and fluid motions generally affect each other about equally [1–11] are the basis for this chapter, with some review and some new developments. The motivation comes from applications as well as F. T. Smith (B) · S. Balta · K. Liu · E. R. Johnson University College London, London WC1E 6BT, UK e-mail: [email protected] S. Balta e-mail: [email protected] K. Liu e-mail: [email protected] E. R. Johnson e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_2

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scientific interest. There have been a fair number of computational, experimental or observational studies in the area [12–22] but comparatively few analytical studies; the latter tend to lead to complementary understanding of parametric and physical effects. Our aim indeed is to combine modelling, analysis, reduced computation and experimental links where possible. The present contribution addresses several different scenarios, for example with viscous or inviscid fluid, with internal or external flow, with a free surface or without. Applications and motivation. The study of interactions between moving solid or flexible bodies (including particles, grains) and the surrounding fluid has many environmental, industrial and biomedical applications. Numerous applications arise across nature such as with falling leaves and moving seeds and coffee grains, not to forget the motion of frozen ice particles and hailstones as well as sedimentation and fluidisation phenomena. Applications also arise in sporting contexts such as running and cycling groups and to some extent in swimming competitions and in surfboarding. The behaviours of various swarms similarly have an interactive fluid-dynamical element to them. Three industrial applications are concerned with the falling of lumps of ice into an engine intake in an aerodynamic safety context, the travel of windblown particles of ice along a wing surface again in the aerodynamic safety context and the falling of rice grains down a chute in a food-sorting context. In addition various disintegration, deposition, liftoff, surface cleansing, oil-well and sequestration modelling applications exist for interactions between solid bodies and fluids. There are also many biomedical applications in principle, for example on travel of solids within vessels of major networks in the human body. Specific applications are to transport of blood clots, embolisation procedures in stroke treatment (transportation of glue), drug-delivery to tumours via a capillary network, the passage of cells through vessels of lateral dimension comparable with the effective cell diameter, and deposition of tiny particles in branching systems [23–30]. One fundamental question is how far and where small objects will travel when transported, which is a global network issue as well as depending on the shapes of the objects and the local vessel shapes. The solid-fluid interactions of interest are with or without side walls being present. One specific example is in food-sorting where grains travel as a monolayer down an inclined chute (which has a free-surface top), fall from the bottom of the chute and then pass an optical system that can detect defective grains. A powerful jet of air is fired to eject a defective grain. Since the grains falling off the chute typically are not uniformly distributed but clustered and inhomogeneous, the air jet removes other grains surrounding the target grain which may not themselves be defective. An industrial goal is to increase uniformity of product feed to reduce the ejection difficulty whilst maintaining a high throughput of grains. The ideal situation is for an evenly spaced and uniformly ordered array of grains to fall down and off the chute such that each grain is aligned with an ejector. This points to a study of air effects on arrays of grains. Numerical estimates on food-sorting in [31, 32] indicate that an inviscid approximation is reasonable as a first-go model. Background studies. The most relevant studies are in [1–10] on interactions for channels, boundary layers, free surfaces, liftoff. We should mention in addition sus-

On Dynamic Interactions Between Body Motion and Fluid Motion

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pension flow studies concerned with relatively sparse grain flows where the interstitial fluid is important for the grain dynamics, such as in aeolian or fluvial transport [1, 33, 34]. These studies address issues such as entrainment which are potentially relevant to some fluid-body interactions of present concern. Other investigations consider oscillations in sedimentation of spheres and fibres, bubbles in fluidized beds, migration of particles, bubble formations and clusters (see [1]). Nonlinear multi-body interactions addressed [35–37] for in-series wakes and internal branching motions are also relevant although they assume fixed bodies in steady flow. The present article. We will begin below with the original food-sorting context in mind before branching out to skimming, liftoff, viscous influences and related configurations. The rapid monolayers in the sorting applications are atypical for a granular flow, and enduring contacts are not as significant here; frequent binary impacts or clashes are more typical. The issues involved lie between or outside the arenas of existing granular flows and suspension flows. The current focus is on substantial interactions in which the fluid flow is at relatively high rates, tending to produce a flow multi-structure. The bodies (grains) respond nonlinearly by means of their own induced motion which in turn affects the fluid flow nonlinearly. In terms of grains falling down and off a chute for example we concentrate first on the fluidbody interactions occurring at the lower end of the chute where the fluid response is effectively inviscid due to the increased velocity of the bodies. The upper part of the chute where viscous or viscous-inviscid behaviour is more appropriate is considered in [31] (see also Sect. 2), while a continuum model for the bulk properties of the grain motion without air effects is in [32]. Similar considerations apply to other contexts. Section 2 describes the model situations of interest, assuming laminar unsteady incompressible fluid flow. These have fluid-body interactions in two spatial dimensions for: one or more bodies inside a channel; skimming of a body along a free surface; rocking of a body on a solid surface; viscous-inviscid effects within a channel or boundary-layer flow. In each case the interaction model and solution structure lead to a nonlinear system of difference, longitudinal-differential and /or integral equations for the motion of one or a finite number of bodies in surrounding fluid or on a free surface. Section 3 then examines more recent developments in the skimming scenario. The phenomenon of take-off of a body from a solid surface is examined in detail in Sect. 4. The influences of many bodies and viscous effects then form the motivation for the work in Sect. 5, with the viscous effects leading to other scales coming into play. Further comments including future research possibilities are made in Sect. 6.

2 The Models We consider the four main model areas in Sects. 2.1, 2.2, 2.3 and 2.4, with the first section giving the typical form of non-dimensionalisation used throughout.

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2.1 Bodies in a Channel Flow The configuration of concern here models grains falling through fluid down a vertical chute as in the introduction but is drawn rotated in Fig. 1 giving thin bodies which travel almost horizontally. The entire motion takes place in a planar Cartesian frame (x ∗ , y ∗ ) as shown, with an asterisk signifying a dimensional quantity. The leading edges of the bodies are aligned with each other and their trailing edges with each other. The bodies form an unknown row-like pattern and they are finite in number. The representative axial extent of the bodies from leading to trailing edge is L ∗1 and the lateral distance in y ∗ is L ∗2 . The typical axial flow speed U ∗ is a prescribed constant. The velocity components are u ∗ , v ∗ in x ∗ , y ∗ respectively, the pressure is p ∗ , t ∗ denotes the time and ρ ∗ is the fluid density. Non-dimensional scaled quantities are used in the frame moving with the bodies at speed U ∗ , such that [(x ∗ − x0∗ )/L ∗1 , y ∗ /L ∗1 , t ∗ U ∗ /L ∗1 ] = [x, β yˆ , t],

(1a)

[u ∗ /U ∗ , v ∗ /U ∗ , p ∗ /(ρ ∗ U ∗ 2 )] = [u n , β vˆn , pn ],

(1b)

where β ≡ L ∗2 /L ∗1 . The location x0∗ is a constant, as is confirmed later by the equations of body motion. The governing equations in full for the fluid are the continuity and Navier-Stokes equations and for the bodies are those of rigid body motion. The fluid and the bodies interact by virtue of the unknown movements of the individual bodies when subjected to fluid dynamic forces and the equally unknown flow of the fluid

Fig. 1 a Sketch of dynamic fluid-body interaction in a channel, with uniform oncoming flow, in nondimensional terms. There are N gaps and N − 1 bodies; in this case N = 4. Representative gap width H ≡ H3 (x, t) = ( f 3− − f 2+ )(x, t) is shown. b Nomenclature for the typical nth body. Here h n (t) is the nondimensional height of the centre of mass and θn (t) is the orientation angle

(a)

(b)

On Dynamic Interactions Between Body Motion and Fluid Motion

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affected by the moving boundaries. When the typical length ratio β is small the fluid flow equations in each gap become those of thin inviscid layers to leading order, ∂u n /∂ x + ∂ vˆn /∂ yˆ = 0,

(2a)

∂u n /∂t + u n ∂u n /∂ x + vˆn ∂u n /∂ yˆ = −∂ pn /∂ x,

(2b)

0 = −∂ pn /∂ yˆ ,

(2c)

in the majority of the flow field. Here u n , vˆn , pn , yˆ are of order unity, and n runs from 1 to N in the successive N gaps for (N − 1) bodies in a row between the side walls yˆ = 0 and yˆ = 1. The integer N ≥ 2. The equations above come from the balances of continuity, longitudinal momentum and lateral momentum respectively. These hold provided not only that β is small but β 2 Re is large, where Re is the characteristic Reynolds number U ∗ L ∗1 /ν ∗ and ν ∗ is the kinematic viscosity of the fluid. The negligible inertial impact in (2c) implies that pn is an unknown function of x, t only. The oncoming fluid motion effectively has u n = 1, vˆn = 0, pn = 0 due to the frame of reference. Effects of gravity, viscosity and wall-contact are neglected. The boundary conditions are the kinematic ones vˆn = ∂ f n± /∂t + u n ∂ f n± /∂ x on yˆ = f n ± (x, t),

(2d)

on the surface of each body given by the unknown position f n ± (x, t), n = 1 to (N − 1), with superscripts (+, −) referring to the upper and lower surface in turn as in Fig. 1, and vˆ N = 0 on yˆ = 1, (2e) vˆ1 = 0 on yˆ = 0, for tangential flow at the straight solid side walls. The bodies are closed in the sense that f n+ (0, t) = f n− (0, t), f n+ (1, t) = f n− (1, t). At the trailing edges where x is unity Kutta conditions apply as the individual gap flows enter into the common wake requiring the pressures across all the gaps to be equal there, p1 = p2 = · · · = p N at x = 1.

(2f)

By contrast the velocities u n at the trailing edges are unequal generally, admitting vortex sheets into the common wake. A significant feature associated with upstream influence is that streamwise jumps (Euler jumps, as in [38]) in pressure must occur at the leading-edge station. The reason stems from the hyperbolic nature of the gap flows in (2a–2c) which indicates zero upstream influence in general and so a possible contradiction with the equipressure requirement (2f) at each trailing edge as the different gap flows usually produce different pressures at the trailing edges if they begin with identical leadingedge pressures. The resolution is provided by a flow-solution discontinuity which can occur in a self-consistent manner only in the vicinity of the leading edges, where all the upstream influence is focused in a sense. It follows that in general the incident

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pressure of zero just ahead of the nth leading edge is different from the two distinct values pn (0+, t), pn+1 (0+, t) (say πn , πn+1 respectively) holding on either side of the nth body just downstream. Instead the scaled Bernoulli quantity p + 1/2 u 2 and the scaled vorticity are conserved across the leading-edge station and give rise to the conditions that p + 1/2 u 2 = 1/2 and ∂u/∂ y is zero at the onset of each gap flow, in view of the incident uniform stream and pressures. The streamwise length scale to smooth out the jumps is of order β in x and the sizes of the velocity and pressure are u ∼ v ∼ 1, p ∼ 1. Hence the leading-edge region is controlled by quasisteady Euler dynamics spanning the channel from wall to wall, with the thin bodies appearing as flat plates aligned with the incident stream, and that scenario leads to conservation of the Bernoulli quantities and (zero) vorticity along each streamline and indeed to Laplaces equation for the scaled stream function. The flow enters and leaves the present Euler region uni-directionally but with an overall displacement of its streamlines accompanied by pressure changes, inside each gap, consistent with the upstream-to-downstream jumps described above, and with smooth attached flow in between such that the stream function is an unknown constant on each of the quasisemi-infinite bodies. Relatively thin viscous boundary layers are generated on every leading-edge surface and these are supposed to remain attached, before forming the beginnings of a relatively small Blasius-like effect on each body over the longer scale of (2a–2f). Substantial separations at the leading edge and elsewhere are discounted. Thus the flow in (2a–2f) remains irrotational virtually everywhere and the scaled vorticity is identically zero. Hence the thin-layer scalings yield the requirement that u n = u n (x, t) must be independent of yˆ . It follows that vˆn varies linearly between its values on the nth and (n − 1)th body surface. Writing + , u n , pn ) (H, u, p) = ( f n− − f n−1

(3)

for each gap, the equations of motion within the fluid gaps become ∂ H/∂t + ∂(u H )/∂ x = 0,

(4a)

∂u/∂t + u ∂u/∂ x = −∂ p/∂ x,

(4b)

i.e. the shallow-water equations, where the influence of the unknown gap width + ) shows up. The Euler jumps local to the leading edges also impose the ( f n− − f n−1 constraints 1 1 at x = 0+, (4c) p + u2 = 2 2 in each gap whereas the Kutta conditions at the successive trailing edges yield each p = πe (t) at x = 1−,

(4d)

for all n with the unknown downstream pressure level πe (t) being independent of n. If for convenience we also define f 0+ = 0, f N− = 1 for the containing wall surfaces

On Dynamic Interactions Between Body Motion and Fluid Motion

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at all t then the overall mass-conservation balance requires N 

+ u n ( f n− − f n−1 ) = 1 at x = 1−,

(4e)

1

in view of the incident conditions ahead of the array of bodies. The above equations (4a–4e) describe the fluid-dynamical part of the interactive motion. In the solid-body part of the motion each body is driven predominantly here by the fluid-dynamical pressure forces acting laterally on either of its surfaces. At this stage it is useful to be more explicit about the bodies, allowing for fixed shapes of overbody and underbody Fn+ (x), Fn− (x) respectively and thus thickness and camber in general. The body surfaces can then be specified simply by   1 θn (t), f n± (x, t) = Fn± (x) + h n (t) + x − 2

(5a)

with h n , θn giving the midpoint of the nth body and the body’s angle of inclination in turn and being unknown functions of time t; see Fig. 1 again. The midpoint positions are the centres of mass for bodies which as here have uniform density distribution. In consequence the equation of lateral motion for each body takes the form Mn

d 2hn = dt 2



1

( pn − pn+1 ) d x.

(5b)

0

Here Mn is the scaled mass of the nth body per unit width normal to the x-y plane, given by Mn ≡ Mn∗ β/(ρ ∗ L ∗1 2 ) where the dimensional body mass Mn∗ might vary over the (N − 1) bodies and the relations (1a and 1b) are taken into account. Similarly the equation of angular motion of each body gives d 2 θn In 2 = dt



1 0



 1 x− ( pn − pn+1 ) d x, 2

(5c)

where the scaled moment of inertia In ≡ In∗ β/(ρ ∗ L ∗1 4 ) and In∗ is the dimensional moment of inertia of the nth body. The central case of (Mn , In ) = (M, I ) being independent of n will be our concern here. The equation of axial body-motion simply confirms that the incident flow velocity, and the location x0∗ , remain constant over the current scales because the axial forces on the body are relatively small. The system controlling the fluid-body interactions is (4a–4e) and (5a–5c) subject to suitable initial conditions, with n running from 1 to N (the number of gaps) in (4a–4d) and from 1 to (N − 1) (the number of bodies) in (5a–5c). The unknown fluid pressures and axial velocities pn , u n for n = 1 to N depend on x, t, and the trailing edge pressure πe (t) is also unknown at each time t, while the unknown body midpoint positions and angles h n , θn respectively for n = 1 to (N − 1) also depend on the scaled time t alone.

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Analytical and computational properties are of interest. Numerical results from a finite-difference approach are given in [1] for a variety of N values and in [3, 10] for various body shapes. Clashes are commonly indicated. Analytically the characteristic clash occurs through a nonlinear solution structure in which the evolving gap width H (x, t) tends to zero at one particular station x in one particular gap within a finite time. The clash at a leading edge has the contributions dh/dt, dθ/dt to the body velocity acquiring the form 

dh dθ , dt dt

 = (a1 , b1 ) + B(t)(a2 , b2 ) + · · · ,

where B(t) = −{ln(t0 − t)}−1 + O(1),

(6a) (6b)

which occurs commonly for thin bodies, as t → t0 −. In (6a and 6b) the constants a1 , b1 , a2 , b2 are O(1), while the irregular behaviour in B(t) implies that the body acceleration becomes large along with the induced pressures locally. A corresponding feature is observed in the numerical results. For thicker bodies the clash occurs in a different form [3], at a station between the leading and trailing edges. Analysis also establishes the large-N form of the system and, for any N value, linear instability of the uniform state with growth rates that are found to agree [1] with the numerical results.

2.2 A Skimming Body The same governing Eqs. (4a and 4b) apply in the water flow beneath a skimming body on a shallow water layer [2]: see Fig. 2. As the depth H (x, t) evolves during a typical skimming event an analogue of (4c) due to [39] holds at the unknown leading contact point x = x1 (t), while (4d) is replaced by p = p0 at x = 1− where p0 is the atmospheric pressure in the air above the water and the trailing edge is assumed sharp, giving a fixed trailing contact point. A free surface evolves in the wake downstream

Fig. 2 Thin body skimming over a shallow liquid layer. Arrows indicate the direction of liquid motion in a frame of reference in which the solid body does not appear to move horizontally. The leading-edge position (i.e. contact point) x1 varies with time t whereas the trailing edge x0 is fixed

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and a splash jet and free surface upstream. The body-motion equations are as in (5a–5c) without the superscripts (±) and without subscripts n, with pn+1 replaced by p0 since the air flow is assumed dynamically negligible, and with the integration range being (x1 , 1) which covers the wetted portion of the body. Solutions are presented in [2, 5, 9] for thin bodies with x1 at 1− and velocity d x1 /dt negative at entry. See also Fig. 2. Depending on the initial conditions the body eventually either becomes totally submerged (when x1 becomes less than the fixed leading edge location, x = 0 say, of the body) or continues to skim off by emerging from the water when x1 becomes 1− after a finite time. The onset of this emergence involves a response in which a behaviour very akin to that in (6a and 6b) recurs. Recent work [8, 9] applies allied ideas to the sinking of the body where the upper surface becomes wetted. The body re-emerges through the free surface in some cases but otherwise usually hits the solid bed under the water layer within finite time in a clashing process that is focussed near the trailing edge.

2.3 Pre-liftoff of a Body From a Solid Surface Here a body is initially at rest or rocking back and forth on a flat horizontal surface or ground. A uniform horizontal flow of fluid is then started impulsively as in Fig. 3. Will the body continue to rock on the surface or instead tend to lift off? This question is addressed [4] by means of (4a and 4b) holding in the two thin gaps of evolving thickness H (x, t) between the underbody and the ground, one gap upstream on the left and one downstream on the right of the moving contact point x = xc (t). The body itself is not necessarily thin. Condition (4c) applies at the left end of the left gap and (4d) with πe (t) atmospheric ( p0 ) at the right end of the right gap, whereas a consistency constraint applies at the unknown position x = x1 (t). With affixes omitted the form (5a) describes H in terms of underbody shape F(x), height factor

Fig. 3 Sketch showing two thin fluid layers, one for 0 < x < x0 (t), the other for x0 (t) < x < 1, beneath a body rocking on the ground with oncoming flow. The contact point x0 (t), the reaction force R(t) and the rotation angle θ(t) vary with time t, Mg is the weight, and CoM is the centre of mass

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h, rotation angle θ analogously and (5b and 5c) describe the underbody motion where pn+1 is p0 and in (5b) a scaled normal reaction R(t) and weight Mg are added in. Properties from analysis and computation [4] show several interesting responses (continued rock or finite-time liftoff where R(t) goes to zero or immediate liftoff being indicated). Further studies have been made in recent Masters projects at UCL.

2.4 Viscous-Inviscid Effects The interaction structure is more complicated when significant viscous effects are present. A central example has a thin body of length comparable with the width of the channel containing it [6], the body being nearly aligned with the oncoming nonuniform fluid flow which is a planar Poiseuille or similar flow u(= u ∗ /U ∗ ) = u 0 (y). See Fig. 4. The dimensional length L ∗1 is now the channel width. Over the axial scale x of O(1) we are led to linearised Euler flow past the body in the form  u 0 (y)

 ∂ ∂  ψ1 = u 0 (y)ψ1 , + ∂x2 ∂ y2

(7a)

since the vorticity is nonzero, where ψ1 is the scaled perturbation stream function. The major boundary conditions with L standing for the non-dimensional length of the body are (7b) ψ1 = 0 at y = 0, 1, no exponential growth far upstream and far downstream,

(7c)

ψ1± = u 0 (y0 )(− f ± + K ± ) at y = y0± for 0 < x < L ,

(7d)

for reasons of tangential flow at the walls, matching upstream and downstream and tangential flow on the overbody and underbody surfaces respectively. The latter surfaces are given effectively by (5a), without subscripts, whereas K ± are unknown functions of scaled time T only. At the channel walls y = 0, 1 there are thin passive

Fig. 4 A freely moving body in channel flow with vorticity [6]. Here the position x(t) is almost constant and the single short arrows indicate a typical particle path in the core of the fluid flow

On Dynamic Interactions Between Body Motion and Fluid Motion

55

viscous layers to reduce the flow velocity to zero at the walls and likewise for viscous layers on the body. It is assumed that the time scale of the body movements discussed below is much larger than the usual flow time scales and dominates the interactions here leaving the flow as quasi-steady. Also the solution is not defined uniquely by the system (7a–7d), an important issue here owing to the fact that arbitrary multiples of u 0 (y) and xu 0 (y) can be added to any solution ψ1 and still leave the system satisfied. To resolve the issue the behaviour over a longer length scale x = O(Re1/7 ) needs accounting for, where Re is based now on the channel width. An expansion analogous to (1b) holds in the inviscid core flow then for 0 < y < 1 but with only small perturbations around u = u 0 (y) which produce an unknown displacement as well as a nonzero normal pressure gradient instead of (2c). A different expansion then holds in each viscous wall layer, such that each layer is controlled by the boundary layer equations ∂U/∂ X + ∂ V /∂Y = 0, (8a) U ∂U/∂ X + V ∂U/∂Y = −∂ P(X, T )/∂ X + ∂ 2 U/∂Y 2 ,

(8b)

where in particular u = Re−2/7 U (X, Y, T ), X = x/Re1/7 , Y = y Re2/7 in the lower layer, essentially the same in the upper layer, and the time scale T is described below. The main constraints on (8a and 8b) require [40] zero slip at Y = 0, matching with the displaced core at large positive Y , satisfying a wall-pressure-difference condition between the two viscous layers, boundedness as X → ±∞ and matching with the linearised Euler flow of (7a–7d) as X → 0± since the body and its nearby flow appear as a point disturbance on the longer length scale. Details given in [6] show how the nonlinear viscous-inviscid system associated with (8a and 8b) acts to determine the correct multiples of u 0 (y) and xu 0 (y) in the near-body system of (7a–7d). Coupled with (7a–7d) are the evolution equations for the movement of the body which are essentially as in (5a–5c) in altered notation. Solutions are presented in [6]. A common finding is instability over the present time scales such that the small angle θ increases exponentially with increasing time T . Another central case has a thin body of length almost comparable with the development length of the channel flow or boundary layer [7]; in the boundary layer setting on an airfoil the development length is usually a significant fraction of the airfoil chord. This case examines a direct link between the moving body shape, acting as a displacement, and the pressure response P in the wall layer(s). The response P then affects the body movement as in (5a–5c), leading once more to dynamic fluid-body interaction. A new form of instability is again found unless mitigation measures are taken [7]. The findings and ideas summarised in Sects. 2.1, 2.2, 2.3 and 2.4 provide the basis for the more recent and continuing studies considered in the following three sections.

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3 Skimming by a Smooth Body The skimming process for a smoothly curved body in general can be decomposed into two consecutive stages [5, 41, 42]: an initial impact stage followed by a planing stage, after which the body either sinks or lifts off from water and thereby completes one skipping (skimming) cycle. We present a shallow water impact and planing model, and then investigate how such a smooth body is able to rapidly transit from its initial impact to a planing motion. We shall also analyse how the presence of an adverse pressure gradient in the trailing separation flow, which is typically associated with flows past a smooth or bluff body, can affect its planing motion. The skimming object of our interest here has an elongated horizontal profile, whose length l¯ is much greater than its thickness; its lower body surface, which may be in contact with water, is smooth and strictly convex. Suppose h¯ is the depth of the water: its shallowness implies h¯ ∼ l¯ with parameter 1. This object could be skimming at an inclined angle, say α, as well as having an angular velocity ω; the angle of inclination, defined as the one made by its major axis and the undisturbed water free surface, is assumed to be small such that α ∼ O( ). Letting (u, ¯ v) ¯ be the body’s horizontal and vertical velocities respectively, in our model of interest the horizontal speed is a magnitude larger than the vertical speed, i.e. v¯ ∼ u. ¯ See Fig. 5 for a depiction of the coordinate system and necessary nomenclature. An upper-half Cartesian coordinate system (x, ¯ y¯ ) is introduced such that its x¯ axis rests at the bottom of the shallow water and points in the opposite direction to the skimming body’s horizontal velocity u. ¯ Its y¯ -axis points upwards and goes through the skimming body’s centre of mass, whose coordinate is (0, y¯m ). In this configuration the coordinate system travels horizontally with the skimming body, the undisturbed water flows in the positive direction of the x-axis ¯ and the body itself has only vertical and angular motions.

Fig. 5 A close-up sketch of a smoothly curved body during the impact stage of a skimming process. x¯1 and x¯2 represent the horizontal positions of the leading and trailing contact edges respectively. h¯ denotes the representative depth of the water layer, the jet root regions 1 and 3 have representative ¯ × O(h), ¯ while the region underneath the impact body has size of O(l) ¯ × O(h). ¯ At scales of O(h) a sufficiently small time, the contact surface of the blunt body can be approximated by a flat plate at the leading order, and x¯1 , x¯2 move away from each other at an extremely large speed

On Dynamic Interactions Between Body Motion and Fluid Motion

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As the body impacts on water, the flow can be approximately divided into three sub-regions: an undisturbed far-stream flow region, where the water is undisturbed ¯ × O(h), ¯ known as the and at rest; a region with elevated free surface of size O(h) “jet-root” or “turn-over” region where the flow separates from the body and splash jets may be emitted; and finally a main flow region trapped under the body and enclosed by the jet-root region. The splash jets are typically thin compared with the main body flow and their effects can be neglected [39, 43]. We let x¯1 , x¯2 be the horizontal positions (called here the leading edge and trailing edge respectively) of the stream turn-over points in the jet-root region, which are not known a priori and would need to be found as part of the skimming system. Once the leading and trailing edge positions are determined together with the known skimming body’s surface function, the total contact area between the body and the fluid can be determined.

3.1 Impact Model Taking (u¯ 0 , v¯0 ) to be the body’s initial horizontal and vertical velocities respectively, the system can be non-dimensionalized as follows: x¯ y¯ , y= , h= ¯l l¯ m¯ ¯ m= , i = ρ l¯4 i; ρ h¯ l¯ x=

u¯ v¯ p¯ h¯ l¯ , , t = , u˜ = , v˜ = , p˜ = ¯l u¯ 0 u¯ 0 u¯ 0 ρ u¯ 20

where m¯ and i¯ are the body’s mass and moment of inertia respectively. Suppose the body’s lower surface is given by η(x), which is smooth and strictly convex, and can be expressed locally in a parabolic form: η(x) = ax 2 + bx + c

(where a < 0).

(9)

The coefficients a, b, c are constants that can be calibrated according to the object’s shape. Notice that to ensure our skimming body has a horizontally elongated profile, the surface coefficients are small: a, b, c ∼ O( ). The water depth under the body can be obtained: h(x, α, t) = ym (t) + xα(t) − η(x),

(x ∈ [x1 (t), x2 (t)]).

(10)

The large differences between the horizontal and vertical scales of our problem can be exploited by introducing the expansion:

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(u, ˜ v, ˜ p, ˜ y, h, a, b, c, α, η) ∼ (1, 0, 0, 0, 0, 0, 0, 0, 0, 0) + (u, V, p, Y, H, A, B, C, θ, T ),

(11)

where the newly scaled variables on the right-hand-side are of order unity and = h¯ 0 /l¯ 1 as before. The water depth and the body surface equations now become: H (x, θ, t) = Ym (t) + xθ (t) − T (x),

(x ∈ [x1 (t), x2 (t)]);

(12a)

(A < 0).

(12b)

T (x) = Ax + Bx + C, 2

For a body skimming at a sufficiently large horizontal velocity, a straight-forward scaling analysis (see [2, 5, 39] for example) shows that pressure force dominates over other effects such as viscosity, gravity and surface tension in the flow. Therefore the Navier-Stokes equations at the leading order can be reduced to: ∂u ∂p ∂u + =− , ∂t ∂x ∂x ∂p = 0. ∂Y

(13a) (13b)

Applying the kinematic boundary condition at the surface of the flow yields this additional relation: ∂H ∂u ∂H + + = 0. (14) ∂t ∂x ∂x By Newton’s third law of reciprocity the skimming body’s vertical and angular momentum equations at the leading order can be written as: 

x2

x1  x2 x1

p(x, t)ds = M

d 2 Ym , dt 2

(15a)

x p(x, t)ds = I

d 2θ , dt 2

(15b)

where M = m, I = i. Notice the horizontal force on the body is only of order , so that in the time scale of our consideration the skimming horizontal velocity is unchanged at the leading order. Imposing the pressure jump conditions as introduced by [2, 39, 42, 44] together with Bernoulli’s principle at the leading and trailing edges we derive the boundary conditions for our impact problem as follows: p(x1 , t) +

    d x1 2 1 d x1 2 1 u(x1 , t) − 1− = , 2 dt 2 dt

(16a)

On Dynamic Interactions Between Body Motion and Fluid Motion

    d x2 2 1 d x2 2 1 u(x2 , t) − 1− p(x2 , t) + = , 2 dt 2 dt    d x1 d x1 1 u(x1 , t) − 1− = 2H (x1 , t)− 2 − 1, dt dt    d x2 d x2 1 u(x2 , t) − 1− = 2H (x2 , t)− 2 − 1. dt dt

59

(16b) (16c) (16d)

Our impact model therefore consists of (12)–(16). We focus on a linearised formulation of this integro-differential problem in the next subsection to gain further insights.

3.2 Linearised Impact Model and Rapid Transition to Planing Motion During the impact phase the solid-liquid contact surface initially goes through a phase of rapid expansion and spray jets are formed at the boundary of the surface. To analyse the impact system behaviour during this phase of rapid contact surface expansion, we focus on a short time after impact such that time t is of order δ where δ 1. Given that the body’s initial vertical velocity is of order unity, we expect the free surface penetration to be small and have the same order as time t. Balancing the terms of the free surface equation (12a) suggests that the model’s horizontal and angular 1 1 scales both evolve on a higher order, specifically x ∼ O(δ 2 ) and θ ∼ O(δ 2 ). From the pressure jump conditions (16a) and (16b) we can deduce that the fluid’s horizontal velocity u evolves on the same scale as x and that the pressure p evolves on the scale of order unity. We therefore asymptotically expand the system variables as follows: t = δ tˆ,

(17a) 1 2

x ∼ x0 + δ xˆ + O(δ),

(17b)

Ym ∼ Y0 + δ Yˆ + O(δ 2 ),

(17c)

θ ∼ θ0 + δ θˆ + O(δ),

(17d)

H ∼ 1 + δ Hˆ + O(δ 2 ),

(17e)

1 2

1 2

u ∼ δ uˆ + O(δ), 1 2

p ∼ pˆ + O(δ ).

(17f) (17g)

Applying asymptotic analysis of such form to our impact model indicates that, unless 3 the body has sufficiently small body mass (M ∼ O(δ 2 )), the force generated by hydraulic pressure underneath the body does not have a significant effect on the body’s momentum in a short time after impact. At the leading order we thus expect the

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body’s vertical and angular velocities to be equal to their initial values: d Yˆ /d tˆ ∼ Vˆ0 and d θˆ /d tˆ ∼ ωˆ 0 . If on the other hand our skimming object has a small mass, for instance M ∼ 3 O(δ 2 ), then its vertical momentum Eq. (15a) has an immediate response to the flow pressure even inside our small time regime in (17). The linearised impact system takes on a form of five differential algebraic equations (DAEs) for five unknowns: ˆ xˆ1 , xˆ2 , Yˆ , fˆ and g: 1 1 Yˆ = A(xˆ12 + xˆ1 xˆ2 + xˆ22 ) − ωˆ 0 (xˆ1 + xˆ2 )tˆ, 3 2 1 d Yˆ 1 , fˆ = − ωˆ 0 (xˆ12 + xˆ1 xˆ2 + xˆ22 ) − (xˆ1 + xˆ2 ) 6 2 d tˆ   ωˆ 0 2 1 d A 3 3 2 ˆ ˆ ˆ (xˆ + xˆ2 ) − (xˆ + xˆ2 )t − (xˆ1 + xˆ2 )Y , f = 2 d tˆ 3 1 2 1   1 d ωˆ 0 3 1 d Yˆ gˆ = − (xˆ1 + xˆ23 ) + (xˆ12 + xˆ22 ) + (xˆ1 + xˆ2 ) fˆ , 2 d tˆ 6 2 d tˆ 2 ˆ ˆ d Y df (6 Mˆ + xˆ13 − xˆ23 ) 2 + 3(xˆ12 − xˆ22 ) + 6(xˆ1 − xˆ2 )gˆ = 0, ˆ dt d tˆ

(18a) (18b) (18c) (18d) (18e)

where Mˆ = δ 2 M. Numerical analysis of this DAEs system indicates that for a body with positive/forward rotation (i.e. ωˆ 0 > 0), a retraction of the trailing edge position could occur inside this small-time regime, see Fig. 6 for a demonstration. It is seen that at the instant of touchdown the speeds at which the leading and trailing edges evolve away from the initial contact point are very large, hence the wetted surface expands rapidly immediately after impact. The “outward expansion” speeds of the two edges decrease as time progresses however, and for a forwardrotating body the trailing edge’s velocity eventually drops to zero (Fig. 6b). Letting tˆc denote the critical time when this phenomenon occurs, before this critical time is reached the fluid is thrown away from the skimming body’s leading edge towards upstream, which is signified by the fluid velocity being negative at this edge as demonstrated in Fig. 6c. At the trailing edge the fluid velocity is initially positive, but as the critical time is approached this gradually decreases to zero, which also corresponds to the trailing edge pressure decreasing to be atmospheric, see Fig. 6d. The vertical centre of mass position Yˆ reaches its minimum shortly before this critical time and begins to move upwards. Hence before the trailing edge pressure drops to be atmospheric the body is already in the early stages of moving upwards in the water. It is evident therefore that the body mass and its angular rotation have an important and immediate effect, influencing the development of the wetted surface and thereby affecting the lift force on the body. The pressure in the fluid is initially high and positive everywhere. In the case of a positive rotation the leading edge separation point sustains the highest pressure, as is demonstrated in Fig. 7a. As time progresses the contact surface grows and pressure 3

On Dynamic Interactions Between Body Motion and Fluid Motion

(a)

61

(b)

1 15

0.5

10

0

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−0.5 −1

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Fig. 6 Various profile plots of a skimming body with a positive rotation ωˆ 0 > 0. The initial vertical velocity of the body Vˆ0 is taken to be −1, and the initial angular velocity ωˆ 0 is taken to be 1. Under these initial settings the trailing edge xˆ2 initially evolves towards the downstream, however at time tˆ ∼ 1.0497 this edge reaches its maximum and begins to retract towards upstream initial contact point. a Evolution of leading and trailing edges xˆ1 , xˆ2 w.r.t. time tˆ. b Evolution of leading and trailing edge velocities ddxˆtˆ1 , ddxˆtˆ2 . c The evolution of fluid velocities uˆ 1 and uˆ 2 . d Evolution of leading and trailing edge pressure profiles. e Vertical evolution of the skimming body’s centre of mass. f Water surface elevation at the leading and trailing edges

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(a)

(b)

2

0

Fluid Velocity

Pressure

1.5 1 0.5 0 −0.5 −2

0.5

=0.0357 =0.2343 =0.6365 =1.0497 −1

0

1

− 0.5

=0.0357 =0.2343 =0.6365 =1.0497

−1

− 1.5 −2

−1

0

1

Fig. 7 The time evolution of the pressure and fluid velocity underneath the skimming body during the impact stage for a positively rotating body. The time begins from shortly after impact tˆ = 0.0357 to the end of impact stage at tˆ = 1.0497 when the trailing edge pressure drops to zero. Notice that the wetted surface area is initially small and increases over time. The initial conditions are Vˆ0 = −1 and ωˆ 0 = 1. a A plot of pressure profiles pˆ underneath the skimming body at different times of the impact stage. b A plot of fluid velocity profiles uˆ underneath the skimming body at different times of the impact stage

decreases everywhere; eventually a negative pressure region begins to develop in a region close to the body’s trailing edge, and in particular an adverse pressure gradient field inside this negative pressure region begins to form. This sub-atmospheric pressure region eventually reaches the trailing edge, at which point the trailing edge fluid velocity drops to zero, i.e. water is no longer ejected away at the trailing edge: see Fig. 7b for an illustration. Once this critical time is reached the skimming body changes from its initial impact phase to a planing phase. In the next subsection we describe the planingphase model.

3.3 Planing Model The change from impact to planing stage is marked by the position of the trailing edge which stops extending further towards the downstream direction, as well as the disappearance of the spray jet at this edge. Our analysis in Sect. 3.2 shows that a region with negative pressure gradient begins to develop near the trailing edge at the end of the impact stage. This phenomenon is consistent with the separation of a high-Reynolds-number incompressible flow passing a bluff body. For the case of a completely submersed body, the presence of a significant adverse pressure gradient field causes the flow to separate from the body, a behaviour which is well analysed using triple-deck theory (see [45–47]) for the laminar or turbulent regime. For the planing case where the body is only partially submersed in water, experiments show that in the region of separation near the trailing edge there is a turbulent mixture of air and water. One can speculate for continuity that there should

On Dynamic Interactions Between Body Motion and Fluid Motion

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be an adverse pressure gradient at the trailing separation edge, at least for a brief period of time, when the body switches from impact to planing motion. Such an adverse pressure gradient, say κ (or κ(t) as a function of time), however, can only be determined via appropriate boundary layer analysis and that is considered beyond the scope of this discussion. A more detailed discussion can be found in reference [47]. Thus under the assumptions of turbulent separation the trailing edge boundary conditions can be prescribed as: p(x2 , t) = 0,

∂ p = κ. ∂ x x=x2

(19)

The governing equations for the fluid flow and the planing body then are very similar to those of the impact model, the differences being that: (1) in our planing model the time t starts at t = t ∗ , where t ∗ is the time at the end of the impact stage; and (2) at the trailing edge the momentum and pressure jump conditions are replaced by the separation conditions of (19). In the sections that follow we pick up our analysis for a light body in Sect. 3.2 and investigate the planing behaviour.

3.4 Linearised Planing Model Under the asymptotic settings of (17) our planing model in Sect. 3.3 reduces to a system of three ODEs with three unknowns xˆ1 , xˆ2 and Yˆ : d 2 Yˆ

+ 3κ( ˆ xˆ1 − xˆ2 )2 = 0, 6 Mˆ + (xˆ1 − xˆ2 )3 d tˆ2   2 ˆ d xˆ1 2 1 2d Y 2 ˆ (xˆ1 − xˆ2 ) + (A xˆ1 − ωˆ 0 tˆxˆ1 − Y ) + κ( ˆ xˆ1 − xˆ2 ) = 0, 2 d tˆ2 d tˆ   d 2 Yˆ d 2 xˆ1 d xˆ1 2 (xˆ1 − xˆ2 ) 2 − (A xˆ12 − ωˆ 0 tˆxˆ1 − Yˆ ) 2 − (2 A xˆ1 − ωˆ 0 tˆ) d tˆ d tˆ d tˆ   d Yˆ d xˆ1 + 2 ωˆ 0 xˆ1 + + κˆ = 0. d tˆ d tˆ 1

(20a) (20b)

(20c)

Here κˆ = δ 2 κ according to the scaling of (17). In Fig. 8 we display solutions to our planing model for a range of prescribed κˆ values. It can be seen that the larger κˆ is, the further the flow is able to attach onto the body, which is represented by the position of xˆ2 , and therefore the greater the contact surface area between water and body. Further analysis (see [9]) reveals that there is a maximum adverse pressure gradient, say κˆ M , that can be sustained at the trailing edge, exceeding which our planing regime breaks down. Under the support of such a κˆ M value the maximum distance between the leading and trailing edge positions

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(a)

(b)

1 0

x1 ( =0)

6

x2 ( =0)

5

x1 ( =0.1)

4

x2 ( =0.1)

−1 −2

x1 ( =0.3)

3

x2 ( =0.3)

2

=0) =0.1) =0.3) =0.5076)

x1 ( =0.5076)

−3

1

x2 ( =0.5076) −4 −5

Y( Y( Y( Y(

0 1

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2.5

3

3.5

4

4.5

−1

5

1

1.5

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3

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Fig. 8 Plot of the leading and trailing edges, as well as the plate’s vertical centre of mass during the planing stage for varying values of trailing edge pressure gradient κ. ˆ The body’s mass Mˆ and rotational velocity ωˆ 0 are both taken to be one. For the case of κˆ = 0 the trailing edge position xˆ2 is not continuous when transition from impact to planing stage. a The evolution of the plate’s leading and trailing edges during the planing stage. b The evolution of the plate’s vertical centre of mass position during planing

can be deduced as follows: √ ˆ 31 . MAX(xˆ2 − xˆ1 ) = [(24 + 18 2) M]

(21)

This limit is dependent only on the body mass, and intuitively the heavier the body, the further apart the leading and trailing edges can be. If on the other hand the adverse pressure gradient is weak, κˆ 1, the body’s planing motion can be divided into three distinct and consecutive phases, which are discussed next. Initial planing phase. Let tˆc denote the time when the body changes to the planing stage. We write ξˆ to be the horizontal distance between the leading and trailing edges: ξˆ = xˆ2 − xˆ1 , and let xˆc1 , Yˆc and Vˆc denote the horizontal position of the leading edge, vertical position of the centre of mass and vertical velocity respectively at time tˆc . Given that the pressure gradient satisfies 0 < κˆ 1, we postulate the following asymptotic expansions: ˆ xˆ1 = xˆc1 + κˆ 2 xˇ1 (tˆ) + O(κ),

(22a)

ξˆ = ξˇ1 (tˆ) + O(κˆ ),

(22b)

Yˆ = Yˆc + Vˆc tˆ + κˆ Yˇ1 (tˆ) + O(κˆ 2 ).

(22c)

1

1 2

Our planing system can then be simplified to a system of differential algebraic equations of the following form:

On Dynamic Interactions Between Body Motion and Fluid Motion

  d 2 d xˇ1 ζ = 0, d tˆ d tˆ

(23a)

ξˇ14 + 2ζ (6M − ξˇ13 ) [6M − ξˇ13 ]

65



d xˇ1 d tˆ

2

+ 12M ξˇ1 = 0,

d 2 Yˇ1 + 3ξˇ12 = 0, dt 2

(23b) (23c)

where the function ζ (tˆ) is given as 2 ζ (tˆ) = xˆc1 + Yˆc + (Vˆc + ωˆ 0 xˆc1 )tˆ.

(24)

Let Yˇ10 , Vˇ10 be the initial value and initial first order derivative for Yˇ1 , and xˇ10 , uˇ 10 be the initial value and initial first order derivative for xˇ1 respectively. We may then apply the following Cauchy conditions: ˇ ˇ Y10 ≡ Y1

tˆ=tˆc

d Yˇ1 ˇ V10 ≡ = 0; d tˆ tˆ=tˆc 1  4 ξˇ10 + 12M ξˇ10 2 uˇ 10 = − . 3 2ζ0 (ξˇ10 − 6M)

= 0,

xˇ10 = 0,

(25a) (25b)

The initial condition for uˇ 10 can be obtained by combining (23b and 23c) and setting tˆ = tˆc . At this point we can write down the solution to xˇ1 : xˇ1 (tˆ) = Φ0 (1 − ζ0 ζ −1 ),

(26)

where Φ0 is given as Φ0 = −

ζ0 Vˆc + ωˆ 0 xˆc1



4 ξˇ10 + 12M ξˇ10 3 2ζ0 (ξˇ10 − 6M)

 21

.

(27)

This explicit solution for xˇ1 leads to the following quartic equation of ξˇ1 from (23b): ζ 3 ξˇ14 − 2Φ02 ξˇ13 + 12Mζ 3 ξˇ1 + 12MΦ02 = 0.

(28)

The formula for finding roots of quartic equations is well known and will not be presented here explicitly. Out of the four possible solutions for ξˇ1 the admissible one should be real, positive and fit the physical context of the system. From (23c) we can obtain the solution for Yˇ1 in a double integral form based on the admissible solution of ξˇ1 and initial conditions (25a):

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F. T. Smith et al. −3

0

7

2.9

6

− 0.02

2.8

5

− 0.04 2.7

4

− 0.06 2.6

− 0.08

3

2.5

− 0.1

2

2.4

− 0.12

1

2.3 1.04

x 10

1.06

1.08

1.1

1.12

1.14

1.1

1.08

1.06

1.14

1.12

0 1.04

1.06

(b) ξˇ1

(a) x ˇ1

1.08

1.1

1.12

1.14

1.16

(c) Yˇ1

Fig. 9 Solutions of the planing system (23) with M = 1, ωˆ 0 = 1, κˆ = 0.1. The values of xˆc1 , xˆc2 , Yˆc and Vˆc are the results from the final stage of impact model (18), their respective values are: −1.9944, 0.9434, −0.4431 and 0.0359

Yˇ1 =

 tˆc

3ξˇ12



ξˇ13 − 6M

d tˆ2 .

(29)

The solutions for xˇ1 , ξˇ1 and Yˇ1 are presented in Fig. 9. The results demonstrate that for a planing body with positive rotation, its leading edge position continues to extend in the upstream direction as with the impact stage case. The trailing edge position on the other hand also begins to move in the upstream direction, and it moves at a greater pace compared with that of the leading edge as demonstrated by the decreasing value of ξˇ1 in Fig. 9b. This signifies that the contact surface between the water and planing starts to decrease. During this phase the planing body continues to emerge from the water as shown in Fig. 9c. Planing phase II. The solution of xˇ1 given in (26) depends inversely on ζ (tˆ), which eventually decreases to zero as time progresses and xˇ1 becomes singular. Suppose tˆN is the time when this singularity occurs. Then from the definition of ζ in (24) tˆN can be deduced as: xˆ 2 + Yˆc , (30) tˆN = − c1 Vˆc + ωˆ 0 xˆc1 which is the time when the second phase of our planing stage begins. In this phase we seek a new asymptotic form as follows: 1

tˆ = tˆN + κˆ 4 t¯,

(31a)

1 4

xˆ1 = xˆc1 + κˆ x¯1 + · · · ,

(31b)

ξˆ = (6M) − κˆ ξ¯1 + · · · , 1 3

3 4

(31c)

Yˆ1 = Yˆc + Vˆc tˆN + κˆ Vˆc t¯ + κˆ Y¯1 + · · · . 1 4

3 4

(31d)

In contrast to the initial planing phase during which the trailing edge evolves on a much larger time scale when compared with that of the leading edge, in the second phase the body’s trailing edge moves on a scale comparable to that of the leading

On Dynamic Interactions Between Body Motion and Fluid Motion

67

3

edge with only an O(κˆ 4 ) adjustment. Substituting these asymptotic expansions into the planing system of (20) yields: ξ¯1 Y¨¯1 + 1 = 0, (32a)

1 2 (6M) 3 Y¨¯1 − (2 xˆc1 + ωˆ 0 tˆN )x¯1 + (ωˆ 0 xˆc1 + Vˆc )t¯ x˙¯12 = 0, (32b) 2

(2 xˆc1 + ωˆ 0 tˆN )x¯1 + (ωˆ 0 xˆc1 + Vˆc )t¯ x¨¯1 + (2 xˆc1 + ωˆ 0 tˆN )x˙¯12 + 2(ωˆ 0 xˆc1 + Vˆc )x˙¯1 = 0. (32c) This system can be solved explicitly via the method of matched asymptotic expansions (see [9]) and the solutions are: (γ 2 t¯2 − 2λ0 Φ0 ζ0 ) 2 − γ0 t , x¯1 = 0 λ0   3 2 ¯2 3 ¯3 2 ¯ ˆ − 23 2 (γ0 t − 2Φ0 ζ0 λ0 ) − γ0 t + 2Φ0 ζ0 γ0 t , Y¯1 = (6 M) 3 λ0 λ20 2 1 2 2 2 ˆ (6 M) 3 λ0 (γ0 t¯ − 2Φ0 ζ0 λ0 ) 2 ξ¯1 = −

2 . 1 2γ02 (γ02 t¯2 − 2Φ0 ζ0 λ0 ) 2 − γ0 t¯ 1

(33a) (33b) (33c)

Here γ0 = ωˆ 0 xˆc1 + Vˆc and λ0 = 2 xˆc1 + ωˆ 0 tˆN . See Fig. 10 for a demonstration of the solutions. The solutions indicate that the leading and trailing edge positions in this phase continue to move in the upstream direction. The pace at which the body emerges from the water significantly increases. It becomes evident that as time progresses our 1 planing system will emerge from the O(κˆ 4 ) regime, which leads to the final large time phase of this planing. Planing phase III. In the final phase of the planing motion the system variables can be expanded asymptotically as 0

0

100

− 0.1 −2

80

− 0.2 − 0.3

−4

60

− 0.4 −6

40

− 0.5 − 0.6

−8

20

− 0.7 −10

0

1

2

3

(a) x ¯1

4

5

− 0.8

0

1

2

3

(b) ξ¯1

4

5

0

0

1

2

3

4

5

(c) Y¯1

Fig. 10 Plots of x¯1 , ξ¯1 and Y¯1 over time t¯ ∈ [0, 5]. The values of xˆc1 , xˆc2 , Yˆc and Vˆc are the results from the final stage of impact model (18), their respective values are: −1.9944, 0.9434, −0.4431 and 0.0359

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tˆ = t˘,

(34a)

xˆ1 = x˘1 ,

(34b)

ξˆ1 = (6M) − κˆ ξ˘1 ,

(34c)

Yˆ1 = Y˘1 .

(34d)

1 3

Substituting these expansions into the planing system of (20) yields the following planing system for phase III: (35a) ξ˘1 Y¨˘1 + 1 = 0, 1 2 (6M) 3 Y¨˘1 − (x˘12 + ωˆ 0 t˘x˘1 + Y˘1 )(x˙˘1 )2 = 0, (35b) 2 1 ¨ (6M) 3 Y˘1 − (x˘12 + ωˆ 0 t˘x˘1 + Y˘1 )x¨˘1 − (2 x˘1 + ωˆ 0 t˘)(x˙˘1 )2 − 2(ωˆ 0 x˘1 + Y˙˘1 )x˙˘1 = 0. (35c) It is difficult to obtain explicit solutions of this coupled non-linear ODE system; numerical solutions are therefore pursued and the results are presented in Fig. 11. We also present a comparison with the numerical solutions to the full planing system (20). The results show that our phase III planing system is able to capture the planing body’s behaviour at large times well. The solutions indicate that during this phase the planing body continues to emerge from the water and leading and trailing edges continue to move in the upstream 1 direction, with the distance between the two edges fixed at (6M) 3 at leading order. In order to see the behaviour of the system at large times, i.e. as t˘ → ∞, we introduce the following scaled large time variables: t˜ = δ t˘,

Y˜ = δ 2 Y˘ ,

ξ˜ = ξ˘ ,

x˜1 = δ x˘1 ,

(b)

(a)

8

0 −1

full

6

−2 4

−3 2

−4 0

full

−5

full −6

1

2

3

4

5

6

−2

1

2

3

4

5

6

Fig. 11 Numerical solutions of the planing system (35) compared with the full planing system (20) for tˆ > tˆN and κˆ = 10−4 . a Comparison of numerical solutions for xˆ1 , xˆ2 . b Comparison of numerical solutions for Yˆ

On Dynamic Interactions Between Body Motion and Fluid Motion

69

where δ 1 and the tilde sign is used to denote variables which are O(1) at large times. Upon substituting the above variables into the phase III planing system (35) we obtain the system: ξ˜1 Y¨˜1 + 1 = 0, (x˜12

(36a)

+ ωˆ 0 t˜x˜1 + Y˜ )x˙˜12 = 0,

(x˜12 + ωˆ 0 t˜x˜1 + Y˜ )x¨˜1 + (2 x˜1 + ωˆ 0 t˜)x˙˜12 + 2(ωˆ 0 x˜1 + Y˜˙ )x˙˜1 = 0.

(36b) (36c)

The solutions here can be written down explicitly with the initial conditions of x˜1 (0) = 0 and Y˜ (0) = 0: 1 x˜1 = − ωˆ 0 t˜, 2 2 ˜ξ = − , ωˆ 02 1 Y˜ = ωˆ 02 t˜2 . 4

(37a) (37b) (37c)

If we could see the planing body lifting off and separating from water, the positions of the leading and trailing edges should eventually coincide as shown by [42]. This is however not the case in our linearised planing model as the large time solutions (37) demonstrate: under our linearised planing regime the body continues to lift upwards as time grows but the distance between the two wetted edges does not decrease. As the body evolves vertically to the squared power of time it will eventually depart from the linearised regime defined in (17), at which point we will need to revert to the full planing system to capture the motion of the planing body.

4 Post-liftoff and Fly-Away of a Body from a Surface In this section, post-liftoff of a thin body and its continued departure from the ground (fly-away) are investigated. We model, analyse and compute the mechanisms for a single body which is initially at rest on a horizontal solid surface and can be lifted off impulsively by a horizontal fluid flow. The aim is to gain an understanding of lift-off followed by a return to the surface or complete fly-away. Criteria for liftoff and fly-away and the ensuing motion are examined. Analysis of the early behaviour when lift-off starts and a numerical study of the ensuing evolution are presented, followed by large-time analysis which shows a critical flow speed for fly-away for any shape of body. The body is assumed much denser than the fluid, implying that the major gravity force acts on the body. Experimental results or observations are many, as in [13, 18, 19]. Applications vary from grain segregation, leaf-blowers, dust loss, dust blowing,

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(a) At time t = 0.

(b) At time t > 0.

Fig. 12 a A sketch of the body at its initial position, the fixed centre of mass (CoM), the contact point x = σ at time t = 0 and the oncoming stream of fluid. b The body position at some time t >0

removal of debris, sand movement on beaches, to aircraft take-off and ski jumping. See for example [15, 16, 22]. The fluid is incompressible and its motion is assumed to be two-dimensional and laminar with uniform density ρ ∗ , where the asterisk (∗ ) refers to a dimensional quantity. We express the motion of the fluid and the immersed thin body (see Fig. 12) with regard to non-dimensional flow velocities (u, v), corresponding Cartesian coordinates (horizontal x, vertical y), time t and pressure p, such that the dimensional versions are u ∗ (u, v), L ∗1 (x, y), L ∗1 t/u ∗ and ρ ∗ u ∗ 2 p, in turn. Here L ∗1 is the length of the body, while u ∗ is the free-stream velocity and the temporal factor L ∗1 /u ∗ is the typical transport time. The Reynolds number Re is large. Also (u, v) is given by (1, 0) in the far field and the leading edge of the body can be taken as origin. A nonlinear evolutionary system for the unknown scaled functions h, θ, u, p is found to govern the interaction. Here x = σ is the prescribed x-location of the centre of mass and the initial contact point with the ground. Also h(t) is the vertical y-location of the centre of mass of the body, while θ (t) is the small angle the body chord makes with the horizontal. First, we present the model in detail, followed by the behaviour for small times. Then we examine the lift-off criterion for different body shapes. If the body does lift off it returns to the ground within a finite time or flies away at large time. We focus on the latter, finding a criterion for fly-away. Finally, we provide conclusions and further discussion.

4.1 Governing Equations and the Parameters The interaction between fluid and body and governing equations are described as follows. The body has a smooth shape and is thin, of vertical scale O(δ) with δ being small. The fixed surface or wall is located at y = 0 on which the body initially is at rest and in contact, directly below its centre of mass whose x-location is x = σ as presented in Fig. 12a. The incoming flow moving from left to right is the uniform stream with (u, v) = (1, 0).

On Dynamic Interactions Between Body Motion and Fluid Motion

71

The scaled body mass and the scaled acceleration due to gravity are represented as M and g respectively in Fig. 12. Also I denotes the moment of inertia given below. The scaled weight of the body is represented as Mg. The flow over the length scale of order unity remains irrotational to leading order almost everywhere and the scaled vorticity is zero under the present assumptions. The velocity u = u(x, t) does not depend on y, a feature which is due to the thin layer dynamics. Then v is forced through continuity to change in y from zero at zero y to a value consistent with the kinematic condition at the unknown position of the moving lower surface of the body. The equations of motion of the body are as in (5b and 5c) such that   L dU B +g = p d x, M dt 0 

d(I Ω B ) = dt



L

(x − σ ) p d x,

(38a)

(38b)

0

where L is the length of the body, which is taken as unity in this section. Also U B = h˙ is the translational velocity and Ω B = θ˙ is the angular velocity. The dimensional mass is ρ ∗ L ∗1 2 Mδ −1 , while the dimensional moment of inertia is ρ ∗ L ∗1 4 I δ −1 . The acceleration due to gravity is δu ∗ 2 gL ∗1 −1 in dimensional terms. The Froude number is (δg)−1 , whereas the Richardson number is δg. Here I < M/4 from its definition. The equations of motion within the fluid gap are shallow-water equations as in (4a and 4b), (39a) Ht + (u H )x = 0, u t + uu x = − px .

(39b)

Here H (x, t) denotes the unknown scaled thickness of the thin gap and depends on the lower surface shape of the body and its orientation given in (39c) below; F − (x) is the prescribed shape of the underbody as in Sect. 2. The kinematic condition yields (39a) while the dominant streamwise momentum balance is given as (39b). Here p is dependent only on x, t by virtue of the normal momentum balance. Changes in the lateral location and orientation of the body lead to the contributions h and θ respectively. These are prescribed at the initial time of zero as in (39d). Thus H (x, t) = −F − (x) + h(t) + (x − σ ) θ (t), 

h(0) = F − (σ ); θ (0) = F − (σ ).

(39c) (39d)

We also have the Euler jump across the leading-edge Euler zone at x = 0+ such that 1 1 p + u2 = 2 2

at x = 0+,

(39e)

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and the Kutta condition at the trailing edge of the body at x = 1 yields p = 0 at x = 1.

(39f)

The constraints (39d) coming from H = 0 at the contact point x = σ , and also ∂ H/∂ x = 0, x = σ , hold for the smooth shapes considered herein. The leadingedge jumps (39e) are required in order to satisfy the equi-pressure condition at the trailing edge (see Sect. 2.1). These conditions are coupled with the body-motion equations in (38a and 38b). The task in general is to solve the nonlinear system (38a and 38b) and (39a–39f) for the unknowns h, θ, u, p.

4.2 Small-Time Behaviour When the fluid flow is begun impulsively or a considerable change in the fluid flow occurs at time t = 0, the body is supposed to be positioned initially on the surface. Then it is assumed to move from rest. For 0 < t 1, two regions are present: one is influenced by the whole underbody for 0 < x < σ, σ < x < 1 and the other is in the neighbourhood of the contact point σ . The task now is to study the mathematical structure of the fluid-body interaction as the lifting begins and the contributions of the inner and outer region dynamics. In the outer region, the appropriate expansions for height and orientation have the patterns above with the typical x ∼ O(1). The small-time perturbations of the vertical location and the orientation of the centre of mass are predicted as (h, θ ) = (h 0 , θ0 ) + t (h 1 , θ1 ) + · · · ,

(40a)



where the dominant terms (h 0 , θ0 ) = (F − (σ ), F − (σ )) are found from (39d) and the contact point x = σ is also the horizontal location of the centre of mass. Hence the gap width, the scaled velocity and pressure H, u, p develop according to (H, u, p) = (H0 (x), u 0 (x), p0 (x)) + t (H1 (x), u 1 (x), p1 (x)) + · · · ,

(40b)

with the leading order term being H0 (x) = −F − (x) + h 0 + (x − σ )θ0 where h 0 and θ0 are given in (40a). The unknown perturbations u 0 , u 1 , p0 , p1 are functions of x to be determined. Substituting (40b) into (39a) and integrating in x leads to the leading order term of the velocity in the outer region, u 0 (x) =

1 θ1 − h 1 (x − σ ) − (x − σ )2 + c1 , H0 (x) 2

(41a)

where c1 is the integration constant. A requirement for definiteness of u 0 at the contact position x = σ is applied in order to keep the related pressure coefficient finite at the initial contact location in accordance with the fluid-body interaction

On Dynamic Interactions Between Body Motion and Fluid Motion

73

structure. Indeed, this indicates that c1 = 0, h 1 = 0 to keep u 0 finite at x = σ since the denominator H0 is then of O((x − σ )2 ) there. Making another substitution into (39a) with (41a) and (40b) and an integration in x yields

2

1 θ1 1 θ2 2 3 − 2h 2 (x − σ ) − (x − σ ) − c2 + (x − σ ) . u 1 (x) = H0 (x) 2 H0 (x)2 2 (41b) Next, integrating (39b) with respect to x at leading order and considering (39e and 39f) yields the leading term in the pressure as  p0 (x) =

− −

x 0x 1

u 1 (x) ˆ d xˆ − 21 u0 (x)2 + 21 ,  x ∈ [0, σ ), 2 2 1 u 1 (x) ˆ d xˆ − 2 u 0 (x) − u 0 (1) , x ∈ (σ, 1].

(42)

Matching below implies that c2 in (41b) is zero. This reduces the local inertial effects. We will study the case θ1 = 0 implying u 0 = 0 in the rest of the section. (See also [10] for the case θ1 = 0 whereby u 0 = 0.) We also observe that 2h 2 ln |x − σ | + π± as x → σ± , B  σ  σ 1 π+ = − u 1 d x, π− = − u1 d x 2 1 0

p0 ∼ where

(43a) (43b)



and B = F − /2 from the expression for the gap width near the original contact point. Figure 13 shows a sample comparison between p0 in (43a) and a numerical solution for the pressure given in [10]. The inner region near the initial contact x = σ is investigated next. For x near the lift-off point, x = σ + tη with η ∼ O(1), the solution takes the form 2h 2 ln(t) + p0∗ (η)) + · · · . (44) (H, u, p) = (t 2 H2∗ (η), u ∗0 (η), B These scalings stem from those in the outer region. Substituting into (39a–39d) and matching yields the leading order terms in the local velocity and pressure expansion such that −2ηh 2 − c0∗ , (45) u ∗0 (η) = Bη2 + h 2 p0∗ (η)

=

p0∗ (0)

+

ηu ∗0 (η)

 − 0

η

1 1 u ∗0 dη − u ∗0 (η)2 + u ∗0 2 (0), 2 2

(46)

where c0∗ , p0∗ (0) are integration constants to be determined. Matching the velocities and pressures in the inner and outer regions yields c2 = 0 in (41b) for u 0 = 0 in (41a). Also c0∗ is determined by

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Fig. 13 Comparison between analytical and numerical solutions of the pressure at time t almost zero

√  σ  σ 2π c0∗ 1 u1 d x − + u 1 d x.  1/2 = − 2 1 0 h 2 F − 

(47)

The pressure coefficient p0∗ (0) can similarly be found, and this completes the velocity and pressure solutions. We note that the main body motion is only controlled by the outer region. The body-movement relations (38a and 38b) give the leading order contributions 

σ

M(2h 2 + g) =

 p0 (x, t) d x +

0

 2I θ2 =

σ

 (x − σ ) p0 (x, t) d x +

0

σ

1

1

p0 (x, t) d x,

(48a)

(x − σ ) p0 (x, t) d x.

(48b)

σ

We need h 2 in (48a) to be positive so that the body can lift off from the surface. Substituting p0 (x) from (42) (for u 0 = 0) into the system (48a and 48b) gives two linear equations for the two unknowns h 2 , θ2 , namely 2Mh 2 = 2I θ2 = with

σ + h 2 I1 + θ2 I2 − W, 2

(49a)

−σ 2 + h 2 I2 + θ2 I3 , 4

(49b)

On Dynamic Interactions Between Body Motion and Fluid Motion





1

(x − σ )3 1 d x, I3 = − H0 (x) 2



(x − σ )4 d x. H0 (x) 0 0 0 (49c) The coefficient I2 is identically zero if the body is symmetric with σ = 1/2. I1 = −2

1

(x − σ )2 d x, I2 = − H0 (x)

75 1

4.3 Lift-Off Criterion The study suggests that the lift-off requirement is simply h 2 > 0. Considering a body having general shape and using the relationship θ2 = (h 2 I2 − σ 2 /4)/β in (49b) where β = (2I − I3 ), (49a) becomes γ h2 =

σ I2 σ 2 −W − , 2 β 4

  I2 with γ = α − 2 , α = (2M − I1 ). β

(50a)

(50b)

We note that I2 can either be negative or positive, while α > 0, β > 0, I1 < 0, I3 < 0. If M and I are sufficiently large that γ > 0, then from (50a) lift-off requires σ2 I2 by virtue of h 2 > 0. The criterion in (50a) becomes Mg < σ2 − 4β Mg
4(ρ B∗ /ρ ∗ ),

(58a)

u ∗ 2 /(h ∗B g ∗ ) > 2(ρ B∗ /ρ ∗ ),

(58b)

[for a symmetric body in the case of (58a)] with g ∗ denoting gravity. This is for a body mass represented as ρ B∗ h ∗B L ∗1 where ρ B∗ is the body density and h ∗B is the mean body thickness. These criteria are on effective Froude numbers and are broadly in line with Shield’s condition [11, 17] in sediment processes; this section shows evolution towards or away from fly-away and determines a precise coefficient (4 or 2) rather than the order of magnitude estimate of Shield. Further agreement with experiments or observations concerns the movement of dust on the surface of Mars as discussed in [4]. The difference between the liftoff and fly-away conditions implies that symmetric bodies for instance which are subject to flow velocities between the two values in (58a and 58b) satisfy the fly-away condition but are unable to lift off, while a body that lifts off may either return to the ground or fly away. The effects of incident shear in the oncoming flow, three-dimensionality and viscous effects have not been considered yet. The normal pressure gradient comes into play considerably on a larger time scale. Several bodies, clashes, reptation, body flexibility, and investigating the influence of the surface shape might also be of further concern.

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5 Many Bodies in Slightly Viscous Fluid Allowing for several plates at small angle of attack in unbounded flow forms a basic part of the extension of the research in Sect. 2.4. The classic exact solution for streaming flow past two cylindrical aerofoils is given by [48] and derived succinctly in [49] where it is extended to include freely propagating point vortices and patches of vorticity within the flow field. Crowdy [50] extended Lagally’s result to flow past a finite stack of cylindrical aerofoils with specified circulations about the aerofoils. Here we consider the case of plates aligned at small angles of attack to the flow. The circulation around each plate is determined uniquely by requiring the flow to leave the trailing edge of the plate smoothly: given the angle of attack of each plate the flow field is unique. This allows the forces on the plates to be computed and thus the subsequent motion of the plates if they are free to move (as in Sect. 2.4).

5.1 A Single Plate Consider the single plate given by y = 0, |x| < 1 in the infinite complex z-plane and take the pre-image of this domain in the complex ζ -plane to be the interior of the unit circle |ζ | = 1, denoted C here. Let what will become the leading edge of the plate, at z = −1, be mapped to the point ζ = 1 on C and the trailing edge, at z = 1, be mapped to the point ζ = −1. Then the conformal mapping between the domains is the Joukowski map z = −(1/2)(ζ + 1/ζ ),

ζ = −z +



z 2 − 1,

(59)

with the point at infinity in the z-plane mapped to the origin in the ζ -plane. We wish to construct the analytic function q(z(ζ )) giving the velocity field q = u + iv that has velocity v = 1 on the plate and vanishes at infinity. Note that it is the analytical function q that is mapped from the ζ - to the z- plane. The corresponding complex velocity potentials do not map to each other with  w(ζ ) =

 q(ζ ) dζ,

W (z) =

 q(z) dz =

q(ζ )(dz/dζ ) dζ.

(60)

Observe that the streamlines for a dipole in an unbounded domain are circles, i.e. the imaginary part of the complex potential for a dipole is constant on circles. If the velocity field q in the ζ -plane is itself taken to be a dipole centred at the leading edge ζ = 1, directed along the ζ axis, i.e. q ∼ i/(ζ − 1), then v = q will be constant on circles through ζ = 1 and, in particular, on the circle C. To construct this velocity field, introduce in the ζ -plane the complex velocity potential for a vortex of strength κ at the leading edge and the corresponding complex velocity

On Dynamic Interactions Between Body Motion and Fluid Motion

81

q = (κ/2π i)(ζ − 1)−1 .

w = (κ/2π i) log(ζ − 1),

(61)

Then q → 0 as ζ → ∞ as required and on C, where ζ = exp(iθ ), q = (κ/4π )[− cot(θ/2) + i],

(62)

showing that v is constant on C and v = 1 there when κ = −4π . Note that u vanishes at the trailing edge θ = ±π and is infinite at the leading edge θ = 0. This completes the determination of the complex velocity and complex velocity potential in the ζ -plane, giving w = 2i log(ζ − 1), q = 2i/(ζ − 1). (63) The corresponding complex velocity and complex velocity potential in the z-plane are  √ √ (64) q(z) = −2i/(1 + z − z 2 − 1) = −i z − 1/ z + 1 − i,  W (z) = −i (1/ζ + 1/ζ 2 ) dζ = i(− log ζ + 1/ζ )   = −i[z + z 2 − 1 − log(z + z 2 − 1)]. (65) Figure 18 gives the streamlines for this flow. The Blasius theorems give the x and y components of force, X and Y , and the moment M about the origin as X − iY =

1 iρˆ 2



1 M = − ρˆ  2

qˆ 2 dz,

 qˆ 2 z dz,

(66)

where ρˆ is the fluid density, qˆ is the total fluid velocity, and the integral is taken around the plate and outside the singularity at the leading edge. Now qˆ = U + q, 1.0

0.5

0.0

- 0.5

- 1.0

-2

-1

0

1

2

Fig. 18 Perturbation streamlines for potential flow past a single plate at small angle of attack

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where U is the oncoming flow so qˆ 2 = U 2 + 2U q + q 2 with the first term being constant and thus making no contribution and the final term negligible in the linear approximation. Hence in the linear approximation  X − iY = iρU ˆ

 q(z) dz,

M = −ρU ˆ 

q(z) zdz.

(67)

Since the velocity fields in ζ and z map to each other, these formulae become immediately 

 X − iY = iρU ˆ

C

q(ζ ) (dz/dζ ) dζ,

M = −ρU ˆ 

C

q(ζ )z(ζ ) (dz/dζ ) dζ,

(68) where C denotes a path around C lying outside the singularity at the leading edge. For the velocity field (63), the residue from the sole enclosed pole at the origin gives X = 0, Y = 2π ρU ˆ , M = π ρU ˆ . It is useful to derive the solution (63) and the transformation (59) from the complex potential, w1 (ζ, σ, τ ) = (κ/2π i) log(ζ − σ eiτ ) − (κ/2π i) log(ζ − σ −1 eiτ ),

(69)

for a point vortex of circulation κ at ζ = σ eiτ , lying inside the circle C when |σ | < 1. For positive κ, differentiating with respect to σ gives a dipole in the arg ζ = τ − π/2 direction [49, for example], w2 (ζ, σ, τ ) = −(κ/2π i)eiτ [1/(ζ − σ eiτ ) + σ −2 /(ζ − σ −1 eiτ )].

(70)

The dipole at the leading edge is obtained by setting τ = 0 and σ = 1 to give w2 (ζ, 1, 0) = iκ/π(ζ − 1),

(71)

which, for κ = 2π , is the required complex velocity (63). Setting τ = π/2 and σ = 0 gives a dipole at the origin, oriented in the arg ζ = 0 direction, which, omitting an additive constant, can be written σ 2 w2 (ζ, 0, π/2) = −(κ/2π )(ζ + 1/ζ ).

(72)

This is the complex potential for uniform flow past the circle but equally provides, for κ = π , the mapping (59) since the streamlines in the ζ -plane map to lines of constant y in the z-plane cut along the plate: the same dipole solution (70) provides the required complex velocity field and the conformal mapping. A similar result was noted by [51] when extending the results of [52] to domains of connectivity three and more, although there the correspondence was between the complex velocity potentials not between the complex velocities as here.

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5.2 A Finite Collection of Plates To generalise the results of Sect. 5.1, consider the unbounded region Dz exterior to a collection of M + 1 plates, denoted D j , j = 0, . . . , M, of bounded extent aligned in the positive x-direction, corresponding to a set of plates at small angle of attack to an oncoming uniform flow. The domain can be scaled so that the plate D0 lies on y = 0 with |x| < 1. The normal velocities on the plates are determined by their individual angles of attack and so are specified as v j , j = 0, . . . , M, where v0 can be scaled to unity by linearity. We follow [50] taking the pre-image domain to be the unit disc in the ζ -plane with M smaller circular discs excised, and denoting the domain by Dζ , the circular boundary of the jth excised circular disc by C j , j = 1, . . . , M and the unit circle, the pre-image of D0 , by C0 . The domain is then specified by the centres δ j and radii ρ j of the circles C j and the results in Sect. 5.1 correspond to the case M = 0. Finding the flow-field and the forces and moments on the plates thus reduces to related sub-problems: finding the conformal map between Dz and Dζ and obtaining the velocity field in Dζ . The mapping between D z and Dζ . The pre-image Dζ is not unique but can be made so, to within a rotation, by specifying the pre-image ζ = β of the point z = ∞, at infinity in Dz . The mapping here will thus depend on β which can be chosen to exploit, for example, symmetry in the domain Dz . The mapping from Dζ to Dz follows from the construction that led to (72). Analogously to (69), the complex velocity potential for a point vortex of strength κ located at ζ0 = σ eiτ can be written [51] as (73) G 0 (ζ, σ, τ ) = (−iκ/2π ) log[ω(ζ, σ eiτ )/σ ω(ζ, σ −1 eiτ )], where ω(ζ, ζ0 ) is the Schottky-Klein prime function associated with the domain. Differentiating (73) with respect to σ gives a dipole with flow in the arg ζ = τ − π/2 direction, w3 (ζ, σ, τ ) = (−iκ/2π )eiτ [Ω(ζ, σ eiτ ) + (1/σ 2 )Ω(ζ, σ −1 eiτ ) − 1/σ ], where Ω(ζ, ζ0 ) =

∂ω(ζ, ζ0 ) 1 , ω(ζ, ζ0 ) ∂ζ0

(74)

(75)

is the logarithmic derivative of ω(ζ, ζ0 ). Setting ζ0 = β gives a dipole at z = ∞, and thus uniform flow, in Dz . It remains to find the positions ζ jl,t = δ j + ρ j exp(iλ jl,t ) on the circles C j that correspond to the leading and trailing edges of the plates. These are the pairs of points where streamlines from infinity meet the circle C j , i.e. the stagnation points on C j and so satisfy dΩ (ζ jl,t , β) = 0. (76) dζ

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The mapping from Dζ to Dz is straightforward given the data δ j and ρ j , with the constant value of the streamfunction w3 on C j giving y j and the values of w3 at the stagnation points giving the ends of the plates. Finding the inverse mapping is more difficult. Given the plate data it is necessary to solve simultaneously the M equations on C j , (77) w3 = y j , and the 2M Eq. (76). This is considered in greater detail in Sect. 5.3. The velocity field in Dζ . The required velocity field associated with each plate is given by a dipole tangent to the plate, i.e. tangential to the circle C j . It is sufficient to construct, for each j, the solution that has a dipole at ζ jl aligned in the λ jl − π/2 direction (i.e. normal to the radius at the stagnation point) with unit imaginary part on C j and zero imaginary part on all other boundaries. Then the solution for arbitrary constant v on each plate follows by linear superposition. From (74), this is simply w3 (ζ, |ζ jl |, λ jl ).

5.3 The Case of Two Plates The simplest non-trivial example is given by two plates. The domain Dζ can then be taken as the annulus q < |ζ | < 1 for some value of the conformal modulus q [53]. Crowdy [50] notes that for this geometry ω(ζ, γ ) = −(γ /C 2 )P(ζ /γ , q),

(78)

where P(ζ /γ , q) = (1 − ζ )

∞  k=1

(1 − q 2k ζ )(1 − q 2k ζ −1 ),

C=

∞  (1 − q 2k ),

(79)

k=1

so the logarithmic derivative (75), and its derivative in (76), are simple sums, allowing the solutions to (76) and (77) to be obtained straightforwardly. Figure 19 shows the annular domain Dζ . The dipole for the mapping lies on ζ = 0 oriented in the ζ direction. Streamlines passing through the stagnation points are shown in bold with the leading edges marked by red crosses and trailing edges by green crosses. (b) The plates in the domain Dz . Streamlines here are simply lines of constant y. Figure 20 gives isolines in the annular domain of v, the velocity component in the y direction in domain Dz . In (a) v = 1 on plate D0 and v = 0 on plate D1 . These isolines are also the streamlines for a dipole at the leading edge of plate D0 oriented along the plate. In (b) v = 0 on plate D0 and v = 1 on plate D1 . These isolines are also the streamlines for a dipole at the leading edge of plate D1 oriented along the plate. The general solution for arbitrary constant v on each plate is a linear combination of these fields.

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(a)

(b)

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2

1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2

-1

0

1

2

Fig. 19 The mapped domains. a The annular domain Dζ . The circular boundaries C0 , C1 correspond to the plates. The dipole lies on ζ = 0 oriented in the ζ direction. Streamlines passing through the stagnation points are shown in bold with the leading edges marked by red crosses and trailing edges by green crosses. b The plates in the domain Dz . Streamlines here are simply lines of constant y

(a)

(b)

Fig. 20 Isolines in the annular domain Dζ of v, the velocity component in the y direction in the domain Dz . a v = 1 on plate D0 and v = 0 on plate D1 . These isolines are also the streamlines for a dipole at the leading edge of plate D0 oriented along the plate. b v = 0 on plate D0 and v = 1 on plate D1 . These isolines are also the streamlines for a dipole at the leading edge of plate D1 oriented along the plate. The general solution for arbitrary constant v on each plate is a linear combination of these fields

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With the velocity field q determined the forces on the plates follow from (68). Since the integrand is periodic over the closed path of integration, the trapezium rule gives exponentially accurate results. To minimise truncation errors the path is taken to lie midway between the given circle and the point, β, at infinity. Once the forces have been determined in Dζ the motion of the plates in the physical plane follows the dynamics in [6]. This gives a set of ordinary differential equations that can be integrated forward in time by standard methods.

6 Further Comments The majority of analytical studies on dynamic fluid-body interactions have been for cases of a single body as in Sects. 3 and 4 of this chapter on a skimming body and on body liftoff respectively and in the references [1–11]. The exceptions are in [1] which concerns a cascade of bodies in channel flow and in the present chapter’s Sect. 5 which describes interactions involving many bodies moving in surrounding potential flow of fluid with none or possibly (by use of simple images) at most one solid fixed wall present. The aim throughout this chapter has been to seek out relatively simple configurations and basic evolution properties first, as the occurrence of important linear and nonlinear effects implies that mathematical accounts are often likely to be insightful. Assumptions made include those of unsteady laminar incompressible fluid motions over medium-to-large ranges of the Reynolds number and in two spatial dimensions but there must also be awareness concerning three-dimensionality, flow separation, transition, turbulence, impacts and rebounds, and so on. We believe the study here may be of interest in terms of mathematical issues, real applications, the science of fluid dynamics with freely moving bodies present and the clear interaction with direct numerical simulations. In reality there are often very complex additional phenomena to deal with in the practical situations of concern as hinted above. Specific issues are connected with the individual sections of the chapter. Thus considerably more detail and analysis of the original multi-body problems addressed in Sect. 2.1 is to be found in [10] especially in terms of the wakes behind the bodies but also in terms of the influences from the presence of flexible patches of wall which respond to the local pressure due to the fluid flow: compare with [7]. More investigation of the skimming problems described in Sect. 2.2 and later in Sect. 3 is given in [9]. It would be interesting to consider adding in the effects of threedimensionality within the various skimming-body analyses conducted to date and the same comment applies to the analysis of liftoff. Both for the skimming scenario in Sect. 3 and the liftoff scenario in Sect. 4 the influence of viscosity also remains to be examined seriously given that the influence is potentially substantial. As far as the beginnings of many-body analysis in Sect. 5 are concerned the basic elements of an account that includes viscous effects can be found in [6] for the channelflow situation, where essentially the same viscous-inviscid interplay over two axial length scales covers the case of many short bodies at small incidence, while the corresponding situation for external flow with a boundary layer brings in triple-deck

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theory. An extension of the study in reference [7] would be necessary to cater for many long bodies freely moving inside a boundary layer or channel flow. There are numerous intriguing follow-on studies which could be undertaken. With that in mind we refer to the earlier remarks on three-dimensional effects, on many bodies being present in the fluid flow and on the influences of different body lengths as well as thicknesses. In addition future investigations could tackle increased connections with experimental work, observations and direct numerical simulations, not to forget full nonlinearity arising with a body in a viscous wall layer or with a body of length comparable with the development length in the oncoming flow, namely of the order of the chord length in a boundary layer and of order Reynolds number multiplied by the channel width in a channel flow. Body movement relative to the nearby wall can also be a key factor [7]. We should perhaps mention that there are many other near-wall phenomena of interest, including problems of flexible walls, dynamic roughnesses on a surface and the impact on airfoil stall. Finally, novel instabilities occur in most of the dynamic fluid-body interactions addressed so far, and these instabilities merit much further study. Acknowledgements Thanks for support are due (FTS) to EPSRC through grant numbers GR/T113 64/01, EP/G501831/1, EP/H501665/1 during part of this research, and (SB) to the Republic of Turkey for financial support. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme. The Mathematics of Sea Ice Phenomena when part of the work on this paper was finalised, supported by EPSRC grant number EP/K032208/1. Thanks are due to Roger Gent and Richard Moser at AeroTex, Rob Lewis at TotalSim, Sarah Bee and Mark Honeywood at Sortex-Buhler and UCL colleagues Robert Bowles, Nick Ovenden and Sergei Timoshin for very helpful discussions on body and particle movement in near-wall shear flow.

References 1. Smith, F.T., Ellis, A.S.: On interaction between falling bodies and the surrounding fluid. Mathematika 56, 140–168 (2010) 2. Hicks, P.D., Smith, F.T.: Skimming impacts and rebounds on shallow liquid layers. Proc. R. Soc. A 467, 653–674 (2011) 3. Smith, F.T., Wilson, P.L.: Fluid-body interactions: clashing, skimming, bouncing. Phil. Trans. R. Soc. A 369, 3007–3024 (2011) 4. Smith, F.T., Wilson, P.L.: Body-rock or lift-off in flow. J. Fluid Mech. 735, 91–119 (2013) 5. Liu, K., Smith, F.T.: Collisions, rebounds and skimming. Phil. Trans. R. Soc. A 372(2020) (2014) 6. Smith, F.T., Johnson, E.R.: Movement of a finite body in channel flow. Proc. R. Soc. A 472(2191), 20160164 (2016) 7. Smith, F.T.: Free motion of a body in a boundary layer or channel flow. J. Fluid Mech. 813, 279–300 (2017) 8. Smith, F.T., Liu, K.: Flooding and sinking of an originally skimming body. J. Eng. Math. 107(1), 37–60 (2017) 9. Liu, J.: Shallow-water skimming, skipping and rebound problems, Ph.D. thesis, University College London (2017) 10. Balta, S.: On fluid-body and fluid-network interactions, Ph.D. thesis, University College London (2017)

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11. Balta, S., Smith, F.T.: Fluid flow lifting a body from a solid surface (in preparation) (2018) 12. Ladd, A.J.C.: Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results. J. Fluid Mech. 271, 311–339 (1994) 13. Foucaut, J.-M., Stanislas, M.: Take-off threshold velocity of solid particles lying under a turbulent boundary layer. Exp. Fluids 20(5), 377–382 (1996) 14. Patankar, N.A., Huang, P.Y., Ko, T., Joseph, D.D.: Lift-off of a single particle in Newtonian and viscoelastic fluids by direct numerical simulation. J. Fluid Mech. 438, 67–100 (2001) 15. Gray, J.M.N.T., Ancey, C.: Multi-component particle-size segregation in shallow granular avalanches. J. Fluid Mech. 678, 535–588 (2011) 16. Virmavirta, M., Kivekäs, J., Komi, P.V.: Take-off aerodynamics in ski jumping. J. Biomech. 34(4), 465–470 (2001) 17. Miller, M.C., McCave, I.N., Komar, P.D.: Threshold of sediment motion under unidirectional currents. Sedimentology 24(4), 507–527 (1977) 18. Owen, P.R.: Saltation of uniform grains in air. J. Fluid Mech. 20(2), 225–242 (1964) 19. Shao, Y., Raupach, M.R., Findlater, P.A.: Effect of saltation bombardment on the entrainment of dust by wind. J. Geophys. Rese.: Atmos. 98(D7), 12719–12726 (1993) 20. Jia, L.-B., Li, F., Yin, X.-Z., Yin, X.-Y.: Coupling modes between two flapping filaments. J. Appl. Mech. 581, 199–220 (2007) 21. Guazzelli, E.: Sedimentation of small particles: how can such a simple problem be so difficult? C.R. Me´canique 334 (2006) 22. Godone, D., Stanchi, S.: Soil Erosion Studies. InTech (2011) 23. Andrew, M., David, M., deVeber, G., Brooker, L.A.: Arterial thromboembolic complications in paediatric patients. Thromb. Haemost. 78(1), 715–725 (1997) 24. Babyn, P.S., Gahunia, H.K., Massicotte, P.: Pulmonary thromboembolism in children. Pediatr. Radiol. 35(3), 258–274 (2005) 25. Baker Jr., W.F.: Diagnosis of deep venous thrombosis and pulmonary embolism. Med. Clin. North Am. 82(3), 459–476 (1998) 26. Gaver, D.P. III, Jensen O.E., Halpern, D.: Surfactant and airway liquid flows. In: Nag, K. (ed.) Lung Surfactant and Disorder: Lung Biology in Health and Disease, vol. 201, p. 191. Taylor and Francis, London (2005) 27. Iguchi, Y., Kimura, K.: A case of brain embolism during catheter embolisation of head arteriovenous malformation. What is the mechanism of stroke? J. Neurol. Neurosurg. Psychiatry 78(1), 81 (2007) 28. Secomb, T.W., Skalak, R., Özkaya, N., Gross, J.F.: Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163, 405–423 (1986) 29. White, A.H.: Mathematical modelling of the embolisation process in the treatment of arteriovenous malformations. Ph.D. Thesis, University of London (2008) 30. Balta, S., Smith, F.T.: Inviscid and low viscosity flows in multi-branching and reconnecting networks. J. Eng. Math. 104(1), 1–18 (2017) 31. Ellis, A.S.: Modelling chute delivery of grains in a food-sorting process, Ph.D. Thesis, University of London (2007) 32. Ellis, A.S., Smith, F.T.: A continuum model for a chute flow of grains. SIAM J. Appl. Math. 69(2), 305–329 (2008) 33. Koch, D.L., Hill, R.J.: Inertial effects in suspension and porous-media flows. Annu. Rev. Fluid Mech. 33, 619 (2001) 34. Willetts, B.: Aeolian and fluvial grain transport. Philos. Trans. R. Soc. Lond. A 356(1747), 2497–2513 (1998) 35. Bowles, R.G.A., Smith, F.T.: Lifting multi-blade flows with interaction. J. Fluid Mech. 415, 203–226 (2000) 36. Purvis, R., Smith, F.T.: Planar flow past two or more blades in ground effect. Q. J. Mech. Appl. Math. 57(1), 137–160 (2004) 37. Smith, F.T., Timoshin, S.N.: Planar flows past thin multi-blade configurations. J. Fluid Mech. 324, 355–377 (1996b)

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38. Smith, F.T., Ovenden, N.C., Franke, P.T., Doorly, D.J.: What happens to pressure when a flow enters a side branch? J. Fluid Mech. 479, 231–258 (2003) 39. Tuck, E.O., Dixon, A.: Surf-skimmer planing hydrodynamics. J. Fluid Mech. 205, 581–592 (1989) 40. Smith, F.T.: Upstream interactions in channel flows. J. Fluid Mech. 79, 631–655 (1977) 41. Howison, S.D., Ockendon, J.R., Oliver, J.M.: Oblique slamming, planing and skimming. J. Eng. Math. 48, 321–337 (2004) 42. Khabakhpasheva, T.I., Korobkin, A.A.: Oblique impact of a smooth body on a thin layer of liquid. Proc. R. Soc. 469(2151) (2013) 43. Wagner, H.: Über Stoß-und gleitvorgänge an der Oberflaäche von Flüssigkeiten (Phenomena associated with impacts and sliding on liquid surfaces). Zeitschr. Math. Mech. 12 (1932) 44. Batyaev, E.A., Khabakhpasheva, T.I.: Initial stage of the inclined impact of a smooth body on a thin fluid layer. Fluid Dyn. 48(2), 211–222 (2013) 45. Stewartson, K., Williams, P.G.: Self-induced separation. Proc. R. Soc. Lond. A 312(1509), 181–206 (1969) 46. Smith, F.T.: The laminar separation of an incompressible fluid streaming past a smooth surface. Proc. R. Soc. Lond. 356(1687), 443–463 (1977) 47. Scheichl, B., Kluwick, A., Smith, F.T.: Break-away separation for high turbulence intensity and large Reynolds number. J. Fluid Mech. 670, 260–300 (2011) 48. Lagally, M.: Die reibungslose Strömung im Aussengebiet zweier Kreise. Z. Angew. Math. Mech. 9, 299–305 (1929) 49. Johnson, E.R., McDonald, N.R.: The motion of a vortex near two circular cylinders. Proc. Roy. Soc. A 460, 939–954 (2004) 50. Crowdy, D.G.: Calculating the lift on a finite stack of cylindrical aerofoils. Proc. Roy. Soc. A 462, 1387–1407 (2006) 51. Crowdy, D.G., Marshall, J.S.: Analytical formulae for the Kirchhoff-Routh path function in multiply connected domains. Proc. Roy. Soc. A 461, 2477–2501 (2005) 52. Johnson, E.R., McDonald, N.R.: Vortices near barriers with multiple gaps. J. Fluid Mech. 531, 335–358 (2005) 53. Nehari, Z.: Conformal Mapping. McGraw-Hill, New York (1952)

Certain Aspects of Problems with Non Homogeneous Reactions Alejandro Omón Arancibia

Abstract This note reviews certain aspects of systems with reaction terms which are non homogeneous, this is nonlinearities such that their value at zero are different from zero. This type of reactions are frequent in problems where temperature is a relevant variable, for example strongly exothermic chemical reaction like a combustion chamber, or a bio-reactor. The topics to be reviewed are far from covering all the aspects to be analyzed in these problems, but despite this they are interesting for a broad audience. Keywords Nonlinear eigenvalue problem · Arrhenius reaction rate · Blow-up 2010 AMS Subject Classification 35J60 · 35P30 · 65N30 To Zoe and Gabriel, the Sources of Inspiration.

1 Introduction The boundary value problem − u = λ f (u) in , u = 0 on ∂ ,

(1)

A. Omón Arancibia (B) Departamento de Ingeniería Matemática, Universidad de La Frontera, Avenida Francisco Salazar 01145, Temuco, Chile e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_3

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and its corresponding initial value problem ∂v = v + λ f (v) in ]0, T [ × , ∂t v = 0 on ]0, T [ × ∂ ,

(2)

v(0, x) = v0 ]0, T [ × ∂ , are both model descriptions of many phenomena. For example, the case f (s) = s(1 − s) is associated to the spreading of a favoured gene in a diploid population, a model that dates since the work of Fisher [28] and Kolmogorov-Petrovskii-Piskounov [43], where traveling wave solutions for (2) were sought. In any of the two previous problems (1) or (2), the term involving the laplacian takes into account the diffusion of the species u in the domain , often an open bounded subset of IR N with regular boundary, while the term f (u) takes into account the reaction rate associated to the available amount of u. For the previous example f (s) = s(1 − s), the reaction is positive for s ∈ ]0, 1[ while out of the interval the reaction is negative; 0 and 1 determine a barrier on whether the reaction term creates or destroys the modelled species u. The boundary condition specifies the interchange of u between  and c and not necessarily needs to be a Dirichlet homogeneous one, while the initial condition specifies the initial amount of u at the beginning of the modelling. In general, the most frequent type of nonlinearities in both (1) or (2) are of the polynomial type, for example f (t) = t p , f (t) = t + t p , or f (t) = t p − t q , which in the three cases also satisfy that f (0) = 0, often known as being homogeneous at zero. There is extended literature on both (1) or (2) with (at least) continuous and homogeneous reaction/nonlinearity f . But not necessarily the nonlinearity needs to be homogeneous at zero, as is the case of f (t) = et , where e0 = 1. The exponential nonlinearity in either (1) or (2) has many motivations, from where two are particularly relevant: first, it is the natural transcendental function (a very mathematical motivation), and second it consists on a first order approximation to an Arrhenius reaction rate. Concerning the first one, is surprising this reason is explicitly given in Ref. [3] by Aris, a well known name in the modelling of chemical reactors (Aris was Chemical Engineer himself). Concerning the second one, Arrhenius reaction rates are fundamental in the modelling of exothermic chemical reaction systems, in particular in Combustion Theory. Just for complementing, the particular case of the exponential nonlinearity in (1) has a strong meaning in Differential Geometry for dimension two (or for surfaces in dimension three), as can be seen in the works of Gauss or Liouville, see for example [50] which is a key reference in Complex Variable application to quadrature. Concerning the Arrhenius reaction rate, roughly speaking, comes from the modelling of a chemical reaction where Temperature is one of the unknowns to model: reacting in a mixture that obeys a set of having a set {Ai }i ∈ I of  species  chemically j j J chemical reactions i ∈I μi Ai −→ i ∈I νi Ai j ∈ J which also depend on the instantaneous temperature T of the reaction, the modelling of the reactive process

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being held in a spatial domain  ⊂ IR n where diffusion of the species take place and also variations of temperature, then the reaction rate for the l−th species is given by r˙l =



  Yi μi ρ f j (T ) mi i∈I

j

j (νl



j μl )

j∈J

(3)

where μlk and νlk correspond to the stoichiometric coefficients as reactant and product respectively of the l-th species in the k−th reaction, ρ is the density of the reacting mixture, Yi is the mass fraction of the i−th species, m i is the mass fraction of a single molecule of the i−th species, and the function f j (T ) is empirical and has a structure of the type f j (T ) = C j T

αj



−E j exp RT

 ,

(4)

where C j , α j and E j are specific values of the j−th reaction, R is the universal gas constant. The expression at (4) is experimental, and it was discovered by the Swedish Chemist Svante Arrhenius, a student of Ludwig Boltzmann, laureate with the Nobel Price in Chemistry the year 1903. It is not difficult to observe that even on a very simple set of reactions, with a few species taking part of it, the system of nonlinear parabolic PDE’s to solve is equal to the number of species - 1, with reaction rates that will have the addition of as many Arrhenius terms as species takes part in the reactions, plus an evolution equation for temperature dynamics, often a nonlinear parabolic one. From both a mathematical and practical point of view, even for a simple chain of (almost) real netreaction, the problem is untractable, therefore simplifications are always introduced. The most common simplification is the decoupling of the chemical kinetics from the thermodynamics, which under other simplifications not mentioned here in detail, finally led to the classical parabolic Gelfand Problem ∂v = v + λ ev in ]0, T [ × , ∂t v = 0 on ]0, T [ × ∂ ,

(5)

v(0, x) = v0 (x) ]0, T [ × ∂ , and under others simplifications it arrives to the evolution Perturbed Gelfand Problem ∂v = v + λ ev/(1+ v) in ]0, T [ × , ∂t v = 0 on ]0, T [ × ∂ , v(0, x) = v0 (x) ]0, T [ × ∂ ,

(6)

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where  > 0 is a small parameter, often associated to the inverse of the activation energy of the reaction in its limit to infinity. For a detailed deduction of the equations that govern a chemical reacting system that is strongly temperature dependent, see any of the classical Refs. [8–18, 32–47]. Having given a basic, and probably not accurate, motivation on why to study any (1) or (2) with nonlinearities that are non homogeneous, this is f (0) = 0, it is given a brief idea of which aspects will be studied. First, in the case of steady problems there will be discussed some aspects on the existence of solutions, and multiplicity; as a complement, some ideas of stability will be also given. As a second part, the case of the evolution problems will be discussed, in particular some ideas concerning blow-up, stability, and certain particular type of solutions: traveling waves. As a third case, although not concerning directly with Partial Differential Equations, it will be studied a model of a one step reactor involving also temperature, whose origin is a system of two nonlinear parabolic equations with a non homogeneous reaction term. Just to conclude, there are given some problems which are of particular interest to the author, who hopes they can be interesting and challenging for young researchers in the topic; up to the author’s knowledge, the problems are open. This note is by far a not detailed presentation of certain topics and results on (1) and (2) which are of personal interest to the author, and under this perspective, the author apologizes on all the mistakes on what will be presented, but also on the choice of what is presented.

2 Steady Problems One of the most difficult problems in nonlinear analysis, is to prove existence of solutions. In this context, to ensure existence of problem (1) for a (at least) continuous nonlinearity f is a hard problem. A very general theory that provides existence of solutions of problems like (1) is the Leray-Schauder Topological Degree, or Topological Degree in infinite dimension, see [13] or [23]. This strong and beautiful theory is the extension of the Degree in finite dimension, but precisely the extension is only possible for Compact Perturbations of the Identity, therefore limited on which type of operators can be studied. Also, the explicit computation of the Degree is not trivial and only in simple cases this can be done, mainly by homotopy with a simpler operator whose Degree is known to be different from zero (often the identity). Variational Methods, which in a very broad sense correspond to see the solutions of problems like (1) as a critical value of a certain functional (often an energy functional), are well extended in proving existence of nonlinear elliptic problems. But variational methods need to ensure convergence of the sequence that extremes the functional and its derivative, this is it is required convergence of the minimizing/maximizing sequence involved, therefore it is fundamental to have available compactness criteria. The well known compactness criteria are the Sobolev’s embedding Theorems, which

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require nonlinearities of the type f (t) = t p or f (t) = t (1 + t p ), which in particular satisfy f (0) = 0. In order to skip any of the two preciously mentioned possible alternatives to prove existence of solutions for (1), first [41] and later [58] extended a very natural idea from Calculus in IR: given {xn }n ⊂ IR a monotonous and bounded sequence, it must converge. One of the beautiful character of previous statement is that ensures convergence, but it is not needed to known (a priori or posteriori) the limit, neither a priori the Cauchy’s criterion: only the monotonous and bounded character in IR are needed for having convergence. In order to extend the previous result to infinite dimensional problems, any of the two previous references defined what are known as sub and super solutions of (1), as a pair of functions u and u which satisfy − u ≤ λ f (u) in ,

(7)

u = 0 on ∂ , and − u ≥ λ f (u) in ,

(8)

u = 0 on ∂ , respectively. When the nonlinearity f is non homogeneous and monotonous increasing, it is shown in [58] that for a given initial sub solution u 0 , the sequence defined by − u n+1 ≤ λ f (u n ) in , u n+1 = 0 on ∂ ,

(9)

generates a monotonous increasing sequence (in the pointwise order), while for an initial super solution u 0 , the sequence defined by − u n+1 ≥ λ f (u n ) in , u n+1 = 0 on ∂ ,

(10)

is monotonous decreasing. If the initial terms satisfy that u 0 ≤ u 0 , both being elements of a Banach or Hilbert infinite dimensional space, often H01 () = W01,2 (), and both sequences {u n } and {u n } converge to the same limit, let us say u, it is held that u 0 ≤ u ≤ u 0 , but also u is a solution of (1). The whole analysis is given in either [41] or [58], and this existence result is often known as the Method of Sub and Super Solutions. In practical terms, one of the main difficulties of this method is to have an initial super solution, as u 0 = 0 is in general a sub solution. Despite of having a first existence result for solutions of (1) with non homogeneous nonlinearities, it is explicitly mentioned now that (1) is a nonlinear eigenvalue problem, this is the unknowns in (1) are two: the function u, but also the values

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λ ∈ IR. Said in a different way: given a fixed value λ > 0, is there at least one u ∈ H01 () such that the pair (λ, u) satisfies (1)? In this context, it is given a second existence result of solutions for (1), always understanding a solution as a pair (λ, u); the set of real values λ > 0 such that there exists a function u, with (λ, u) satisfying (1) is known as the spectrum of the problem. For problem (1) with nonlinearities f such that f (0) = 0, the Bifurcation Theory of Crandall-Ravinowitz, see [20], provides existence results of non trivial solutions. But for the type of non linearities under attention in this note, this theory fails as f (0) = 0. As the condition f (0) = 0 is not held, the frame that ensures existence of non trivial solutions for elliptic problems with non linearities such that f (0) > 0 is given by the Implicit Function Theorem, see [2]. A particular well developed application of the Implicit Function Theorem is Ref. [22], motivated by a catalyst model. The general idea in the Implicit Function Theorem is to study the set of solutions of the equation F(λ, u) = u + λ f (u) = 0, with F : H01 () −→ L 2 (), where under some satisfied hypotheses there is ensured the existence of a nontrivial curve ∂F is not invertible, this of solutions starting from (λ, u) = (0, 0) up to when Fu = ∂u is when the linear weighted-eigenvalue problem − ψ = λ f  (u) ψ in ,

(11)

ψ = 0 on ∂ , has a nontrival solution ψ, see Ref. [6]; for existence of solutions of (11), see [14]. For the case of nonlinearities f which are continuous, convex increasing and such that f (0) > 0, it is not so difficult to prove that the spectrum of (1) is bounded, this is there exists a λ∗ > 0 such that for λ ∈ ]0, λ∗ [ problem (1) has at least one solutions, while for λ > λ∗ problem (1) has no solution; λ∗ is known as the bending value, and it is classical to present plots on IR + × L ∞ (), known as Bifurcation Diagrams, which within others show the multiplicity of solutions (associated to each λ); see Ref. [49]. Probably the most relevant non homogeneous nonlinearity in (1) is f (t) = et , known as the Gelfand Problem. As was earlier said, the relevance of this problem is that it gives a first order description of the temperature evolution in a highly exothermic reaction. Although references in this context date up to the beginning of the 20−th century, there are two quite relevant ones concerning this problem: the first one is the book by Franz-Kamenetskii [29], and the second one is the monograph [33] by Gelfand, where for the first time there was established formal multiplicity results for (1) in radial geometry for dimension two and three. The impact of the results in [33] were so relevant that because of them the problem was baptized as Gelfand Problem in his honour. It is stated in [33] that for dimension three, there exists a numerable family of balls (equivalently of radii) such that the chemical reaction is self sustained in the vessel. In particular, this type of results was (is) very well known from experimental data, but not formally established at all before Gelfand’s work. The Gelfand Problem is nowadays in the front line of research in non linear elliptic partial differential equations concerning, for example, problems

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in Differential Geometry (Conformal Radius), Applied Mathematics (Combustion Theory or better Charts concerning Cartography), or Partial Differential Equations itself (Hardy Inequalities, stability of solutions); information about it can be found in any of the two Refs. [8–27]. Less studied but as relevant as the exponential nonlinearity, the so called Perturbed Gelfand Problem to (1) with nonlinearity f : IR + ∪ {0} −→ IR + given   corresponds t with  > 0. Even more, skipping many details, a nonlinby f (t) = ex p 1 + t earity that takes into account the proper structure of the Arrhenius reaction would be f : IR + ∪ {0} −→ IR + with f (t) = (1 +  t) p et/(1+t) where now p is a Rational number. Notice that again in any of the two previous nonlinearities f (0) > 0. In the case of the Perturbed Gelfand problem, it has unbounded spectrum, this is given any λ > 0 the boundary value problem − u = λ eu/(1+u) in  u=0

(12)

on ∂ ,

has at least one solution; this comes from the fact that the nonlinearity is bounded in IR + . There are lots of questions concerning (12), in which seem to be relevant the qualitative behaviour of the solution set of (12) as  0+ . To be more specific, does the solution set of (12) converges in some sense to the solution set of the Gelfand Problem? The answer is not clear at all, although in the case  the unit ball in dimension two [25] has given results on this, and earlier [5]. Despite previous works, the very interesting and earlier Ref. [17] studied problems motivated from chemical reactors and gave criteria for convergence of the perturbed solutions, in the perturbation parameter  > 0, to the solutions of the limit problem (or the convergence of the bifurcation branches of the perturbed problems to the limit bifurcation branches); but Ref. [17] went even beyond and it also explicitly stated convergence is not always held, this is there are bifurcations branches which do not converge to the limit problem. This last situation is critical in the problem − u = λ (1 +  u) p eu/(1+u) in  u=0

(13)

on ∂ ,

with p < 0. Some theoretical results, the same as numerical results too, are given in the unpublished manuscript [55]. This reference also gives information on the first eigenvalue of the corresponding linearized problems, which up the the knowledge of the author is not reported in the literature. Defining λ as the smaller bending point and λ the bigger bending point, there are given now estimations for both λ and λ in the next Theorem 1 Given 
1 and  > 0 in (16). Then, its spectrum satisfies that λ ≤ λ1 (α − 1)

 α α  1 α , e x ∗ ()

(17)

where λ1 is the first eigenvalue of the laplacian and x ∗ () is a point bound that depends on  such that x ∗ () −→ 1 as  0+ uniformly in α > 1. A very relevant aspect when the nonlinearity f in (1) is convex increasing and non negative in the positive semi-axis, like in the Gelfand Problem, is that the first eigenvalue of the linearized at a minimal solution u (this is, the first eigenvalue of (11) at u = u ) satisfies that λ1 < λ, while the linearized at a non minimal solution satisfies that λ1 > λ, and at the first bending it is held that λ1 = λ (therefore the failure of the Implicit Function Theorem), see for example [6–64]. In this context, minimal solutions are said to be stable; all the eigenvalues of (11) at u = u are above of λ); while non minimal solutions are said to be unstable: at least the first eigenvalue of (11) at u = u is below λ. Lately, we will try to connect this definition of stability with the classical one for a dynamics, in particular at the end when some proposed problems will be posed. Just to conclude this section on steady problems, there are given a series of different plots concerning numerical bifurcation branches of problem (1), and also the computation of the corresponding Morse Index in order to verify the consistency of the numerics. The integration of the PDE was done with Finite Elements using polynomials of first degree, all implemented in a Newton’s Algorithm for the solution of the corresponding nonlinear system, and then iterate on the nonlinear eigenvalue λ; the nonlinear term was approximated by a Trapezium approximation (Figs. 1, 2, 3, 4 and 5).

Certain Aspects of Problems with Non Homogeneous Reactions

99

140

120

100

80

60

40

20

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 1 Bifurcation diagram for solutions of (12):  the unit ball,  = 0.1, lower (minimal) and middle branch; λ in the x-axis and u(0) in the y-axis 2500

2000

1500

1000

500

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 2 Bifurcation diagram for solutions of (12):  the unit ball,  = 0.1, middle and upper branch; λ in the x-axis and u(0) in the y-axis

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5

4

3

2

1

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 3 First eigenvalue of the linearized problem:  = 0.1; λ in the x-axis and μ1 in the y-axis 4.5 4 3.5 3

u(0)

2.5 2 1.5 1 0.5 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

λ

Fig. 4 Bifurcation diagram for (16): p = 2 and  = 0.1, minimal bending at 1.5202926 and non minimal bending at 1.091109

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4.5 4 3.5

μ1(λ)

3 2.5 2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

λ

Fig. 5 First eigenvalue of the linearized problem: p = 2 and  = 0.1

3 Evolution Problems Solutions of (1) are a particular case of solutions of (2): they are solutions that do not evolve in time. In this context it is natural to expand the knowledge concerning solutions of (2). A first natural question concerning (2) independently of its non linearity is to give an existence theorem even local in time, this is a result that ensures the existence of a solution u(t, x) for at least a short period of time. In general, results of this type are based in extensions to infinite dimension vectorial spaces of the classical results for ODE’s in finite dimensional spaces. Notice that the well (and pioneering in time) established existence result by Euler’s Polygonal approximation is not extended to infinite dimensional spaces by the lack of a compactness argument in order to have convergence. Within frequent aspects studied in (2) are the existence of Blow-up, estimation of blow-up time, estimation of the blow-up set, and stability of global orbits. As was pointed at the beginning of this chapter, steady solutions are a particular type of solutions of (2), as also are only time dependent solutions, this is solutions of the initial value problems x  (t) = λ f (x(t)) t > t0 x(t0 ) = x0 ,

(18)

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implicitly defined by the relation x(t) x(t0 )

ds = λ (t − t0 ), f (s)

which has as a consequence: if the nonlinearity f is such that

(19)

∞ t0

dx f (x)

< ∞, then

solutions of (18) are local in time. For example, in the case of f (t) = et it is almost

∞ direct to see that e−x d x = e−x0 < ∞, therefore the right side in (18) must be t0

defined up to a finite time t; this is seen from solving explicitly the integration, which gives the explicit expression for the solutions   −x0  e + t0 − t x(t) = − ln λ . λ

(20) −x0

From (20) it is direct to see that the solution is defined only for t0 < t < e λ + t0 , interval that varies in term of three parameters: t0 , λ and x0 . An always powerful pair of techniques always relevant in the study of blow-up are the Parabolic Maximum Principle and Comparison Theorem. Concerning the second one, a sub solution of (2) is a function v(t, x) that satisfies ∂v ≤ v + λ f (v) in ]0, T [ × , ∂t v ≤ 0 on ]0, T [ × ∂ ,

(21)

v(0, x) ≤ v0 ]0, T [ × ∂ , and a super solution v is a function that satisfies ∂v ≥ v + λ f (v) in ]0, T [ × , ∂t v ≥ 0 on ]0, T [ × ∂ ,

(22)

v(0, x) ≥ v0 ]0, T [ × ∂ , Clearly a solution v(t, x) will satisfies v ≤ v ≤ v. When in (2) the nonlinearity f is continuous, convex increasing and positive, always having the exponential nonlinearity as the best representative, multiplying the parabolic PDE by ψ1 ≥ 0 in  the first eigenfunction of the Laplacian, by Jensen’s Inequality it is obtained the ordinary differential inequality

Certain Aspects of Problems with Non Homogeneous Reactions

d dt



v(t, x) ψ1 (x) d x ≥

−λ1



103

⎛ ⎞ v(t, x) ψ1 (x) d x + λ f ⎝ v(t, x) ψ1 (x) d x ⎠





(23) whose qualitative behaviour also gives relevant information on the blow-up of the solution. Although the qualitative behaviour of the non linear ODE defined by (23) is not direct to understand, it is reasonable to assume that under certain values of λ and the initial condition  v0 (x) ψ1 (x) d x, it can blow-up; calling Tsuper −sol this eventual blow-up time, in the particular case of the exponential nonlinearity it is held that Theorem 3 Given f (t) = et in (2), it is held that 

TBlow−up

e−x0 + t0 , Tsuper −sol ∈ λ

 (24)

Previous result can be directly extended to nonlinearities like f (t) = (1 + αt)β , with α ≥ 0 and β ≥ 1. Essentially, Theorem 3 gives an upper and lower bound for the blow-up time of the solutions of (2) with respect to two comparison functions (solutions of simpler problems). Better blow-up estimations are strongly dependent on the specific nonlinearity in (2). In the description of Blow-Up profiles, since the pioneering work by Jean Leray in the Navier-Stokes system [37], similarity has always played  an important role, this is in the study of solutions with the structure v(t, x) = v trq , where q often takes value a half. In the aim of extending this idea to the prediction of the blow-up profile to the particular case of the exponential nonlinearity, and also in the spirit of giving better descriptions by asymptotic expansions, first Kassoy [40] gave an expansion through a transformation in the internal layer, and by this some interesting predictions/conjectures were given. The situation changed dramatically when Dold [24] proposed a new self-grouping kernel which improved all previous work concerning similar description of the blow-up profile; Dold’s work up to now has not being improved in the asymptotic profile description, and his kernel has inspired important works, being a significance advance from all previous results. Concerning the blow-up set, this is an open problem in many fronts, as any estimation of it is strongly influenced by the nonlinearity, but also on the geometry of  itself. One of the first results concerning this is in [30], who proved that for problem (2) posed in  a ball, and under the assumption that v(t, x) = v(t, r ), the blow-up is only at the center of the ball, which in some sense was expected from the whole symmetry of the domain and the classical work [34]. For others previously mentioned nonlinearities, the situation is different. For example, in the case of the Perturbed Gelfand Problem, this is f (t) = et/(1+t) the problem does not present blow-up in lots of regimes (of initial conditions), as the nonlinearity is bounded, which in a sense to be discussed later makes the problem stable.

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But when (2) has a nonlinearity of the type f (t) = (1 + t)q et/(1+t) the situation is almost unknown (not reported in the literature at all), and with unexpected cases. In the case of α > 1, it is proved in the manuscript [55] the next result for the initial value problem ∂v = v + λ (1 + v)α ev/(1+ v) in ]0, τ [ × , ∂t v(0, x) = v0 (x) in , v = 0 on ]0, τ [ × ∂ ,

(25)

Theorem 4 Let α > 1 and  > 0 in (25). Let also v0 ≥ 0 being the initial condition in (25) such that h 0 = v0 ψ1 d x is well defined, where ψ1 corresponds to the   α first eigenfunction of the laplacian, such that h 0 satisfies λ αe h α−1 − λ1 > 0, 0 with λ1 the first eigenfunction of the laplacian. Then, the solution of (25) blows-up earlier than    α λ αe h α−1 1 0  α ln TBlow−U p = t0 + . (26) λ1 (α − 1) λ αe h α−1 − λ1 0 In the same manuscript there are presented results for the case α < 0 in (25); by calling q = −α, problem (25) is written now as ev/(1+ v) ∂v = v + λ in ]0, T [ ×, ∂t (1 +  v)q v = 0 on ]0, T [× ∂ , v(0, x) = v0 (x) in ,

(27)

it is first proved that Theorem 5 Given q = 1, 2 in (27), with  > 0 such that  q < 1. Then, the spectrum of the steady problem defined by (27) is bounded. It is a direct consequence of previous Theorem that Lemma 1 Given α = 1, 2 in (27), and  > 0 such that α  < 1. Take now λ > 0 such that λ > 38e λ1 for α = 1, and λ > 3e λ1 for α = 2. Then, given any initial condition for (27), the solution of it must blow-up.

3.1 Lyapunov Functions and Stability Stability is a relevant point in the study of a (continuous or discrete) dynamic, where stability is understood as: already known a solution of a particular problem with a

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certain set of inputs, then does the solution changes a lot or a few under small changes in the input set? If the changes are small it is assumed stability, while for big changes it is assumed instability. The key idea in this topis is by Alexander Lyapunoff in 1892, who in his doctoral thesis Probléme Général de a stabilité du Mouvement introduced what we know today as Lyapunoff Functions, whose main characteristic is that they do not increase, or strictly decrease through the trajectories of the dynamics (they decrease through the solution in time); an equilibrium (steady state) will be Lyapunov stable if for any initial condition in a neighbor of the equilibrium, the solution starting at the initial condition is such that the Lyapunov function decreases through the orbit of the solution, therefore converges to the equilibrium. The problem with this technique is that in general it is difficult to have a Lyapunov function, except for certain well known type of evolution systems, for example Gradient Systems. Also, when the system has an Energy, this corresponds to a Lyapunov function.

u In the case of problems like (2), defining F(u) = f (s) ds, which is associated 0

, and under certain technical aspects to a Potential Energy, multiplying (2) by ∂u ∂t skipped here for example, ∂u ∈ L 2 (]0, T [, H01 () ∩ C([0, T ], L 2 ())), it is obtained that ∂t ⎞ ⎛  2 2  ∂u  |∇u| d   dx = − ⎝ d x − λ F(u) d x ⎠ ,  ∂t  dt 2





from where it is direct to observe that identifying

(28)





|∇u|2 2

d x as the kinetic energy,

the mechanical energy (28) associated to the dynamics defined by (2) decreases through the solutions/orbits; this is, there is a natural Lyapunov function for (2). On the opposite, although there is a natural Lyapunov function, the proof that it decreases through the orbits defined by the global solutions of (2) is in practical terms impossible, as this would require the knowledge of the semi-group of the problem, defined as φt (·) = et (+λ f ) (·) ∞ k  t ( + λ f )k . = k! k=0

(29)

Notice that in previous series, f is a nonlinear operator. Despite the impossibility of giving an explicit stability criteria, even having a Lyapunov function, it is at least possible to give an instability criteria, giving a partial description of the unstable manifold of (2) for certain nonlinearities f . As an instability result, it is stated in [54] for general nonlinearities that

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Theorem 6 Given f ∈ C(IR) a nonlinearity in (2) such that f (0) = 0, and F(t) as previously defined, both satisfying the inequality 2F(t) − f (t)t ≤ 0 ∀ t ∈ IR.

(30)

Then, for any initial condition v0 ∈ H01 () of (2), with |F(v0 )| ∈ L 1 (), satisfying the inequality

|∇ v0 |2 d x − 2 λ



F(v0 ) d x < 0,

(31)



the solution v of (2) blows-up. In the particular case of f (t) = et , where f (0) > 0, it is proved in Ref. [54] that Theorem 7 Let λ be in the spectrum of the steady problem associated to (2) with f (t) = et .

Given v0 ∈ H01 () with ev0 d x < ∞ and satisfying 

− λ e || + 2 λ 

ev0 d x −

|∇v0 |2 d x > 0,

(32)



then the solution v of (2) with initial condition v0 blows-up, where || corresponds to the Measure of  in the Lebesgue sense. In the case of (32), it only reflects the fact that the energy of global solutions, in particular steady solutions, is bonded from below and if there is given an initial condition v0 with initial energy below the global lower bound, the solution must blow-up by the lack of a steady state where to converge. The two previous results are continuations of earlier works, see [31–53]. The limitations of Lyapunov Function method’s is directly seen in the case of (27): only in the case q = 2 it is held that d dt



⎞ ⎛ d ⎝ |∇v|2 v2 dx = − d x − λ ev/(1+ v) d x ⎠ , 2 dt 2 

(33)



while for any other value of q there are not known Lyapunov functions. In general

t F(t) = f (s) ds is not analytic. 0

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107

4 A Particular Type of Solution: Traveling Waves Traveling waves are a common type of response to different and complex phenomenologies, or a relevant type of solutions in very different phenomenologies in nature. At the beginning of this note, it was explicitly mentioned the classical references by Fisher and Kolmogorov-Petrovskii-Piskounov, concerning the spreading of a gene in a population. The type of solutions sought in both works was a traveling wave, this is a solution of with the structure v(t, x) = v(x + ct), therefore the (IVP) corresponds to the nonlinear ODE cvz − vzz = λ f (v) u ∈ IR,

(34)

with z = x − ct. Two important observations need to be done while seeking this type of solutions: the first one is that c, the speed of propagation of the traveling wave, is also an unknown of the problem and needs to be sought; notice that a priori there are more unknowns than equations, but this is sorted in different ways. Second, for having a traveling wave, at z ± ∞ which are the points the traveling wave connects, it is needed they corresponds to steady solutions of (34) or equivalently they solve f (t) = 0. For the example of Fisher and Kolmogorov-Petrovskii-Piskounov, f (s) = s(1 − s), therefore f (s) = 0 has two solutions: s = 0 and s = 1. Then, the traveling wave corresponds to the profile given by the initial condition that moves from z = ∞ at height 1 to z = −∞ at height 0. Existence of traveling wave solutions have been a very important focus of attention since the early works [28, 43]. Within relevant literature it must be also mentioned [4]. As was earlier said, the existence of traveling wave solutions implies the existence of the speed c of the traveling wave, but a priori the problem has more unknowns than equations, which is an important difficulty to sort. Two of the main techniques which have been very fruitful in other applications for sorting this are Perturbation Techniques and Asymptotic Analysis. The idea behind them is very simple: first, there are solved two problems neglecting the nonlinearity f , one involving z = −∞ and the other involving z = ∞; then, taking in a different scale the whole problem involving the nonlinearity, in a narrow layer. Finally, it is matched the outer problem, this is the two solutions which neglected the nonlinearity, with the inner problem which did not neglected the reaction. In order to “glue” the two solutions, it is needed that the velocity satisfies certain restrictions, therefore the value of the velocity is fixed, then the whole traveling wave solutions is determined. A good and detailed developed of the method of Perturbation and Matching for the Fisher-KolmogorovPetrovskii-Piskounov problem is given in the book [52], which is summarized now. By seeking a solution f (z) = u(t, x), with z = x + ct, the original problem studied by Fisher u t = D u x x + k u (1 − u)

(35)

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becomes into D f  − c f  + k f (1 − f ) = 0 lim f (z) = 0

(36)

z −∞

lim f (z) = 1

z∞

which written as a first order system in the unknowns f  = F and D F  = cF − k f (1 − f ) leads to the phase-plane equation dF c F − k f (1 − f ) = d f DF

(37)

whose critical points are ( f, F) = (0, 0) and ( f, F) = (1, 0) respectively. By √ linearizing around each one, (0, 0) is an unstable node only if c ≥ cmin = 2 D k, and (1, 0) is a saddle for any c ≥ 0. Therefore, in order to have a traveling wave solution, corresponding to a trajectory in the phase portrait that joins the two critical points defined by √ (37), it is necessary that the wave moves at a speed c which satisfies c ≥ cmin = 2 D k. √ Under the change of variables t  = k t and x  = x/ D k, the phase-plane equation becomes dF cF − f (1 − f ) = , df F

(38)

φ − f (1 − f ) dφ = , df F

(39)

which becomes into 

where  = c−2 and φ = c F. Now, by seeking a solution given by the expansion φ( f, ) = g0 ( f ) +  g1 ( f ) + 2 g2 ( f ) + . . ., which leads to a sequence of solutions that its three first terms are given by g0 = f (1 − f ), g1 = f (1 − f )(1 − 2 f ), g2 = ddf ( f 2 (1 − f )2 (1 − 2 f )). From an analysis of the solution F = c−1 φ, the   slope of the traveling wave in a first order satisfies s = 4c1 + O c15 . In order to have a “better resolution” at z = 0, where f = 1/2, it is proposed now the expansion f (z) = h(ξ) with ξ = cz = 1/2 z, and expanding h(ξ, ) = h 0 (ξ) +  h 1 (ξ) + 2 h 2 (ξ) + . . ., it is obtained that      4e−z/c 1 e−z/c 1 1 − ln + O f (z) = (1 + e−z/c )−1 + 2 −z/c 2 −z/c 2 c (1 + e ) (1 + e ) c4 (40)

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and in order to match, or glue, with the previous expansion in √ the intermediate regime, it is needed that c ≥ cmin = 2, recovering the constant D k by returning on the change of variable with the primes. In the case of the type of nonlinearities under attention in this note, for example problems involving exponential nonlinearities, the situation becomes critical, as f (s) = es has no steady solutions, therefore a priori is not possible to seek a traveling wave connecting z = ∞ and z = −∞ with a profile that moves from right to left. In some sense this could have been predicted from the fact that the ODE x  = λ e x has no steady solutions, as was previously discussed. On the other side, for problem with exponential nonlinearities, or with Arrhenius type of nonlinearities, experiments show the existence of traveling wave solutions, corresponding to steady planar premixed flames; see for example the excellent Ref. [36] concerning the case of combustion waves. Just skipping details, very well explained in any of the references, the simplest combustion reaction of a premixed planar one-dimensional flame (Pre Mixed-PlanarFlame, PM-P-F) leads to the system ⎫ −T  + cT  = Y f (T ) ⎪ ⎪ ⎬ −dY  + cY  = −Y f (T ) PM − P − F T (−∞) = 0 T (∞) = 1 ⎪ ⎪ ⎭ Y (−∞) = 1 Y (∞) = 0 where f (T ) is an Arrhenius function of Temperature. In PM-P-F, T represents the temperature and Y the normalized amount of premixed fuel, c the speed of the wave (also an unknown), and d is the inverse  of the  Lewis number. Because of the existence of an Arrhenius expression exp − R ETig , with E a distinctive energy of the reaction involved, R the universal constant of gases, and finally Tig an specific ignition temperature, the left limit at z = −∞ is never zero, therefore from the mathematical point of view PM-P-F, a prior, does not have solution. This difficulty, very well identified in the literature, is known as the Cold Boundary Difficulty in English literature and La Difficulté de la Frontiére Froide in French literature. A naive, also intuitive, but overall a good solution in order to get a nonlinearity that allows traveling wave solutions, which as was earlier said are observe in experiments, is to redefine f (T ) such that f (T ) = 0 if T ≤ θ and f (T ) > 0 when T > θ, where θ is a reference temperature, plus maintain the strongly temperature dependence in the non zero part: this corresponds to define a cut-off function in temperature. A well developed discussion, justification, and also report of numerical experiences on this can be seen in [11], which is full of deep ideas and strong mathematical tools and concepts.

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Previous to [11], Ref. [10] studied the existence of traveling wave solutions for the particular case Le = 1, which leads to the scalar problem − (k(u)u  ) + c(u) = g(u) on IR

(41)

u(−∞) = 0 u(∞) = 1, with k ∈ C 1 and strictly positive, and g such that g(u) = 0 for u ∈ in [0, θ] and g(u) > 0 for u > θ. Without taking into account the cold boundary difficulty, the reaction g has a structure of the type  1 u−1 . g ∝ ex p  1 +  α(u − 1)/ 

(42)

In order to solve the cold boundary difficulty the authors used a Heaviside function as a cut-off at temperature θ. Then, they proved the existence of traveling wave solutions in bounded increasing intervals, and took limit in the sequence of solutions; of course the existence of a limiting solution is by nothing trivial, as compactness arguments were not direct. The limiting problem in fact corresponded to a free boundary problem with a Dirac Mass in the reaction at a certain point completely determined by u(0) = θ. Always in the context of Combustion Theory, important results on existence of traveling wave solutions for cylindrical domains are available. For example, Ref. [12] did very important work concerning existence of solutions but also stability of the profile. This topic is still very relevant, as was earlier said, experimental results show the existence of this type of traveling wave solutions. Always in the frame of traveling wave solutions, book [63] needs a special reference. This book not only recompile the most relevant results on the topic, but also it gives new results, and detailed develop of important techniques just referenced, for example certain asymptotic expansions. Within the results presented at the beginning of the book there are existence results for traveling wave solutions, in particular by Degree Theory; in next chapter there will be referenced some results involving Degree Theory. Also stability is deeply studied in one of the chapters. One of the most relevant achievements of the book is that it studied not only scalar problems, but systems of PDE’s and their traveling wave solutions. A final comment to be done concerning the seek of the speed at which the traveling wave moves, is that its existence can be proved by variational methods, this is it satisfies a certain min-max principle, which is coherent with the fact that there exists a minimal speed at which the traveling wave exists, in general the observed speed in experiments. Asymptotic Analysis and Matching Techniques have played a fundamental role in traveling wave problems, but also in some other problems concerning non homogeneous reaction, as can be seen in [39] or the previously quoted referenced [36].

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5 A Particular Planar System from Chemical Reactors The classical text on Nonlinear Analysis [23] ends at section (30.3) with the planar system  ⎫  x1 ,⎬ x1 (t) = −x1 − β (x1 − x 1 ) + B Da (1 − x2 )n ex p 1+ x1   (I ) x1 ⎭ x2 (t) = . −x2 + Da (1 − x2 )n ex p 1+ x1 as an example concerning Hopf’s bifurcation, this is bifurcation of periodic solutions. It is not difficult to observe that system (I) is again non homogeneous. In (I) the unknown x1 represents a perturbed dimensionless temperature and x2 a dimensionless and normalized reactive species that lies in [0, 1], x 1 is a certain reference temperature (a cooling temperature acting as a control),  is a dimensionless parameter associated to the activation energy of the reaction, B and β are also parameters, and Da is the well known Damköhler number. When  0, a good approximation of (I) is given by  x1 (t) = −x1 − β (x1 − x 1 ) + B Da (1 − x2 )n e x1 , (I I ) x2 (t) = −x2 + Da (1 − x2 )n e x1 . The number n ∈ !N represents, in either (I) or (II), stoichiometric values of the reactants: the case n = 1 is simply a reaction of the type A → B, while for n ≥ 2 it can be assumed a catalytic reaction, for example of the type A + A → 2 A (the case n = 2). The origin of (I) comes from a first order expansion, averaged in space, of a pair of two nonlinear parabolic equations, with Arrhenius dependence on temperature and with nonlinear boundary conditions, see [19–57]. The qualitative behaviour of both (I) and (II) is really interesting, and still up to today not totally known, except from local descriptions around equilibria. In particular, the qualitative behaviour of (II) even in the case n = 1 is almost unknown. For simplicity, most of the results to reference in this section will be for (II) with n = 1, although the proofs/ideas behind what will be presented can be directly extended to the case n ≥ 2. The case of (I) even with n = 1 increases the difficulty on the analysis because of the parameter  > 0. Before giving the most relevant results, it is noticed that equilibria of (II) with n = 1 can be reduced to seek the solutions of the nonlinear scalar equation in the unknown x2 B B B x2 e− 1+β x2 = Da (1 − x2 ), 1+β 1+β

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as the second equation for the corresponding component of the equilibrium is B x1 = 1+β x2 .

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Also, the Jacobian matrix of the field in the right side of (II) with n = 1 corresponds to   −(1 + β) + B Da(1 − x2 )e x1 −B Da e x1 . A= Da (1 − x2 )e x1 −1 − Da e x1 from where it is obtained that det (A) = (1 + β) + (1 + β) Da e x1 − B Da (1 − x2 ) e x1 , and under the assumption that (x1 , x2 ) = n = 1, it is held that



B x ∗ , x2∗ 1+β 2

det (A) = 1 + β + (1 + β) =

(44)

 is an equilibrium of (II) with

x2∗ − B x2∗ 1 − x2∗

B (x2∗ )2 − B x2∗ + (1 + β) , 1 − x2∗

(45)

from where it is obtained a first existence result given by Theorem 8 Given x 1 ≥ 0, β > 0 and B > 0, then system (II) with n = 1 satisfies that its Topological Degree is such that d L−S (F, , (0, 0)) = 1,

(46)

where F :  −→  (x1 , x2 ) −→ F(x1 , x2 )

(47)

with F(x1 , x2 ) = (−(1 + β)x1 + B Da (1 − x2 )e x1 , −x2 + Da (1 − x2 )e x1 ) corresponding to the vector field of the system, and  = IR + × ]0, 1[. Proof See [1]. Theorem 8 allows to conclude that system (II) with n = 1 has always at least one equilibrium in  = IR + × ]0, 1[. This is important, as independently of the parameters to be chosen, there is always at least one equilibrium where to, eventually, global solutions can converge. On the other side, the exact number of equilibria strongly depend on the relation between the involved parameters, which is presented in the next

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Theorem 9 Given x 1 ≥ 0, β > 0 and B > 0, then system (II) with n = 1 can have B is smaller, equal or bigger one, two or three equilibria, depending on whether 1+β that 4. Proof See [1].  Theorems 8 and 9 are very related. The Topological Degree can be seen as the sum of the local index, so in the case of having one equilibrium it is held that d L−S (F, , (0, 0)) = 1, while in the case of having two equilibria is is held that d L−S (F, , (0, 0)) = 0 + 1 = 1, and finally in the case of three equilibria it is held that d L−S (F, , (0, 0)) = 1 + (−1) + 1 = 1, where in each case the values in the right of d L−S (F, , (0, 0)) correspond to the local index at the corresponding equilibrium. From general properties of the Topological Degree, in the case of having three equilibria, the one identified with being “the middle one” is known to be a locally saddle point for the dynamic, then unstable. As was said in the beginning of this section, systems (I) and (II) are well known cases of planar systems developing Hopf’s bifurcation, this is periodic solutions. Despite that, except from some computations in [57], based on Dulac’s criteria, the proof of the existence of periodic solutions for either (I) or (II) is still an open problem. For a proof of existence of periodic solutions for system (II) with n = 1, again see [1]. The case of system (II) with n ≥ 2 is almost not studied and no results on multiplicity are reported in the literature. As was earlier said, this case can be seen as a first approximation to a catalytic reaction. In this context, almost all the techniques and results reported in Ref. [1] can be extended, with some changes which explicitly take into account in particular the value n ≥ 2. The extensions of Theorems 8 and 9 are given by Theorem 10 Given x 1 ≥ 0, β > 0 and B > 0, then n ≥ 2 in system (II) with n ≥ 2 satisfies that its Topological Degree is such that d L−S (F, , (0, 0)) = 1,

(48)

F :  −→  (x1 , x2 ) −→ F(x1 , x2 )

(49)

where

with F(x1 , x2 ) = (−(1 + β)x1 + B Da (1 − x2 )n e x1 , −x2 + Da (1 − x2 )n e x1 ) corresponding to the vector field of the system, and  = IR + × ]0, 1[. Theorem 11 Given x 1 ≥ 0, β > 0 and B > 0, then system (II) with n ≥ 2 can B is smaller, equal or have one, two or three equilibria, depending on whether 1+β √ 2 bigger that ( n + 1) .

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As was already said, the proofs are direct extensions of the case n = 1. In particular, also in the case of having three equilibria with n ≥ 2, the “middle” one is a saddle. Now, in the case of system (I) the situation is very unknown, despite of the value n ∈ IN ; reference gives some very simple results concerning periodic orbits, mainly based on local linearization around an equilibria. But a detailed description of multiplicity and an accurate local classification of the dynamic around equilibria is far from being known. Some of the ideas/techniques previously mentioned concerning Theorems 8 and 9 can be extended, but with far more difficulty. Although not given explicitly as a theorem, it is held for system (I) that its Topological Degree at zero is one, this is there exists at least one equilibrium. The exact multiplicity is also known, the same as the local index. The set of Figs. 6, 7, 8 and 9 give numerical simulations in the phase portrait x1 − x2 ∈ IR + ×]0, 1[ that show limits cycles for different values on the involved parameters of system (II). It is interesting to notice the scale in the x2 −axis, where in Figs. 6, 7 and 9 the limit cycle is local around the bigger equilibrium and the convergence to the cycle is by an increasing spiral from inside. The case of Fig. 8 is different, as the convergence to the limit cycle is from outside (starting at the unbounded region) by a rolling spiral; also, in this case the limit cycle encloses the three equilibria. 1

0.98

0.96

0.94

0.92

0.9

0.88

0

2

4

6

8

10

12

Fig. 6 Limit cycle (phase portrait) for (I): n = 2, β = 155, B = 194.45, Da = 273.202691,  = 0, x 1 = 0

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0.95

0.94

0.93

0.92

0.91

0.9

0.89

3

2.5

2

1.5

1

0.5

0

Fig. 7 Limit cycle (phase portrait) for (I): n = 2, β = 155, B = 194.45, Da = 297.382293,  = 0, x 1 = 0 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0

2

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8

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14

16

18

20

Fig. 8 Limit cycle (phase portrait) for (I): n = 4, β = 5, B = 27.662546, Da = 0.34469417,  = 0, x 1 = 0

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0.95

0.9

0.85

0.8

0.75

0.7

0.65

0

1

2

3

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5

6

7

8

9

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Fig. 9 Limit cycle (phase portrait) for (I): n = 4, β = 5, B = 27.772546, Da = 10.13144703,  = 0, x 1 = 0

Extension of previous model to the reaction A → B → C, this is a reactor with three species and strongly dependent on temperature, are given in [19], but most of the description on the qualitative behaviour of solutions rely on asymptotic expansions. The author is already working in the extension of previous results, in particular the ones involving the computing of the Topological Degree to this three species system.

6 Final Comments As was said in the introduction, this note presented only some ideas concerning reaction-diffusion problems where the nonlinearity is non polynomial and non homogeneous. There are many other ideas which are not even mentioned in the note, and the author apologizes for this. In order to conclude the note, the author proposes three questions/problems, which up to him are relevant and important to study: 1. it is well known that for steady problems, there is often one solution known as minimal solution which is also defined as stable. The notion of stability for the steady solutions is the one defined by the value of the first eigenvalue of the linearized problem with respect to the nonlinear eigenvalue. Is there a formal relation between this notion of stability for the steady problem equivalent with the classical notion of stability for evolution problems?, in particular if there exists a Lyapunov function? The same questions with respect to unstable or non

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minimal solutions. The evidence suggest this is true, despite the author does not know of any result on this; 2. can be given a detailed description of the stable or unstable manifold of the evolution problem?, this is can be totally described the set of initial condition that develop or not develop blow-up?; this question for certain nonlinearities has been answered partially by the author; 3. can there be obtained more information on stability of steady solutions of either (1) or (2), the same as of system (I) even in the case of existence of limit cycles, by the Leray-Schauder Degree?

References 1. Aguirre, P., Catañeda, A., Omón Arancibia, A. , Robledo, G.: On a planar system from chemical reacting modelling: a new geometrically/qualitative description (submitted manuscript) 2. Ambrosetti, A., Prodi, G.: A Primer of Nonlinear Analysis. In: Cambridge Studies in Advance Mathematics, vol. 34. Cambridge University Press (1993) 3. Aris, R.: Some characteristic nonlinearities of chemical reaction engineering. In: Bishop, A.R., Campbell, D.K., Nicolaenko , B. (eds.) Nonlinear Problems: Present and Future. North-Holland (1982) 4. Aronson, D.G., Weinberger, H.F.: Multidimensiona diffusion arraising in population genetic. Adv. Math. 30, 33–76 (1978) 5. Ash, E., Eaton, B., Gustafson, K.: Counting the number of solutions in combustion and reactive flow problems. Z. angew. Math. Phys. 41, 558–578 (1981) 6. Bandle, C.: Existence theorems, qualitative results and a priori bounds for a class of nonlinear Dirichlet problems. Arch. Rat. Mech. Anal. 58, 219–238 (1975) 7. Bazley, N., Wake, G.: The disappearence of criticality in the Theory of Thermal Ignition. Z. angew. Math. Phys. 29, 971–975 (1978) 8. Bebernes, J., Eberly, D.: Mathematical problems from combustion theory. In: Applied Mathematical Sciences, vol. 83. Springer, Berlin (1989) 9. Bellman, R., Bentsman, J., Meerkov, S.: Vibrational control of systems with Arrhenius dymanics. J. Math. Ann. Appl. 91, 152–191 (1983) 10. Berestycki, H., Nicolaneko, B., Scheurer, B.: Travling wave solutions to a combustion model and their singular limit. SIAM J. Math. Anal. 16, 1207–1242 (1985) 11. Berestycki, H., Larrouturou, B., Joquejoffre, J.M.: Mathematical investigation of the cold boundary difficulty. In: Dynamical issues in combustion theory, the IMA volumes in Applied Mathematics, vol. 35. Springer, Berlin (1987) 12. Berestycki, H., Larroauturou, B., Nirenberg, L.: A nonlinear elliptic problem describing the propagation of curved premixed flame. In: Mathematical Modeling in Combustion Theory, NATO Advance Research Work Series E. Applied Science, vol. 140 (1987) 13. Berger, M., Berger, M.: Perspectives in Nonlinearities: An introduction to Nonlinear analysis. In: Benjamin Cummings Mathematical Lectures Notes (1968) 14. Bers, L., John, F., Schechter, M.: Partial Differential Equations. Wiley & Sons. Inc, New York (1964) 15. Boddington, T., Gray, P., Wake, G.: Criteria for thermal explosions with and without reactant consuption. Proc. R. Soc. Lond. A 357, 403–422 (1977) 16. Börsch-Supan, W.: On the stability of bifurcation branches in thermal ignition. Z. angew. Math. Phys. 35, 332–344 (1984)

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A Survey on the Melnikov Theory for Implicit Ordinary Differential Equations with Applications to RLC Circuits Michal Feˇckan

Abstract Our recent results are presented on the development of the Melnikov theory in investigation of implicit ordinary differential equations with small amplitude perturbations. In particular, the persistence of orbits connecting singularities in finite time is studied provided that certain Melnikov like conditions hold. Achievements on reversible implicit ordinary differential equations are also considered. Applications are given to nonlinear systems of RLC circuits. Keywords Implicit ordinary differential equations · Impasse points · IK-singularities · RLC circuits 2010 MSC 34A09 · 37C60 · 47N70

1 Introduction Implicit ordinary differential equations (IODEs for short) find applications in a large number of physical sciences and have been studied by several authors [16, 19, 20, 23–27]. In particular, IODEs naturally arise in modelling nonlinear RLC circuits as it is demonstrated also in this paper. We survey in this paper our recent results on IODEs with small amplitude perturbations by using the Melnikov theory. In particular, the persistence of orbits connecting singularities in finite time is studied This work was supported by the Slovak Research and Development Agency (grant number APVV14-0378) and the Slovak Grant Agency VEGA (grant numbers 2/0153/16 and 1/0078/17). M. Feˇckan (B) Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia e-mail: [email protected] M. Feˇckan Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_4

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provided that certain Melnikov like conditions hold. Implicit differential equations have been studied also in many other papers [11, 12, 15, 18, 22] but the results of this paper do not seem to be covered by them. On the other hand, those results deal with more general implicit differential systems using analytical, topological and numerical methods. In the terminology of [27], we study persistence of global solutions terminating in finite time either to I–singularities or to IK–singularities. Related results are given in [9, 10]. We note that the Melnikov method and its extensions are mainly used to prove existence of chaotic orbits in dynamical systems [1, 21, 31]. We apply this method to study IODEs, since it is a natural way to handle our studied problems. Moreover, our kind of problems is different to former ones on the existence of chaos. Consequently, our results are original which have not yet been studied by other researchers. The paper is organized as follows. Sections 2 and 3 deal with nonlinear RLC circuits. Section 4 studies weakly coupled nonlinear RLC circuits. IODEs in arbitrary finite dimensional spaces are considered in Sect. 5. The so called bly-like catastrophe is demonstrated in Sect. 6 for reversible IODEs. A general theory for connecting IK singularities and impase points is presented in Sect. 7. Many concrete examples are given for illustrating theoretical results. We summarize the content of this paper in Sect. 8.

2 Nonlinear RLC Circuits: Homoclinic Case For motivation, we consider a nonlinear RLC circuit given by u + Li + Ri = e(t), i =

dQ = C(u) , dt

where L, R are the self-inductance and the ohmic resistance, respectively, and C is the nonlinear capacitance, which gives LC(u) + RC(u) + u = e(t). Linear theory is well-known with C(u) = C0 u for some C0 > 0. Recently several papers appeared with nonlinear C(u) [17, 28] for coupled circuits on lattices. In this part, we sketch our approach on the concrete equation (u2 + u) + u + εγ(u2 + u) + εh(t + α) = 0,

(2.1)

where h ∈ Cb2 , i.e., a function with bounded derivatives on R up to the second order, including for example h(t) = sin t, γ, α are parameters and ε is a a small parameter. Since (2.1) has a form (2u + 1)u + 2u2 + u + εγ(2u + 1)u + εh(t + α) = 0,

Melnikov Theory for IODEs

123

we see that (2.1) is an IODE with a singularity at u0 = 21 . Setting v = (u2 + u) and expanding derivatives in (2.1), we arrive at the system v 2u + 1 v  = − u − ε [γv + h(t + α)] u =

(2.2)

whose unperturbed associated equation (i.e., with ε = 0) v 2u + 1 v  = −u

u =

(2.3)

1 1 (3 − t 2 ), vh (t) = 36 t(t 2 − 9), 0 ≤ |t| < 3. We note that has the solution uh (t) = 12 2 3 2 3u + 4u + 3v is a first integral of (2.3) and (see Fig. 1)

  lim (uh (t), vh (t)) = − 21 , 0 .

t→±3∓

Hence, in (2.2), we set u(t) = U (s) and v(t) = V (s) with s = ϕ(t) = 13 arctanh 3t 1 −1 and |t| < 3. Note that ϕ(0) = 0, ϕ (t) = 9−t (s) = 3 tanh 3s and (2.2) is 2, t = ϕ d transformed into ( ˙ = ds )

Fig. 1 Contour plot of function 3u2 + 4u3 + 3v 2

0.5

0.0

0.5

1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

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M. Feˇckan

V sech2 3s 2U + 1 V˙ = −9U sech2 3s − 9ε sech2 3s [γV + h(3 tanh 3s + α)] U˙ = 9

(2.4)

whose unperturbed associated equation V sech2 3s 2U + 1 V˙ = −9U sech2 3s

U˙ = 9

  tanh 3s has a homoclinic solution Uh (s) = 41 1 − 3 tanh2 3s , Vh (s) = − 23 1+cosh . Now, 6s we look for solutions of the perturbed Eq. (2.4) that are orbitally close to (Uh (s), Vh (s)). Thus, we change (U, V ) with two other variables, say x = U − Uh , y = V − Vh and arrive at the differential equation x(x tanh 3s + y) 18(y + x tanh 3s) = 6x tanh 3s + 6y − 24 2 3 + 4x cosh 3s 4x + 3 sech2 3s 2 2 2 3 y˙ = − 9x sech 3s − 9ε sech 3s[γ(y − 4 sech 3s tanh 3s) + h(3 tanh 3s + α)]. x˙ =

tanh 3s+y) is not quadratic in (x, y) uniformly with reUnfortunately, the term 24 x(x 4x+3 sech2 3s spect to s since its limit as s → ±∞ is 6(y ± x)). So, since Uh (s) + 21  23 sech 6s and Vh (s)  ∓ 23 sech 6s, as |s| → ∞, we set, instead U = x sech 6s + Uh , V = y sech 6s + Vh , and search for a bounded solution of equation

24x(y + x tanh 3s) 4x + 3(1 + tanh2 3s) 18ε [γ(y sech 6s y˙ = −9x sech2 3s + 6y tanh 6s − 1 + sech 6s − 43 sech2 3s tanh 3s) + h(3 tanh 3s + α)],

x˙ = 6(tanh 3s + tanh 6s)x + 6y −

(2.5)

this means that, we look for a solution (U, V ) of Eq. (2.4) whose U -component tends to − 21 as s → ∞ at the exponential rate e−6|s| . Note that for x > − 43 , we have 24x(y+x tanh 3s) is now quadratic 4x + 3(1 + tanh2 3s) > 0 and then the expression − 4x+3(1+tanh 2 3s) in (x, y). The linearization of Eq. (2.5), with ε = 0, along x = 0, y = 0 is x˙ = 6(tanh 3s + tanh 6s)x + 6y y˙ = −9x sech2 3s + 6y tanh 6s

(2.6)

whose limiting equations as s → ±∞ are, respectively, x˙ = 12x + 6y y˙ = 6y

(2.7)

Melnikov Theory for IODEs

and

125

x˙ = −12x + 6y y˙ = −6y.

(2.8)

Equation (2.7) has the unstable equilibrium (0, 0) while (0, 0) is stable for Eq. (2.8). As a consequence, Eq. (2.6) has an exponential dichotomy on both R+ and R− with projections respectively P+ = 0 and P− = I [2, 21]. Since RP+ = N P− = {0}, where R and N denote the range and kernel/null space, we see that x = 0, y = 0 is the unique solution of (2.6) bounded on R and the adjoint equation x˙ = − 6(tanh 3s + tanh 6s)x + 9y sech2 3s y˙ = −6x − 6y tanh 6s

(2.9)

∗ has an exponential dichotomy on both R± with projections (I − P+ ) = I and ∗ (I − P− ) = 0. Hence, all solutions of system (2.9) are bounded on R and actually decay to zero exponentially fast. Then, from the general theory, we deduce that the inhomogeneous system

x˙ = 6(tanh 3s + tanh 6s)x + 6y + εh1 (s) y˙ = −9x sech2 3s + 6y tanh 6s + εh2 (s) with h1 (s), h2 (s) ∈ Cb0 (R) has a (unique) bounded solution near (0, 0) if and only if 

∞ −∞

h1 (s)xia (s) + h2 (s)yia (s)ds = 0, i = 1, 2,

(2.10)

where (x1a (s), y1a (s)) and (x2a (s), y2a (s)) being two independent (bounded) solutions of (2.9). In order to find (x1a (s), y1a (s)) and (x2a (s), y2a (s)) and then an explicit expression of (2.10), we need to construct the fundamental matrix of Eq. (2.9). But we do not go details and refer to [5] for obtaining the following Melnikov function: 24 8 M1 (α, γ) = − (2 sin 3 + 3 cos 3) cos α − γ 9 35 . M2 (α) = κ sin α, κ = −2.43515, (2 sin 3 + 3 cos 3), and M(α, γ) has the simple zero (α, γ) = (0, γ0 ) with γ0 = − 35 27 where 1 sin 3 + 6 cos 3 + γ(−2 cos 3 + 3 sin 3) + sin 3(−4 − 3 Ci(6) 3  . + ln 216 + 2 Si(6)) + cos 3(3 + 2 Ci(6) − ln 36 + 3 Si(6)) = −2.43515.

κ=

So, we have the following result:

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M. Feˇckan

Theorem 2.1 Let h(t) = sin t. Then, there exist ε0 > 0 and C 1 -functions α(ε) and γ(ε) defined on (−ε0 , ε0 ) such that for 0 = |ε| < ε0 , γ = γ(ε) and α = α(ε), v 2u + 1 v  = − u − ε [γv + h(t + α)] u =

has a solution (uε (t), vε (t)) near (uh (t), vh (t)) in the interval t ∈ (−3, 3). Moreover, α(0) = 0, γ(0) = γ0 , and lim uε (t) = − 21 , lim vε (t) = 0. |t|→3

|t|→3

Now we extend the result for Eq. (2.1) to the more general equation (f (u) + u) + εγ(f (u) + u) + u + εh(t + α, u, ε) = 0

(2.11)

where h(t, u, ε) ∈ Cb2 and f (u) is C 3 and at least quadratic at the origin. Setting v = (u + f (u)) , we arrive at the system v 1 + f  (u) v  = −u − ε [h(t + α, u, ε) + γv] u =

(2.12)

whose unperturbed associated equation (i.e., with ε = 0) u =

v 1 + f  (u) v  = −u

(2.13)

is equivalent to dv u(1 + f  (u)) =− ⇐⇒ vd v + u(1 + f  (u))du = 0. du v So the solutions of (2.13) are level curves of the Hamiltonian 1 2 (u + v 2 ) + 2



u

τ f  (τ ) d τ .

0

We assume the following conditions hold: (i) there exists u0 ∈ R such that f  (u0 ) + 1 = 0; (ii) u0 f  (u0 ) < 0; Let H(u, v) be the Hamiltonian  H(u, v) := u2 + v 2 − u02 + 2

u

u0

ξf  (ξ) d ξ = v 2 + 2



u u0

ξ(1 + f  (ξ)) d ξ

Melnikov Theory for IODEs

127

then (u0 , 0) belongs to the 0−level curve of H(u, v) (i.e., H(u0 , 0) = 0) and     0 2u0 (1 + f  (u0 )) = . ∇H(u0 , 0) = 0 0 The Hessian matrix of H at (u0 , 0) is HH (u0 , 0) =

  2u0 f  (u0 ) 0 , 0 2

hence, (u0 , 0) is a saddle for the Hamiltonian H(u, v). Next, we assume, without loss of generality, that (iii) in the domain  = {(u, v) : 1 + f  (u) > 0}, there is a branch of the level set H(u, v) = 0 connecting (u0 , 0) with itself. Note, however, that (u0 , 0) is not a fixed point of Eq. (2.12) since at this point 1+fv (u) is not defined. Hence, in general, this branch consists of a solution (u(t), v(t)) that will reach (u0 , 0) in finite time. We also note that in the time interval there should be a time t0 such that u (t0 ) = 0 i.e., v(t0 ) = 0. We have u (t0 ) =

u(t0 ) v  (t0 ) =− . 1 + f  (u(t0 )) 1 + f  (u(t0 ))

Now, u(t0 ) = 0 (since otherwise Cauchy uniqueness theorem implies (u(t), v(t)) = (0, 0)) and then u(t0 )u (t0 ) < 0. Shifting time, we can assume that t0 = 0 and denote with (uh (t), vh (t)) the solution connecting (u0 , 0) with itself and such that vh (0) = 0. Then, since (uh (−t), −vh (−t)) satisfies the same Cauchy problem (with initial time t = 0), we get uh (−t) = uh (t), vh (−t) = −vh (t), and there will be T > 0 such that lim (uh (t), vh (t)) = (u0 , 0).

t→±T

We set uh (0) = uh0 , vh (0) = 0. Possibly changing u(t) with u(−t) and v(t) with −v(−t), we may assume that vh (t) > 0 on (−T , 0). Note that there cannot be another zero of vh (t) in (−T , 0), since then (uh (t), vh (t)) would be periodic. Indeed, if vh (t1 ) = 0 for some t1 ∈ (−T , 0), then uh (−t1 ) = uh (t1 ) and vh (−t1 ) = −vh (−t1 ) = 0, so vh (−t1 ) = vh (t1 ). The uniqueness of the Cauchy problem gives the 2t1 -periodicity of uh (t) and vh (t). Then, from (2.12), we have that, in the time interval (−T , 0], uh (t) is increasing from u0 to uh0 and

u (1 + f  (uh )) uh (1 + f  (uh )) = 1. = h

uh vh −2 u0 ξ(1 + f  (ξ))d ξ

u Since g(u) := −2 u0 ξ(1 + f  (ξ))d ξ > 0 on (u0 , uh0 ) and 0 = vh (0) = g(uh0 ), we must have u0 < 0 < uh0 and

128

M. Feˇckan

 T =2

1 + f  (u)

du. u −2 u0 ξ(1 + f  (ξ))d ξ

uh0

u0

Setting u(t) =

uh (t)



t 0

1 + f  (uh0 ) 1 + f  (uh0 ) , d τ + [1 + f  (uh (τ ))]uh (τ )2 [1 + f  (uh (t))]uh (t)

(2.14)

we can state the following [5] Theorem 2.2 Assume that conditions (i)–(iii) hold and let h(t, u, ε) be a C 2 function bounded with its derivatives. Then, there exists T > 0 such that Eq. (2.11) with ε = 0 has a bounded solution (uh (t), vh (t)), t ∈ (−T , T ) such that lim uh (t) = u0 ,

t→±T

lim vh (t) = 0.

t→±T

Moreover, if the Melnikov vector M(γ, α) = −

1 T2



T −T



vh (t)[h(t + α, uh (t), ε) + γvh (t)] [1 + f  (uh (t))]u(t)h(t + α, uh (t), ε)

 dt

(here u(t) is as in (2.14)) has a simple zero at (α0 , γ0 ), there exist ε0 > 0 and C 1 functions α(ε) and γ(ε) defined on (−ε0 , ε0 ) such that for 0 = |ε| < ε0 , γ = γ(ε) and α = α(ε), Eq. (2.11) has a solution (uε (t), vε (t)) near (uh (t), vh (t)) in the interval t ∈ (−T , T ) and lim vε (t) = 0. lim uε (t) = u0 , t→±T

t→±T

Furthermore, α(0) = α0 , γ(0) = γ0 . Remark 2.3 System (2.12) can be rewritten as A(x)˙x = F(x, ε) with

(2.15)

    u 1 + f  (u) 0 x := , , A(x) := 0 1 v   v F(x, ε) := . −u − ε [h(t + α, u, ε) + γv]

  Since det A(x∗ ) = 1 + f  (u0 ) = 0 and (det A) (x∗ ) = f  (u0 ) = 0 for x∗ = uv0 , v ∈ R, the line u = u0 consists of noncritical 0-singularities of (2.15) (see [27, p. 163]). / RA(x∗ ) and (det A) (x∗ )w = 0 for any 0 = w ∈ N A(x∗ ) Next, conditions F(x∗ , ε) ∈ (see [27, (4.25), (4.26)]) now read v = 0. So all 0-singularities x∗ with v = 0 are impasse points for (2.15) (see [27, p. 163]). Hence by [27, Theorem 4.7], for any ε,

Melnikov Theory for IODEs

129

(2.15) has local solutions tending to any impasse points x∗ whose derivatives blow u0  ∗ up (see Fig. 1). On the other hand x0 := 0 is an I–singularity of (2.15) (see [27, p. 166]). So, in this paper, we studied I–singularities as classified in [27] assuming that for ε = 0, (2.15) has a global solution (uh (t), vh (t)) tending to x0∗ with bounded derivatives. Under some generic Melnikov conditions, we have proved the persistence of such kind of solutions. This is opposite to impasse points for ε = 0 which are pairwise connected and these connections persist for ε = 0 without additional conditions. Finally, following the classification of singular points given in [27, p. 168], one may go on as follows: if f  (u0 ) = 0, then x∗ is a K–singularity for any v = 0 while x0∗ is a IK–singularity (see Sect. 7).

3 Nonlinear RLC Circuits: Heteroclinic Case Now we consider IODE  u−

u3 3



  u3 + εγ u − + u + εh(t + α, u, ε) = 0 3

 where γ, α ∈ R and ε ∈ R is a small parameter. Like above, setting v = u −

(3.1) u3 3



=



(1 − u )u , we get the system 2

(1 − u2 )u = v v  = −u − εγv − εh(t + α, u, ε) whose unperturbed associated equation reads: (1 − u2 )u = v v  = −u.

(3.2)

Equation (3.2) has the solution:   2 − t2 1 t 2 vh (t) = √ = √ 1 − √ 2 2 2 2

t uh (t) = √ 2 for t ∈] −



2,



2[. Note that lim√ (uh (t), vh (t)) t→± 2

= (±1, 0)

√ √ 2 and |uh (t)| < 1 for t ∈] − 2, √ √ 2[. Moreover 1 − u vanishes at (±1, 0) and 1 − 2 uh (t) > 0 for any t ∈] − 2, 2[.

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M. Feˇckan

Equation (3.2) has a first integral H (u, v) = 2(u2 + v 2 ) − u4 that gives the conservative (non singular) equation: u˙ = v v˙ = −u(1 − u2 )

(3.3)

d here ˙ = ds . So Eqs. (3.2) and (3.3) have the same orbits in the region u2 = 1 (see Fig. 2). Note that the points (±1, 0) are hyperbolic fixed points of Eq. (3.3), the Jacobian matrix at these points being:   01 . 20

Given a solution of (3.2) with |u(t)| < 1 in a certain interval, a solution of (3.3) is obtained by means of the change of time: s = ϑ−1 (t),

ds 1 = dt 1 − u(t)2

Note that different solutions give a different time change. When (u(t), v(t)) = (uh (t), vh (t)) we obtain: 

t 0

1 τ= 1 − uh2 (τ )

Fig. 2 Contour plot of function 2(u2 + v 2 ) − u4



t 0

√   √ 2 2+t 1 t d τ = 2 arctanh ln = √ √ √ 2 − τ2 2 2−t 2

Melnikov Theory for IODEs

for t ∈] −



2,



131

2[. So

√   √ t 1 2+t s = √ ln √ = 2 arctanh √ , 2 2−t 2

t=





s 2 tanh √ 2



and we obtain the heteroclinic solution of (3.3):    

  s s s 1 1 Uh (s) = tanh √ , Vh (s) = √ 1 − tanh2 √ = √ sech2 √ . 2 2 2 2 2 Now we extended our consideration to general equation (u − f (u)) + u + εγ(u − f (u)) + εh(t + α, u, ε) = 0

(3.4)

where f (u) is a C 2 function such that f (u) = o(u) as u → 0 and h(t, u, ε) is a C 2 function periodic in t. Setting (u − f (u)) = v in Eq. (3.4), we obtain (1 − f  (u))u = v v  = −u − εγv − εh(t + α, u, ε)

(3.5)

with the unperturbed associated equation (1 − f  (u))u = v v  = −u

(3.6)

We assume the following conditions hold: (A1 ) f , h ∈ C 2 , and their second order derivatives are uniformly continuous and locally bounded w.r. to (u, ε) uniformly w.r. to t ∈ R; (A2 ) f  (u± ) = 1 and u± f  (u± ) > 0; (A3 ) system U˙ = V (3.7) V˙ = −(1 − f  (U ))U has a solution (Uh (s), Vh (s)), s ∈ R s. t. (Uh (s), Vh (s)) → (u± , 0) as s → ±∞; (A4 ) 1 − f  (Uh (s)) > 0 for any s ∈ R. Of course, Remark 2.3 can be applied also there. Note (u± , 0) are hyperbolic equilibria of Eq. (3.7). Indeed the Jacobian matrix at these points is  J± :=

0 1 u± f  (u± ) 0



132

M. Feˇckan

  andhas the ± ). Note that an eigenvector of −μ+  eigenvalues ±μ± , μ± := u±f (u 1 1 and an eigenvector of μ− is . Now summarize some important is −μ+ μ− observations in the following (see [6]) Lemma 3.1 Suppose Eq. (3.6) has a solution (uh (t), vh (t)) in the interval ]T− , T+ [, such that limt→T± (uh (t), vh (t)) = (u± , 0), 1 − f  (uh (t)) > 0 for any t ∈]T− , T+ [ and the integrals 



0

dτ 1 − f  (uh (τ ))

diverge (to ±∞). Then

 s = s(t) = 0

t

dτ 1 − f  (uh (τ ))

(3.8)

is a strictly increasing function. Let t = ϑ(s) be its inverse. Then Uh (s) = uh (ϑ(s)), Vh (s) = vh (ϑ(s)) is a solution of Eq. (3.7) such that lims→±∞ (Uh (s), Vh (s)) = (u± , 0). So (Uh (s), Vh (s)) is a heteroclinic connection to (u± , 0). Moreover 

s

ϑ(s) =

(1 − f  (Uh (τ ))d τ .

(3.9)

0

Conversely, if (Uh (s), Vh (s) is a solution of Eq. (3.7) such that 1 − f  (Uh (s)) > 0 for any s ∈ R and lims→±∞ (Uh (s), Vh (s)) = (u± , 0), and we define ϑ(s) as in (3.9) with the inverse s(t) = ϑ−1 (t), then uh (t) = Uh (s(t)), vh (t) = Vh (s(t)) is a solution of Eq. (3.6) in the interval ]T− , T+ [, tending to (u± , 0) as t tends to T± and 1 − f  (uh (t)) > 0 for any t ∈]T− , T+ [. Here 

±∞

T± =

(1 − f  (Uh (τ ))d τ , −∞ < T− < 0 < T+ < ∞.

0

Moreover,

  s(t) satisfies (3.8). Finally, there exist δ± > 0 and eigenvectors v± ∈ 1 of the eigenvalues −μ+ and μ− of J± such that ∓μ±

Melnikov Theory for IODEs

133

    U˙ h (s)  −μ+ s  −(μ+ +δ+ )s  − v e ) as s → +∞ +  V˙h (s)  = O(e     U˙ h (s)  μ− s  (μ− +δ− )s  ) as s → −∞,  V˙h (s) − v− e  = O(e and

   Uh (s) − u+   = O(e−μ+ s ) as s → +∞    Vh (s)    Uh (s) − u−   = O(eμ− s ) as s → −∞.    Vh (s) μs

2 2e Let ϕ(s) = eμ+ s +e = eμs sech(μ∗ s), with μ = −μ− s = (μ+ +μ)s e +e−(μ− −μ)s μ+ +μ− ∗ μ = 2 . Now we have the following

μ− −μ+ 2

and

Theorem 3.2 Let v1 (t) and v2 (t) be two independent bounded solutions of the linear equation   vh (t)f  (uh (t)) − [1−f  (u (t))]2 1  h vj (t) = vj (t), 1 − 1−f  (u 0 h (t)) which are exists (see [6]) and suppose the Melnikov vector  M(γ, α) :=

T+

vj (t)∗



T−

0 γvh (t)+h(t+α,uh (t),0)



dt

has a simple zero at some (γ0 , α0 ). Then there exist ρ > 0, ε0 > 0 and C 1 functions γ = γ(ε), α = α(ε), such that γ(0) = γ0 , α(0) = α0 and for any |ε| < ε0 Eq. (3.5) with γ = γ(ε), α = α(ε), has a unique solution (u(t, ε), v(t, ε)) such that (u(t, ε), v(t, ε)) → (u± , 0) as t → T± at the same speed as (uh (t), vh (t)) and sup [|u(t, ε) − uh (t)|ϕ(s(t))−1 < ρ,

t∈]T− ,T+ [

sup |v(t, ε) − vh (t)|]ϕ(s(t))−1 < ρ

t∈]T− ,T+ [

where s(t) is given by (3.8). Finally, we apply the above result to Eq. (3.1). Note, in this case we have √ t 2 − t2 uh (t) = √ , vh (t) = √ , T± = ± 2 2 2 2 So the Melnikov vector is:  M(γ, α) :=

√ 2 √ − 2

vj (t)∗

 2 2−t √ 2 2

0 √ γ+h(t+α,t/ 2,0)

 dt

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M. Feˇckan

v1,2 (t) being two independent solutions of the adjoint linear system: f  (uh (t))vh (t) 2t w1 + w2 = − w1 + w2 [1 − f  (uh (t))]2 2 − t2 1 2 w1 = 2 w1 w2 = − 1 − f  (uh (t)) t −2

w1 = −

(3.10)

Now, the linear system (the linearization of (3.6) along (uh (t), vh (t))): 2t 2 w1 + w2 2 2−t 2 − t2  w2 = −w1 w1 =

(3.11)

√ √ has the solution ( 2uh (t), 2vh (t)) = (1, −t). Another independent solution is 

 t 1 1 t , w1 (t) = + √ arctanh √ 2 2 − t2 2 2 

 t 1 t w2 (t) = . 1 − √ arctanh √ 2 2 2 Thus a fundamental matrix of (3.11) is: ⎛ W (t) = ⎝

1

√1 2

 arctanh

−t 1 −

√t

√t

 2

⎞ t − t 2 −2   ⎠

arctanh 2

√t 2

from which we get the fundamental matrix of (3.10):   ⎛ t t 2 − t 2 ⎝ 1 − √2 arctanh √2  X (t) = t 2 − √12 arctanh √t 2 t 2 −2

⎞ t

⎠ 1

and then in M(γ, α) we can take:   ⎞ ⎛   t t 2 − t 2 ⎝ 1 − √2 arctanh √2 ⎠ 2 − t2 t v1 (t) = (t) = , v . 2 t 1 2 2 − √12 arctanh √t 2 t 2 −2 As a consequence     √2 2 − t2 2 − t 2 2 t − √1 arctanh √t t t −2 2 2 M(γ, α) = √ √ γ + h(t + α, √ , 0) dt 2 2 2 2 − 2 1    1   √ 16 −t − √ (1 − t 2 ) arctanh t 0 γ + h( 2t + α, t, 0)dt = 1 2(1 − t 2 ) 15 −1

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If h(t, u, 0) = h0 (t) is an even function we have 

1 −1



√  t + (1 − t 2 ) arctanh t h0 (t 2) dt = 0

and  1  −1

 1   √ √ t + (1 − t 2 ) arctanh t h0 (t 2) dt = 2 t + (1 − t 2 ) arctanh t h0 (t 2) dt > 0

if h0 (t) > 0 in ]0, (γ0 , 0), where

0



2[. So if these conditions hold, we see that M(γ, α) has a simple zero at γ0 = −

 1 √ 15 √ 2 (1 − t 2 )h0 (t 2) dt. 16 −1

4 Weakly Coupled Nonlinear RLC Circuits In this section we study coupled IODEs such as A0 (x1 )x1 = f (x1 ) + εg1 (t, x1 , x2 , ε, κ) A0 (x2 )x2 = f (x2 ) + εg2 (t, x1 , x2 , ε, κ)

(4.1)

with x1 , x2 ∈ R2 , det A0 (x0 ) = 0 = (det A0 ) (x0 ), f (x0 ), gj (t, x0 , x0 , ε, κ) ∈ RA0 (x0 ) and other assumptions that will be specified below. Let us remark that (4.1) is a form of (2.15) with, among other things,   A0 (x1 ) 0 , x = (x1 , x2 ) A(x) = 0 A0 (x2 ) hence det A(x) = det A0 (x1 ) det A0 (x2 ) clearly satisfies det A(x0 , x0 ) = 0, (det A) (x0 , x0 ) = 0 and (det A) (x0 , x0 ) = 0. Thus (x0 , x0 ) is not a I −point. Multiplying the first equation by the adjugate matrix adj A0 (x1 ) and the second by the adjugate matrix adj A0 (x2 ) we obtain the system ω(x1 )x1 = F(x1 ) + εG 1 (x1 , x2 , t, ε, κ) ω(x2 )x2 = F(x2 ) + εG 2 (x1 , x2 , t, ε, κ).

(4.2)

We assume that ω(x) := det A0 (x), F(x) and G j (x1 , x2 , t, ε, κ) satisfy the following assumptions

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(C1) F ∈ C 2 (R2 , R2 ), ω ∈ C 2 (R2 , R) and the unperturbed equation ω(x)x = F(x)

(4.3)

possesses a noncritical singularity at x0 , i.e. ω(x0 ) = 0 and ω  (x0 ) = 0. (C2) F(x0 ) = 0 and the spectrum σ(F  (x0 )) = {μ± } with μ− < 0 < μ+ , and x = F(x) has a solution γ(s) homoclinic to x0 , that is lim γ(s) = x0 , and ω(γ(s)) = 0 s→±∞

for any s ∈ R. Without loss of generality, we may, and will, assume ω(γ(s)) > 0 for any s ∈ R. Moreover G i ∈ C 2 (R6+m , R2 ), i = 1, 2 are 1-periodic in t with G i (x0 , x0 , t, ε, κ) = 0 for any t ∈ R, κ ∈ Rm and ε sufficiently small. (C3) Let γ± be the eigenvectors of F  (x0 ) with the eigenvalues μ∓ , resp. Then ∇ω(x0 ), γ±  > 0 (or else ω  (x0 )γ± > 0).   γ(s) From (C2) we see that (s) := is a bounded solution of equation: γ(s) x1 = F(x1 ), x2 = F(x2 ) and that x0 persists as singularity of Eq. (4.2). So this section is a continuation of the above sections, but here we study more degenerate IODE. Next, setting as above 

s

θ(s) :=

ω(γ(τ ))d τ

(4.4)

0

and xh (t) = γ(θ−1 (t)), it is easily seen that xh (t) satisfies ω(x)x = F(x) whose linearization along xh (t) is ω(xh (t))z  = F  (xh (t))z − F(xh (t))

ω  (xh (t))z ω(xh (t))

(4.5)

with the adjoint equation ω(xh (t))v  =

ω  (xh (t))∗ F(xh (t))∗ v − F  (xh (t))∗ v. ω(xh (t))

Note xh (t) is defined on the interval ]T− , T+ [ for  T± := 0

±∞

ω(γ(τ ))d τ < ∞

(4.6)

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137

and lim xh (t) = x0 .

t→T±

Now we are ready to state the following result [4] Theorem 4.1 Let m = 3 and M(α, κ) be given as ⎛

T+

G 1 (xh (t), xh (t), t + v1∗ (t) ω(xh (t))

α, 0, κ)



dt ⎟ ⎜ ⎟ ⎜  T− ⎜ T+ G 1 (xh (t), xh (t), t + α, 0, κ) ⎟ ⎜ ∗ dt ⎟ v (t) ⎟ ⎜ ω(xh (t)) ⎟ ⎜  T− 2 M(α, κ) = ⎜ T+ ⎟, (x (t), x (t), t + α, 0, κ) G ⎟ ⎜ 2 h h ∗ ⎜ dt ⎟ v (t) ⎟ ⎜ T− 1 ω(x (t)) h ⎟ ⎜  T+ ⎠ ⎝ (x (t), x (t), t + α, 0, κ) G 2 h h ∗ dt v2 (t) ω(xh (t)) T−

(4.7)

where v1 (t), v2 (t) are two independent bounded solutions of the adjoint Eq. (4.6), which do exist. Suppose that α¯ and κ¯ exist so that M(α, ¯ κ) ¯ = 0 and

∂M (α, ¯ κ) ¯ ∈ GL(4, R). ∂(α, κ)

Then there exist ε¯ > 0, ρ > 0, unique C 1 −functions α(ε) and κ(ε) with α(0) = α¯ and κ(0) = κ, ¯ defined for |ε| < ε, ¯ and a unique solution (x1 (t, ε), x2 (t, ε)) of Eq. (4.2) with α = α(ε), κ = κ(ε), 0 < |ε| < ε, ¯ such that sup |xj (t + α(ε), ε) − xh (t)|ϕ(θ−1 (t))−1 < ρ, j = 1, 2

T− 0 such that the equation sin α − χ sin(α + ) = 0

(4.16)

has a simple zero α0 with sin α0 < 0, cos α0 > 0 and cos(α0 + ) > 0, formulas (4.15) give a simple zero (α0 , γ1,0 , γ2,0 , λ0 ) of (4.14) with positive γ1,0 , γ2,0 , λ0 , and

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Theorem 4.1 can be applied to (4.8). If χ cos  = 1 then (4.16) is equivalent to tan α =

χ sin  . 1 − χ cos 

(4.17)

Hence assuming  ∈]π, 2π[ and χ cos  < 1, the right hand side of (4.17) is negative, and then  π  χ sin  ∈ − ,0 (4.18) α0 = arctan 1 − χ cos  2 satisfies sin α0 < 0 and cos α0 > 0. Since α0 satisfies (4.16) and sin α0 < 0, condition cos(α0 + ) > 0 is equivalent to tan(α0 + ) < 0. Then using (4.17), we derive 3π sin  ,  = . (4.19) 0 > tan(α0 + ) = cos  − χ 2 When cos  3π   < 0, then (4.19) is not satisfied, since sin  < 0. So we take  ∈ , 2π and (4.19) gives also χ < cos . Clearly χ < cos  implies 1 > χ cos . 2 Summarizing we see that for any fixed χ and  satisfying

3π , 2π , 0 < χ < cos ,  ∈ 2 

(4.20)

the Melnikov function (4.14) has a simple zero (α0 , γ1,0 , γ2,0 , λ0 ) given by (4.15) and (4.18), and γ1,0 > 0, γ2,0 > 0, λ0 > 0. Hence in the region given by (4.20) we apply Theorem 4.1 to (4.8) with parameters γ1 , γ2 , λ near γ1,0 , γ2,0 , λ0 determined by (4.15) and (4.18), i.e., a12 χ sin   , a11 1 + χ2 − 2χ cos  a22 (1 − χ cos ) , γ1,0 = −  a21 1 + χ2 − 2χ cos  a22 (cos  − χ) γ2,0 = −χ  a21 1 + χ2 − 2χ cos  λ0 = −

(4.21)

for any fixed  and χ satisfying (4.20). Summarizing, we get the following: Theorem 4.2 For any fixed , χ satisfying (4.20) and then α0 , γ1,0 , γ2,0 , λ0 given by (4.18) and (4.21), there is an ε0 > 0 and smooth functions α, γ1 , γ2 , λ :] − ε0 , ε[→ R with α(0) = α0 , γ1 (0) = γ1,0 , γ2 (0) = γ2,0 , λ(0) = λ0 , such that for any ε ∈] − ε0 , ε0 [\{0}, system (4.8) with γ1 = γ1 (ε), γ2 = γ2 (ε), λ = λ(ε), possesses a unique solution (u1 (ε, t), u2 (ε, t)) on ] − 3 + α(ε), 3 + α(ε)[ such that

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143

  lim sup uj (ε, t + α(ε)) − ε→0 t∈]−3,3[   lim sup uj (ε, t + α(ε)) + ε→0 t∈]−3,3[

  −1 t 2   1 9 − t2 1− =0 4 3  t  = 0. 6

Of course, solutions given by Theorem 4.2 vary smoothly with respect (, χ) satisfying (4.20). Missed in the above analysis is the second possibility when   ∈]0,  π[. Then sin  > 0 and (4.17) is negative if κ cos  > 1, so we get  ∈ 0, π2 and κ > 1. Then inequality of (4.19) is satisfied since κ > 1 > cos . So we conclude that the result of Theorem 4.2 is valid also for  π , κ cos  > 1,  ∈ 0, 2 a12 χ sin   λ0 = , a11 1 + χ2 − 2χ cos  a22 (1 − χ cos )  , γ1,0 = a21 1 + χ2 − 2χ cos  a22 (cos  − χ) . γ2,0 = χ  a21 1 + χ2 − 2χ cos 

5 General Dimensional IODEs In this section, we consider a general IODEs such as A(x)x = f (x) + εg(x, t, ε, κ),

(5.1)

where x ∈ Rn and A(x) is a n × n−matrix of constant rank. Equations like (5.1) are usually handled by multiplying them by the adjugate matrix adj A(x) (transpose of the matrix of cofactors). It is obtained IODE

where

ω(x)x = F(x) + εG(t, x, λ, ε),

(5.2)

ω(x) = det A(x), F(x) = adj A(x)f (x), G(x, t, ε, κ) = adj A(x)g(x, t, ε, κ).

(5.3)

Assuming that A ∈ C 2 (Rn , L(Rn )), f ∈ C 2 (Rn , Rn ) and g ∈ C 2 (Rn+m+2 , Rn ) is 1periodic in t, then ω ∈ C 2 (Rn , R), F ∈ C 2 (Rn , Rn ) and G ∈ C 2 (Rn+m+2 , Rn ) is 1periodic in t.

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As above, we shall look directly at Eq. (5.2), without considering condition (5.3) assuming ω ∈ C 2 (Rn , R), F ∈ C 2 (Rn , Rn ) and G ∈ C 2 (Rn+m+2 , Rn ) is 1-periodic in t. We suppose the following conditions hold. (C1) The unperturbed equation associated to (5.2): ω(x)x = F(x) possesses noncritical singularities at x± , i.e. ω(x± ) = 0 and ω  (x± ) = 0. (C2) x± are hyperbolic equilibria for the ordinary differential equation x˙ = F(x),

˙=

d ds

(5.4)

and the spectra σ(F  (x± )) have the same number of eigenvalues with negative real parts counted with multiplicities, say k− . Moreover a solution γ(s) of (5.4) exists that is heteroclinic to x± , that is lim γ(s) = x± , and ω(γ(s)) = 0 for s→±∞

any s ∈ R. Finally, G(t, x± , λ, ε) = 0 for any t ∈ R, λ ∈ R and ε sufficiently small. Without loss of generality, we may, and will, assume ω(γ(s)) > 0 for any s ∈ R. (C3) We have: lim

s→∞

1 ln |γ(s) − x+ | = −μ+ < 0, s

lim

s→−∞

1 ln |γ(s) − x− | = μ− > 0, s

where −μ+ and μ− are simple eigenvalues of F  (x+ ), F  (x− ) respectively and all the other eigenvalues of F  (x+ ) have real parts either positive or less than −μ+ and similarly, all the other eigenvalues of F  (x− ) have real parts either negative or greater than μ− . (C4) Let v+ , resp. v− , be an eigenvector of the eigenvalue −μ+ , resp. μ− , of F  (x+ ), resp. F  (x− ). Then ω  (x± )v± = 0 (or else ∇ω(x± ), v±  = 0). Without loss of generality we can assume that ω  (x± )v± > 0. γ  (s) (C5) ω(γ(s)) is the unique bounded solution, up to a multiplicative constant, of the linear system

 F(γ(s))  ω  (γ(s))x  ω (γ(s)) x = F  (γ(s))x − F(γ(s)). x = F (γ(s)) − ω(γ(s)) ω(γ(s)) 

Now we make the time change:  t = α + ϑ(s), ϑ(s) =

s

ω(γ(σ))d σ.

0

Note that ω(γ(s))  ω  (x± )[γ(s) − x± ], as s → ±∞, and hence ϑ(s) is a bounded function on R with limits:

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145

lim ϑ(s) = T± .

s→±∞

Shifting time s we can also assume T+ = T = −T− . As in [3] we make the change of variables: x(α + ϑ(s)) = γ(s) + ϕ(s)y(s) = x± + ϕ(s)[y(s) + η± (s)] ± , ϕ(s) = e−μ− s2+eμ+ s = eμs sech(μ∗ s) with μ∗ = where η± (s) = γ(s)−x ϕ(s) ∗ ∗ μ+ + μ = μ− − μ and show that:

μ− −μ+ 2

(5.5) and μ =

ω(γ(s) + ϕ(s)y(s)) ≥ ϕ(s)ω  (x± )v± as s → ±∞ (that is for |s|  1). So ω(γ(s) + ϕ(s)y(s)) ≥ M ϕ(s), for any s ∈ R and some M > 0 provided |y(s)| < δ for some δ > 0. Plugging (5.5) into (5.2) we arrive at the equation: F(γ) ϕ ω(γ) F(γ + ϕy) − − y ϕω(γ + ϕy) ϕ ϕ ω(γ) +ε G(α + ϑ(s), γ + ϕy, λ, ε) ϕω(γ + ϕy)

y =

(5.6)

The linearization of (5.6) with ε = 0 at y = 0 is  F(γ(s))ω  (γ(s)) ϕ (s) y y = F (γ(s)) − − ω(γ(s)) ϕ(s) 



and, arguing as in [3], we prove that it has an exponential dichotomy on both half lines R± with projections P+ and P− such that: dimRP+ = k− − 1, dimN P− = n − k− − 1. From [3] we know that RP+ ∩ N P− = {0} and also dimRP+ + N P− = n − 2 so the adjoint variational system: 

ϕ (s) ω  (γ(s))∗ F(γ(s))∗ − y y = − F  (γ(s))∗ − ω(γ(s)) ϕ(s)

(5.7)

has a two dimensional space of bounded solutions. Let ψ1 (s), ψ2 (s) be a basis for such a space. Then from the Lyapunov-Schmidt method we derive the following result:

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Theorem 5.1 Suppose conditions (C1)–(C5) hold and λ ∈ R. Then equation (5.7) has a two dimensional space of bounded solutions. Let ψ1 (s) and ψ2 (s) be a basis for the space of bounded solutions of (5.7). If the Poincaré-Melnikov function: 

⎛





⎜ M1 (α, λ) −∞ := ⎜ ⎝ ∞ M2 (α, λ) −∞

⎞ ψ1∗ (s) G(γ(s), α + ϑ(s), λ, 0) ds⎟ ϕ(s) ⎟ ∗ ⎠ ψ2 (s) G(γ(s), α + ϑ(s), λ, 0) ds ϕ(s)

(5.8)

has a simple zero at (α0 , λ0 ), then there exist ρ > 0, ε¯ > 0 and C 1 -functions λ = λ(ε), α = α(ε), such that λ(0) = λ0 , α(0) = α0 and for any |ε| < ε0 Eq. (5.6) with λ = λ(ε), α = α(ε) has a unique bounded solution y(s, ε) such that sups∈R |y(s, ε)| < ρ. Moreover lim sup |y(s, ε)| = 0. ε→0 s∈R

Concerning the original equation, setting x(t, ε) as in (5.5) with t = α + ϑ(s) and α = α(ε), we obtain a solution of (5.2) such that   sup x(t, ε) − γ(ϑ−1 (t − α(ε)) ϕ−1 (ϑ−1 (t − α(ε))) < ρ.

t∈(−T ,T )

Finally, putting s = ϑ−1 (t) in the Melnikov vector, (5.8) becomes: 



⎛

T+

⎜ T M1 (α, λ) − := ⎜ ⎝  T+ M2 (α, λ) T−

⎞ ψ1∗ (ϑ−1 (t)) G((t), α + t, λ, 0) dt ⎟ ϕ(ϑ−1 (t))ω((t)) ⎟ ∗ −1 ⎠ ψ2 (ϑ (t)) G((t), α + t, λ, 0) dt ϕ(ϑ−1 (t))ω((t))

where (t) = γ(ϑ−1 (t)). It has been proved in [3, p.1173] that ψj (s) = ϕ(s)vj (ϑ(s)), where vj (t) is a solution of ω((t))v˙ =

ω  ((t)) F((t))∗ v − F  ((t))∗ v ω((t))

(5.9)

such that {v1 (t), v2 (t)} is a basis for the space of solutions of (5.9) that are bounded on ]T− , T+ [. So: ⎛

⎞ G((t), α + t, λ, 0) dt ⎟ ⎜ T M1 (α, λ) ω((t)) ⎟  −T+ := ⎜ ⎝ M2 (α, λ) G((t), α + t, λ, 0) ⎠ dt v2∗ (t) ω((t)) T−





T+

v1∗ (t)

which is consistent with the expression given in Theorem 3.2.

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147

6 Blue Sky-Like Catastrophe for Reversible IODEs We consider IODE

A(x)x = f (x, ε),

(6.1)

where A ∈ C 2 (Rn , L(Rn )) and f ∈ C 2 (Rn+m , Rn ) under the assumption that an involution T ∈ L(Rn ) exists such that (R) A(T x) = T A(x)T and f (T x, ε) = −T f (x, ε) for any x ∈ Rn and ε ∈ Rm . Recall that a map T : Rn → Rn is an involution if T 2 = I.

(6.2)

Note that if x(t) solves (6.1) then x˜ (t) := T x(−t) is also its solution. So assumption (R) is a generalization of reversibility to IODE, just take A(x) = I (see [29, 30]). A standard way for handling (6.1) is multiplying it by the adjugate matrix adj A(x). Then Eq. (6.1) becomes: ω(x)˙x = F(x, ε), where

ω(x) = det A(x), F(x, ε) = adj A(x)f (x, ε).

(6.3)

Consequently, in this section we focus our study on IODE of the form ω(x)˙x = F(x, ε)

(6.4)

for general ω ∈ C 2 (Rn , R) and F ∈ C 2 (Rn+m , Rn ) under the symmetry assumption (H1) ω(T x) = ω(x), F(T x, ε) = −T F(x, ε) for any x ∈ Rn , ε ∈ Rm . Next, from assumption (R) it is easy to deduce for (6.3) that ω(T x) = det A(T x) = det(T A(x)T ) = det T det A(x) det T = (det T )2 det A(x) = det A(x) = ω(x), since (6.2) implies (det T )2 = 1. Similarly, since det T T −1 = adj T , we derive F(T x, ε) = adj A(T x)f (T x, ε) = − adj(T A(x)T )T f (x, ε) = − adj T adj A(x) adj T T f (x, ε) = − det T adj T adj A(x)f (x, ε) = − det T 2 T −1 adj A(x)f (x, ε) = −T adj A(x)f (x, ε) = −T F(x, ε). Summarizing, we see that ω and F given by (6.3) satisfy (H1), i.e., (R) implies (H1) for (6.3). Moreover we also assume that the following conditions hold: (H2) x0 is a noncritical singularity of Eq. (6.4) [i.e., ω(x0 ) = 0 and ω  (x0 ) = 0] and the associated unperturbed problem

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M. Feˇckan

dx := x˙ = F(x, 0) ds

(6.5)

has x0 as hyperbolic equilibrium [i.e., F(x0 , 0) = 0 and all eigenvalues of the matrix Fx (x0 , 0) have nonzero real parts]. Furthermore equation x˙ = F(x, 0) has a homoclinic solution γ(s), [i.e., a non constant solution γ(s) such that lim γ(s) = x0 ]. Moreover x0 , γ(0) ∈ Fix T [i.e., T x0 = x0 and T γ(0) =

s→±∞

γ(0)], and F(x0 , ε) = 0 for any ε small. (H3) Fx (0, 0) has a simple positive eigenvalue μ+ , with eigenvector v+ , and a simple negative eigenvalue μ− , with eigenvector v− , and the real parts of the other eigenvalues are either greater than μ+ or less than μ− . Moreover ω  (x0 )v+ = 0, ω(γ(s)) > 0 for any s ∈ R and γ(s) is in a general position, i.e., 1 log |γ(s)| = μ± . s

lim

s→∓∞

Let ϕ(s) = eμ+ s +1 eμ− s and without loss of generality we assume that x0 = 0. It is easy to check that from (H1) it follows: if x(t) is a solution of (6.4), then T x(−t) is also a solution of (6.4). Then we recall [30] the following Definition 6.1 A given solution x(t) of (6.4) defined in a symmetric interval I is T -symmetric (or T -reversible) if x(−t) = T x(t), for any t ∈ I. Because of the uniqueness of solutions, x(t) is T -symmetric if and only if x(0) = T x(0). We set:  ϑ(s) :=

s

ω(γ(τ ))d τ and xh (t) := γ(ϑ−1 (t)).

0

It has been observed in [3] that the integrals  T± :=

±∞

ω(γ(τ ))d τ

0

are convergent. Then xh (t) is a solution of the unperturbed IODE ω(x)˙x = F(x, 0)

(6.6)

in the interval ]T− , T+ [. Note that ϑ(s) is increasing. Moreover, we have 



0



ω(γ(τ ))d τ = − ω(γ(−τ ))d τ T− = −  ∞ 0  ∞ −∞ ω(T γ(τ ))d τ = − ω(γ(τ ))d τ = −T+ . =− 0

0

Melnikov Theory for IODEs

149

Then T+ = −T− := T > 0. Similarly, we show that ϑ(−s) = −ϑ(s) and then T xh (t) = xh (−t) for all t ∈] − T , T [. Next, set A(∞) := Fx (0, 0) + μI +

μv+ ω  (0) ω  (0)v+

and A(−∞) = −T A(∞)T . We assume (H4) 1. The equation 

y =−

Fx∗ ((γ(s), 0)

 ϕ (s) ω  (γ(s))∗ F(γ(s), 0)∗ − I y − ω(γ(s)) ϕ(s)

(6.7)

has a 1−dimensional space of bounded and anti-reversible solutions on R, i.e. Ty(s) = −y(−s) for all s ∈ R. 2. The equation  A(−∞)y for s < 0,  y = A(∞)y for s > 0 has a 1−dimensional linear space of exponentially unbounded and reversible solutions on R. Let yb (s) be a nonzero, bounded and anti-reversible solution of (6.7), and let  b := 0



ϕ−1 (s)yb∗ (s)Fε (γ(s), 0)ds.

where ∗ is meant with respect to the scalar product on Rn that makes T unitary. Note yb (s) are uniquely defined, up to a multiplicative nonzero constant. Then so is b ∈ Rm . Next, let Fix T := {x ∈ Rn | T x = x}. In this section we look for T −symmetric solutions x(t) of (6.4) on [−, ] for some 0 <  < T and ε small which are near to xh (t) on [−, ] and such that x() − xh () ∈ Fix T .

(6.8)

Note a solution x(t) of Eq. (6.4) is 2-periodic if and only if x() = x(−), and for T -symmetric solutions this equality reduces to: x() ∈ Fix T . Thus looking for T -symmetric and 2-periodic solutions of Eq. (6.4) amounts to searching for solutions of Eq. (6.4) such that x(0), x() ∈ Fix T .

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As a consequence condition (6.8) does not give periodic solutions of Eq. (6.4). However, since T xh () = T γ(ϑ−1 ()) = γ(−ϑ−1 ()) = γ(ϑ−1 (−)) = xh (−), if a T -symmetric solution x(t) of Eq. (6.4) satisfies (6.8) we get: x() − x(−) = xh () − xh (−) = γ(ϑ−1 ()) − γ(ϑ−1 (−)) → 0 as  → T . Thus the closer  to T the smaller x() − x(−). Because of this reason we call a solution x(t) of (6.4) satisfying (6.8), T -reversible and shadowed 2-periodic solution. Moreover, splitting x(t),  x(t) = x(t) − xh (t), x(t) = xh (t) + we see that (6.8) holds for a T −symmetric x(t) if and only if  x(t) is 2−periodic. We also look for such solutions that  x is small. This fact justifies the notation of shadowed 2-periodic solution near xh (t). Now we are ready, to state the objective of this section [7]. Theorem 6.2 Suppose that conditions (H1)-(H4) hold and the vector b is not zero. Then there exist a positive constant χ∗ and neighborhood U of ε = 0 ∈ Rm and a hypersurface  ⊂ U such that for any ε ∈  Eq. (6.4) has a T −symmetric solution x(t, ε) such that sup x(t, ε) − xh (t)ϕ(ϑ(t)−1 ) = O(ε). t∈]−T ,T [

Moreover, for any 0 <  < T sufficiently close to T and any sufficiently small φ1 in a one dimensional subspace of Rn , there is a C 1 −smooth 2−parametric family of surfaces ,φ1 ⊂ U such that for any ε ∈ ,φ1 , thereis a shadowed  symmetric ∗ −1

2−periodic orbits of (6.4). Furthermore, ,φ1 is O e−χ uniformly with respect to  and φ1 .

ϑ ()

−near to 

Corollary 6.3 Suppose condition (H1)-(H4) hold. Then if  β :=

T

ω(xh (t))−1 vb (t)∗ Fε (xh (t), 0)dt.

(6.9)

0

is nonzero, the conclusion of Theorem 6.2 holds. Recall that he linearization of (6.6) along xh (t) is v˙ =

  1 ω  (xh (t)) Fx (xh (t), 0) − F(xh (t), 0) v, ω(xh (t)) ω(xh (t))

and its adjoint equation is

(6.10)

Melnikov Theory for IODEs

v˙ =

1 ω(xh (t))

151



 ω  (xh (t))∗ F(xh (t), 0)∗ − Fx (xh (t), 0)∗ v. ω(xh (t))

(6.11)

Motivated by the above sections, we consider the planar system: v + ε1 h1 (u, v) 2u + 1 v˙ = −u + ε2 h2 (u, v)

u˙ =

which can be written in the form of (6.4) as (2u + 1)˙u = v + ε1 (2u + 1)h1 (u, v) (2u + 1)v˙ = (2u + 1)(−u + ε2 h2 (u, v)), so

x = (u, v)∗ ∈ R2 , ω(u, v) = 2u + 1, F(u, v, ε) = (v + ε1 (2u + 1)h1 (u, v), (2u + 1)(−u + ε2 h2 (u, v)))∗ , T (u, v) = (u, −v)∗ .

Note that T is unitary with respect to the standard scalar product in R2 . We suppose that h1 , h2 are C 2 -smooth and satisfy (E) h1 (u, −v) = −h1 (u, v), h2 (u, −v) = h2 (u, v). Then (H1) holds. The unperturbed associated Eq. (6.5) u = v v  = − (2u + 1)u has the solution ∗    1 2 s 3 4 s 1 − 3 tanh , −6 csch s sinh γ(s) = 4 2 2 with (see Fig. 1)

∗  1 lim γ(s) = x0 = − , 0 . t→±∞ 2

Clearly (H2) is satisfied. Then it can be checked that ω(γ(s)) =  Fx (x0 , 0) =

3 2

sech2

s 2

> 0. From

 01 , 10

we get μ± = ±1 and v± = (1, ±1)∗ . Since ω  (x0 )v+ = 2 > 0, condition (H3) holds as well. Moreover, (H4) trivially holds. Next, we have

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 ϑ(s) =

s

 ω(γ(τ ))d τ =

0

s

0

3 τ s sech2 d τ = 3 tanh , 2 2 2

so T± = ±3 and xh (t) = γ(ϑ−1 (t)) = (uh (t), vh (t)) = Thus

  2 ∗ t 1 t2 . 1 − ,t −1 4 3 9

 1 9 − t 2 , ω  (xh (t)) = (2, 0)∗ , 6   ∗ F(xh (t), 0) 1  0 1 = −2t, t 2 − 3 . Fx (xh (t), 0) = t 2 −2 0 ω(xh (t)) 12 3 ω(xh (t)) =

So, now (6.10) has the form 2t 6 z1 + z2 2 9−t 9 − t2 z˙2 = −z1 .

z˙1 =

(6.12)

The adjoint system of (6.12) is (see (6.11)) 2t ζ1 + ζ2 −9 6 ζ1 ζ˙2 = 2 t −9

ζ˙1 =

t2

with the fundamental matrix solution Z

−1∗

9 − t2 (t) = 9 

Thus we take vb (t) =



3−t 2 3−t 2 ln 3+t 3 8 3−t 2 1 6−t − 23 t 23 9−t 2 − 4t



+ 34 t ln 3+t 3−t

 .



t 2 −9 t 2 −3 ln 3+t − 43 t 9 8 3−t 2 6−t 1 + 36 t(t 2 − 9) ln 3+t 6 3−t

.

Then (6.9) has the form    3 t2 − 9 t2 − 3 3 + t h1 (uh (t), vh (t)) ln − t , β= 9 8 3−t 4 0     2 1 6−t 3+t h2 (uh (t), vh (t)) + t(t 2 − 9) ln dt . 6 36 3−t 

3

For concreteness, we take h1 (u, v) = av and h2 (u, v) = bu for a, b ∈ R \ {0}, then we get

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153

  3+t 2 − 6t dt, β= t(t − 9) (t − 3) ln 3−t 0     3 b 3 3+t dt = (−a, b). (3 − t 2 ) 6(6 − t 2 ) + t(t 2 − 9) ln 432 0 3−t 8 

a 2592



3

2

2

Applying Corollary 6.3, we get Corollary 6.4 Given the system v u˙ = 2u+1 + ε1 av v˙ = −u + ε2 bu

there exists a curve  ⊂ R2 passing through 0 ∈ R2 with a slope ab at (ε1 , ε2 ) = (0, 0) such that for any (ε1 , ε2 ) ∈  the conclusion of Theorem 6.2 holds with any fixed 0 < χ∗ < 1.

7 Connecting IK Singularities and Impasse Points In this section, the persistence of solutions are studied under nonautonomous perturbations of IODEs connecting either impasse points (see [27, p. 163]) with IKsingularities (see [27, p. 168]) or two impasse points. Important parts of the paper [8] are applications of the theory to concrete perturbed fully nonlinear RLC circuits.

7.1 Connecting Impasse Points with IK-singularities We consider IODE ω(x)x = F(x) + εG(x, t, ε, κ), x ∈ Rn , κ ∈ Rm

(7.1)

for ω ∈ C 2 (Rn , R), F ∈ C 2 (Rn , Rn ) and G ∈ C 2 (Rn+m+2 , Rn ) with bounded derivatives. First we suppose (C1) The unperturbed (7.1):

ω(x)x = F(x)

(7.2)

possesses noncritical singularities at x0 and x1 , i.e. ω(xi ) = 0 and ω  (xi ) = 0, i = 1, 2. (C2) The ODE: x˙ = F(x) (7.3)

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has the hyperbolic equilibrium x0 , i.e. F(x0 ) = 0 and the spectrum σ(DF(x0 )) has no eigenvalues on the imaginary axis, and a solution γ(s) on (−∞, 1] such that lim γ(s) = x0 , and ω(γ(s)) = 0 for any s ∈ (−∞, 1) with γ(1) = x1 . s→−∞

Moreover G(x0 , t, ε, κ) = 0 for any t ∈ R, κ ∈ Rm and ε sufficiently small. (C3) It results: 1 ln |γ(s) − x0 | = μ+ , lim s→−∞ s where μ+ is a simple positive eigenvalue of F  (x0 ) with the corresponding eigenvector γ− and all the other eigenvalues of F  (x0 ) have real parts greater than μ+ . ˙ < 0. (C4) ∇ω(x0 ), γ−  · ∇ω(x1 ), γ(1) Let k− be the number of eigenvalues of F  (x0 ) with negative real parts counted with multiplicities. Then F  (x0 ) has n − k− eigenvalues with positive real parts counted with multiplicities. According to arguments of [3, p. 1169], the linear system 

F(γ(s))ω  (γ(s))  − μ+ I y y˙ = F (γ(s)) − ω(γ(s)) has an exponential dichotomy on R− with a projection P− such that rank P− = k− + 1 or dim N P− = n − k− − 1. We have the following result [8]. Theorem 7.1 Assume conditions (C1)–(C4) hold and that ω(γ(s)) > 0 for any s ≤ 1. Let  s θ(s) := ω(γ(τ ))d τ , (7.4) 0

then γ(θ−1 (t)) satisfies Eq. (7.2) for t ∈ (−T∗ , 0], where  T∗ =

0

−∞

ω(γ(τ ))d τ .

In [3, p. 1164] it has been observed that T∗ is finite. Then given a compact subset K ⊂ Rm there exist η0 > 0 and ε0 > 0 such that for any (η, ε, κ) ∈ N P− × R × K satisfying |η| ≤ η0 , |ε| ≤ ε0 , there exists a unique continuous function x(t, η, κ, ε) defined for t ∈ [−T∗ , T¯ ], with T¯ > 0 and satisfying Eq. (7.1) and ω(x(t, η, κ, ε)) > 0 for any t ∈ (−T∗ , T¯ ) as well as ω(x(−T∗ , η, κ, ε)) = ω(x(T¯ , η, κ, ε)) = 0. Moreover lim |x(t, η, κ, ε) − x0 | = 0 t→−T∗

and

sup |x(t, η, κ, ε) − γ(θ−1 (t))| → 0

−T∗ 0 and ∇ω(x1 ), γ(s ˙ 1 ) < 0. (H3) ∇ω(x−1 ), γ(s We have the following [8] Theorem 7.2 Assume conditions (H1)–(H3) hold. Then for any (ξ, ε) such that |ξ − x1 | + |ε| is sufficiently small and ω(ξ) = 0, there exists a unique continuous function x(t) = x(t, ξ, ε, κ) on the interval [¯t , s1∗ ], for some ¯t < s1∗ , that satisfies Eq. (7.1) as well as ω(x(t)) > 0 for t ∈ (¯t , s1∗ ), x(s1∗ ) = ξ and ω(x(¯t )) = 0. Moreover, lim

sup |x(t) − γ(θ¯−1 (t))| = 0

|ξ−x1 |+|ε|→0 t∈[¯t ,s∗ ] 1

uniformly for κ in compact sets of Rm for ¯ := s∗ − θ(s) 1



s1∗

ω(γ(τ ))d τ .

(7.6)

s

Furthermore, ¯t is a C 2 -function of the parameters ξ, ε, κ with the property ¯t →

s∗ s1∗ − s∗1 ω(γ(τ ))d τ as |ξ − x1 | + |ε| → 0 uniformly for κ in compact sets in Rm . −1

Similar the following result holds [8].

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Theorem 7.3 Assume conditions (H1)–(H3) hold. Then for any (ξ, ε) such that |ξ − x−1 | + |ε| is sufficiently small and ω(ξ) = 0, there exists a unique continuous ∗ ∗ , ˜t ], for some ˜t > s−1 , that satisfies function x(t) = x(t, ξ, ε, κ) on the interval [s−1 ∗ ∗ ∗ Eq. (7.1) in (s−1 , ˜t ) with x(s−1 ) = ξ and such that ω(x(t)) > 0, for t ∈ (s−1 , ˜t ) and ω(x(˜t )) = 0. Moreover, lim

sup |x(t) − γ(θ˜−1 (t))| = 0

|ξ−x−1 |+|ε|→0 t∈[s∗ ,˜t ] −1

˜ := s∗ + uniformly for κ in compact sets of Rm for θ(s) −1

s

ω(γ(τ ))d τ . Further s∗ ∗ more, ˜t is a C 2 -function of parameters ξ, ε, κ such that ˜t → s−1 + s∗1 ω(γ(τ ))d τ −1 as |ξ − x−1 | + |ε| → 0 → 0 uniformly for κ in compact sets in Rm . ∗ s−1

Of course the sign of ω(γ(s)) in the interior of the interval is not crucial. Sim∗ , s1∗ ). However in this case the ilar statements hold when ω(γ(s)) < 0 for s ∈ (s−1 ¯ ˜ functions ϑ(s), θ(s) and θ(s) are decreasing and then time t has to be reversed as above. Suppose that γ(s) is a solution of Eq. (7.3) defined in the interval (−∞, T ], T > 1, such that ω(γ(1)) = ω(γ(T )) = 0, ω(γ(s)) > 0 for s < 1, ω(γ(s)) < 0 for 1 < s < T . Then we can apply both Theorems 7.1 and 7.2 (this last with time reversed since ω(γ(s)) < 0 for 1 < s < T ) and prove the existence of ρ > 0 such that for any η ∈ N P− with |η| < ρ and ξ such that ω(ξ) = 0 and |ξ − γ(1)| < ρ, there are two solutions of Eq. (7.1), x(t, η, κ, ε) and x(t, ξ, ε, κ), the first defined for t ∈ (−T∗ , T¯ ) and satisfying sup |x(t, η, κ, ε) − γ(θ−1 (t))| → 0 t∈(−T∗ ,T¯ )

as |ε| + |η| → 0 and the second one defined for t ∈ (1, ¯t ) and satisfying sup |x(t, ξ, ε, κ) − γ(θ¯−1 (t))| → 0

t∈(1,¯t )

¯ are as in (7.4) and (7.6), respectively. as |ε| + |ξ − γ(1)| → 0, where θ(s) and θ(s)

7.3 Fully Nonlinear RLC Circuits Motivated by [17, 28], a fully nonlinear RLC circuit is described by the second order ordinary differential equation u + L(v) + R(v) = e(t), v = C(u) ,

(7.7)

Melnikov Theory for IODEs

157

where L, R are the nonlinear self-inductance and the ohmic resistance, respectively and C is the nonlinear capacitance. u is the potential/voltage, q = C(u) is the charge is the current intensity. We suppose that L, R, C and e are C 3 -smooth. and v = dq dt We can write Eq. (7.7) as a system in R2 : L (v)v  = −u − R(v) + e(t),

(7.8)

C  (u)u = v. In this section, instead of (7.8) we consider the perturbed problem L (v)v  = −u − εR(v) + εe(t),

(7.9)

C  (u)u = v,

that is we assume that the external e.m.f. and the ohmic resistance are small. We also assume that v0 , u0 ∈ R \ {0} exist such that L (v0 ) = 0, L (v0 ) = 0 and C  (u0 ) = 0, C  (u0 ) = 0. Setting ε = 0 we get the unperturbed system: L (v)v  = −u, C  (u)u = v,

(7.10)

which has almost the same phase portrait as v˙ = −C  (u)u, u˙ = L (v)v with˙ =

d . ds

(7.11)

Indeed, system (7.11) has the Hamiltonian: 

v

H(v, u) =

zL (z)dz +

0



u

zC  (z)dz

0

and it is easy to check that H(v, u) is constant along trajectories of Eq. (7.10). Next, Eq. (7.11) has four equilibria: e1 = (0, 0), e2 = (v0 , 0), e3 = (0, u0 ) and e4 = (v0 , u0 ). The Jacobian matrix of system (7.11) at a point (v, u) is  J (v, u) = Thus:



0 −C  (u) − uC  (u)   0 L (v) + vL (v)

0 −C  (0) J (0, 0) =  0 L (0)







0 −C  (0) J (v0 , 0) =  0 v0 L (v0 )



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and  J (0, u0 ) =

0 −u0 C  (u0 )  0 L (0)



 J (v0 , u0 ) =

0 −u0 C  (u0 )  0 v0 L (v0 )



Hence the characteristic polynomial at the fixed points are respectively: p(0,0) (λ) = λ2 + C  (0)L (0) p(v0 ,0) (λ) = λ2 + v0 C  (0)L (v0 ) p(0,u0 ) (λ) = λ2 + u0 C  (u0 )L (0) p(v0 ,u0 ) (λ) = λ2 + u0 v0 C  (u0 )L (v0 ). We suppose L (0) = 0 and C  (0) = 0. Thus the stability of the fixed points depend on the signs of u0 , v0 , C  (0), L (0), C  (u0 ), L (u0 ). Let ω(v, u) = C  (u)L (v). Certainly ω(ei ) = 0 for i = 2, 3, 4, while ω  (e2 ) = 0 and ω  (e3 ) = 0, but ω  (e4 ) = 0. We suppose that Eq. (7.11) has a solution connecting the fixed point e3 (at t = −∞) with another point in the manifold ω(v, u) = 0. We want to study persistence of this kind of solutions. This situation arises, for example, if Eq. (7.11) has heteroclinic connection between the fixed points e3 , e2 crossing the set S := {(v, u) | ω(v, u) = 0}, where we assume that 0 is a regular value of ω. In particular, if this happens, it must be:  H(e2 ) = 0

v0

zL (z)dz =



u0

zC  (z)dz = H(e3 ).

0

So let a heteroclinic orbit γ(t) intersect the set S transversally at two points (v1∗ , u1∗ ) and (v2∗ , u2∗ ) with γ(tj∗ ) = (vj∗ , uj∗ ), j = 1, 2, t1∗ < t2∗ . Results of Sect. 7.1 can be used to study the persistence of the branch {γ(t) | t ≤ t1∗ } to a solution of Eq. (7.9) tending to the fixed point e3 as t → −∞ (or +∞ possibly reversing time) and hitting S at finite time. Section 7.2 can be applied to prove a result concerning the middle part {γ(t) | t1∗ ≤ t ≤ t2∗ }. We do not go into details and refer the reader to Sect. 4 in [8].

8 Conclusions The Melnikov method and its extensions are mainly used to prove existence of chaotic orbits in dynamical systems [1, 21, 31]. But we have observed during the study of certain problems in IODEs that this method is also useful and a natural approach to such kind of results [3–8]. In particular, the persistence of orbits connecting singularities in finite time is studied provided that certain Melnikov like conditions hold. In the terminology of [27], we study persistence of global solutions terminating in finite time either to I–singularities or to IK–singularities. We survey in this paper these our recent results on IODEs with small amplitude perturbations by using the Melnikov

Melnikov Theory for IODEs

159

theory with applications to nonlinear RLC circuits. Our results are original which have not yet been investigated by other researchers. Our future plan in this direction is twofold: to provide numerical computations supporting theoretical results and to study finite time connections of higher degenerate singularities.

References 1. Awrejcewicz, J., Holicke, M.M.: Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods. World Scientific Publishing Co., Singapore (2007) 2. Battelli, F., Lazzari, C.: Exponential dichotomies, heteroclinic orbits, and Melnikov functions. J. Differ. Equ. 86, 342–366 (1990) 3. Battelli, F., Feˇckan, M.: Melnikov theory for nonlinear implicit ODEs. J. Differ. Equ. 256, 1157–1190 (2014) 4. Battelli, F., Feˇckan, M.: Melnikov theory for weakly coupled nonlinear RLC circuits. Bound. Value Probl. 2014, 101 (2014) 5. Battelli, F., Feˇckan, M.: Nonlinear RLC circuits and implicit ODEs. Diff. Integr. Equ. 27, 671–690 (2014) 6. Battelli, F., Feˇckan, M.: On the existence of solutions connecting singularities in nonlinear RLC. Nonlinear Anal. 116, 26–36 (2015) 7. Battelli, F., Feˇckan, M.: Blue sky-like catastrophe for reversible nonlinear implicit ODEs. Discrete Contin. Dyn. Syst. Ser. S 9, 895–922 (2016) 8. Battelli, F., Feˇckan, M.: On the existence of solutions connecting IK singularities and impasse points in fully nonlinear RLC circuits. Discrete Contin. Dyn. Syst. Ser. B 22, 3043–3061 (2017) 9. Boichuk, A.A., Samoilenko, A.M.: Generalized Inverse Operators and Fredholm Boundary Value Problems. VSP, Utrecht-Boston (2004) 10. Boichuk, A.A., Samoilenko, A.M.: Generalized Inverse Operators and Fredholm BoundaryValue Problems, 2nd edition, Inverse and Ill-Posed Problems Series, vol. 59. De Gruyter, Berlin (2016) 11. Feˇckan, M.: Existence results for implicit differential equations. Math. Slovaca 48, 35–42 (1998) 12. Frigon, M., Kaczynski, T.: Boundary value problems for systems of implicit differential equations. J. Math. Anal. Appl. 179, 317–326 (1993) 13. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, Boston (2007) 14. Hartman, Ph: Ordinary Differential Equations. Wiley, New York (1964) 15. Heikkilä, S., Kumpulainen, M., Seikkala, S.: Uniqueness and comparison results for implicit differential equations. Dyn. Syst. Appl. 7, 237–244 (1998) 16. Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations, Analysis and Numerical Solution. European Math. Soc, Zürich (2006) 17. Lazarides, N., Eleftheriou, M., Tsironis, G.P.: Discrete breathers in nonlinear magnetic metamaterials. Phys. Rew. Lett. 97, 157406 (2006) 18. Li, D.: Peano’s theorem for implicit differential equations. J. Math. Anal. Appl. 258, 591–616 (2001) 19. Medved’, M.: Normal forms of implicit and observed implicit differential equations. Riv. Mat. Pura ed Appl. 10, 95–107 (1992) 20. Medved’, M.: Qualitative properties of generalized vector fields. Riv. Mat. Pura ed Appl. 15, 7–31 (1994) 21. Palmer, K.J.: Exponential dichotomies and transversal homoclinic points. J. Differ Equ. 55, 225–256 (1984) 22. Rabier, P.J.: Implicit differential equations near a singular point. J. Math. Anal. Appl. 144, 425–449582 (1989)

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23. Rabier, P.J., Rheinboldt, W.C.: A general existence and uniqueness theorem for implicit differential algebraic equations. Differ. Integr. Equ. 4, 563–582 (1991) 24. Rabier, P.J., Rheinbold, W.C.: A geometric treatment of implicit differential-algebraic equations. J. Differ. Equ. 109, 110–146 (1994) 25. Rabier, P.J., Rheinbold, W.C.: On impasse points of quasilinear differential algebraic equations. J. Math. Anal. Appl. 181, 429–454 (1994) 26. Rabier, P.J., Rheinbold, W.C.: On the computation of impasse points of quasilinear differential algebraic equations. Math. Comput. 62, 133–154 (1994) 27. Riaza, R.: Differential-Algebraic Systems, Analytical Aspects and Circuit Applications. World Scien. Publ. Co., Pte. Ltd., Singapore (2008) 28. Veldes, G.P., Cuevas, J., Kevrekidis, P.G., Frantzeskakis, D.J.: Quasidiscrete microwave solitons in a split-ring-resonator-based left-handed coplanar waveguide. Phys. Rev. E 83, 046608 (2011) 29. Vanderbauwhede, A.: Heteroclinic cycles and periodic orbits in reversible systems. In: Wiener, J., Hale, J.K. (eds.) Ordinary and Delay Differential Equations, Pitman Res. Notes Math. Ser., vol. 272, pp. 250–253. Longman Sci. Tech., Harlow (1992) 30. Vanderbauwhede, A., Fiedler, B.: Homoclinic period blow-up in reversible and conservative systems. Z. Angew. Math. Phys. (ZAMP) 43, 292–318 (1992) 31. Wiggins, S.: Chaotic Transport in Dynamical Systems. Springer, New York (1992)

Numerical Solution of Space-TimeFractional Reaction-Diffusion Equations via the Caputo and Riesz Derivatives Kolade M. Owolabi and Hemen Dutta

Abstract The present chapter considers the numerical solution of space-timefractional reaction-diffusion problems used to model complex phenomena that are governed by dynamic of anomalous diffusion. The time- and space-fractional reaction-diffusion equation is modelled by replacing the first order derivative in time and the second-order derivative in space respectively with the Caputo and Riesz operators. We propose some numerical approximation schemes such as the matrix method, average central difference operator and L2 method. To give a general twodimensional representation of the analytical solution in terms of the Mittag-Leffler function, we apply the Laplace transform technique in time and the Fourier transform method in space. The effectiveness and applicability of the proposed methods are tested on a range of practical problems that are current and recurring interests in one, two and three dimensions are chosen to cover pitfalls that may arise. Keywords Caputo fractional derivative · Fractional reaction-diffusion equations · Numerical simulation · Left- and right- Riemann-Liouville fractional derivatives · Riesz fractional derivative 2010 Mathematics Subject Classification 26A33 · 65L05 · 65M06 · 93C10

K. M. Owolabi (B) Faculty of Natural and Agricultural Sciences, Institute for Groundwater Studies, University of the Free State, Bloemfontein 9300, South Africa e-mail: [email protected] K. M. Owolabi Department of Mathematical Sciences, Federal University of Technology, PMB 704, Akure, Ondo State, Nigeria H. Dutta Department of Mathematics, Gauhati University, Guwahati 781014, India e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_5

161

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K. M. Owolabi and H. Dutta

1 Introduction Fractional calculus is characterized as as extension of ordinary differentiation to arbitrarily noninteger order case. Its study has gained an appreciable significance and popularity in the last few years due to its robust applications in areas of engineering and science. Fractional calculus has been used to model physical phenomena through fractional differential equations. Over the years, the study of nonlinear problems of traveling wave solutions played an effective role in studying nonlinear real-life processes. Nowadays, many other applicable areas of fractional calculus are encountered in various application fields such as biology, chemistry, electricity, mechanics, economics, geology and medicine, notably the signal-image processing and control theory. The primary topics include the continuous time random walk (CTRW), anomalous diffusion, vibration and control, fractional Brownian motion, power law, fractional filters, biomedical engineering, Riesz potential, fractals, fractional neutron point kinetic model, nonlocal phenomena, memory-dependent scenario, computational fractional derivative equations, porous media, fractional variational principles, fractional predator-prey dynamics, fractional phase-locked loops, fractional transforms, fractional wavelet, acoustic dissipation, analysis of singularities and integral representations for fractional differential equations, non-Fourier heat conduction, geophysics, special functions, viscoelasticity, rheology, relaxation, creep, fluid dynamics, chaos and groundwater scenarios [3–7, 19, 34, 48]. An excellent literature on the application of fractional calculus can be found in [20, 27, 41, 44]. The fractional reaction-diffusion model gives a useful illustration of dynamics in complex phenomena which are characterized by non-exponential relaxation and anomalous diffusion [1, 2, 13, 18, 26]. A lot of successful mathematical equations were modelled in [20, 29, 41, 44] to describe the limiting distribution of a specified stochastic based on fractional derivatives. A typical Caputo fractional reaction-diffusion equation has the form [33] C

Dtα u(x, t) = d

∂ 2 u(x, t) + f (u(x, t)), α > 0. ∂x2

(1.1)

The symmetric Riesz fractional reaction-diffusion equation is given as (see Owolabi and Atangana [32, 38]) ∂u(x, t) ∂ β u(x, t) =d + f (u(x, t)), ∂t ∂xβ

1 < β ≤ 2.

(1.2)

Recently, authors [15, 42] developed a method based on matrix technique to solve space-time fractional diffusion equation C α 0 Dt u(x, t)

=d

∂ β u(x, t) , 1 < β ≤ 2. ∂xβ

(1.3)

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There has been a number of appreciable interest over the years in formulating numerical schemes for their solution. Some numerical schemes for solving fractional advection-dispersion problems in one dimension with the Riemann-Liouville fractional derivative has been reported in [24] on a closed domain. Based on the Riemann-Liouville derivative, Sousa and Li [45] reported a weighted second-order order scheme based on finite difference to establish an unconditionally stable method for fractional diffusion equation in one dimension. Meerschaert et al. [23, 25] formulated a two dimensional difference schemes for the solution of fractional diffusion equation. In [47], numerical solution of fractional diffusion equation via an extrapolated Crank-Nicolson scheme in one dimension was discussed. Both the Crank-Nicolson approximation and Richardson extrapolation methods were combined by Tadjeran and Meerschaert [46] to solve two-dimensional problem of fractional order. In their work, a second-order accuracy finite difference method that is unconditionally stable was obtained. Yang et. al [52] as well as Ding and Zhang [12] independently proposed some numerical schemes for fractional partial differential equation (FPDE) with the Riesz space derivatives. Authors [9, 10, 35, 36, 38, 40] proposed the Fourier spectral method to solve higher order Caputo fractional reaction-diffusion systems in space. Recently, a number of numerical methods have been developed in [8, 16, 17, 33, 51, 53, 54] for the approximation of time-fractional differential equations. However, based on the author’s knowledge, there are few studies on numerical treatment of the Caputo time and Riesz space fractional diffusion-like equations, numerical solution of nonlinear Caputo time and Riesz space fractional reaction-diffusion equations in high dimensions is still poorly understood and less investigated. In this chapter, an extension is given to the solution of nonlinear space-timefractional reaction-diffusion problems in one and high dimensions. A space-timefractional reaction-diffusion model is obtained from the classical reaction-diffusion equation when the first-order (time) derivative operator is replaced by the Caputo fractional derivative of order α ∈ (0, 1] and second-order (space) derivative operator is given by the Riesz fractional derivative of order β > 0, for 0 < β ≤ 2, we obtain C α 0 Dt u

 =d

∂β u ∂β u + β + ··· ∂xβ ∂y

 + F(u),

(1.4)

where C0 Dtα u is the Caputo derivative operator of order α in subdiffusive medium 0 < β β α < 1, and ∂∂xβu + ∂∂yβu + · · · represent the Riesz derivatives of order β ∈ (0, 2] which correspond to superdiffusive process in x, y, . . . dimensions. We recall a classical reaction-diffusion equation when α = 1 and β = 2. The remainder part of this work is organized as follows. A quick tour of some useful preliminaries and definitions of fractional calculus is considered in Sect. 2. Different numerical methods for the discretization of Caputo (in time) and Riesz space fractional reaction-diffusion equation. In Sect. 4, extension is given to the application of the proposed methods when applied to solve space-time fractional reaction-diffusion systems in one and two dimensions. Conclusion is drawn in Sect. 5.

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2 Basic Properties and Definitions of Fractional Calculus In this section, we present a brief tour of some basic notations, properties and definitions of the fractional calculus that are used to formulate equations in this work. Some of the derivatives considered are the Caputo, left- and right- sides Riemann-Liouville and Riesz operators [27, 41, 44]. In addition, we recall the Fourier transform as well as the Laplace transform for the Caputo derivative. Definition 2.1 A real-valued function u(x), for x > 0 is defined in space Cμ , μ ∈ R, if there exist a real positive number, say n(> μ) : u(x) = xn u1 (x) where u1 (x) ∈ C[0, ∞), and is defined in space Cμm if u(m) ∈ Cμ , m ∈ N . For α > 0, for a given real-valued function u(x, t) on R+ = (0, ∞), the left-sided Riemann-Liouville fractional operator of order α is given as [20] ∂n 1 = (n − α) ∂xn



x

u(ξ) d ξ, n − 1 < α ≤ n, for n ∈ N . α−n+1 a (x − ξ) (2.1) The right-sided Riemann-Liouville fractional operator of order α is defined as [20, 41] RL α a Dx u(x, t)

−∂ n 1 · (n − α) ∂xn



b

u(ξ) d ξ, n − 1 < α ≤ n, n ∈ N , (ξ − x)α−n+1 x (2.2) where (·) stands for the Gamma function, and the case n = 2 which correspond to superdiffusive process will be considered here. It should be noted that when α = n = 2 yields a classical second-order partial derivative ∂ 2 u(x, t)/∂x2 . RL α x Db u(x, t)

=

Definition 2.2 For a real valued function u(x, t) of the class S of decreasing test functions on R, the Fourier transform is defined as uˆ (ω) = F [u(x); ω],  +∞ = eiωx u(x)dx, ω ∈ R,

(2.3)

−∞

and the corresponding inverse Laplace transform u(x) = F −1 [ˆu(x); ω],  +∞ 1 = e−iωx uˆ (ω)d ω, x ∈ R. 2π −∞ Definition 2.3 The Riesz space fractional derivative is given as [21, 32]

∂α u u(x, t) ∂x2

(2.4)

of order α ∈ (1, 2]

RL α  1 ∂α RL α u(x, t) = − πα a Dx u(x, t) + x Db u(x, t) , 2 ∂x 2 cos 2

(2.5)

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where Ra L Dxα u(x, t) and Rx L Dbα u(x, t) are the left and right Riemann-Liouville derivatives as defined above. Let α > 0, u ∈ Co∞ (),  ⊂ R. The Fourier transforms for the left- and rightsided Riemann-Liouville fractional derivatives hold   L Dxα u(x) = (iω)α uˆ (ω) , F R−∞ F

RL x

α D∞ u(x) = (−iω)α uˆ (ω) ,

where uˆ (ω) denotes the Fourier transform of u.  e−ωx u(x)dx. uˆ (ω) = R

Given n > α be an integer, the Caputo (time-fractional) derivative of order α > 0 is defined as ⎧ 1 t n (t − ξ)n−α−1 ∂ u(x,t) d ξ, n − 1 < α < n ⎨ α 0 (n−α) ∂t n ∂ u(x, t) C α D = = (2.6) 0 t ⎩ ∂ n u(x,t) ∂t α , α = n ∈ N. ∂t n The fractional derivative of function u(x) in the sense of Caputo is given as α

CD u(x) = D

n−α

1 Du(s) = (n − α)



x

(x − s)n−α−1 u(n) (s)ds

(2.7)

0

n . for n − 1 < α ≤ n, n ∈ N , x > 0, u ∈ C−1 The following lemma gives some basic properties for the Caputo fractional operator.

Lemma 2.4 If n − 1 < α ≤ n, n ∈ N and u ∈ Cμm , μ ≥ −1, then, Dα (c) = 0, (c is a constant)  (γ+1) γ+α (x) , γ≥α−1, α γ D (x) = 0,(γ−α+1) γ≤α−1, Dα I α u(x) = u(x), I α Dα u(x) = u(x) −

n−1  k=0

u(k) (0)

xk , x > 0. k!

(2.8)

Definition 2.5 The Laplace transform of the Caputo derivative of fractional order 0 < α ≤ 1 is given by L

C 0

 Dtα u(t) = pα U (p) = pα−1 u(0),

(2.9)

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where u(p) is the Laplace transform of u(t). Definition 2.6 The one-parameter Mittag-Leffler function is defined by the series expansion [5, 39, 42] Eα (z) =

∞  k=0

zk , α > 0. (αk + 1)

(2.10)

Lemma 2.7 ([12, 43]) A tridiagonal matrix of order n − 1 is given in general form as ⎞ ⎛ b a 0 ··· 0 0 ⎜ c b a 0 ··· 0 ⎟ ⎟ ⎜ ⎜ 0 c b a 0 · · ·⎟ ⎟ ⎜ ⎜ .. ⎟ , (2.11) D = ⎜ ... . . . . . . . . . . ⎟ ⎟ ⎜ ⎜· · · 0 c b a 0 ⎟ ⎟ ⎜ ⎝ 0 ··· 0 c b a ⎠ 0 0 ··· 0 c b the eigenvalues and eigenvectors of matrix D are given by  λi = b + 2a and



 c  21

⎜ ⎜ ⎜ ξi = ⎜ ⎜ ⎝  n−1 c 2

 ca 22 a

a

  iπ c cos , i = 1, 2, . . . , n − 1, a n

sin

 iπ 

 n  sin 2iπ n .. .

sin

(n−1)iπ n

⎞ ⎟ ⎟ ⎟ ⎟ , i = 1, 2, . . . , n − 1, ⎟ ⎠

that is, Dξi = λi ξi , i = 1, 2, . . . , n − 1. In addition, D is diagonalizable and E = (ξ1 , ξ2 , . . . , ξ1 n − 1) diagonalizes D, that is, E−1 DE = , where  = diag(λ1 , λ2 , . . . , λ1 n − 1). For the purpose of numerical simulation in high dimensions, we need define the Kronecker matrix product. Suppose A is n × m and B be p × q matrices defined as ⎛

a11 ⎜a21 ⎜ A=⎜ . ⎝ ..

a12 a22 .. .

··· ··· .. .

⎛ ⎞ b11 a1m ⎜b21 a2m ⎟ ⎜ ⎟ .. ⎟ , B = ⎜ .. ⎝ . . ⎠

an1 an2 · · · anm

b12 b22 .. .

··· ··· .. .

⎞ b1q b2q ⎟ ⎟ .. ⎟ , . ⎠

bp1 bp2 · · · bpq

(2.12)

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the inner product of A and B is expressed as the Kronecker product ⎛ ⎞ a11 B a12 B · · · a1m B ⎜a21 B a22 B · · · a2m B⎟ ⎜ ⎟ A⊗B=⎜ . .. . . .. ⎟ . ⎝ .. . . . ⎠ an1 B an2 B · · · anm B

(2.13)

3 The Caputo and Riesz Fractional Derivatives Approximation Here, various numerical approximation for the Caputo time fractional derivative and the Riesz space operator via the left- and right-sided Riemann-Liouville derivatives are proposed.

3.1 The Matrix Method Given the general one-component time-space fractional reaction-diffusion equation C α 0 Dt u

=d

∂β u + F(u). ∂xβ

(3.1)

By adopting the matrix method as discussed in [42], we discretize (3.1) as

(α) An ⊗ Bm − d (Bn ⊗ Cm(β) ) unm = fnm ,

(3.2)

with n is the number of time steps and m denotes the number of discretization interval for spatial variable x. Next, we consider the nodes (ih, jk), for j = 1, 2, . . . , n, which correspond to the time i-th approximation node, where i = 1, 2, . . . , m. The αth order Caputo time derivative of a function u(x, t) can be approximated using 

  T α α α α α = A(α) ui,n ui,n−1 ui,n−2 · · · ui,2 ui,1 , ui,n−1 ui,n−2 · · · ui,2 ui,1 ui,n n

(3.3)

where

A(α) n

⎛ (α) a1 ⎜ 0 ⎜ 1 ⎜ ⎜ 0 = α⎜ k ⎜··· ⎜ ⎝ 0 0

a2(α) a1(α) 0 ··· ··· 0

··· a2(α) a1(α) ··· 0 ···

··· ··· a2(α) ··· 0 0

(α) an−1 ··· ··· ··· a1(α) 0

⎞ an(α) (α) ⎟ an−1 ⎟   ⎟ ··· ⎟ α j α , j = 1, 2, . . . , n. ⎟ , a = (−1) ··· ⎟ j j ⎟ a2(α) ⎠ a1(α) (3.4)

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α For the spatial discretization, the left-sided Riemann-Liouville (RL a Dx u(x, t)) oper(α) ator can be discretized using a triangular strip matrix denoted by Bm as

T   T (α) (α) um−2 · · · u1(α) u0(α) = Bm(α) um um−1 um−2 · · · u1 u0 , um(α) um−1

(3.5)

where ⎛

(α) ⎜b0

a1(α)

..

b(α) 0

b(α) 1

.

..

.

b(α) m−1

⎞ b(α) m

⎟ ⎟ ⎟ . b(α) m−1 ⎟   .. ⎟ (α) (α) . . ⎟ α (α) j α . . 0 b b Bm , j = 1, 2, . . . , n, ⎟ , bj = (−1) 0 1 j .. .. .. ⎟ ⎟ . . ⎟ ··· ··· . (α) ⎟ · · · 0 0 b(α) b 0 1 ⎠ 0 ··· 0 0 b(α) 0 (3.6)  with point x = jh, for j = 0, 1, 2, . . . , m, where h is the spatial step size and αj = ⎜ ⎜ 0 ⎜ 1 ⎜ ⎜ = α⎜ 0 h ⎜ ⎜ ⎜··· ⎜ ⎝ 0 0

..

.

..

α(α−1)···(α−j+1) . j!

In a similar fashion, we approximate the right-sided Riemann-Liouville operator (Rx L Dbα u(x, t)) with a lower triangular matrix, represented by Cm(α) as T   T (α) (α) um(α) um−1 um−2 · · · u1(α) u0(α) = Cm(α) um um−1 um−2 · · · u1 u0 ,

(3.7)

where Cm(α) is given as the transpose of Bm(α) . Conveniently, one can get the numerical approximation for the Riesz fractional derivative of order α ∈ (0, ] as the combination of (3.5) and (3.7). Another way of approximating the Riesz operator is by using the finite difference scheme as discussed in [28] T   T (α) (α) um(α) um−1 um−2 · · · u1(α) u0(α) = Em(α) um um−1 um−2 · · · u1 u0 ,

(3.8)

where ⎛

Em(α)

c0(α) ⎜ (α) ⎜ c1 ⎜ (α) ⎜ c2 1 ⎜ .. = α⎜ h ⎜ ⎜ . ⎜ (α) ⎜cm−1 ⎝ cm(α)

c1(α) c2(α) c0(α) c1(α) c1(α) c0(α) .. .. . . .. (α) . c 2

c3(α) c2(α) c1(α) .. . c1(α)

(α) . . . c2(α) cm−1

⎞ · · · cm(α) (α) ⎟ · · · cm−1 ⎟ (α) ⎟ · · · cm−2 ⎟ ⎟ . ⎟, · · · .. ⎟ ⎟ (α) (α) ⎟ c0 c1 ⎟ ⎠ c1(α) c0(α)

(3.9)

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169

and   (−1)j (α + 1) cos πα 2 , j = 0, 1, 2, . . . , m. (α/2 − j + 1)(α/2 + j + 1)

cj(α) =

To show the applicability of the matrix method, we set the reaction term as F(u) = κu(x, t)p which makes (3.1) to correspond to one-dimensional nonlinear space-time fractional reaction-diffusion version of the Kierstead, Slobodkin and Skellam (KiSS) model [39], where κ denotes the growth rate of plankton, p > 0 ∈ Z+ and d represents the turbulent diffusivity. See [30, 31] for extensive numerical and theoretical studies. We seek the solution of (3.1) on interval x ∈ [0, L], using an homogeneous Dirichlet boundary conditions taken in the form u(0, t) = u(L, t) = 0, with initial condition u(x, 0) = 0. Figures 1 and 2 show various distributions of the species for some values of α and β. α=0.50, β=1.45

α=0.50, β=1.65

α=0.89, β=1.93

100

100

100

200

200

200

300

300

300

400

400

400

500

500

500

600

600

600

10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 100

10 20 30 40 50 60 70 80 90 100

Fig. 1 Snapshots of one-dimensional space-time fractional KiSS equation (3.1) showing effects of parameters α ∈ (0, 1) and β ∈ (1, 2) for κ = 0.5, d = 0.1 and p = 2.0. Simulation final time t = 600 and x ∈ [0, L], L = 100 1

α=0.83, β=1.75 α=0.75, β=1.65 α=0.50, β=1.50 α=0.35, β=1.50

0.9 0.8 0.7

u(x,t)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.6

0.4

0.8

1

x

Fig. 2 Numerical results showing subdiffusive (0 < α < 1) and superdiffusive (1 < β < 2) distributions of nonlinear fractional KiSS model (3.1) for t = 2 and L = 1. Other parameters are given in Fig. 1

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For the second case, we consider F(u) = u − u3 in (3.1). This equation is known Allen-Cahn equation which arise from the modelling of phase transition or in the study of stationary/steady waves, for instance, the Schrödinger equation. This equation was initially introduced in the study of phase transition problems. In addition, it has been used to discuss the evolution of a diffuse phase boundary, concentrated in a small bounded region of size d . It also arise from phase transition in the study of materials science. The steady states u = ±1 are attracting, the solution also tend to show a flat surface areas around these values disjointed by interfaces as d → 0 [49]. Figure 3 shows the distribution of the fractional Allen-Cahn equation at different instances of α(0, 1) and β ∈ (1, 2) which represent subdiffusive and superdiffusive scenarios in x ∈ [−L,  L] for L = 1, with parameter d = 0.01 subject to initial value [cos(πx) − 1]. For the classical case as shown in Fig. 3d, we u(x, 0) = 21 sin 3πx 2 observed that the initial condition gives rise to an intermediate unstable state, as a result of rapid transition to solution with one interface. As the value fractional powers α and β are increasing or tend to classical cases, the unstable interface is widely pronounced as evident in Fig. 3c. (a) α=0.35, β=1.35

(b) α=0.50, β=1.45

1.5

1

1

u(x,t)

u(x,t)

0.5 0 −0.5

0.5 0 −0.5

−1 100

−1 100 1

1

0.5

50

0.5

50

0

0

−0.5

t

0

−1

−0.5

t

x

−1

x

(d) α=1.00, β=2.00

1

1

0.5

0.5

u(x,t)

u(x,t)

(c) α=0.89, β=1.90

0

0 −0.5

0 −0.5

−1 100

−1 100 1 0.5

50

0

1 0.5

50

0

−0.5

t

0

−1

−0.5

x

t

0

−1

x

Fig. 3 Metastability of the fractional Allen-Cahn equation for varying values of α and β. Parameters are d = 0.01, x = [−L, L], L = 1. Simulations runs for t = 100 and N = 200

Numerical Solution of Space-Time-Fractional Reaction-Diffusion …

171

3.2 The Caputo Derivative Approximation In this chapter, we follow Owolabi [33] and utilize the difference scheme to discretize the time-fractional derivative term  tn+1 1 ∂ α u(x, tn+1 ) ∂u(x, τ ) 1 = dτ α ∂t (1 − α) 0 ∂τ (tn+1 − τ )α n  (s+1)  1 1 ∂u(x, τ ) = dτ (3.10) (1 − α) s=0 s ∂τ (ts+1 − τ )α n  (s+1)  1 ∂u(x, τs ) 1 · dτ, ≈ (1 − α) s=0 s ∂τ (ts+1 − τ )α we simply approximate the first order time derivative in (3.10) by u(x, τs+1 ) − u(x, τs ) ∂u(x, τs ) = + O(). ∂τ τ

(3.11)

So that  n  ∂ α u(x, tn+1 ) u(x, ts+1 ) − u(x, ts ) (s+1) 1 1 ≈ dτ α ∂t α (1 − α) s=0  (t s+1 − τ ) s  n  u(x, tn+1−s ) − u(x, tn−s ) (s+1) d ξ 1 = (1 − α) s=0  ξα s

(3.12)

 −α   n+1−s −α (u( n + 1) − un ) + u − un−s [cs ] , n ≥ 1 (2 − α) (2 − α) s=1 n

=

−α (u1 − u0 ) (2 − α) n   −α  n+1 −α   n+1−s u = − un + cs u − un−s , n ≥ 1 (2 − α) (2 − α) s=1 −α (u1 − u0 ), (2 − α)

n=0

n = 0,

where cs = (s + 1)1−α − s1−α , s = 0, 1, 2, . . . , n, and u0 = u(x, 0) = ϕ(x).

3.3 The Riesz Derivative Approximation By taking the mesh points xn = a + nh, n = 0, 1, 2, . . . , N , and tm = mk, m = 0, 1, 2, . . . , M , where h = (b − a)/N and k = T /M , where h and k are the corresponding spatial step-size and temporal step-size.

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So the Riesz fractional derivative of order β ∈ (1, 2) can be approximated in the following ways. Here, we consider two different methods for the numerical approximation of the Riesz operator.

3.3.1

L2 Discretization Approach

For convenience, we first truncate the infinite interval x ∈ [−∞, ∞] with finite case x ∈ [a, b]. Recall that   ∂ β u(x, t) β RL β RL u(x, t), = −ϑ D + D β a x x b ∂|x|β where ϑβ = β

1 2 cos πβ 2

(3.13)

,

β

and Ra L Dx u(x, t) and Rx L Db u(x, t) are the left-side and right-side fractional RiemannLiouville operators of order β(1 < β < 2) which can be written respectively as RL β a Dx u(x, t)

1 

∂ n u(a, t) xn−β (n + 1 − β) ∂xn n=0  x 2 ∂ u(τ , t) 1 + (x − τ )1−β d τ , (2 − β) a ∂τ 2

=

(3.14)

and RL β x Db u(x, t)

1  (b − x)n−β ∂ n u(b, t) (n + 1 − β) ∂xn n=0  b 2 ∂ u(τ , t) 1 + (τ − x)1−β d τ . (2 − β) x ∂τ 2

=

(3.15)

Hence, the first order approximation method for both the left-side and right-side of the Riemann-Liouville operators become [11, 52] RL β a Dx u(xξ , t) =



(β 2 − 3β + 2)u(x0 , t) (2 − β)[u(x1 , t) − u(x0 , t)] + ξβ ξ β−1 ⎫ ξ−1 ⎬  (β) + dn [u(xξ−n+1 , t) − 2u(xξ−n , t) + u(xξ−n−1 , t)] + O(h), ⎭ 1 (3 − β)hβ

n=0

(3.16)

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and 

(β 2 − 3β + 2)u(xσ , t) (2 − β)[u(xσ , t) − u(xσ−1 , t)] + (σ − ξ)β (σ − ξ)β−1 ⎫ −ξ−1 ⎬  (β) + dn [u(xξ+n−1 , t) − 2u(xξ+n , t) + u(xξ+n+1 , t)] + O(h), ⎭

RL β x Db u(xξ , t) =

1 (3 − β)hβ

n=0

(3.17) where dn(β) = (n + 1)2−β − n2−β , n = 0, 1, . . . , ξ − 1, or n = 0, 1, . . . , σ − ξ − 1. By using these formulas in (3.13) we have ∂ β u(xξ , t) −ϑβ = ∂|x|β (3 − β)hβ " 2 (β − 3β + 2)u(x0 , t) (2 − β)[u(x1 , t) − u(x0 , t)] × + ξβ ξ β−1 # ξ−1  (β) + dn [u(xξ−n+1 , t) − 2u(xξ−n , t) + u(xξ−n−1 , t)]

(3.18)

n=0



(β 2 − 3β + 2)u(x , t) (2 − β)[u(x , t) − u(x−1 , t)] + ( − ξ)β ( − ξ)β−1 #$ −ξ−1  (β) + dn [u(xξ+n−1 , t) − 2u(xξ+n , t) + u(xξ+n+1 , t)] + O(h)

+

n=0 (β)

where dn

3.3.2

remains as earlier defined.

Average Central Difference Operator

Following the approximation technique reported in [14], we propose the lemma that follows. Lemma 3.1 Let β > 0, u(x, t) ∈ C0∞ (R), the left and right Riemann-Liouville operators in x−direction have the following Fourier transforms Fx



Fx

RL β −∞ Dx u(x, t)

RL x



= (iω)β uˆ (ω, t),

β D∞ u(x, t) = (−iω)β uˆ (ω, t),

(3.19)

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where uˆ (ω, t) is the Fourier transform of u(x, t) w.r.t x;  uˆ (ω, t) =

exp(−iωx)u(x, t)dx. R

The left fractional central difference operator is given by Tuan and Gorenflo [50] as CD

β D− u(x, t)

=

∞  n=0

ωn(β) u

 % "  β , t . x− n− 2

(3.20)

In similar fashion, we define the right fractional central difference approximation scheme as  % "  ∞  β CD β (β) , t . (3.21) D+ u(x, t) = ωn u x + n − 2 n=0 Next, we define the average fractional operator as β

± u(x − δ, t) =

1 1 [x ± (δ − (β/2)), t] + [x ± (δ + (β/2)), t]. 2 2

(3.22)

Based on Eqs. (3.20), (3.21) and (3.22), we obtain the respective left and right average central (AC) difference operators as β



 β D− u(x, t) ,   "    % ∞  β β − u x − j − , t , (−1)j = j 2 j=0   ∞ β 1 [u(x − j, t) + u(x − (j − β)h, t)], (3.23) = (−1)j j 2 j=0 β

AC

D− u(x, t) = −

AC

D+ u(x, t) = +

CD

and β



 β D+ u(x, t) ,   "    % ∞  β β + u x + j − , t , (−1)j = j 2 j=0   ∞ β 1 [u(x + j, t) + u(x + (j − β)h, t)]. (3.24) = (−1)j 2 j=0 j β

CD

Numerical Solution of Space-Time-Fractional Reaction-Diffusion …

175

(b)

(a)

0

−3

10

10

AC−method L2−method

−4

10

AC−method L2−method

−2

10

−5

10

−4

10

10

L∞

|E|

−6

−7

−6

10

10

−8

−8

10

−9

10

10

−10

10

−12

−10

10

−4

10

−3

10

−2

10

−1

10

10

0

10

0

50

100

150

200

N

Time step

Fig. 4 Convergence in time (a) and space (b) results for one-dimensional fractional KiSS model (3.1) at α = 0.55, β = 1.55 and t = 1. Other parameters and initial data are given in Fig. 1

Before advancing to the next section, we quickly justify the efficiency of the proposed approximation methods by considering (3.1) with F(u) = u2 and report the norm of relative errors defined by & 'n

2 j=1 (uj − Uj ) 'n 2 j=1 (Uj )

|E| =

(3.25)

and the maximum norm-error L∞ given as L∞ = max |uj − Uj |, j

(3.26)

where n is the number of interior points, uj and Uj are the computed and exact values of u at point j. We use a gold-standard run computed at t = 1/2048. Figure 4 shows the trade-off between the L2 method and average central difference method in conjunction with the Caputo time fractional finite difference scheme. Plots (a) and (b) corresponds to time and space convergence.

3.4 Two-Dimensional Solution with Laplace and Fourier Transforms Define u(x, y, t) to be a real-valued function which argument t ∈ R denotes the time variable and (x, t) ∈ [−L, L] × [−L, L], L > 0 represents the space variable. To provide the analytical solution for form (3.1), we apply the Laplace and Fourier transforms methods. For the sake of simplicity, we relax d = 1 and set F(u) = 0 to reduce the equation to time-space fractional diffusion problem

176 C α 0 Dt u(x, y, t)

K. M. Owolabi and H. Dutta

∂ β u(x, y, t) ∂ β u(x, y, t) + , 0 < α ≤ 1, 1 < β ≤ 2, (3.27) ∂xβ ∂xβ

=

with initial and boundary conditions u(x, y, 0) = u0 (x, y),

lim u(x, y, t) = 0.

x,y→±∞

(3.28)

Let us assume that the solution of (3.27) takes the series form u(x, y, t) =

∞ ∞  

χnm (t)ei(nx+my) , i =

√ −1.

(3.29)

n=−∞ m=−∞

Next, the left- and right-sided Riemann-Liouville derivatives is computed as RL β −∞ Dx u(x, y, t)

" %  x 1 u(ξ, y, t) ∂ = dξ , ∂x (2 − β) −∞ (x − ξ)β $ (  x ∞ ∞   1 einξ ∂ imy = χnm (t)e dξ , β ∂x (2 − β) n=−∞ m=−∞ −∞ (x − ξ) =

∞ ∞   χnm (t)eimy inx e (2 − β)(in)β , (2 − β) n=−∞ m=−∞ ∞ ∞  

=

χnm (t)(in)β ei(nx+my) ,

(3.30)

n=−∞ m=−∞

and for the right-side, we have d ∂ RL β (−1) x D−∞ u(x, y, t) = ∂x

d = (−1) dx = =

"

dx (

 ∞ 1 (2 − β) x ∞  1

(2 − β) ∞ 

∞ 

n=−∞ m=−∞ ∞ ∞  

% u(ξ, y, t) d ξ , (x − ξ)β  −∞ ∞  χnm (t)eimy

n=−∞ m=−∞

x

$ einξ dξ , (x − ξ)β

χnm (t)eimy inx e (2 − β)(−in)β , (2 − β) χnm (t)(−in)β ei(nx+my) .

(3.31)

n=−∞ m=−∞

This implies that ∞ ∞     ∂ β u(x, y, t) 1   = − χnm (t)ei(nx+my) (in)β + (−in)β . (3.32) β ∂x 2 cos βπ n=−∞ m=−∞ 2

Numerical Solution of Space-Time-Fractional Reaction-Diffusion …

177

Similarly, we can evaluate ∂ β u(x, y, t)/∂yβ as ∞ ∞     ∂ β u(x, y, t) 1   = − χnm (t)ei(nx+my) (im)β + (−im)β . β ∂y 2 cos βπ n=−∞ m=−∞ 2

(3.33) On using (3.29) and (2.3) in (3.27), we obtain C α 0 Dt χnm (t)

=−

1   (in)β + (−in)β + (im)β + (−im)β χnm (t). (3.34) βπ 2 cos 2

By taking the Laplace of both sides of the above equation, we get pα χnm (p) − pα−1 χnm (0) + Gχnm (p) = 0,

(3.35)

where G denotes G=

1   (in)β + (−in)β + (im)β + (−im)β χnm (t). 2 cos βπ 2

For the sake of computation, we consider the main branch of G as G = |n|β + |m|β and by applying the inverse Laplace transform of (3.35) becomes   χnm (t) = χnm (0)Eα −G t α ,

(3.36)

where Eα remains the Mittag-Leffler function. In the same manner to the solution of u(x, y, t), we can express the initial function u0 (x, y) in terms of Fourier series u0 (x, y) =

∞ ∞  

u0nm (0)ei(nx+my) ,

(3.37)

n=−∞ m=−∞

where u0nm

1 = (2π)2



L



−L

L −L

u0 (x, y)e−i(nx+my) dxdy

are the Fourier coefficients. In (3.29), if we take t = 0 and get χnm = u0nm (0), then the closed form solution is given as u(x, y, t) =

∞ ∞   n=−∞ m=−∞

  u0nm (0)Eα −G t α ei(nx+my) .

(3.38)

178

K. M. Owolabi and H. Dutta

4 Numerical Experiments The aim of this section is to further apply the algorithms proposed to solve a range of fractional time-space problems.

4.1 Example 1 Let us first consider the Arneodo’s differential system [22] α 0 Dt u(t) α 0 Dt v(t) α 0 Dt w(t)

= v(t), = w(t),

(4.1) (4.2)

= −φu(t) − ϕv(t) − ψw(t) + σu3 (t),

(4.3)

  where 0 Dtα is the Caputo fractional-order derivative C0 Dtα . The value of dimensionless parameters utilized here are φ = −4.5, ϕ = 3.5, ψ = 1, σ = −1 subject to initial conditions u(0) = −0.2, v(0) = 0.5 and w(0) = 0.2. For this particular example, we use the numerical scheme (3.12) to obtain simulation results in Figs. 5 and 6. The plots in Fig. 5 rows 1–3 correspond to an arbitrarily chosen values α = 0.81, α = 0.89 and α = 0.93 respectively at t = 400. At α = 0.81, α = 0.89 and α = 0.93 we obtain the time series simulation results in Fig. 6 at time t = 100. Again, we reformulate the time Caputo Arneodo’s fractional system (4.1) in the form ∂ β u(x, t) + v(x, t), ∂|x|β ∂ β v(x, t) α + w(x, t), (4.4) 0 Dt v(x, t) = d2 ∂|x|β ∂ β w(x, t) α − φu(x, t) − ϕv(x, t) − ψw(x, t) + σu3 (x, t), 0 Dt w(x, t) = d3 ∂|x|β α 0 Dt u(x, t)

= d1

where di , i = 1, 2, 3 are the diffusion coefficients, the fractional operators 0 Dtα , 0 < C α  ∂β α < 1 and ∂|x| and Riesz derivative defined by β are the Caputo derivative 0 Dt (2.5). In the experiment as shown in Fig. 7, we utilized zero-flux boundary condition and set d = 0.07, d2 = 0.01, d3 = 0.02, φ = −4.5, ϕ = 3.5, ψ = 1, σ = −1. The initial conditions are given above. The main interest of this work is not really in one dimensional simulations, they are easily undertaken by applying method of lines in time, in conjunction with spatial adaptive schemes, or even just simple fractional Crank-Nicholson and other family of finite difference schemes. Unfortunately, simulations in two space dimensions, based upon conventional ideas become more time consuming. However, an idea

Numerical Solution of Space-Time-Fractional Reaction-Diffusion …

1

0.6

0.8

0.5

0.6

0.4

0.4

0.3

0.5

0.2

0.2

v(t)

0

w(t)

w(t)

179

0.1

−0.5

0

−1 1

−0.4

−0.2 0.5 0

0.5

0

−0.5 −0.5

v(t)

1

1.5

2

−0.6

−0.3 −0.4 −0.2

u(t)

0 −0.2

−0.1

0

0.2 0.4 0.6 0.8

1

−0.8 −0.4

1.2 1.4 1.6

−0.2

0

2

4

1.5

3

4

1

2

0.5

1

0 −2 −4

0

w(t)

v(t)

w(t)

2

−0.5

0 −2

v(t)

0

−4 −1

1

3

2

0

−3

−2

−4 0

−5 −2.5 −2 −1.5 −1 −0.5

3

2.5

2

1.5

1

0.5

8 6

v(t)

−5

0 −1

0

v(t)

−2

−5 −4

0

−5 −4

0

−4 −6

−4

u(t)

2

1.5

−2

−3

4

2

1

2

−2

−10 5

0.5

4

2 1

0

w(t)

w(t)

5 4 3

5

0

v(t)

u(t)

10

0.6

−1

−1.5

−2.5 −0.5

u(t)

0.4

−2

−1

−6 2

0.2

v(t)

u(t)

−3

−2

−1

0

1

2

3

−8 −5

4

0

5

v(t)

u(t)

Fig. 5 Numerical results for the time Caputo Arneodo’s fractional chaotic system (4.1) for different instances of α. Rows 1–3 correspond to α = 0.81, α = 0.89 and α = 0.93 respectively at t = 400 α=0.75

1.5 1

α=0.85

4

u v w

5

u,v,w

u,v,w

u,v,w

1

0

0 −1 −2

−0.5

−3

−1

−5 20

40

60

time

80

100

0 −5 −10

−4 0

u v w

10

2

0.5

α=0.95

15

u v w

3

0

20

40

60

time

80

100

−15

0

20

40

60

80

100

time

Fig. 6 Numerical results for the time Caputo Arneodo’s fractional chaotic system (4.1) showing spatiotemporal and chaotic evolution of time versus (u(t), v(t), w(t)) for different values of α at t = 100

well-established in the Laplace and Fourier transforms community can be applied to remove the issue of stiffness associated with fractional reaction-diffusion system, and thereby permit the use of larger time-steps in conjunction with an explicit timesolver; which in-turn would enhance the Matlab routines to run very fast on standard computer machine.

180

K. M. Owolabi and H. Dutta

(a)

(b)

(c)

(d)

0.35 u v w

0.3

(u,v,w,t)

0.25

0.2

0.15

0.1

0.05

0 0

1

2

3

4

5

x

Fig. 7 Surface plot evolution of the time-space Caputo-Riesz Arneodo’s fractional system (4.4) at α = 0.90, β = 1.53 with final simulation time t = 3 and N = 200

It is in high dimensions that the methodology ideas reported in previous sections can become of serious value. We choose a couple of nontrivial examples taken from the classical reaction-diffusion literature, which we formulate here in time-space noninteger order reaction-diffusion scenarios to illustrate the numerical algorithms.

4.2 Example 2 For the second example, let us consider the two-dimensional time-space-fractional nonlinear Schrödinger equation which combines the Caputo time-fractional derivative of order α ∈ (0 < α ≤ 1) with the Riesz space-fractional derivative of order β ∈ (1 < β ≤ 2), given by [21, 32]  ∂β ∂β u(x, y, t) +d + ∂|x|β ∂|y|β + ρf (u(x, y, t)) = 0, (x, y) ∈ , t ∈ (0, T ],

iC0 Dtα u(x, y, t)



with the respective initial and boundary conditions

(4.5)

Numerical Solution of Space-Time-Fractional Reaction-Diffusion …

181

u(x, y, 0) = u0 (x, y), (x, y) ∈  × ∂, u(x, y, t) = 0, (x, y) ∈ , t ∈ [0, T ], where  ⊂ R2 is the domain of computation, ∂ denotes the boundary, parameters √ ∂β ∂β ρ and d are real constants. It should be noted that, i = −1, ∂|x| β , ∂|y|β are the Riesz space fractional derivatives in x and y directions, and the initial function u0 (x, y) is known to be sufficiently smooth function. In the simulation framework, we set  = [0, 5] × [0, 5], d = 0.1, ρ = 2, T = 0.6, and u(x, y, t = 0) = sech(x)sech(y) exp(i(x + 1)) to obtain the two-dimensional results for different instances of fractional powers α and β as shown in Fig. 8. For three-dimensional case as displayed in Fig. 9 for different instances of α and β, we consider 

 ∂β ∂β ∂β +d + + u(·, t) ∂|x|β ∂|y|β ∂|z|β + ρf (u(·, t)) = 0, (x, y, z) ∈ , t ∈ (0, T ],

iC0 Dtα u(·, t)

(4.6)

and set f (u) = ρ|u|2 on a computational domain  = [−L, L]3 , L = 40. Here the computational size L is chosen bigger enough for the solution waves to have enough room to propagate. In the experiment, we utilized the initial condition % γ 2  γ 2 γ 2  . + y− + z− u(x, y, z, 0) = 0.5 exp 0.5 x − 2 2 2 "



4.3 Example 3 For the third and last example, let us consider the nonlinear time-space fractional reaction-diffusion system [37]:  ∂β ∂β u(·, t) + f (u, v), = d1 + ∂|x|β ∂|y|β   β ∂ ∂β C α v(·, t) + g(u, v), + 0 Dt v(·, t) = d2 ∂|x|β ∂|y|β C α 0 Dt u(·, t)



(4.7)

where u(·, t) corresponds to u(x, y, t) ∈ T =  × [0, T ] and u(x, y, z, t) ∈ T =  × [0, T ] in two and three spatial dimensions, with nonlinear kinetics f (u, v) = u(1 − u) −

uv , u+ϕ

g(u, v) =

φuv − ψv. u+ϕ

(4.8)

182

K. M. Owolabi and H. Dutta α=0.55, β=1.25 0.5

0.5 1

0

1.5

y

2 −0.5

2.5 3

−1 3.5 4

−1.5

4.5 5

1

3

2

4

5

x α=0.57, β=1.45 0.6

0.5

0.5

1

0.4 1.5 0.3

y

2

0.2

2.5

0.1

3

0

3.5

−0.1

4

−0.2

4.5

−0.3

5

−0.4 1

2

3

4

5

x α=0.67, β=1.89 1 0.5 1 1.5 0.5

y

2 2.5 3

0

3.5 4 4.5 5

−0.5 1

3

2

4

5

x α=0.81, β=1.83 0.6 0.5

0.5

1

0.4

y

1.5

0.3

2

0.2

2.5

0.1 0

3

−0.1

3.5

−0.2

4

−0.3

4.5 5

−0.4 1

2

3

4

5

x

Fig. 8 Two-dimensional evolution of fractional nonlinear Schrödinger equation (4.5) for different values of α and β at d = 0.1, ρ = 2, T = 0.6, and N = 100

Numerical Solution of Space-Time-Fractional Reaction-Diffusion …

183

Fig. 9 Three-dimensional evolution of fractional nonlinear Schrödinger equation (4.6) for different values of α and β with d = 0.1, ρ = 2 at T = 10

In 2D, we simulate with initial conditions [36]:    u(x, y, 0) = 1 − exp −20 (x − 0.5)2 + (y − 0.5)2 ,    v(x, y, 0) = exp −20 (x − 0.5)2 + (y − 0.5)2 ,

(4.9)

and homogeneous Dirichlet boundary conditions on ∂ for  ∈ [−L, L], L = 10, with ϕ = 1.025, φ = 0.2, ψ = 0.5, d1 = 1 and d2 = 1.5e − 04. Two-dimensional distribution of complex (spiral) spatiotemporal patterns are displayed in Fig. 10. A close observation shows that the distribution of species u(x, y, t) and v(x, y, t) in 2D are similar. Hence, we continue our analysis with distribution of v−species as reported in Fig. 11, where the first- and second-columns correspond to (α = 0.99, β = 1, 67) and (α = 0.94, β = 1, 45). The last column is obtain for α = 1.00, β = 2.00. At this point, it should be mentioned that pattern generation in subdiffusive (0 < α < 1) and superdiffusive (1 < β < 2) regimes are almost the same as in the classical reaction-diffusion case (α = 1, β = 2). In three dimensions, we experiment (4.7) with two boundary conditions. In Fig. 12, the uppper and lower-rows correspond to homogeneous Neumann and Dirichlet boundary conditions mounted at the close ends of domain  ∈ [0, 5]3 and  ∈ [−10, 10] respectively, subject to initial functions    u(x, y, 0) = 1 − exp −20 (x − 0.5)2 + (y − 0.5)2 + (z − 0.5)2 ,    v(x, y, 0) = 0.25 exp −20 (x − 0.5)2 + (y − 0.5)2 + (z − 0.5)2 , with ϕ = 1.0, φ = 0.2, ψ = 0.5, d1 = 1 and d2 = 1.5e − 04.

(4.10)

184

K. M. Owolabi and H. Dutta v (α=0.95, β=1.5)

u (α=0.95, β=1.50) −10

−10 0.8

−8

−8 0.7

−4

0.6

−4

−2

0.5

−2

0

0.6

−6

y

y

−6

0.4

0.5

0.4

0 2

2

0.3

0.3

4

4 0.2

6

0.1

8 0

−5

−10

0.2

6 8

0.1

x u (α=0.97, β=1.63)

−10

0

−5

−10

5

x v (α=0.97, β=1.63)

5

−10

0.8

−8

−8 0.7 0.6

−4

−2

0.5

−2

0

0.4

0

2

y

y

−4

0.5

0.4

2

0.3

0.3

4

4 0.2

6

−5

−10

0

x u (α=1, β=1.53)

0.2

6

0.1

8

8

5

0.1

−5

−10

−10

5

0

x v (α=1, β=1.53)

−10

0.8

−8

0.7

−8 0.7

−6

0.6

−4

−2

0.5

−2

0

0.6

−6

−4

y

y

0.6

−6

−6

0.4

2

0.5

0.4

0 2

0.3

0.3

4

4 0.2

6

−10

−5

0

0.2

6

0.1

8

8

0.1

−10

5

−5

0

x

5

x

Fig. 10 Numerical solution to (4.7) in 2D for different values of α and β with ϕ = 1.025, φ = 0.2, ψ = 0.5, d1 = 1, d2 = 1.5e − 04 and t = 1000 α=0.99, β=1.67 0.7

−6

0.6 0.5 0.4

2

0.3

4

0.5

−4

2

0.3

4

0.2

6

8

0.1

8

0.5

−2 0.4

0

6

0.6

−6

−2

y

0

0.6

−8

−6 −4

−2

α=1.0, β=2.0

−10

−8

−4

y

α=0.94, β=1.65

−10

−8

0.2

y

−10

0.4

0 2

0.3

4 6

0.2

8 0.1

0.1 −10

−5

0

x

5

−10

−5

0

x

5

−10

−5

0

x

5

Fig. 11 Distribution of v for some values of α and β. Other parameters are given in Fig. 10

Numerical Solution of Space-Time-Fractional Reaction-Diffusion …

185

Fig. 12 Three-dimensional results for system (4.7) for different values of α and β with ϕ = 1.0, φ = 0.2, ψ = 0.5, d1 = 1, d2 = 1.5e − 04 and t = 10. For upper-row (α = 0.99, β = 1.15) and lower-rows at(α = 0.91, β = 1.53)

5 Conclusion This chapter deals with the solutions of time and space fractional reaction-diffusion problems. In formulation of the model, we replace the the first- and second-order partial derivatives in time and space with the Caputo and Riesz fractional derivatives. We first assume the Fourier expansion series to be the solution of the equation, and later apply the Laplace and Fourier transforms with respect to time and space to give a clear idea of the analytical solution. A number of methods of approximation techniques is proposed. In the numerical experiment, we give an extension to the solution of time and space fractional diffusion equation by adding the nonlinear source time. Simulations are performed in one, two and three dimensions to examine the dynamic richness of pattern formation in Caputo time and Riesz space fractional reaction-diffusion equations. Numerical results for some values of parameters α and β which correspond to subdiffusive and superdiffusive processes were displayed in the figures. In addition, we observed from the simulation results that pattern formation in standard reaction-diffusion equations are similar to their fractional or noninteger order counterparts.

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An Extended Langhaar’s Solution for Two-Dimensional Entry Microchannel Flows with High-Order Slip R. Rasooli and B. Çetin

Abstract The tremendous advances in micro-fabrication technology have brought numerous applications to the field of micro-scale science and engineering in recent decades. Microchannels are inseparable part of microfluidic technology which necessitate knowledge of flow behavior inside microchannels. For gaseous flows, the mean free path of a gas is comparable with characteristic length of a microchannel due to the micro-scale dimension of the channel. So, no-slip velocity assumption on the boundaries of channel is no longer valid, and a slip velocity needs to be defined. Although rigorous modeling of rarefied flows requires molecular solutions, researchers proposed use of slip models for applicability of the continuum equations. In slip-flow regime (i.e. Knudsen numbers up to 0.1), well-known Maxwell’s firstorder slip model is applicable. For higher Knudsen numbers, higher-order slip models can be implemented to extend the applicability limit of the continuum equations. In the present study, Langhaar’s assumptions for entrance region of two-dimensional microchannels (microtube, slit-channel and concentric annular microchannel) have been implemented using high-order slip models. Different slip models proposed in the literature have been used and velocity profile, entrance length and apparent friction factor have been obtained in integral forms. Keywords Microchannel flow · High-order slip · Langhaar’s solution

1 Introduction In recent decades, progresses in micro-fabrication technology have open up countless opportunities for many applications as an extension of theoretical micro-scale science by enabling fabrication of micro-scale devices using photolithographic [1], high precision mechanical machining [1, 2] and laser machining methods [3, 4]. Through R. Rasooli · B. Çetin (B) Microfluidics and Lab-on-a-chip Research Group, Mechanical Engineering Department, ˙I.D. Bilkent University, 06800 Ankara, Turkey e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_6

189

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these progresses, even fabrication of structures composed of tubes with diameters of nanometers have become possible [5]. Microchannels and microtubes are inseparable part of microfluidic devices, and play a crucial role which necessitates fundamental knowledge of flow behavior at micro-scale for effective and optimal design of these devices. Many theoretical investigations have been made on the physics of fluid flow in channels and ducts at micro-scale [6–8]. One important effect at micro-scale is rarefaction. As characteristic length (L) of flow approaches to the mean-free-path (λ) of the fluid, continuum approach fails to be valid and fluid flow modeling moves from continuum to molecular modeling. Rarefaction is characterized by Knudsen number (Kn = λ/L). Continuum hypothesis applies for Kn  10−3 . For 10−1  Kn  10, the regime is known as transition regime in which continuum equations fail to model the fluid flow. In this regime, molecular models such as DSMC and MD, or solutions of Boltzmann Transport Equation is required. The regime where 10−3  Kn  10−1 is slip-flow regime in which continuum equations need to be modified through velocity-slip and temperature-jump boundary conditions to take into account molecular interactions of fluid particles with the solid bodies within the Knudsen layer [6]. Following a non-dimensionalization with a reference length and velocity scale, the general form of a velocity-slip boundary condition for an isothermal flow can be written as [8–11]: Us − Uwall = A1 Kn(∂n U )wall + A2 Kn 2 (∂n2 U )wall ,

(1)

where (∂n ) is gradient in the normal direction of the solid boundary, A1 and A2 are the slip-coefficients. Many researchers proposed to employ different coefficients based on kinetic theory of rarefied gases [9, p. 74]. For a fully-diffuse reflection, proposed coefficients in the literature are tabulated in Table 1. The first term is called as Maxwell’s first-order slip, and inclusion of the second term turns the boundary condition into a second-order one. Approximation of Boltzmann Transport equation up to O(Kn) (i.e. first order in Kn) results in compressible form of Navier-Stokes equations which require only the first-order boundary conditions. Approximations with higher order terms result in higher order equations such as Burnett and Woods equations. However, researchers showed that application of the second-order boundary condition with Navier-Stokes equations [7, 12–15] or first-order Maxwell boundary

Table 1 Coefficients for different slip models Model A1 First-order Model Schamberg Model Cercignani Model Deissler Model Hsia Model Mitsuya Model

1.0 1.0 1.1466 1.0 1.0 1.0

A2 0 –5π/12 –0.9756 –9/8 –1/2 –2/9

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condition with second-order quasi-hydyrodynamic equations [7] may extend applicability of the slip-flow regime up to Kn ≈ 0.25. The second-order term in the boundary condition may introduce some numerical difficulties associated with accurate calculation of the high-order derivatives, especially for complex geometries [9]. Therefore, some studies introduced higher-order accurate boundary conditions which include only first-order derivatives [16, 17]. Considering diffuse reflections of gas molecules, Beskok and Karniadakis [16] proposed a general velocity-slip boundary condition (will be referred as Beskok Model hereafter) as: Us − Uwall =

Kn (∂n U )wall 1 − bKn

(2)

where b is a general slip coefficient. Strictly speaking, parameter b is a function of Kn. For a general choice, Eq. (2) is first-order in Kn. However, for a specific choice of the parameter b, the boundary condition can be transformed into a second-order one. For slip-flow regime, Beskok and Karniadakis [16] derived a condition to ensure second-order accuracy of (2) as:  b=

1 ∂n2 Uo 2 ∂n U o

 wall

,

(3)

where Uo is the velocity-profile corresponding to a no-slip case (i.e. Kn = 0). For pressure-driven flows in microchannels, on top of rarefaction effect, lowMach-number compressibility effect comes into picture due to high viscous resistance leading to a nonlinear streamwise pressure variation [18–20]. That is to say, the fluid flow requires compressible modeling to be able to observe this non-linear streamwise pressure. However, considering short microchannels and/or microchannel flow with low inlet/exit pressure ratio, compressibility effect may be negligible, and the fluid flow can be modeled as incompressible. Moreover, length-over-diameter ratio is typically large for microchannels, which implies entrance effects are negligible. However, the entrance effects may become quite significant for microchannels and nanotubes with small length-over-diameter ratio and micropores as a part of larger scale porous structures. Study of the entrance effects requires solution of a hydrodynamic entrance problem. The hydrodynamic entrance problem for an incompressible fluid flow was well studied for macroscale several decades ago [21–24]. Same problem has been recently re-visited for microscale [8, 13, 25, 26] for different geometries. Although the hydrodynamic entrance problem is not a true boundary layer problem, boundary layer idealizations, which neglect axial diffusion of momentum and radial pressure gradient, are reasonable approximations for laminar flow problems in ducts [21, 22, 26]. Strictly speaking, this type of idealization leads to a solutions independent of Reynolds number (Re) and suitable for high Re flows. To see the whole picture, full Navier-Stokes equations with presence of axial momentum diffusion and radial pressure gradient need to be solved for hydrodynamic entrance (i.e. entry flow) problems. Solution of full Navier-Stokes equations in the entrance region leads to a

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peculiar behavior. Velocity overshoots with two symmetric velocity maxima off the centerline was observed both in numerical [25, 27] and experimental [28] studies. Main reason of this velocity overshoot is result of the sudden velocity change at the inlet of a channel due to the no-slip boundary condition.The velocity overshoot is significant especially for strictly uniform velocity inlet which is actually an idealization for practical applications. It was observed that the velocity overshoot is much weaker for irrotational inlet velocity profiles, or for inlet velocity profiles which are approximated to a uniform one. The overshoot is only significant at the vicinity of the channel entrance. The zone where this overshoot can be observed is a function of Re. Therefore, if measurement data is not gathered close to the inlet of a channel, the velocity overshoot cannot be detected experimentally [29]. One would expect a weaker velocity overshoot for the flows in a microchannel, since sudden change of the velocity at the inlet due to no-slip boundary condition also reduces with the presence of slip-velocity at the channel wall [25]. At macroscale, following boundary layer idealizations, analytical solutions were derived employing different methods such as matching method (based on perturbation analysis), integral method, linearization methods [30, pp. 68–73]. Matching and integral method results in discontinuous solutions for the velocity gradients and pressure distributions. Alternatively, linearization methods yield continuous solutions, but transverse velocity components may not be predicted rigorously. Three different linearization procedures are possible: (i) Langhaar’s linearization [21], (ii) Targ’s linearization [30] and (iii) Sparrow’s linearization [22]. The latter two require solution of an eigenvalue problem which may have some convergence issues especially for regions close to the inlet. Langhaar’s linearization, on the other hand, offers an integral type solution with a satisfactory solution at the centerline, in the vicinity of the inlet and far downstream. At macroscale, Langhaar’s linerization was implemented for the different geometries such as tube [21], annular tube [31], parallel plate and rectangular ducts [30, 32]. Many researchers devoted great effort on experimental investigations and proposed different techniques and experimental protocols [18, 19, 33, 34], but still measurement of different flow parameters to understand fluid physics is a challenging task. At this point, analytical and numerical models serve as a basis for fundamental understanding of the phenomena despite the fact that they require some empirical input parameters in the form of slip-coefficients. In addition, analytical and numerical models serve as a fast way to investigate different scenarios. Present note reports an analytical solution of a 2D incompressible, isothermal flow in a developing region of a microchannel considered both in cylindrical and cartesian coordinates (see Fig. 1) based on Langhaar’s linearization. For rarefaction effect, different second-order models are included in the analysis. Moreover, the general velocity-slip boundary condition proposed by Beskok and Karniadakis [16] for which the general slip-coefficient is evaluated with high accuracy based on a no-slip case, is implemented. Although second-order methods extend the applicability of the continuum equations, theoretical background of the models is still a debate [14, 15], and an active research field. The current model is presented with second-order models available in the literature. In the light of new studies, the coefficients of the slip models may be revised

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a

r(y)

x (a) Microchannel in cylindrical coordinates (microtube with a radial coordinate r) and cartesian coordinates (slit-channel with a transverse coordinate y)

r R1

R2

x (b) Concentric annular microchannel

Fig. 1 Schematics of the 2-dimensional microchannel problems

[15, 35] and so does the current analytical model. By no means, the authors would like to question the validity of the higher order models considered, rather the extension of Langhaar’s solution including higher order terms is presented in this note to obtain integral form of the developing velocity profiles inside microchannels. Furthermore, hydrodynamic entrance length and apparent friction factor which are quantities of interest for many engineering calculations are also presented based on the results derived with lineraization. Analytical nature of the model enables a fast evaluation of the velocity field without any need for calculation of eigenvalues. Furthermore, it also enables implementation of Beskok general slip-model with high accuracy. The proposed model may be extended for the exploration of heat transfer problems at the incompressible limit (which has been studied in the literature [36–39] for microchannels in the combined entrance region without any need for numerical calculation of the flow field.

2 Mathematical Modeling 2.1 Velocity Profile Assuming an incompressible fluid with constant properties, following Prandtl boundary layer idealizations, the equation of motion in the axial direction (x being the axial direction) together with continuity equation can be written as: ∇ · u = 0,

(4)

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u · ∇ux =

1 ∂x p + ν∇ 2 ux , ρ

(5)

where u is the velocity field, p is the pressure field, ux is the velocity component in the axial direction, ρ is the density and ν is the dynamic viscosity. Moreover, following boundary layer idealization, (i) the diffusive term in the axial direction can be neglected compared to the components in the transverse direction, (ii) the transverse velocities within the channel can be assumed to be small compared to the axial velocity, and (iii) the pressure gradient in the axial direction can be taken as a function of axial coordinate only [21, 32].

2.1.1

Microtube and Slit-Channel

With the following dimensionless parameters λ=

x ul r p ux ,η= , q= , σ= , p˜ = , Um Um a ReDh · Dh 1/2ρUm2

(6)

where Um is the mean velocity (which is also the uniform inlet velocity), ul is the velocity in the transverse direction, a is the scale for the channel height (a = R for a microtube and half of the channel height for a slit-channel) and ReDh is the Re based on hydraulic diameter, Eqs. (4) and (5) can be non-dimensionalized as: 2

4k−2 λ

  ReDh ∂ ηqk + k = 0, ∂σ q ∂q

k−2 ∂λ

  1 ∂ ∂ p˜ ∂λ ∂λ ∂λ + 2k−2 ReDh η = −22k−5 + k qk , ∂σ ∂q ∂σ q ∂q ∂q

(7)

(8)

where k = 0 corresponds to flow in a slit-channel (i.e. 2D problem in cartesian coordinates) and k = 1 corresponds to the flow in a microtube (i.e. 2D problem in cylindrical coordinates). Parameter k is defined to derive the velocity profile on a common ground for both coordinate systems. Following Langhaar’s linearization [21, 32], the convective terms on the right hand side can be replaced by (γ 2 λ), and Eq. (8) can be written as:   ∂ p˜ 1 ∂ k ∂λ q − γ 2 λ = 22k−5 . qk ∂q ∂q ∂σ

(9)

The solution can be expressed as a combination of modified Bessel’s and hyperbolic functions as: λ(γ, q) = A(γ)G1 (γq) + B(γ)G2 (γq) + C(γ)

(10)

An Extended Langhaar’s Solution for Two-Dimensional Entry …

where

G1 (γq) = k I0 (γq) + (1 − k) cosh(γq) G2 (γq) = k I1 (γq) + (1 − k) sinh(γq) 22k−5 ∂ p˜ C(γ) = − 2 γ ∂σ

195

(11)

Coefficients A(γ), B(γ) and C(γ) can be obtained using the boundary condition at the channel wall: λ(s) (γ) = −22−k A1 Kn

∂ 2 λ  ∂λ  + 42−k A2 Kn 2 2   ∂q q=1 ∂q q=1

λ(g) (γ) = −22−k

Kn ∂λ   1 − bKn ∂q q=1

(12)

(13)

where the superscript (s) stands for the second-order slip in generic form, and (g) stands for the general-slip model (i.e. Beskok Model). The symmetry boundary condition at the centerline together with the continuity equation reads as: 

1

qk λdq = 2−k .

(14)

0

Note that, b is a function of (σ) in the entrance region, and reaches a value of −2.0 at the region far from the entrance (i.e. fully-developed region). The velocity profile can be obtained as: λ(γ, q) = where

α1 γ n G1 (γ) + α2 G2 (γ) − γα1n G1 (γq) α1 γ n G1 (γ) + (α2 − 2k α1n )G2 (γ)

α1 = (1 − n)γ (1 − 42−k A2 γ 2 Kn 2 ) + n(1 − bKn), α2 = (1 − n)γ 2 (22−k A1 Kn + k4k A2 Kn 2 ) + 22−k nγ 2 Kn.

(15)

(16)

n = 0 corresponds to a second-order slip, and n = 1 corresponds to a general-slip model. The coefficient b can be derived using Eq. (3) as: b = −2γ

G1 (γ) + 2k G2 (γ)

(17)

To complete the solution, the relation between the parameters γ and σ needs to be determined. For this purpose, Eq. (8) can be integrated over the cross-section as:

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1

0

    1 ˜ ∂λ k k−2 ∂λ k−2 2k−5 ∂ p 4 λ 2 + 2 ReDh η q dq = − qk dq ∂σ ∂q ∂σ 0    1 1 ∂ k ∂λ q qk dq + k ∂q 0 q ∂q

(18)

The first integral can be simplified using continuity equation, Eq. (7), as: 

1

 4

k−2

0

  1 ∂λ ∂λ k k−2 k−2 d q dq = 4 λ λ2 qk dq + 2 ReDh η ∂σ ∂q dσ 0

(19)

The second integral can be simplified by writing equation of motion for the central core of the channel:   1 ∂ 2 λ  ∂ p˜ 22k−5 d λ20 22k−5 qk dq = − 2 , (20) − ∂σ k + 1 dσ ∂q q=0 0 where the subscript (“0 ”) refers to the values at the centerline. Introducing two new functions F and G as:  1 λ20 , λ2 qk dq − F(γ) = 2 k +1 0 (21)    ∂λ  ∂ 2 λ  − G(γ) = 25−2k   ∂q q=1 ∂q2 q=0 Equation (18) can be written as: F  (γ) dσ = . dγ G(γ)

(22)

Functions F(γ) and G(γ) are written in closed form as:   S2 1 2 2n 2 1−k n 2 − 2Sα1 G2 + γ α1 G1 − G2 + G1 G2 k +1 2 γ   1 S − γα1n 2 − k +1 H

2 F(γ) = 2 H



G(γ) = 25−2k where

γ 2 α1n (γ − G2 ) H

S(γ) = γ n α1 G1 + α2 G2 H(γ) = S − 2k α1n G2

(23)

(24)

(25)

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197

By integrating Eq. (22), the relation between σ and γ can be defined as:  σ=−

∞ γ

F  (γ) ˜ d γ. ˜ G(γ) ˜

(26)

Once the γ value is selected, Eqs. (15) and (26) defines the velocity profile within the microchannel. Using Eqs. (7) and (22), the lateral velocity can be obtained as: η=−

1 G(γ) ∂ Q(γ, q) 4ReDh F  (γ) ∂γ

(27)

where the function Q(γ, q) is expressed as: Q(γ, q) =

2.1.2

qS(γ) − 2k α1n G2 (γq) H(γ)

(28)

Concentric Annular Microchannel

With the following dimensionless parameters λ=

ux ul r x p ,η= , q= ,σ= , p˜ = , Um Um R2 Re · R2 1/2ρUm2

(29)

where Um is the mean velocity (which is also the uniform inlet velocity), ul is the velocity in the transverse direction, R1 is the inner radius, R2 is the outer radius and Re is the Reynolds number based on the outer radius of channel, Eqs. (4) and (5) can be non-dimensionalized for a concentric annular microchannel as:

λ

∂λ Re ∂(ηq) + =0 ∂σ q ∂q

(30)

  ∂λ ∂λ 1 ∂ p˜ 1 ∂ ∂λ + Re η =− + q ∂σ ∂q 2 ∂σ q ∂q ∂q

(31)

Similarly, by employing Langhaar’s linearization method [21, 32], the left hand side term (convective term) can be replaced by (γ 2 λ), and Eq. (31) reads as:   ∂λ 1 ∂ p˜ 1 ∂ q − γ2λ = . q ∂q ∂q 2 ∂σ

(32)

The solution is obtained as a combination of modified Bessel’s functions as: λ(γ, q) = A(γ)I0 (γq) + B(γ)K0 (γq) + C(γ)

(33)

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Coefficients A(γ), B(γ) and C(γ) can be found using the boundary condition at the channel wall,     ∂λ ∂λ λ(f ) (q = 1) = −2Kn , λ(f ) (q = m) = 2Kn (34) ∂q q=1 ∂q q=m λ(g) (q = 1) = −2

Kn 1 − b1 Kn



∂λ ∂q



, λ(g) (q = m)

=2

q=1

Kn 1 − b2 Kn



∂λ ∂q

 q=m

(35) and the continuity equation reads as: 

1

qλdq =

m

1 (1 − m2 ). 2

(36)

where m is the radii ratio of the channel (m = R1 /R2 ). Note that only the first-order and Beskok model is considered for the concentric tube without loss of generality of the modeling. Velocity profile can be obtained using Eqs. (33)–(36). To discuss the second-order effectsClosed form solution of the velocity profile can be found in the Appendix. The coefficient b can be written both for the inner and outer walls as:  b1 = 2



ux,0



 , b2



ux,0

= −2

q=m



ux,0

 (37)



ux,0

q=1

and can be written in closed form as: 2 2γ 2 m [K0 (γ)I0 (γm) − I0 (γ)K0 (γm)] − γm [K0 (γ)I1 (γm) + I0 (γ)K1 (γm)] − 1 m

2 γ [K0 (γ)I0 (γm) − I0 (γ)K0 (γm)] b2 = 2 +1 γI1 (γ)K0 (γm) + γK1 (γ)I0 (γm) − 1 b1 =

(38)

Furthermore, the asymptotic values of b1 and b2 in far distance downstream (i.e. fully-developed region) can be obtained as: (fd )

m2 (1 + 2 log m) − 1

m 1 + m2 (2 log m − 1)

b1

=2

(fd ) b2

m2 + 2 log m − 1 =2 2 m − (1 + 2 log m)

(39)

To obtain the relation between parameters σ and γ, the axial momentum equation, Eq. (31), should be integrated over the cross-section of the channel as: 

1

m

      1  1 ∂λ 1 ∂ p˜ ∂λ ∂λ 1 ∂ λ − + Re η qdq = qdq + q qdq, (40) ∂σ ∂q 2 ∂σ ∂q m m q ∂q

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199

The first integral can be simplified using the continuity equation as: 

1 m

   1 ∂λ ∂λ d λ + Re η qdq = λ2 qdq, ∂σ ∂q dσ m

(41)

Assuming the central core of the velocity is not affected by viscous effects, the first integral at the right hand side can be simplified as: 

1



m

    1 1 ∂ p˜ 1 d 1 − m2 ∂ 2 λ 2 ¯ − qdq = , λ qdq − 2 ∂σ 2 dσ m 2 ∂q2 q= m+1

(42)

2

where the bar values refer to the values at the mean radius of the pipe. The last integral can also be readily evaluated as: 

1 m

      ∂λ ∂λ 1 ∂ ∂λ q qdq = −m , q ∂q ∂q ∂q q=1 ∂q q=m

(43)

Substituting all these equations in the Eq. (40) and re-arranging results in: d dσ



1



λ¯ 2 λ − 2



m

 qdq =

2

∂λ ∂q



 −m q=1

∂λ ∂q

 − q=m

1 − m2 2



∂2λ ∂q2

 q= m+1 2

(44)

Modifying functions F(γ) and G(γ) for a concentric channels as: 

 λ¯ 2 F(γ) = λ − qdq 2 m       ∂λ 1 − m2 ∂ 2 λ ∂λ −m − G(γ) = ∂q q=1 ∂q q=m 2 ∂q2 q= m+1 

1

2

(45)

2

Equation (44) can be written in a similar form as: dσ F  (γ) = . dγ G(γ)

(46)

Integratio of Eq. (46), the relation between σ and γ can be defined:  σ=−

∞ γ

F  (γ) ˜ d γ. ˜ G(γ) ˜

(47)

By assigning a value to parameter γ and using Eq. (46), the value of dimensionless axial coordinate (σ) can be obtained, and finally the velocity profile is known using closed form solution available in the Appendix.

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2.2 Pressure Drop and Friction Factor 2.2.1

Microtube and Slit-Channel

To evaluate the pressure drop along the channel, Eq. (8) needs to be integrated over the cross-sectional area as:  1 d ∂λ  22k−5 ∂ p˜ = 4k−2 λ2 qk dq − (48) −  , k + 1 ∂σ dσ 0 ∂q q=1 and then, integrated in the σ-direction: 

1

˜p = 2(k + 1)

 qk λ2 dq − 2 + 25−2k (k + 1)

0

∞ γ

∂λ  F  (γ) ˜ d γ˜  ∂q q=1 G(γ) ˜

(49)

One important parameter for the engineering calculations is the Fanning friction factor and the apparent friction factor. The fanning friction factor can be defined as:  f · ReDh = −2

3−k



∂λ ∂q

,

(50)

q=1

using the derived velocity profiles, the Fanning friction factor can be written as: −γ 2 α1n G2 (γ) H(γ)

f · ReDh =

(51)

The apparent friction factor which considers frictional loss together with the effect of acceleration in the developing region can be derived as: fapp · ReDh =

2.2.2

˜p 4σ

(52)

Concentric Annular Microchannel

Equation (31) can be integrated over the cross-sectional area of the channel as: 1 ∂ p˜ d (1 − m2 ) = − 4 ∂σ dσ



1



∂λ λ qdq + m ∂q



m

 −

2

q=m

∂λ ∂q

 (53) q=1

Integrating the above equation along the axial direction of channel results in: ˜p =

     1  ∞   F (γ) ˜ 4 4 ∂λ ∂λ 2 dq − 2 − d γ˜ qλ − m ∂q q=m ∂q q=1 G(γ) ˜ 1 − m2 m 1 − m2 γ

(54)

An Extended Langhaar’s Solution for Two-Dimensional Entry …

201

The Fanning friction factor based on the inner and outer radii can be derived as: 2 f · Re = − 1−m



1

2 f 2 · Re = − 1−m



∂λ ∂q ∂λ ∂q

 

q=m

(55)

q=1

Furthermore, the apparent friction factor can be calculated as: fapp · ReDh =

˜p 4σ

(56)

where ˜p is given in Eq. (49).

3 Results and Discussion The mathematical modeling is coded by the help of the Mathematica® . One critical point in the evaluation is the improper integrals given in Eqs. (26), (47), (49) and (54). The evaluation of these integrals may be problematic as (σ → ∞). Although these integrals need be integrated to infinity, it is known that the flow is fully-developed after a certain σ value. Once the flow is fully-developed, the integrals can be evaluated without any problem since the fully-developed velocity profile is known.

3.1 Verification of the Model Centerline velocity which is an important parameter in determination of entrance length and boundary layers growth is calculated and compared with available data in the literature for continuum gas flow in Fig. 2 to verify the accuracy of the present model. For comparison, full solution of Navier-Stokes equation together with continuity using finite element method based simulation environment COMSOL Multiphysics® , an analytical method based on eigenvalue expansion [26] and numerical result based on finite difference method from the literature [30] are included in Fig. 2. Although not presented, COMSOL model and Liu’s results [30] can predict the velocity overshoot. However, a linearization is used in the solution of Duan and Muzychka [26] and their results cannot predict the overshoot. All solutions merge to the same centerline velocity in the fully-developed limit. COMSOL and Liu’s results are very close to each other, and the present study and the results by [26] are on two opposite side of the numerical curves. The discrepancy is in the entrance region. All the results are within ±3% uncertainty for the calculation of the centerline velocity regardless of the linearization. A recent experimental data obtained by

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Fig. 2 Development of centerline velocity for continuum gas flow Fig. 3 Comparison of velocity profile in entrance region with experimental data for a macrotube

laser doppler velocimetry [29] is also compared against the result of present study for a macro-scale tube at different axial locations. The experimental data is taken at the locations above and below the centerline. As seen from the figure, the results are in good agreement. Following these verifications, it is evident that Langhaar’s linearization can be implemented for modeling of velocity field within the entrance region with a reasonable accuracy (Fig. 3).

3.2 Velocity Profile and Entrance Length 3.2.1

Microtube and Slit-Channel

The developing velocity profiles within the entrance region in a microtube and slitchannel is shown in Fig. 4 for different slip models. At σ = 0 which corresponds to

An Extended Langhaar’s Solution for Two-Dimensional Entry …

203

Fig. 4 Developing velocity profile within the microchannel

the inlet, a uniform velocity profile is obtained which is an inherent assumption of the boundary layer idealizations. As σ increases, the boundary layers develop, and at a certain distance downstream, they merge and fully-developed velocity profile in the form of a parabola is achieved. In the slip-flow regime, the parabola is more flat than that of a continuum flow due to the reduction of shear stresses on the solid walls with increased Kn. Although the fully-developed profiles are similar for different models, the velocity profiles within the entrance region are quite different at some certain σ values which leads to quite different pressure drop and friction factor characteristics. It is also worth to mention that our model cannot capture the velocity overshoot as expected. Since the sudden change of the velocity at the inlet is less pronounced for higher rarefaction, the overshoots are expected to be weak. One parameter which is important to characterize the flow in the developing region is the entrance length, which can be defined as the required distance to downstream for which the centerline velocity reaches to 99% of the fully-developed value. The effect of slip velocity and Kn on the entrance length is illustrated in Fig. 5 with different models. All models except Beskok model predict slower velocity development with increasing Kn. However, Beskok model predicts a reduction in entrance length after a certain Kn. This behavior can be attributed to the reduced friction on the solid walls with increased rarefaction (i.e. high Kn). Furthermore, the fully-developed velocity profile becomes closer to a uniform one increased rarefaction. Therefore, a shorter entrance length is quite expected.

3.2.2

Concentric Annular Microchannel

The velocity profiles within the entrance region of a concentric annular microchannel with a radii ratio of m = 0.5 is given Fig. 6 for Kn = 0.01. The cross-sections are chosen in the vicinity of inlet (σ = 7.4 × 10−5 ) and at sufficiently far downstream (σ = 1.7 × 10−2 ) to ensure fully-developed velocity profile. Since the velocity

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Fig. 5 Variation of entrance length versus Kn for different slip models

Fig. 6 Developing velocity profiles in a concentric annular microchannel (Kn = 0.01, m = 0.5)

gradients are different on the inner and outer walls which results in different slip velocities and shear stresses, the slip-velocity on the inner and outer walls are different in magnitude. Furthermore, different slip velocities on the solid walls cause non-symmetrical distribution of velocity profile around the central core. The discrepancy among two different slip models rises in the entrance region of channel and reduces drastically in the fully-developed region. The non-dimenisonal entrance length is presented for different slip models and radii ratios m = 0.1, m = 0.5 and m = 0.8 in Fig. 7. A lower radii ratio is a case close to the microtube, and a higher radii ratio is a case closer to the slit-channel. As expected, considering the results in Fig. 5, the entrance length decreases with increasing radii ratio, and the first-order model predicts a longer entrance region.

An Extended Langhaar’s Solution for Two-Dimensional Entry … Fig. 7 Variation of entrance length versus Kn using different slip models

205

First-order Model

0.05

Beskok Model

L + (L / R Re) 2 e

0.04 0.03 0.02 0.01 0

0

0.05

m = 0.1

0.10

Kn

m = 0.5

0.15

0.20

m = 0.8

3.3 Pressure Drop and Friction Factor 3.3.1

Microtube and Slit-Channel

Pressure drop in the form of an apparent friction factor is shown in Fig. 8 both for microtube and slit-channel using different slip models and Kn. Apparent friction is also compared with available data in the literature for continuum gas flow and at the limit of the slip-flow regime. In continuum gas flow, the present model predicts apparent friction in good agreement with the data in literature. For Kn = 0.10, the data of [25], which uses a first-order slip, is also included for microtube and slitchannel. For slit-channel, the result for a rectangular channel with an aspect ratio of 2.0 is included. The discrepancy between the present model and the data of [25] is due to the velocity overshoot. That is the reason, the discrepancy exists at the vicinity of the inlet, and the results converge to the same curve after a certain σ. It can be also seen that the discrepancy diminishes as Kn increases, and the curves merge at a location closer to the inlet. The figure also shows a reduction in apparent friction with increased Kn which can be explained by considering the reduction of shear stresses on the solid walls. The high-order models deviate from the first-order model as Kn increases. Moreover, the asymptotic limit of the each curve is different. Again, the difference in this asymptotic value also increases with increasing Kn. Although the asymptotic values are close to each other, the apparent friction factor in the entrance region is quite different for different slip models. The apparent friction factor is more sensitive to the used slip model for shorter microchannels.

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Fig. 8 Apparent friction factor for different Kn and models

3.3.2

Concentric Annular Microchannel

Variation of apparent friction along the channel for concentric annular microchannel is illustrated in Fig. 9. Continuum regime (Kn = 0) together with two different Kn are considered for three different radii ratios. Reduction of the apparent friction can be observed as Kn number increases. For continuum regime flow (Kn = 0) and Beskok model, the apparent friction has a monotonic increase by approaching to inlet; however, the first-order model asymptotically reaches a constant value at the vicinity of the inlet. Far away from the inlet, all the curves reaches an asymptotic value which corresponds to the fully-developed value.

Fig. 9 Apparent friction factor for different slip models and Kn

An Extended Langhaar’s Solution for Two-Dimensional Entry …

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4 Concluding Remarks Measurement of different flow parameters to understand fluid physics at micro-scale is still challenging. Therefore, analytical and numerical models serve as a basis for fundamental understanding of the phenomena. Present study presents an extended Langhaar’s solution for the solution of a 2D incompressible, isothermal flow in a developing region of a microchannel considered both in cylindrical and cartesian coordinates. The analytical model is verified for the flows at macro-scale, and validity of the certain assumptions have been discussed. The hydrodynamic entrance length and apparent friction factor which are the quantities of interest for many engineering calculations are presented. To extend the applicability of the model, different second-order models together with the general velocity-slip boundary condition are included in the analysis. Although the applicability of the second-order models used in this study may be questionable, the present analytical model is a general mathematical tool to model the microchannel flows with high-order slip models. The model can easily be revised by the help of recent findings as long as the general form of the second-order boundary conditions is implemented. The analytical nature of the presented model enables the implementation of different slip flow models, and a fast evaluation of the velocity field which can be extended for the exploration of the combined entrance heat transfer problems for microchannels.

Appendix: Velocity Profile for Concentric Annular Microchannel Velocity profile is expressed as λ(γ, q) = A(γ)I0 (γq) + B(γ)K0 (γq) + C(γ)

(57)

Using the first-order slip model coefficients A, B and C can be defined as: (f )

A=

(f )

(f )

A1

B1

C1

A2

B2

C2

, B= (f )

, C= (f )

(f )

(58)

Coefficient A can be expressed as: (f )

A1 = γ(m2 − 1)1 (f )

A2 = [1 I1 (γm) − 2 I0 (γm) − 2I1 (γ)] 1 − [1 K1 (γm) − 2 K0 (γm) − 2K1 (γ)] 2

(59)

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where

1 = 2γKn(m2 − 1) [K1 (γm) + K1 (γ)] + K0 (γm) − K0 (γ) 2 = 2γKn(m2 − 1) [I1 (γm) + I1 (γ)] − I0 (γm) + I0 (γ)

(60)

1 = 2γ Kn(m − 1) + 2m, 2 = γ(m − 1) 2

2

2

Coefficient B can be expressed as: (f )

B1 = γ(m2 − 1)2 (f )

B2 = 3 I1 (γm) + 4 I0 (γm) + 5 I1 (γ) + 6 I0 (γ) − 4/γ

(61)

where 3 = −4 K1 (γ) + 1 K0 (γ) 4 = (5 + 2)K1 (γ) − 2 K0 (γ) 5 = 4 K1 (γm) + 3 K0 (γm) 6 = 1 K1 (γm) + 2 K0 (γm) 3 =2γ 2 Kn(m2 − 1) + 2, 4 = 4γKn(m + 1) [γKn(m − 1) + 1] , 5 = 1 − 2m

(62) Coefficient C can be expressed as: (f )

C1 = [5 I1 (γm) − 2 I0 (γm)] 7 − [5 I1 (γ) − 2 I0 (γ)] 8 (f )

(f )

C2 = −B2 where

7 = 2γKnK1 (γ) − K0 (γ) 8 = 2γKnK1 (γm) + K0 (γm)

(63)

(64)

Using general slip model, coefficients A, B and C can also be expressed as: (g)

A=

(g)

(g)

A1

B1

C1

A2

B2

C2

, B= (g)

, C= (g)

(g)

(65)

Coefficient A can be expressed as: (g)

A1 = (1 − m2 )9   (g) (66) A2 = 6 I1 (γm) + (m2 − 1)I0 (γm) − 2/γ [mI1 (γm) + I1 (γ)] 9   2 + 6 K1 (γm) − (m − 1)K0 (γm) − 2/γ [mK1 (γm) + K1 (γ)] 10

An Extended Langhaar’s Solution for Two-Dimensional Entry …

209

where 2γKn 2γKn K1 (γm) − K0 (γm) + K1 (γ) + K0 (γ) b1 Kn − 1 b2 Kn − 1 2γKn 2γKn 10 = − I1 (γm) − I0 (γm) − I1 (γ) + I0 (γ) b1 Kn − 1 b2 Kn − 1 2γKn(m2 − 1) 6 = b1 Kn − 1 9 =

(67)

Coefficient B can be expressed as:   (g) B1 = 5 b˜ 2 I1 (γm) + b˜ 1 I1 (γ) + 2 b˜ 1 b˜ 2 [I0 (γm) + I0 (γ)] B2 = 11 I1 (γm) + 12 I0 (γm) + 13 I1 (γ) + 14 I0 (γ) − 4b˜ 1 b˜ 2 /γ (g)

(68)

where     11 = 2γKn 2b˜ 1 m + 2b˜ 2 − 5 K1 (γ) + b˜ 2 2b˜ 1 m − 5 K0 (γ)   12 = b˜ 1 2b˜ 2 − 5 K1 (γ) − 2 b˜ 1 b˜ 2 K0 (γ)     13 = −2γKn 2b˜ 1 m + 2b˜ 2 − 5 K1 (γm) + b˜ 1 2b˜ 2 − 5 K0 (γm)   14 = b˜ 2 2b˜ 1 m − 5 K1 (γm) + 2 b˜ 1 b˜ 2 K0 (γm)

(69)

b˜ 1 =b1 Kn − 1, b˜ 2 = b2 Kn − 1 The coefficients b1 and b2 are defined in Eq. (38). Coefficient C can be expressed as:     (g) C1 = 5 I1 (γm) + b˜ 1 2 I0 (γm) 15 + 5 K1 (γm) + b˜ 1 2 K0 (γm) 16 (g)

(g)

C2 = −B2

(70)

where 15 = 2γKnK1 (γ) + b˜ 2 K0 (γ) 16 = −2γKnI1 (γ) + b˜ 2 I0 (γ)

(71)

Fully-developed velocity profile using first-order slip model slip model can be written as: (f )

λfd =

Q1 Q2

(72)

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where 



Q1 = 2(m − 1) 8Kn 2 (1 − m2 ) + 2Kn(m3 + 1) − 2Knq2 (m + 1) + log



Q2 = (m2 − 1) 8Kn 2 (m + 2) + m2 (1 − 2Kn) − 2Kn



qm(m+1)(1−m+4Kn) mm(1−q



2 +4Kn)

  + (1 − m)(m2 + 1) + 8Kn(m2 − m + 1) log mm

(73) and for general slip model, fully-developed velocity can be written as: (g)

λfd =

S1 S2

(74)

where 



S1 = 2(m2 − 1) log qm b˜ 1 b˜ 2 (m2 − 1) + 4Kn(b˜ 1 m − b˜ 2



  + log mm b˜ 2 b˜ 1 (q2 − 1) + 4Kn   + 2Kn b˜ 1 (1 − q2 ) + b˜ 2 mKn(m2 − q2 ) + 4Kn(m2 − 1) − m3 (m + 1)

(75)

S2 = 8Kn 2 log mm b1 m3 (mb2 + 1) + b2 (1 − b1 )

− Kn 2 (m2 − 1) (m2 − 1)(b1 b2 m + 2b1 − 16) − 2b2 m(m2 + 3) + 4b1 m2

× Kn 2(1 − m5 ) + 8m2 (m − 1) − 8 log mm (1 + m3 ) + 6m(m3 − 1)   5   + log mm −m − m(m2 − 1)2 [1 − b1 b2 Kn] (76) The coefficients b1 and b2 are defined in Eq. (39), and b˜ 1 = b1 Kn − 1 and b˜ 2 = b2 Kn − 1.

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30. Shah, R.K., London, A.L.: Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data. Academic Press, Cambridge (1978) 31. Sugino, Eitaro: Velocity distribution and pressure drop in the laminar inlet of a pipe with annular space. Bull. JSME 5(20), 651–655 (1962) 32. Han, L.S.: Hydrodynamic entrance lengths for incompressible laminar flow in rectangular ducts. J. Appl. Mech. 27(3), 403–409 (1960) 33. Harley, J.C., Huang, Y., Bau, H.H., Zemel, J.N.: Gas flow in micro-channels. J. Fluid Mech. 284, 257–274 (1995) 34. Arkilic, E.B., Breuer, K.S., Schmidt, M.A.: Mass flow and tangential momentum accommodation in silicon micromachined channels. J. Fluid Mech. 437, 29–43 (2001) 35. Cercignani, C., Lorenzani, Silvia: Variational derivation of second-order slip coefficients on the basis of the boltzmann equation for hard-sphere molecules. Phys. Fluids 22(6), 062004 (2010) 36. Cetin, B., Yazicioglu, A.G., Kakac, S.: Slip-flow heat transfer in microtubes with axial conduction and viscous dissipation-An extended Graetz problem. Int. J. Therm. Sci. 48, 1673–1678 (2009) 37. Cetin, B., Bayer, O.: Evaluation of Nusselt number for a flow in a microtube using second-order slip model. Therm. Sci. 15(Suppl. 1), 103–109 (2011) 38. Çetin, Barbaros: Effect of Thermal Creep on Heat Transfer for a Two-Dimensional Microchannel Flow: An Analytical Approach. J. Heat Transf. 135(10), 101007–101008 (2013) 39. Çetin, B., Zeinali, S.: Analysis of heat transfer and entropy generation for a low-Peclet-number microtube flow using a second-order slip model: an extended-Graetz problem. J. Eng. Math. 89(1), 13–25 (2014)

Dynamics of Solitons in High-Order Nonlinear Schrödinger Equations in Fiber Optics Gholam-Ali Zakeri

Abstract In optical fibers, the higher order nonlinear Schrödinger equation (NLSE) with cubic-quintic nonlinearity describes the propagation of extremely short pulses. We construct kink, bright and dark solitons of a generalized higher order NLSE in a cubic-quintic non-Kerr medium by applying proposed modified extended mapping method. These obtained solutions have key applications in physical science and engineering. Moreover, we also present the formation conditions on solitary wave parameters in which kink, dark and bright solitons can exist for this media. We graphically illustrate the collision of the constructed soliton solutions that help realize the physical phenomena of NLSE. We also outline descriptions of various issues on integrability. We discuss the stability of the model in normal dispersion and anomalous regime by using the modulation instability analysis. Many other types of such models arising in applied sciences can also be solved by these reliable, powerful and effective methods.

1 Introduction Nonlinear Schrödinger equation (NLSE) has a wide range of applications in engineering and physical science. Nonlinear fiber optics has been one of the most rapidly growing research area in the last several decades [1–4]. Optical solitons play a prominent role in the field of fiber communication technology. In conservative systems, light travels as a stable non-diverging beam due to a balance between dispersion and nonlinearity. This localized wave takes a bell-shape intensity profile in transverse direction which is now referred as a bright soliton. The existence of solitons in fiber was proposed by Hasegawa and Tappert in a sequence of papers [5, 6] during 1973. They have shown that the nonlinear dependence of the index of refraction on intensity makes it possible the transmission of picosecond optical pulses without G.-A. Zakeri (B) Department of Mathematics, and Interdisciplinary Research Institute for the Sciences (IRIS), California State University - Northridge, Northridge, CA 91330-8313, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_7

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any distortion in dielectric fiber waveguides with group velocity dispersion. Later, in 1980, these phenomena for the 7-picosecond duration pulses were verified by Mollenauer, Stolen, and Gordon in [7]. On the other hand, the development of doping silica fibers with rare-earth materials such as erbium and ytterbium during the 1990s enhanced this rapid growth in the field of fiber optics [8]. Furthermore, applications of erbium and ytterbium in fiber amplifiers and lasers were a major step forward in the redesign of fiber optics, and in the propagation of optical solitons, breathers, and optical rogue waves whose existence resulted from nonlinear effects in optical fibers [9, 10]. During the early 2000s, several kinds of new fibers such as micro-structured fibers, holey fibers, photonics crystal fibers have been developed and are considered as highly nonlinear fibers. These nonlinear effects stemmed and enhanced to a point that they can be observed even in a fiber as short as a few centimeters in length. Dispersive properties of these newly developed fiber optics are quite different from those of the conventional fibers. In 2007, “optical rogue waves” were introduced as a new subfield in optical field [11], emerged as a fast growing new subfield of research. The optical rogue waves are optical pulses whose amplitude or intensity is much higher (at least by a factor 2) than of the surrounding pulses, for this reason some authors refer to them as extreme wave rare events [12]. These optical rogue waves have been generated by lasers, and observed in wide aperture cavities, in plasmas and in a variety of other optical systems. For an over view of development in optical rogue waves, see [13, 14]. In the following we describe some of the most commonly versions of NLSE that are applicable. The fundamental NLSEs that has a bright soliton solution is given by 1 iψx + ψtt + |ψ|2 ψ = 0, 2

(1)

where ψ is the optical pulse propagating in a nonlinear fiber, x represents normalized propagation distance and t is the retarded time. The existence of a bright soliton solution of (1) stems from a balance between the dispersion term, ψtt and the cubic nonlinearity, |ψ|2 ψ. This “balance” leads to the existence of two different types of solutions on which the modulus of the complex-valued solution, |ψ(x, t)| could be independent of t called a stationary wave, or if |ψ| is a function of x − ct, describes a moving wave, where c is a real constant. In optical fiber, t is related to real time T = t + x/vg , where vg is the group velocity of electromagnetic waves traveling along the optical fiber. There are other types of optical solitons that are described by extended NLSE or system of such equations. These extended NLSEs are constructed by including higher-order nonlinearities such as quintic |ψ|4 ψ, septic |ψ|6 ψ, . . ., and higher-order dispersions such as ψttt , ψtttt , . . .. Each of these terms has certain effects and causes a different balance on the soliton solutions. There are other types of nonlinear dispersive terms that effect on soliton produces time dependent phase, and causes a different balance in existence of soliton called shock waves. These nonlinear terms are given by i|ψ|2 ψt , or i(|ψ|2 )t ψ which describe different self-steepening.

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An actual fiber optic is not homogeneous due to an imperfect manufacturing process such as diameter fluctuations, a variation in the lattice parameters defining the medium, and so on [15–17]. This non-uniformity in the core medium of a real fiber influences several factors such as loss/gain, dispersion, phase modulation, and so on [18]. Under such non-uniformities, the (1+1)-dimension NLSE used in the field of soliton management [19, 20] in a Kerr-type nonlinear medium is given by iψx + f (x)ψtt + g(x)|ψ|2 ψ + (α(x) + iγ(x))ψ = 0,

(2)

where ψ(x, t) is the complex envelope of the optical field, of x and t are acting as a transverse coordinate. Here, f (x) is the dispersion parameter and g(x) is the nonlinearity management parameter that corresponds to the intensity-dependent refractive index change, α(x) represents the external potential applied and γ(x) is the dissipation coefficient or linear gain parameter. The first extended NLSE with higher order nonlinearity includes a quintic term as follows: (3) iψx + d2 ψtt + r3 |ψ|2 ψ + r5 |ψ|4 ψ = 0. From optical fiber view, the inclusion of the quintic nonlinearity term, |ψ|4 ψ, in (1) is to establish a better delicate balance between ψtt , and |ψ|4 ψ on one hand, and, the quintic term |ψ|4 ψ on the other hand stabilizing the soliton. If all coefficients in (3) are constant, then mathematically, (3) possesses two types of solitons given by algebraic and hyperbolic expressions. For very short pulses such as picosecond, it requires that we consider the effects of third and fourth dispersion terms to maintain the delicate balance as given by iψx + d2 ψtt − id3 ψttt + id4 ψtttt + r3 |ψ|2 ψ + r5 |ψ|4 ψ = 0.

(4)

Depending on the specific application, one requires the inclusion of one or all certain higher-order nonlinearity or dispersive or self-steeping terms: iψx + d2 ψtt + r3 |ψ|2 ψ + r5 |ψ|4 ψ + is1 (|ψ|2 )t ψ = 0,

(5)

iψx + d2 ψtt + r3 |ψ|2 ψ + id3 ψttt + is1 (|ψ|2 )t ψ + is2 (|ψ|2 ψ)t = 0,

(6)

or

or iψx + d2 ψtt − id3 ψttt + d4 ψtttt + r3 |ψ|2 ψ + r5 |ψ|4 ψ + is1 (|ψ|2 )t ψ + is2 (|ψ|2 ψ)t = 0, (7)

or iψx + d1 ψt + d2 ψtt − id3 ψttt + d4 ψtttt + r3 |ψ|2 ψ + r5 |ψ|4 ψ + r6 ψ + is1 (|ψ|2 )t ψ + is2 (|ψ|2 ψ)t = 0.

(8)

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It has been shown that in the above hierarchies the soliton solutions of for example (7) do not include the soliton solutions of (5) by setting certain parameters in solutions of (7) to get those of (5). Each equation listed above is applicable to certain physical set up and has its own set of properties, e.g., (6) is Painlevé, while (5) does not have such a property. This shows that the solution of each equation is not a subset of solutions of another equation with less nonlinear terms [21–24]. Another approach to build an extended NLSE for a given application is to consider additional terms in a form of perturbation to (1) given by 1 iψx + ψtt + |ψ|2 ψ = R[ψ(x, t); x, t]. 2

(9)

Various types of perturbation terms, R, have been investigated in recent years to analyze the dynamics of propagation of solitons for a variety of applications, e.g., in [25] the authors used R = −i|ψ|2 ψ and in [26], R = −i|ψ|4 ψ to study pulse propagation through an optical waveguide in the presence of Kerr nonlinearity and quintic weak loss, in [27], R is let to be f (x, t) − iβψ(x, t) − δψ(x, t) where f is considered as a damping effect and the added linear term for stabilizing the driven soliton, and in [28], the authors used R = α(c1 |ψ|4 ψ + c2 ψ 2 ψtt∗ + c3 ψ ∗ ψt2 + c4 |ψt |2 ψ + c5 |ψ|2 ψtt + c6 ψtttt ) to consider a wider class of physically relevant perturbations, where α is a small parameter. While in [29], a two-dimensional generalized version of [28] is considered that analyzes the dynamics of soliton to the Hyperbolic NLSE for an arbitrary number of collective coordinates, also, see [30]. For a good comprehensive survey of the results obtained for solitons and complex nonlinear medium see [31]. There is a vast amount of literature devoted to the study of various aspects of NLSE such as analytical solutions, integrability, symmetries, and qualitative behaviors of solitons in nonlinear optics. It is important to mention that a close parallel to the topic of this chapter is Bose-Einstein condensates (BECs) with higher-order interactions. The following are samples of articles that deal with higher-order nonlinearity in BECs, [32–34]. For the sake of completeness and being very brief, we have given only reference to the most relevant work. This chapter is organized as follows. In Sect. 2, we give an outline of integrability, Lax pair, and its connection to Darboux transformation. In Sect. 3, we discussed several methods for obtaining analytic solutions of a given NLSE, such as similarity reduction via two-step transformations method, Hirota method via bilinear transformation, and via N-soliton interactions bilinear operator. In Sect. 4, modulational instability in a simple form is discussed, and finally in Sect. 5, we gave a brief account of stability analysis.

2 Integrability The inverse transformation method when used to solved the initial value problem of the Korteweg-de Vries (KdV) equation with self-consistent sources governing the

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interaction between the long and short capillary-gravity waves [35] was an inspiration to development of modern integrability of systems of partial differential equations (PDEs). In 1968, Peter Lax gave an elegant reformulation of the above mentioned KdV solution [36] by using two commuting operators that are now called a Lax pair. Lax observation led to many generalizations to investigate the integrability of many applicable nonlinear PDEs. Considering two linear problems [37] 

   ∂ ∂ − U F = 0, − V F = 0, ∂t ∂x

(10)

where U (t, x, λ) and V (t, x, λ) are n × n matrices of elements of a Lie algebra depending on the spectral parameter λ. A compatibility condition for (10) is obtained by cross-differentiating of (10) over the zero curvature of the submanifold to get Ut − Vx + [U, V ] = 0,

(11)

where [U, V ] = U V − V U . A good introduction on the sufficient conditions for integrability of a system can be found in [38], in addition, some of the later work can be found in [39–41]. Integrable systems can be used to obtain other nonlinear integrable systems as shown in [40]. It has been shown that there is algebraic connection between symmetries and integrability [42], the relationships among inverse scattering method, Bäcklund transformation and the existence infinite number of localized conservation laws have been established. For example, in [43] the authors have shown that any nonlinear PDEs that can be solved exactly by the inverse scattering, yield the N-soliton solutions, and such equation has an infinite number of conversation laws. In addition, such nonlinear equations have Bäcklund transformations which transform the equations to themselves. There is also a relation between conservation laws and the bilinear forms [44], and it is known that the Hirota method is used to derive Lax pair, bilinear forms, Bäcklund transformations and N-soliton [40]. Although many authors have made important contributions to the development of methods to calculate infinite local conservation laws for NLSE, we mention a few, in particular, see [45–49]. PDEs that are integrable by the inverse transform method in (1+1)-dimension possess an infinite hierarchy of local conservation laws. This means that a PDE in the form of ∂ρ ∂ρ + c(ρ) = 0, (12) ∂t ∂x can be rewritten as

∂ ∂ f1 (ρ) + f2 (ρ) = 0, ∂t ∂x

(13)

where f1 (ρ) and f2 (ρ) are called conserved densities. In fact any choice of f1 (ρ), f2 (ρ) that satisfy f2 (ρ) − c(ρ)f1 (ρ) = 0 works [50]. For example for KdV ut + uxxx − 6uux = 0,

(14)

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the first few local conservation laws are ut + (uxx − 3u2 )x = 0, (u )t + (2uuxx ux − 2

(ux2

ux2

− 4u )x = 0,

+ 2u )t + (2uxxx ux − 3

(15) 3

2 uxx

+ 6u uxx − 2

(16) 12uux2

− 9u )x = 0. 4

(17)

The existence of nontrivial local conservation laws (i.e., those beyond the conservation of mass, momentum and energy) is a good indication of integrability and can be used as definition of integrability. It has been shown that Lax pair can be used to obtain local conservation laws [41]. Moreover, Lax pair can be used to obtain Darboux transformations [51]. Although many authors have made important contributions to development of methods to calculate infinite local conservation laws for NLSE, we mention a few, in particular, see [52–55]. Because of the importance role of Lax pair in analyzing integrating of an NLSE, in the following, we give few examples with some details on obtaining Lax pair. For each example we use the compatibility condition (11). Example 1 Let U (x, t, λ) and V (x, t, λ) be two 2 × 2 matrices that are expanded in terms of parameter λ as follows. U = U0 + λU1 , V = V0 + λV1 + λ2 V2 .

(18)

Substituting (18) into (11) we get a third degree polynomial of λ. To have this as an identity, each coefficient of powers of λ must be zero which gives the following four equations: U0t − V0x + U0 V0 − V0 U0 = 0,

(19)

U1t − V1x + U0 V1 + U1 V0 − V0 U1 − V1 U0 = 0, −V2x + U0 V2 + U1 V1 − V1 U1 − V2 U0 = 0,

(20) (21)

U1 V2 − V2 U1 = 0.

(22)

Since we have four equations and five unknown matrices, we can select a basic choice for U1 , then from (22) we can find V2 , we let    i 0 a2 b2 . U1 = , V2 = c2 d2 0 −i 

(23)

Substituting into (22) gives b2 = c2 = 0 and a2 + d2 = 0. Let us choose a2 = i and d2 = −i. Equations (20) and (21) suggest that we can choose U0 = V1 . Setting an arbitrary matrix for V1 as function of (x, t), then from (19), we get 

    ipq ipx  i 0 0 p(x, t) −2 V2 = , U0 = V1 = , V0 = iq2x ipq . 0 −i q(x, t) 0 − 2 2

(24)

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Now substituting these into the compatibility condition (11), we have a coupled of equations for p(x, t) and q(x, t) that can be used to determine the Lax pair in this case. ipxx − ip2 q = 0, 2 iqxx qt − + ipq2 = 0. 2

pt +

(25) (26)

If we can choose p and q in such a way that lead to fundamental nonlinear Schrödinger equation then the obtained Lax pair represents the fundamental NLSE (1). To this end, we let p = iφ∗ and q = iφ, the above two equations becomes complex conjugate of each other, we have 1 iφt + φxx + |φ|2 φ = 0. 2

(27)

A local conservation law form of fundamental NLSE (1) is 

i (φφ∗x − φx φ∗ ) (φφ )t + 2 ∗

 = 0.

(28)

x

So, to accommodate for higher order terms one must either consider a higher order polynomial expansion for V or to introduce more unknown functions in constructing U and V . We illustrate this process in the next two examples. Example 2 Obtaining Hirota equation that plays an important role in optics. We build upon the work done in Example 1, using the same U0 , U1 for p = iφ∗ and q = iφ. To take into the account for higher order of nonlinearity, we add a cubic term in the expansion of V , U = U0 + λU1 ,

V = V0 + λV1 + λ2 V2 + λ3 V3 .

(29)

Substituting (29) into (11) we get a fourth degree polynomial of λ that reduces to: U0t − V0x + U0 V0 − V0 U0 = 0,

(30)

U1t − V1x + U0 V1 + U1 V0 − V0 U1 − V1 U0 = 0, − V2x + U0 V2 + U1 V1 − V1 U1 − V2 U0 = 0, − V3x + U0 V3 + U1 V2 − V2 U1 − V3 U0 = 0,

(31) (32) (33)

U1 V3 − V3 U1 = 0.

(34)

Following the same procedure as in Example 1, beginning with the last equation and working toward the first one, we get

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   φ∗ f1 (t) 0 f3 (t) (f1 (t) − f2 (t)) 2 , V2 = φ V3 = , 0 f2 (t) (f (t) − f2 (t)) f4 (t) 2 1   2 φ∗x φ∗ − |φ|4 (f1 (t) − f2 (t)) (f (t) − f (t)) − (f (t) − f (t)) 3 4 1 2 2 4 , V1 = φ φ∗ |φ|2 (f (t) − f4 (t)) − 4x (f1 (t) − f2 (t)) (f1 (t) − f2 (t)) 2 3 4   a −c∗ (35) V0 = 0 ∗0 , c0 a0 where a0 and c0 are given as follows: 1 (f3 (t) − f4 (t))(|φ|2 − i(f1 (t) − f2 (t))(φφ∗x − φ∗ φx )), 4 1 iφx c0 = (f1 (t) − f2 (t))(2|φ|2 φ + φxx ) − (f3 (t) − f4 (t)). 8 4

a0 =

(36)

In addition, we get two equations that are complex conjugate of each other, one of these equations is   1 2 ∗ (f3 (t) − f4 (t))(2|φ| φ + iφxx ) + (f1 (t) − f2 (t)) 3|φ| φx + φxxx + iφx = 0. 2 (37) If we let f1 (t) − f2 (t) = 8iα and f3 (t) − f4 (t) = 2i, (27) reduces to the well-known Hirota equation: 2

1 |φ|2 φ + iφt + 6iα|φ|2 φx + φxx + iαφxxx = 0. 2

(38)

Thus the Lax pair given here represents Hirota equation of optics. Example 3 The localized soliton solutions generated by a couple of Hirota equations play an important role in optics. These solitons illustrate the transmission of waves when pulse lengths become comparable to the wavelength and the inclusion of higher order nonlinear effects are considered [56]. Because of their important roles in various applications several authors have obtained their Lax pair, the classical Darboux transformation, the Painlevé analysis, analytic solutions in the form of bright and dark solitons, and the lower order dark rogue wave solutions. These equations are given  1 iut + uxx + |u|2 + |v|2 u + iα[uxxx + 6|u|2 ux + 3(|v|2 ux + uv ∗ vx )] = 0, 2  1 ivt + vxx + |u|2 + |v|2 v + iα[vxxx + 6|v|2 vx + 3(|u|2 vx + vu∗ ux )] = 0, 2

(39)

where u(x, t) and v(x, t) are complex-valued functions representing smooth envelop, α is small real parameter that controls the third dispersion uxxx , vxxx , the self-steeping and inelastic Raman scattering nonlinear terms.

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In this case, the Lax pair, U, V are 3 × 3 and are given by U = U0 + λU1 ,

V = V0 + λV1 + λ2 V2 + λ3 V3 .

(40)

Following the same procedure as outlined in Example 1, we have [57] ⎡

⎡ ⎤ −2i 0 0 1 ⎣ 1 0 i 0 ⎦ , V3 = U1 , U1 = 12α 16α 0 0 i ⎤ ⎡ u v − iux − 2α − ivx i(|u|2 + |v|2 ) − 2α 1 1 1 ⎣ u∗ − iux∗ −i|u|2 −ivu∗ ⎦ , (41) V2 = U1 + U0 , V1 = 2α 8α 16α 4 v∗ ∗ ∗ − ivx −iuv −i|v|2 , 2α ⎤ ⎡ α(a1 + a2 ) + ia20 αa3 − iu2x αa4 − iv2x ∗ 2 ∗ ⎥ ⎢ V0 = ⎣ −αa3∗ − iu2x −αa1 − i|u| αa5 − ivu2 ⎦ , 2 2 ∗ iv ∗ −αa4∗ − 2x −αa5∗ − iuv2 −αa2 − i|v| 2 ⎤ 0 −u −v U0 = ⎣ u ∗ 0 0 ⎦ , v∗ 0 0

where a0 = |u|2 + |v|2 , a1 = uux∗ − u∗ ux , a3 = uxx + 2a0 u, a4 = vxx + 2a0 v,

a2 = vvx∗ − v ∗ vx , a5 = u∗ vx − vux∗ .

(42)

Now, substituting U, V into the compatibility condition, Ut − Vx + [U, V ] = 0, we get four equations in which two of them complex are conjugate of the other two equations which are the coupled Hirota equations as given above. Algorithm for finding subsection in an algorithmic vector  U, V : To summarize  this j notations for U = nj=0 λj Uj and V = m j=0 λ Vj where there are m + 1 equations, assume that n < m, and all Uk = 0 for k > n then Uj , Vj ’s are (m + 1) × (m + 1) matrices, and assume that n < m, and they can be found from the following [Un , Vm ] = 0, k  [Uj , Vm−j−k ] = 0,

for k = 1, . . . , n − 1 (43)

j=1

Ukt − Vkx +

k 

[Uj , Vk−j ] = 0, for k = m, m − 1, . . . , 1.

j=0

Then we proceed by solving these equations one at a time from first toward the last one. For the most practical problems it is sufficient to choose n = 1 and select U1 as the zero seed then use the above algorithm to find Vj ’s.

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Lax pair and Darboux transformation Essentially, Darboux transformations are based on the existence of Lax pair for a system of equations which encode the NLSE under consideration in the form of their compatibility conditions. The Darboux transformations are obtained based on algebraic manipulations which can be iterated step by step to generate infinitely many nontrivial solutions with a simple seed solution of a Lax pair. For the system φx = U φ, φt = V φ, then U and V , the Lax pair satisfy the compatibility condition Ut − Vx + [U, V ] = 0. The Darboux transformation associated with above spectral problem is φˆ = T φ (44) where the requirement on transformation T is found as follows φˆ x = Tx φ + T φx = Tx φ + T U φ = (Tx + T U )φ, φˆ t = Tt φ + T φt = Tt φ + T U φ = (Tt + T U )φ.

(45)

Now setting ˆ φˆ t = Vˆ φ, ˆ φˆ x = Uˆ φ,

(46)

where Uˆ and Vˆ have exactly the same functional form as U , and V , respectively. Then the requirement on T is as follows Tx + T U = Uˆ T , Tt + T V = Vˆ T .

(47)

Now, cross-differentiating equations in (47), and under the assumption that T is a non-singular matrix, we get Uˆ t − Vˆx + [Uˆ , Vˆ ] = T (Ut − Vx + [U, V ]) T −1 .

(48)

In order to continue with higher order iterations of (44), it is common to adopt the following notations: φ[j] = T [j]φ[j − 1], φ[j]x = U [j]φ[j], φ[j]t = V [j]φ[j],

(49)

ˆ φ[0] = φ and the assumption of T [j] being nonsinguwhere j = 1, 2, . . ., φ[1] = φ, lar guarantees that that all eigenfunctions φ[j]’s will be nonzero. Then (48) becomes as U [j]t − V [j]x + [U [j], V [j]] = T (U [j − 1]t − V [j − 1]x + [U [j − 1], V [j − 1]]) T −1 .

(50) Equation (50) provides an algorithmic approach to find the n-fold Darboux transformations for a NLSE when its Lax pair is known and U [j], V [j] have the same functional forms as U [j − 1], V [j − 1]. For a detail step by step calculations see e.g. [49, 57–60].

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Painlevé analysis An algorithmic approach to test the integrability of a nonlinear model is the Painlevé analysis. This test consists of verifying that the solutions of the model can be written as Painlevé expansions (which is a generalized Laurant expansion). To better explain the process, we use the coupled Hirota equations (39), together with their complex conjugates. Then we let u∗ = w, v ∗ = z to get  1 iut + uxx + |u|2 + |v|2 u + iα[uxxx + 6|u|2 ux + 3(|v|2 ux + uzvx )] = 0, 2  1 ivt + vxx + |u|2 + |v|2 v + iα[vxxx + 6|v|2 vx + 3(|u|2 vx + vwux )] = 0, 2 (51)  1 −iwt + wxx + |u|2 + |v|2 w + iα[wxxx + 6|u|2 wx + 3(|v|2 wx + wvzx )] = 0, 2  1 −izt + zxx + |u|2 + |v|2 z + iα[zxxx + 6|v|2 zx + 3(|u|2 zx + zuwx )] = 0. 2

We let u(x, t) =

∞ 

uj ((x, t)φ(x, t)j+a ,

v(x, t) =

∞ 

j=0

w(x, t) =

∞  j=0

vj (x, t)φ(x, t)j+b ,

j=0

wj (x, t)φ(x, t)

j+c

,

z(x, t) =

∞ 

(52) zj (x, t)φ(x, t)

j+d

j=0

where uj , vj , wj , zj , φ are analytic functions of (x, t) near manifold M = {(x, t) : φ(x, t) = 0} which is often called a movable singularity manifold. Now using a leading order analysis [61] by substituting u = u0 φa , v = v0 φb , w = w0 φc , and z = z0 φd into (52) then since u0 , v0 , w0 , z0 are nonzero, by setting their coefficients to zero, we get a = b = c = d = −1 and u0 v0 + w0 z0 = −1 and u0 , v0 , w0 , z0 are obtained each as an expression of φ and its partial derivatives. On the other hand, by substituting the complete series expressions (52), one gets a system of equations in the following form: F(uj−1 , vj−1 , . . . , u0 , v0 , . . . , φ, φx , . . .)(uj , vj , wj , zj )T = (0, 0, 0, 0)T ,

(53)

where F is a non-singular 4 × 4 matrix. Then we check the conditions under which it becomes singular. For (51), F is singular [56] when j = −1, 0, 0, 0, 1, 2, 2, 3, 4, 4, 4, 5. The values of j are called the resonances [62]. There are three steps in the Painlevé process: the leading order analysis, determination of resonant points and finally verification of compatibility conditions. The resonance at j = −1 corresponds to the arbitrary (undefined) singularity of manifold M . For relations on φ for j = 2, 2, 3, 4, 4, 4, 5 satisfy the compatibility condition and introduce sufficient number of arbitrary functions. Thus (39) possess the Painlevé property. For a detailed example of step by step calculation see [63].

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G.-A. Zakeri

3 Methods of Solutions There are many approaches to obtain analytic solutions and analyze the locally coherent structure of soliton solutions of PDEs, and in particular of an extended NLSE. The methods that have been proposed to solve these nonlinear evolution equations are the similarity reduction methods [64], the Hirota bilinear method [65], the Darboux transformation [66] and its various modification such as Darboux dressing technique [67], the Bäcklund transformation [43, 68], inverse scattering transformation [35], the Painlevé analysis method [69], and in particular in [61], it has been shown that the Painlevé property may provide a unified description of integrability. Originally solitons have been developed and observed in nonlinear and dispersive systems that are autonomous, which means that time or space variable, in the case of spatial and temporal optical solitons, has only played the role of independent variable. On the other hand, the solitary waves in non-autonomous nonlinear and dispersive systems can propagate in the form of non-autonomous solitons. Here, we try to briefly describe approaches that can be used to solve a non-autonomous equation and make some observations on the limitations and their properties.

3.1 Similarity Reduction Method One approach to solve a non-autonomous NLSE containing higher order terms with variable coefficients is to use a similarity reduction method in two steps. In the first step the goal is to convert the equation into a constant coefficients one, and then in the second step to use a certain transformation to obtain desire class of soliton solutions. Although one can use the general Lie group approach outlined in [64], most authors use a specific transformation as the similarity reduction variable. The main idea of this approach is to utilize a transformation to convert the non-autonomous NLSE to a constant coefficients NLSE and then utilize different specific functional forms to seek for specific type of class of solutions as illustrated below. First, we use a transformation in the following general basic form ψ(x, t) = (X (x, t), T (x, t))

(54)

and by reassigning the coefficients of various terms in the NLSE under consideration, involving  to various parameters. This results in obtaining a system of PDEs about the coefficients of NLSE and the resulting NLSE in terms of  is a constant coefficients equation. Various modifications of (54) have been used, for example: ψ(x, t) = (X (x, t), T (x, t))eiϕ(x,t) ,

(55)

ψ(x, t) = ρ(x, t)(X (x, t), T (t))eiϕ(x,t) ,

(56)

or

Dynamics of Solitons in High-Order Nonlinear …

225

and so on, where ρ and ϕ are real-valued functions. In Eq. (56), ρ is positive-valued function, and represents a multiplier to soliton amplitude. To better illustrate this approach, let us consider the following NLSE iψt + f (x, t)ψxx + g(x, t)|ψ|2 ψ = h(x, t)|ψ|4 ψ + i[s1 (x, t)(|ψ|2 ψ)x + s2 (x, t)(|ψ|2 )x ψ] + [u(x, t) + iv(x, t)]ψ, (57) where f (x, t), g(x, t), h(x, t), s1 (x, t), s2 (x, t), u(x, t) and v(x, t) are real-valued functions and represent the coefficients of dispersion, cubic and quintic nonlinearities, two different self-steepening coefficients, and two linear terms that can use to stabilize the soliton, or in some applications can represent gain or loss in external potential terms, respectively. Now, substituting (56) into (57), we obtain the following NLSE: iT + w1  + w2 XX + w3 ||2  − w4 ||4  − iw5 ||2 X − iw6  2 ( ∗ )X = 0, (58) where wj ’s are constants and from setting the resulting coefficients, we get the following PDEs to be solved. ρxx − ϕt − f ϕ2x − u, ρ w2 T  = fXx2 , w3 T  = ρ2 (s1 ϕx + g), w4 T  = ρ4 h,

(60) (61) (62)

w5 T  = ρ2 Xx (s2 + 2s1 ), w6 T  = ρ2 Xx (s2 + s1 ),

(63) (64)

w1 T  = f

(59)

where “prime” means derivative w.r.t t. Equation (57) has other terms that for simplicity of Eq. (58) we set those coefficients to zero for which we get Xt + 2fXx ϕx = 0,

(65)

(ρ Xx )x = 0, ρρt + f (ρ2 ϕx )x − vρ2 = 0, 3 s2 = − s 1 . 2

(66) (67)

2

(68)

Equation (68) indicates that (57) has a similarity reduction solution in the form of (56), when the coefficients of two different self-steepening are related by Eq. (68). From combining Eq. (63) and (64) and (68), we get w6 = −w5 .

(69)

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G.-A. Zakeri

Thus Eq. (58) has only 5 constants to be determined. Since the system of Eqs. (59)–(68) does not have a unique solutions, through this process, we can only capture a certain classes of solutions of (57). Equations (65)–(67) must be solved for X (x, t), ρ(x, t) and ϕ(x, t). A solution of (59)–(68): We assume that ρ(x, t) = ρ1 (t), f (x, t) = f1 (t), and v(x, t) = v1 (t). Because of the complexity of these PDEs, we look for a solution for which X assumes the following form: X (x, t) = a(t)x + b(t),

(70)

where 1/a(t) is the width of the localized solitary wave, and its center-of-mass is located at −b(t)/a(t). Under these assumptions, we get ϕ(x, t) =

b (t) −a (t) 2 x − x + c0 (t) 4f1 (t)a(t) 2f1 (t)a(t)

(71)

Let the coefficients of x2 and x in Eq. (71) be represented as c2 (t), the chirp function, and c1 (t), respectively. We obtain ρ1 (t) =



η(t)e1/2

t

v1 (s)ds

,

(72)

a(t) = η(t)e− 0 v1 (s)ds , η(t)v1 (t) − η  (t) , c2 (t) = f1 (t)η(t)

(73)

0

t

(74)

where η(t) is arbitrary function of t and c1 (t) must satisfies the following equation: [c2 (t) + 4c22 (t)f1 ]x2 + [c1 (t) + 4c2 (t)c1 (t)f1 ]x + c12 (t)f1 + c0 (t) − ϕt − ϕ2x f1 = 0.

(75) Equation (62) implies that h(x, t) must be independent of x. Our calculations show that Eq. (57) can be reduced to a constant coefficients, if it is in the following form: iψt + f (t)ψxx + g(x, t)|ψ|2 ψ = h(t)|ψ|4 ψ + is1 (x, t)[(|ψ|2 ψ)x − 3/2(|ψ|2 )x ψ] + [u(x, t) + iv(t)]ψ,

(76)

where g(x, t) = s1 (x, t)(xg0 (t) + g1 (t)) + g2 (t) for some g0 (t), g1 (t) and g2 (t) functions of t. One of the factors that influences the evolution of optical solitons is the second-order dispersion parameter, f (t), that we have shown it must be a function of t only, when variable coefficients are applicable, while for nonlinear terms such as cubic term parameter can be a function of both x, and t, and the quintic parameter a function of t only. For optical solitons their existence is a result of balance between the group velocity dispersion and nonlinear effects. Now to obtain solitary wave solution of (58), as the second step of the process, we use the following expression to obtain kink solitons of (58).

Dynamics of Solitons in High-Order Nonlinear …

 (X , T ) = m0

e2μ b0 + b1 e2μ

227

1/2

ei(m1 X −m2 T ) ,

(77)

where μ = m3 X − m4 T , m0 , . . . , m4 are real numbers, c0 is a complex number that controls that amplitude soliton. A direct substitution of (77) into (58) one can obtain all the parameter values as follows: m4 = 2m1 m3 w2 , m2 (m21 − m23 )w2 − w1 ,  |m0 |2 w3 − 4b1 m23 w2 m0 η(t) w4 , b = , m1 = 1 2|m0 |2 w6 m3 3w2

(78)

where c3 and b0 are arbitrary constants. In this case the solution of (57) is given by (56). To obtain bright solitary wave solutions, we let  (X , T ) = c0

eμ b0 + b1 eμ + b2 e2μ

1/2

ei(c1 X −c2 T ) ,

(79)

and to obtain dark solitary wave solution of (58), we let  (X , T ) = c0

b2 + 2b3 cosh μ b0 + 2b1 cosh μ

1/2

ei(c1 X −c2 T ) ,

(80)

in a similar manner as we did for kink soliton, we can find the parameter relations for the bright and dark solitary waves. Special sets of solutions: There are a number of free parameters in the obtained solution that each specific choice leads to a certain class of solutions. In the following we list a few examples: (a) A moving bright solitary wave is obtained by selecting ρ(x, t) = a(t)exp[−k12 (b(t) + xa(t))2 ], a(t) = 2b(t), η(t) = 3b(t), b(t) = [1 + cos2 (αt)]−1/2 ,

(81)

here α, , k1 are free real positive numbers. (b) For a breathing solitary waves solution, we use the same ρ as the above case, but we let a(t) = 1 + k2 cos(αt) + k3 sin(t), b(t) = 0, η(t) = 3a(t), here α, k2 , k3 are free real positive numbers.

(82)

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G.-A. Zakeri

(a) a moving bright soliton

(b) a moving dark soliton

(c) a dark breathing soliton

Fig. 1 Illustrating typical moving bright, moving M-dark, and breathing M-dark solitary waves given by (79) and (80). Since the cross-section (for t fixed) of the solitons in (b) and (c) are in the shape of M, they are referred as M-dark. Data used for panel a are α = 1,  = 0.5, k1 = 1.5 for panel b data are α = 1, k2 = 1.2, k3 = −1.4 and data for panel c are α = 1.5,  = 0.7, λ = 1.2

(a) a moving kink with amplitude decay soliton

(b) kink resonance solitary wave

Fig. 2 Illustrating a moving kink with decaying amplitude after kink occurs and a kink that causes resonance after kink occurs as given by (77) and (78). Data used for panel a are α = 1.1, λ = 0.5,  = −0.5, and for panel b are α = 1.5, λ = −0.2,  = 0.7

(c) For a quasi-periodic that causes resonance in the solitary wave, we select ρ(x, t) = [a(t) + λa(t)exp[−k12 (b(t) + xa(t))2 ]]−1/2 , a(t) = [1 + (1 + sin(αt))2 ]−1 , b(t) = 0, η(t) = 3a(t), T (t) = e−3t .

(83)

In Figs. 1 and 2, we have illustrated some samples of the above solutions as outlined above.

3.2 Hirota Method and Bilinear Transformation Let us consider propagation of ultrashort soliton pulses in an inhomogeneous optical fiber media given by iψz + iw1 (z)ψt + w2 (z)ψtt + iw3 (z)ψttt + f (z)|ψ|2 ψ + ig(z)|ψ|2 ψt + ih(z)ψ = 0,

(84)

and for the propagation of femtosecond soliton pulses in the inhomogeneous optical fiber media given by

Dynamics of Solitons in High-Order Nonlinear …

229

iψz + w2 (z)ψtt + iw3 (z)ψttt + f (z)|ψ|2 ψ + ig(z)|ψ|2 ψt + ih(z)ψ = 0,

(85)

where ψ(z, t) is the complex envelope of the electrical field, z, and t are direction of propagation, and the retarded time, respectively. For both Eqs. (84) and (85), w1 , w2 , w3 are the coefficients of dispersion terms, f , g, h representing the Kerr nonlinear parameter, the time-delaying effect parameter, and the inhomogeneous parameter related to the phase modulation and gain/loss parameter, respectively. To obtain the bilinear forms associated with Eq. (85). We let ψ=

u(z, t) exp(−i v(z, t)

 h(z)dz)

(86)

where u(z, t) is a complex-valued function and v(z, t) is a real-valued function. This particular choice of ψ given in Eq. (86) simplifies the resulting equation. The bilinear form for Eq. (85) is as follows:   Dz − iw2 (z)Dt2 + 2w3 (z)Dt3 + f (z)Dz3 − ig(z)Dz2 Dt u · v = 0, Dt2 v · v = 2u∗ · u,

(87)

here, Dz and Dt are the bilinear derivative operators defined by  Dzm Dtn u

·v =

∂ ∂ −  ∂z ∂z

m 

∂ ∂ −  ∂t ∂t

n

u(z, t)v(z  , t  )|z =z,t  =t ,

(88)

where z  and t  are formal variables, m and n are non-negative integers. The main idea here is to bilinearize Eq. (85) through certain transformation, by the truncated parameter expansion at different levels, a series of solutions, called the N-solitons for Eq. (85) can be obtained. Using Eqs. (87), the N-soliton solutions of Eq. (85) can be obtained from the following expansions, u(z, t; ) = u1 (z, t) + 3 u3 (z, t) + 5 u5 (z, t) + . . . , v(z, t; ) = 1 + 2 v2 (z, t) + 4 v4 (z, t) + 6 v6 (z, t) + . . . ,

(89)

where  is a formal expansion parameter, uj (z, t)’s are complex-valued and vj (z, t)’s are real-valued functions to be determined. Let us consider few examples. Example 1 To get 1-soliton (N = 1) solution of Eq. (85), we set all coefficients of powers of  ≥ 3 to be zero in (89) to get the following: |a|2 ∗ eθ+θ , ∗ 2 (k + k )  θ = kt − (2k 3 + ik 2 ) w3 (z)g(z)dz,

u(z, t : ) = aeθ , v(z, t : ) = 1 + 2

where a and k are complex constants, with  = 1, one-soliton is given as

(90)

230

G.-A. Zakeri  ψ(z, t) =

    a2 w2 (z) θ − θ∗ 1 |a|2 exp sech [θ + θ∗ + ln( )] −ih(z)dz. (91) 2f (z) 2 2 (k + k ∗ )2

Example 2 For 2-soliton (N = 2) solution of Eq. (85), we choose all coefficients of  ≥ 5 to be zero in (89) to get  ψ(z, t) =

2w2 (z) u1 + u3 f (z) 1 + v2 + v4

 −ih(z)dz,

(92)

where, ∗



u1 = a1 eθ1 + a2 eθ2 , u3 = a121 eθ1 +θ2 +θ1 + a122 eθ1 +θ2 +θ2 , v2 = a11 eθ1 +θ1 ∗ + a12 eθ1 +θ2 ∗ + a21 eθ2 +θ1 ∗ + a22 eθ2 +θ2 ∗ , ∗



v4 = a1212 eθ1 +θ2 +θ1 +θ2 , a12j =

a1 aj∗ (k1 − k2 )2

, a1212 = 2

(k1 + kj∗ )2 (k2 + kj∗ )  θj = kj t − (2kj3 + ikj2 ) w3 (z)g(z)dz,

|a1 |2 |a2 |2 (k1 − k2 )2 (k1∗ − k2∗ )2 , (k1 + k1∗ )2 (k2 + k1∗ )2 (k1 + k2∗ )2 (k2 + k2∗ )2 (93)

where aj , θj , and kj are complex constants for j = 1, 2. This process can be continued to obtain N-solitons, for detail calculations of the general formulas, see [70].

3.3 N-Soliton Interactions via Bilinear Operators In order to better illustrate the procedure of bilinear operators for obtaining N-soliton solutions, we consider the following NLSE: 1 iψt + ψxx + ψ|ψ|2 − iα3 (ψxxx + 6ψx |ψ|2 ) 2 (94) ∗ 2 + γ(ψxxxx + 6ψx2 ψ ∗ + 4ψ|ψx |2 + 8ψxx |ψ|2 + 2ψxx ψ + 6ψ|ψ|4 ) = 0. This method is applicable to problems arising in any media that deals with interactions of biological system, anisotropic Heisenberg ferro-magnetic spin chain with possibly different magnetic interactions, the alpha helical proteins, and so on [71]. Here, ψ(x, t) is the slowly varying envelope of wave propagating in the x direction and time t. In optical fiber problems the role of x and t is reversed. α3 and γ are real values of parameters effecting third and fourth order dispersion terms as indicated in Eq. (94). Let ψ = g(x, t)/f (x, t) where g is a complex-valued and f is a real valued

Dynamics of Solitons in High-Order Nonlinear …

231

function, we apply the differential operator (88) introduced by Hirota [65] on (94), we get   1 3 2 f Dx (g · f ) − f 2 gDx2 (f · f ) + f 2 g 2 g ∗ + γ f 3 Dx4 (g · f ) 2   2 2 2 4 − f gDx (f · f ) + 6g Dx2 (f · f ) − 6fDx2 (g · f ) Dx2 (f · f ) + 6g ∗ Dx2 (g · f )    + 4gDx (g · f ) Dx (g ∗ · f ) + 8gg ∗ fDx2 (g · f ) − gDx2 (f · f ) + 2g 2 fDx2 (g ∗ · f )    −g ∗ Dx2 (f · f ) + 6g 3 g ∗2 − iα3 f f 2 Dx3 (g · f ) − 3Dx (g · f ) Dx2 (f · f )  +6gg ∗ Dx (g · f ) = 0.

if 3 Dt (g · f ) +

(95)

It is known that a “balance” between 21 f 2 gDx2 (f · f ) from dispersion term and f 2 g 2 g ∗ from the cubic nonlinearity, guarantees the existence of bright-soliton [72]. This requirement leads to (96) Dx2 (f · f ) = 2gg ∗ , in addition, using the algebraic relation [73],  2  2  2 Dx (f · f ) Dx (f · f ) Dx4 (f · f ) = + 3 , f2 f2 f2 xx

(97)

Equation (95) reduces to   f iDt + 1/2Dx2 + γDx4 − iα3 Dx3 (g · f ) − 3g ∗ Dx2 (g · g) = 0.

(98)

Now, by letting Dx2 (g · g) = sf ,

(99)

where s(x, t) is an auxiliary function, we can simplify (98) to the following:   iDt + 1/2Dx2 + γDx4 − iα3 Dx3 (g · f ) − 3g ∗ s = 0.

(100)

Equations (96), (99)–(100) form a system equations for f , g, s whose solutions are N-solitons. To determine f , g and s, we expand f , g, s in terms of a formal parameter  and then let  to be 1: g=

N  n=1

2n−1 g2n−1 , f = 1 +

N  n=1

2n f2n , s =

N 

2n s2n .

(101)

n=1

Now using the expansion of s into the auxiliary equation (99), we get the following recursive relations for each n in 1 < n ≤ N :

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G.-A. Zakeri

  2 s2 = 2 g1xx g1 − g1x s2n =

2n 

(102)

    2 g(2j−1)xx g2n−2j+1 − g(2j−1)x g(2n−2j+1)x − s2j f2n−2j .

(103)

j=1

Then the N-soliton solutions of (94) are given by N ψ=

1

n=1 g2n−1 .  + Nn=1 f2n

(104)

Example 1 Let N = 1 for 1-soliton, g = g1 , f = 1 + 2 f2 and s = 2 s2 . Substituting these into (96), (99) and (100) and equating like powers of , we get ∗

g1 = a1 eθ1 , f2 = a11 eθ1 +θ1 ,   θ1 = k1 x + α3 k13 + i(γk14 + 1/2k12 ) t + θ10 ,

(105)

here k1 and θ10 are some complex constants. Thus the 1-soliton solution of (94) is given by ψ(x, t) =

 √  η1 g1 = √ eiIm(θ1 ) sech Re(θ1 ) + ln a11 , 1 + f2 2 a11

(106)

where a1 = ηη ∗ /(k1 + k1∗ )2 , and η is a free complex constant. Example 2 Let N = 2 for 2-soliton, g = g1 + 3 g3 , f = 1 + 2 f2 + 4 f4 and s = 2 s2 + 4 s4 . Substituting these into (96), (99) and (100) and equating like powers of , we get g1 =

2  i=1

f2 =

2 

ai eθi , g3 =

2 



a12i eθ1 +θ2 +θi ,

i=1

∗ ∗ ∗ aij eθi +θj , f4 = a1212 eθ1 +θ2 +θ1 +θ2 , (107)

i,j=1

1 g1 + g2 θ2 = k2 x + (2α3 k23 + i(2γk24 + k22 ))t + θ20 , ψ(x, t) = , 2 1 + f2 + f4 where θ1 as given in (105) and θ20 is a constant complex number. We can continue in this manner and obtain the expressions for 3−, 4−, . . . Nsolitons. For a more complete list of these N-solitons see [70, 71]. Analyzing these N-solitons, they reveal that collisions among solitons take different forms, such elastics and non-elastics depend on the values of γ and α3 . For example, collisions corresponding to negative values of α3 have more “breathing” phenomena or“ripples” after head-on collision compared with positive values of α3 when all other parameters are kept the same. In the case of 2-solitons, soliton velocities

Dynamics of Solitons in High-Order Nonlinear …

(a) solitons before collision

(b) solitons collision

233

(c) after collision

Fig. 3 Illustrating dynamics of collision of a forward moving soliton into a standing soliton which causes ripples during elastics collision

(a) solitons before collision

(b) solitons collision

(c) after collision

Fig. 4 Illustrating dynamics of collision of two solitons moving toward each other in an elastic collision without causing any ripples during collision

(a) solitons before collision

(b) solitons collision

(c) after collision

Fig. 5 Illustrating dynamics of collision of three solitons moving toward each other in an elastic collision without causing ripples during collision

v1 and v2 depend on the wave numbers, k1 , k2 , α3 and γ. The head-on collision occurs when v1 v2 < 0 and overtaking occurs when v1 v2 > 0. If one of the solitons is standing, say S2 , i.e., v2 = 0 then the collision is elastic and S1 passes through S2 without any changes. These phenomena are illustrated in Figs. 3, 4, 5, 6 and 7. Data used in Fig. 3: θ1 = 0.5, θ2 = 0.7, γ = 0.25, α3 = −1, k1 = −0.5 + i, k2 = 0.5 − 0.5i. For Fig. 4, same data as Fig. 3 except k1 = 0.75 + 0.25i, k2 = 1/3 + 2/3i. Data for Fig. 5: θ1 = 0.2, θ2 = 0.5, γ = 0.25, α3 = −1, k1 = −0.5 + i, k2 = 0.5 − 0.5i. For Fig. 6, we used θ1 = 0.2, θ2 = 0.5, θ3 = 0.7, γ = 0.01, α3 = 1, k1 = 0.25 + 0.09i, k2 = 0.4 + 0.07i, k3 = 0.8 + 0.05i. For Fig. 6, we used θ1 = 0.5, θ2 = 0.7, θ3 = 0.1, θ4 = 0.3, γ = 0.01, α3 = 1, k1 = −0.6 + 0.2i, k2 = 0.6 − 0.3i, k3 = 0.8 + 0.5i, k4 = −0.8 − 0.4i.

234

G.-A. Zakeri

(a) solitons before collision

(b) solitons collision

(c) after collision

Fig. 6 Illustrating dynamics of collision of four solitons moving toward each other in an elastic collision without causing no ripples during collision

(a) 2-soliton collision

(b) 3-solitons collision

(c) 4-soliton collision

Fig. 7 Plots of wave amplitude squared with respect to x and t corresponding to the same data as in Figs. 3, 4, 5 and 6. In an elastic head-on collision, ripples or breathing occurs. However, stable solitons maintain their shape and amplitude after collision, which serves as evidence of their stability

4 Modulational Instability Historically, modulational instability (MI) of a traveling wave was observed in a sequence of experiments by Benjamin and Feir over several years in early 1960’s. In their experiments, they assumed that the Stokes wave was stable, however, over and over, Benjamin-Feir observed a new kind of instability which was the occurrence of sidebands resulting from perturbations. These observations were also noticed around the same time in nonlinear optics. MI occurs when a plane wave is traveling under the influence of nonlinearity together with dispersions. MI occurs when a strong harmonic carrier wave and small sidebands displaced a little in frequency which leads to the breakup of plane wave into a train of ultra-short pulses. This is a symmetry breaking instability that a small perturbation on top of a constant amplitude carrier plane wave experiences exponential growth. Many nonlinear systems exhibit a type of instability that leads to modulation of the steady state as a result of an interplay between nonlinearity and dispersive effects. MI is the cause for various physical interesting effects such as formation of envelope solitons in electrical transmission lines, nonlinear optical fibers, dielectric media, and the break-up of monochromatic ocean waves. The effect of MI is considered to be the main mechanism of rogue wave formation in all media. MI is an indispensable mechanism for the understanding of some relevant dynamical processes in generation and propagation of solitary waves, matter-wave transport, atomic number squeezing. It is of great interest to find under

Dynamics of Solitons in High-Order Nonlinear …

235

which conditions isolated pulses could be formed during the evolution of the wave in the system governed by a NLSE. We establish the conditions under which a uniform wave train governed by NLSE will become unstable to a small initiated perturbation. For a more recent account of MI to get a broader picture of importance of MI analysis see [74] and reference therein. For an illustrating example of investigating MI, let us consider the following simpler version of cubic-quintic of (8) which describes a homogeneous Gross-Pitaeviskii (GP) equation [75]. GP equation has many applications in areas such as in nonlinear optics and more recently has been used to study in Bose-Einstein condensate (BEC) to study nonlinear patterns which can exist in them, including dark and bright solitons. We let iψt + r1 ψx + r2 ψxx + r3 |ψ|2 ψ + r4 ψ + r5 |ψ|4 ψ = 0,

(108)

where r1 is group velocity, r2 is second order dispersion coefficient, r3 is controlling cubic nonlinearity, r4 is controlling linear gain and acts as stabilizing the soliton propagation, and r5 is controlling the quintic nonlinearity effect. The main idea of the MI investigation is to add perturbations to the amplitude of a plane wave with the frequencies close to the carrier wave frequency and then use a linearized analysis to show that for certain parameter values, the perturbations grow exponentially. Let’s consider the following plane wave ψ0 = Aei(kx−wt) ,

(109)

where amplitude A, wave number k and the angular frequency w are real constants. Substituting (109) into (108) requires that r5 A4 + r3 A2 + r4 − r2 k 2 = 0, w = r1 k.

(110)

Now, we perturb the amplitude as follows ψ = [A + u(x, t)]ei(kx−wt) ,

(111)

where  is a small positive parameter, and |u(x, t)|  1 representing small perturbation to the amplitude of plane wave (109). Substituting (111) into (108) and linearizing the resulting equation, we get iut + r1 ux + r2 uxx + 2ir2 kux + A2 (r3 + 2r5 A2 )(u + u∗ ) = 0.

(112)

For (112), we seek a solution in the form of ∗

u = u1 ei(κx−τ t) + u2∗ e−i(κx−τ t) ,

(113)

where, κ and τ are wave number and frequency of low-frequency perturbations modulating the carrier signal, respectively. u1 and u2 are complex constants and their

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Fig. 8 Profile of modulational gain as a function of κ for a fixed dispersion coefficient and two fixed values of k = 8, 15 as solid and dashed lines, respectively. On the horizontal axis κ looks like an “K”

modulus represent wave amplitude. Substituting (113) into (112), we get (2r5 A4 + r3 A2 − r2 (2kκ + κ2 ) + iτ )u1 + (r3 + 2r5 A2 )A2 u2 = 0, (r3 + 2r5 A2 )A2 u1 + (2r5 A4 + r3 A2 + r2 (2kκ − κ2 ) − iτ )u2 = 0.

(114)

Equation (114) constitute a homogeneous system, for any non-trivial solutions, its determinant must be zero which gives τ = −ir2 (2kκ + κ2 ).

(115)

The quantity τ depends on the values of the parameters given by dispersion r2 , and k and κ the wave numbers of plane wave and perturbed wave. The spectrum gain can be positive or a negative number. Negative value indicates the growth rate Imτ which means the stability of the system since exp[(Imτ )t] vanishes as t → ∞ and the system remains stable under modulation of the frequency. On the other hand, positive values of spectrum are signature of instabilities and the small perturbation increases without any limit as time t increases [76, 77]. Such a situation is referred to the modulationally unstable and a typical of such symmetry breaking is illustrated in Fig. 8. Here we have shown that when a continues plane wave is subject to modulation under the effect of nonlinearity in combination with dispersion. In modulation instability process, weak perturbation imposed on a plane wave state grow exponentially due to the interplay between these nonlinear and dispersive effects, which is shown in Fig. 8.

5 Stability Analysis For all applications only stable solutions are of practical use physical experiments. Small fluctuations of amplitude of soliton results from inhomogeneity in manufacturing fiber optics makes it necessary to completely analyze the stability of obtained solutions of NLSE. There are several approaches to linear stability analysis. One approach to seek a soliton solution of an interested NLSE in the following form: ψ(x, t) = u(x)eiφ(x,t) ,

(116)

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where u(x) is a localized function of x representing amplitude and φ(x, t) is a realvalued function of propagation. These solitons can be evaluated by direction substitution, squared operator iteration or the Newton-conjugate-gradient method, just to name a few [78]. Once the conditions on u(x) and φ(x, t) are determined, to analyze the linear stability, we perturb (116) as follows:  ∗  ψ(x, t) = u(x) + v(x)eλt + w(x)eλ t eiφ(x,t) ,

(117)

where  is a small parameter, and |v|, |w|  |u|. Substituting (117) into the NLSE and keeping terms of order , we get an eigenvalue problem for v, w, in general, in the following form,  i

ab cd



v w



 =λ

 v , w

(118)

where a, b, c, d are some differential operators. If eigenvalues, λ have positive real parts, then the obtained solitary wave is linearly unstable; otherwise, it is linearly stable. The next step in the process of determining stability of a NLSE is to include the effect of nonlinear terms that are neglected in linear stability analysis. Those nonlinear terms control the interactions of higher order terms that play an important role in the physics of the problem. So it is necessary to consider the NLSE using some numerical methods by introducing a small random noise in the initial condition and study its effect of the solution. One of the most straight forward method is to use split-step Cranck Nicolson method. For some specific examples of the above process see [75]. Example 1 Consider the following general inhomogeneous NLS equation iψt + f (x, t)ψxx + g(x, t)|ψ|2 ψ + ν(x, t)ψ + iΓ (x, t)ψ = 0.

(119)

In the case f (x, t) is a constant, Eq. (119) is applicable to the mean field GPE, where ν(x, t)is the trapping potential and Γ (x, t) has small value in general describes the gain or loss of atoms in the condensate. Linear stability analysis. Solutions of Eq. (119) can be expressed as ψ(x, t) = (x, t)eiφ(x,t) .

(120)

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Upon substituting this expression in Eq. (119), we obtain the following equations: −φt + f (x, t)(xx − φ2 ) + g(x, t)||2  + ν(x, t) = 0, t + f (x, t)(2x φx + φxx ) + Γ (x, t) = 0.

(121)

If  is time independent, then the solution given by (120) describes the plane wave solutions. Under this condition, the second differential equation in (121) can be integrated with respect to x, provided that Γ (x, t)/f (x, t) is time independent. In this case, we get     1 Γ 2 dx dx. (122) φ=− 2 f This shows that φ is time independent and thus all plane wave solutions of (119) in the form of (120) are linearly stable. On the other hand, if Γ (x, t)/f (x, t) is time dependent, then we introduce an amplitude perturbation to the solution of (120) and we write ψ(x, t) = ((x) + ψ(x, t))eiφ(x,t) . (123) Here, we assume that   1, i.e., ψ is a small perturbation added to the amplitude . Substituting Eq. (123) into Eq. (119) and using ψr = Re(ψ) and ψi = Im(ψ) and keeping terms of order of O(), we find the following linearized equation: Ut = JLU,

(124)

where  U (x, t) =

     0 1 ψr (x, t) −L2 −S ,J = ,L = ψi (x, t) S L1 −1 0 L1 = φt − ν − g 2 − f ∂xx , L2 = 2g 2 − L1 , S=f

(φ2x

− φxx − 2φx ∂x ) − Γ,

(125) (126)

where L1 and L2 are Hermitian Schrödinger operators and S is the anti-Hermitian operator. Now, let the solution of Eq. (124) be in the form of U (x, t) = eλt U˜ (x) = eλt (ψ˜ 1 (x), ψ˜2 (x))T . Substituting this into Eq. (124), we get the following eigenvalue JLU˜ = λU˜ .

(127)

Since S is anti-Hermitian operator, the eigenvalues of Eq. (127) are complex. If the real part of each eigenvalue of (127) is zero, Re(λ) = 0, i.e., all λ are purely imaginary or zero, then the corresponding solutions is stable. In contrast, if at least one of the eigenvalues has a positive real part, Reλ > 0, then instability occurs. We note that Eq. (119) is phase invariant and remains unchanged. Now, using Noether’s theorem [79], and the same procedure as in [80], the zero mode of Eq. (127) which corresponds

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to λ = 0 is obtained from letting U˜ = (0, (x))T which implies L1 (x) = 0 and S(x) = 0. This implies that zero is an eigenvalue of Eq. (127). The remaining eigenvalues can be obtained numerically using “Fourier collocation method for the whole spectrum,” for details see Sect. 7.3.1 of [78]. We note that L1 (x) = 0 is a Hermitian operator of the Sturm-Liouville type and in the case of periodic coefficient functions, all its eigenvalues are real. On the other hand, from Eq. (125), we get L1 + L2 = 2g 2 that if g > 0 then its eigenvalues are positive and L1 + L2 is a positive definite operator and hence the Hermitian operator H = (L1 + L2 )1/2 L1 (L1 + L2 )1/2 can be constructed and thus (H + λ2 )ζ = 0. Now, if we let μ0 = inf ζ =1 < ζ H ζ > and μ ≥ 0 and λ01 = inf ζ =1 < ζ L1 ζ >= 0, be the ground state eigenvalue of L1 then λ2 ≤ 0. Under these conditions the eigenvalues of Eq. (127) are either purely imaginary or zero and the solutions in the form of (120) are stable. If μ0 < 0 and λ0 = 0 then λ2 > 0 and in this case the solutions of Eq. (125) are unstable because of the presence of at least one positive real eigenvalue. If g < 0, then L1 + L2 is not a positive definite operator, and since λ1 is zero, the eigenvalues of L1 are non-positive. Similar arguments apply to S(x) = 0, and the sign of f φx − f (ln(φx ))x − Γ /φx determines the sign of the eigenvalues of S. Numerical simulation for stability For the above linear stability analysis: by keeping only terms of O(), we have neglected the effect of nonlinear terms to get (124). Those nonlinear terms describe the interactions of the higher order terms that play important roles in the physics of the problem. Thus for analyzing the stability of solutions of (119), we use the split-step Cranck-Nicholson: where the spatial derivative parts are carried out by Fourier transforms. The time evolution consisting of the Cranck-Nicholson integration method is calculated over a large region in the (x, t)-plane to minimize the error caused by the boundaries of the region. Thus for each case, we run the numerical simulations over one and a half times the region to be tested. The method is based on a finite-difference method. In order to obtain the numerical solutions, we introduce a 5% white noise in the form of random perturbation in the initial condition. We consider the propagation of an initial curve φ(x, 0) with its related perturbed curve φpert (x, 0) = φ(x, 0)(1 + 0.05˜x) over a region corresponding to an exact solution (where x˜ is a random variable over the interval [0, 1]). For the sake of simplicity, we take all of the coefficients in equation (119) to equal cos 3t. In each case, we take the exact solution at time t = 0 to be the initial value curve for that case. For all numerical simulations we have used x = 0.01, and t = 0.01. Various numerical simulations show that the obtained numerical solutions remain stable under small amounts of white noise. In Fig. 9, panels a, b, c, and d represent samples of three different snap shots for numerical solutions for dark, bright, kink, and anti-kink, respectively. %5 random noise was introduced in the initial data, and we allowed the program to run for a long time to make sure that instability does not occur. Our numerical solutions show that the obtained solutions were stable.

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(a) snap shots of a dark soliton

(b) snap shots of a bright soliton

(c) snap shots of a kink soliton

(d) snap shots of an antikink soliton

Fig. 9 Plots of wave amplitude squared with respect to x and t corresponding to Eq. (119), using %5 random noise. The initial noise introduced does not grow over a long time (t = 15, for t = 0.01)

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MHD Mass Transfer Flow Past an Impulsively Started Semi-Infinite Vertical Plate with Soret Effect and Ramped Wall Temperature N. Ahmed

Abstract An exact solution to the problem of a hydromagnetic natural convective mass transfer flow of an incompressible viscous electrically conducting non-Gray optically thin fluid past an impulsively started semi-infinite vertical plate with ramped wall temperature in presence of appreciable radiation, thermal diffusion and uniform transverse magnetic field is presented. The magnetic Reynolds number is assumed to be small enough to neglect the induced hydromagnetic effects. Closed form Laplace Technique is adopted to get the exact solutions of the resultant non-dimensional governing equations. The influences of thermal radiation, ramped parameter, magnetic field, thermal diffusion and time on the flow and transport characteristics are studied graphically. Keywords Optically thin · Thermal radiation · Thermal diffusion · Natural convection AMS Subject Classification 76W05 PACS 44.27. +g

Nomenclature B B0 C Cp C∞ Cw

Magnetic flux density; Strength of the applied magnetic field, Tesla or Weber ; m2 ; Molar species concentration, kmol 3 m Specific heat at constant pressure, kgJK ; Concentration far away from the plate, kmol ; m3 Species concentration at the plate, kmol ; m3

N. Ahmed (B) Department of Mathematics, Gauhati University, Guwahati 781014, India e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_8

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DM DT ebλ g g Gr, Gm J Kλ, M p Pr q qr qr Q Ra Sc Sr t t0 T Tw T∞ u, x  U0     x ,y ,z

2

Mass diffusivity, ms ; 2 Molar thermal diffusivity, m Kkmol ; s Planck function; Gravitational acceleration vector; Acceleration due to gravity, sm2 ; Thermal Grashof number; Solutal Grashof number; Current density vector; Absorption coefficient; m1 Magnetic parameter; Pressure, mN2 ; Prandtl number; Fluid velocity vector; Radiative heat flux vector; Radiation heat flux, mW2 ; Radiation parameter; Ramped parameter; Schmidt number; Soret number; Time, s; Characteristic time, s; Temperature, K; Isothermal temperature, K; Temperature far away from the plate, K; Component of fluid velocity, ms ; Plate velocity, ms ; Cartesian coordinate system, (m, m, m);

Greek symbols σ ρ ρ∞ μ κ β β υ ϕ,

Electrical conductivity, kg ; m3

1 ; (Ohm×m)

Fluid density, Fluid density far away from the plate, mkg3 ; kg Ns or m Coefficient of viscosity, ms 2; W Thermal conductivity, m K ; Coefficient of thermal expansion, K1 ; 1 Coefficient of solutal expansion, kmol ; 2 m Kinematic viscosity, s ; Viscous dissipation of energy per unit volume,

J ; m3

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Subscripts w Refers to physical quantities at the plate; ∞ Refers to physical quantities far away from the plate;

1 Introduction MHD is a branch of Physics which is concerned with the motion of electrically conducting fluids in presence of magnetic field. MHD pumps, MHD generators and MHD flow meters are some examples of MHD principles. Dynamo and motor is a classical example of MHD principle. The problems dealing with convection in hydromagnetic flows have got importance in Geophysics, Astrophysics, Plasma Physics, Missile technology etc. The MHD principles also find its applications in medicine and biology. The present form of MHD is based on the pioneer contributions of several notable authors like Alfven [9], Cowling [10], Shercliff [28], Ferraro and Plumpton [16] and Crammer and Pai [12]. The natural flow arises in fluid when the change in temperature as well as species concentration causes density variation leading to the existence of buoyancy forces acting on the fluid. Free convection or natural convection is a process of heat or mass transfer in natural flow. The heating of rooms and building by use of radiator is an example of heat transfer by natural convection. On the other hand, the principles of mass transfer are relevant to the working of systems such as a home humidifier and the dispersion of smoke released from a chimney into the environment. The evaporation of alcohol from a container is an example of mass transfer by natural convection. Radiation is also a heat transfer process through electromagnetic waves. Convective flows in presence of radiation are encountered in different industrial and environment processes. At the high operating temperature, the radiation effect can be quite significant. Many processes in engineering and industrial areas occur at high temperatures, and the knowledge of radiation heat transfer becomes very important for design of pertinent equipment. Heat transfer problems become more fruitful from the physical point of view when the simultaneous mass transfer effects on flow are also taken into account. Several authors have carried out model studies on the problems of free convective hydrodynamic and hydromagnetic flows under different flow geometries and physical conditions taking into account of thermal radiation. Some of them are Takhar et al. [35], Mansour [18], Raptis and Perdikis [25], Mankinde [20], Samad and Rahman [27], Prasad et al. [23], Mbeledogu et al. [21], Orhan and Ahmet [22], Seth et al. [34], Ahmed [4, 6], Ahmed et al. [5], Ahmed and Dutta [2, 3]. An analytical solution to the problem of unsteady MHD freeconvection flow with Hall effects of a radiating and heat absorbing fluid past a moving vertical plate with variable ramped temperature was obtained by Seth et al. [29]. Seth et al. [30] investigated analytically the problem of heat and mass transfer effects on unsteady MHD natural convection flow of a chemically reactive and radiating

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fluid through a porous medium past a moving vertical plate with arbitrary ramped temperature. In the cases, when both heat and mass transfer occur simultaneously in a moving fluid, the relations connecting the fluxes and driving potentials are of a more intricate nature. It is seen that a mass flux can be generated not only by composition gradient but by temperature gradient as well. The effect concerned with the mass flux under temperature gradient is termed as the Soret effect. The experimental investigation of this effect was first performed by renowned Chemist Charles Soret in 1879 and so this effect is known as the Soret effect in the honour of his name. Soret effect is concerned with the method of separating havier gas molecules from lighter ones by ramped temperature gradient over a given volume of a gas containing particles of different manners. Soret effect is also termed as thermal diffusion effect. Roughly speaking, thermal diffusion effect deals with the mass flux as a result of temperature difference. Comprehensive literature on various aspects of thermal diffusion and diffusion-thermo on different kinds of mass transfer related problems can be found in Eckert and Drake [15], Kafoussias and Williams [17], Postelnicu [24], Ahmed [4, 6, 8], Ahmed and Sengupta [7]. Laplace transform technique is an integral transform technique. It is generally used in case of the problems dealing with impulsively started flow. It also finds its application in case of small Reynolds number flow problems. That is for slow motion or creeping motion. Solutions obtained under the scheme of Laplace transform are generally exact or of closed form. Therefore it does not require stability analysis for checking of validation. Owing to this fact, a good number of researchers have carried out their research works by adopting this method in different kinds of flow problems under different physical and geometrical conditions. Some of them are Ahmed [6], Ahmed et al. [5], Ahmed and Dutta [3], Seth et al. [29–33], Das and Jana [14], Agarwalla and Ahmed [1], Mahanthesh et al. [19], Das et al. [13] and Shah et al. [26]. As the present author is aware, no attempt has been made till now to investigate the problem of natural convective hydromagnetic flow of an electrically conducting viscous incompressible non-Gray optically thin fluid past an impulsively started semi-vertical infinite plate with ramped wall temperature in presence of thermal diffusion, thermal radiation, and uniform transverse magnetic field taking the characteristic time independent of flow properties. Such an attempt has been made in the present work. It is worthwhile to mention that Equations and Solutions of the problem in hand in absence of mass transfer effect are consistent to those of the work of Ahmed and Dutta [3].

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2 Gray and Non-gray Gases The optimal thickness of a material is defined by   I0 , τ  loge I¯ where I0 is the original intensity of the beam of light and I¯ is the intensity of light after passing through the material. A gas is called optically thin if τ  1 and it is said to be optically thick if τ  1. Optical thickness is a dimensionless quantity and it measures the capacity of a particular material. A gas is said to be a Gray gas if the optical thickness τ of the gas is independent of the wave number of electromagnetic radiation. Otherwise, the gas is said to be non-Gray. In general, the commonly found atmospheric gases are non-Gray. In Cogley’s [11] model, the rate of radiation heat flux qr in optically thin non-Gray gas near equilibrium is specified by ∇ . qr  4I (T − T∞ ), where 

∞ I 

(K λ )w 0

∂ebλ ∂T

 dλ. w

3 Inverse Laplace Transforms of Some Special Functions   I. L −1 e−as f (s)  H (t − a)F(t − a), √

y2 II. L −1 e− s y  √y 3 e− 4t , 2 πt 2

−at e −1 √ 1  √πt , III. L √ s+a

er f c( at ) √ IV. L −1 s √1s+a  , −√ s y

a

y e  er f c 2√t , V. L −1 s −√ s y



y2 VI. L −1 es √s  2 πt e− 4t − y er f c 2√y t , −√s y − y2 (4t+y2 )

  4t VII. L −1 es 2 √s  πt e s − 16 y 6t + y 2 er f c 2√y t , −y√sξ 



√ 2  2 y ξ − y4tξ VIII. L −1 e s 2  λ(ξ, y, t)  t 1 + y2tξ er f c y2√ξt − √ , e πt

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N. Ahmed

 L

−1



e−

s

IX.  L

√  s+ηy ξ

−1



e−

√ s+η ξy

s2

 √  ξy √ 1 √ξηy e er f c √ + ηt  ψ(ξ, η, y, t)  2 2 t √   √ ξy √ +e− ξηy er f c √ − ηt , 2 t   f (ξ, η, y, t)

√   √  t y ξ y √ξ η y ξ √  er f c √ + η t + √ e 2 4 η 2 t √    √  √ y ξ t y ξ √ − √ e−y ξη er f c √ − η t , + 2 4 η 2 t    √  √ 2 −y s+a η + at t e η t(1 − 4at) L −1 ϕ(η, a, t) −  ϕ(η, ¯ a, t) √ s3 4a 8a a XI. ηt 2 − √ e−(η +at ) ; 2a π √ √   √  √  where η  2√y t , ϕ(η, a, t)  e2η at er f c η + at + e−2η at er f c η − at and √ √   √  √  ϕ(η, ¯ a, t)  e2η at er f c η + at − e−2η at er f c η − at . 

X.

4 Heaviside’s Unit Step Function, Error Function, and Complementary Error Function Heaviside’s unit function is defined by  H (t − a) 

0, t < a . 1, t > a

The Error function is defined as 2 er f (t)  √ π

t

e−u du. 2

0

The complementary Error function is defined as 2 er f c(t)  1 − er f (t)  1 − √ π

t 0

2 2 e−u du  √ π

∞ t

e−u du. 2

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5 Some Properties of Error Function and Complementary Error Function I. er f  (0)  II. III. IV. V. VI. VII. VIII.

√2 , π  − √2π ,

er f c (0) er f (0)  0, er f c(0)  1, er f (x) + er f (−x)  0, er f c(x) + er f c(−x)  2, er f c(x) − er f c(−x)  −2er f (x), −z 2 er f c (z)  − 2e√π ,

IX. er f  (z) 

2

−z 2e √ π

,

X. er f c (z) + er f c (−z)  −4



1 −z 2 e . π

6 Mathematical Formulation of the Problem Equations governing the motion of an incompressible, viscous, electrically conducting radiating fluid in the presence of a uniform transverse magnetic field taking into account of the effect of thermal diffusion are: Continuity equation ∇ · q  0,

(1)

Magnetic field continuity equation ∇ · B  0,

(2)

Ohm’s law for moving conductor J  σ (E + q × B), MHD momentum equation with buoyancy force   ∂q ρ + (q · ∇)q  −∇ p + J × B+ρ g+μ ∇ 2 q, ∂t 

(3)

(4)

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Energy equation  ρ Cp

 ∂T J2 2 − ∇ · qr , + · ∇)T  κ∇ T + ϕ + (q ∂t  σ

(5)

Species continuity equation ∂C + (q · ∇)C  D M ∇ 2 C + DT ∇ 2 T. ∂t 

(6)

All the physical quantities are defined in the Nomenclature. Consider a transient natural convective hydromagnetic mass transfer flow of an incompressible, viscous and electrically conducting optically thin non-Gray fluid past a semi-infinite vertical plate in the presence of a transverse magnetic field of uniform strength B0 . Initially the plate and surrounding fluid were at rest at the same temperature T∞ with constant concentration C∞ at all the points in the fluid. At time t   0+ , the plate starts to move suddenly in its own plane with speed U0 . The concentration near the plate rises to Cw and the plate temperature is instantly  raised linearly to Tw + (Tw − T∞ ) tt0 for 0 < t  ≤ t0 , and the constant temperature  Tw (Tw > T∞ ) is maintained at t > t0 . In order to get the mathematical model of the theoretical problem idealized, impose the following restrictions: I. All the fluid properties are constant except the density in the buoyancy force term. II. Viscous and Ohmic dissipation of energy are negligible. III. Magnetic Reynolds number is small enough to neglect the induced magnetic field in comparison to the applied magnetic field. IV. Flow is parallel to the plate. V. Plate is electrically non-conducting. VI. Radiation heat flux in the direction of the plate is negligible in comparison to that in the normal direction. VII. No external electric field is applied for which the polarization voltage is neg

ligible leading to E  0.   Introduce a rectangular Cartesian space-time coordinate system x  , y  , z  , t  with X axis along the plate in the upward vertical direction, Y axis normal to the plate directed into the fluid region and Z axis along the width of the plate. Let q  u  , 0, 0 the fluid  velocity and B  (0, B0 , 0) be the applied magnetic field at the point denote x  , y  , z  , t  in the fluid. The radiation heat flux vector qr is given by qr  (0, qr , 0).  Equation (1) gives ∂∂ux   0, which in turn yields,   u  u y, t  .

(7)

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253

The assumption (VII) and Eq. (3) lead the momentum Eq. (4) to take the form ρ

∂u  ∂ 2u ∂p  μ − ρ g −  − σ B02 u  . ∂t  ∂ y 2 ∂x

(8)

and 0

∂p . ∂ y

(9)

Equation (9) yields that p is independent of y  and so, along a normal to the plate, pressure near the plate is the same as that far away from the plate. Based on this fact, Eq. (8), for the fluid region far away from the plate reduces to ∂p  −ρ∞ g. ∂x By eliminating

∂p ∂x

(10)

from Eqs. (8) and (10), we derive ρ

∂u  ∂ 2u  μ + (ρ∞ − ρ)g − σ B02 u  . ∂t  ∂ y 2

(11)

Equation of state according to classical Boussinesq approximation is   ¯ − C∞ ) . ρ∞  ρ 1 + β(T − T∞ ) + β(C

(12)

Elimination of ρ∞ from Eqs. (11) and (12) gives 2  ∂u  ∂ 2u ¯ − C∞ ) − σ B0 u .  υ 2 + gβ(T − T∞ ) + g β(C  ∂t ∂y ρ

(13)

The assumption II and Cogley’s [11] assumption lead the Eq. (5) to transform to ρ Cp

∂T ∂2T  κ − 4I (T − T∞ ). ∂t  ∂ y 2

(14)

The species continuity Eq. (6) becomes ∂C ∂ 2C ∂2T  D M 2 + DT 2 .  ∂t ∂y ∂y

(15)

The relevant initial and boundary conditions (Ref. 2, 30) to be satisfied by the Eqs. (13), (14) and (15) are u   0, T  T∞ , C  C∞ ∀ y  > 0, t  ≤ 0,

(16)

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N. Ahmed

u   U0 , T  T∞ +

Tw − T∞  t , C  Cw at y   0, 0 < t  ≤ t0 , t0

(17)

u   U0 , T  Tw , C  Cw at y   0, t  > t0 ,

(18)

u  → 0, T → T∞ , C → C∞ as y  → ∞, t  > 0.

(19)

Now, for the sake of normalization of the flow model, the following nondimensional quantities are introduced. u θ

  ¯ u T (Tw −T∞ ) , y  Uy0 t0 , t  tt0 , Gr  υ gβ(TUw3−T∞ ) , Gm  υ gβ(CUw3−C∞ ) , Sr  Dυ(C , U0 w −C ∞ ) 0 0 2 μ Cp σ B0 υ U02 t0 T −T∞ C−C∞ 4I υ υ , φ  Cw −C∞ , Pr  κ , Q  ρ C U 2 , M  ρ U 2 , Sc  D M , Ra  υ . Tw −T∞ p 0 0

(20) In normalized form, the Eqs. (13), (14) and (15) reduce to 1 ∂ 2u ∂u  + RaGrθ + RaGmφ − Ra Mu, ∂t Ra ∂ y 2

(21)

1 ∂ 2θ ∂θ  − Ra Qθ, ∂t Ra Pr ∂ y 2

(22)

∂φ 1 ∂ 2 φ Sr ∂ 2 θ + .  ∂t RaSc ∂ y 2 Ra ∂ y 2

(23)

The initial and boundary conditions in non-dimensional form are u  0, θ  0, φ  0 ∀ y ≥ 0, t ≤ 0,

(24)

u  1, θ  t, φ  1 at y  0, 0 ≤ t ≤ 1,

(25)

u  1, θ  1, φ  1 at y  0, t > 1,

(26)

u → 0, θ → 0, φ → 0 as y → ∞, t > 0.

(27)

7 Method of Solution On taking Laplace Transforms of the Eqs. (21), (22) and (23) and conditions (24)–(27), the mixed initial and the boundary value problem reduce to a boundary value problem governed by the equations  d 2 u¯  ¯ − M Ra 2 + s Ra u¯  −Ra 2 Grθ¯ − Ra 2 Gmφ, dy 2

(28)

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255

d 2 θ¯ − Ra Pr(Ra Q + s)θ¯  0, dy 2 2¯ d 2 φ¯ ¯  −SrSc d θ , − RaSc φ dy 2 dy 2

(29) (30)

subject to the conditions u¯ 

 1 ¯ 1 1 , θ  2 1 − e−s , φ¯  at y¯  0, s s s

(31)

u¯  0, θ¯  0, φ¯  0 as y¯ → ∞.

(32)

The solutions of the Eqs. (28)–(30) subject to the conditions (31) and (32) are as follows  √ 1 θ¯  2 1 − e−s e− Pr Ra(Ra Q+s)y , s √

φ¯  φ¯ 1 e−

s RaScy



+ φ¯ 2 e−

Pr Ra(Ra Q+s)y

(33) ,

⎧ √ √ √ − Ra(M Ra+s)y − Pr Ra(Ra Q+s)y − RaScs y ⎪ u ¯ e + u ¯ e + u ¯ e ⎪ 1 2 3 ⎪ ⎪ ⎪ for Pr √1, Sc  1, Pr  Sc √ ⎪ ⎪ √ ⎪ ⎪ − Ra(M Ra+s)y ⎪ u ¯ e + u¯ 5 e− Pr Ra(Ra Q+s)y + u¯ 6 e− Ra Pr s y 4 ⎪ ⎪ ⎪ ⎪ for Pr √1, Sc  1, Pr  Sc√ ⎪ ⎪ √ ⎨ − Ra(M Ra+s)y + u¯ 8 e− Ra(Ra Q+s)y + u¯ 9 e− RaScs y ; u¯  u¯ 7 e ⎪ for Pr  1,√Pr  Sc ⎪ √ √ ⎪ ⎪ − Ra(M Ra+s)y ⎪ u ¯ e + u¯ 11 e− Pr Ra(Ra Q+s)y + u¯ 12 e− Ras y ⎪ 10 ⎪ ⎪ ⎪ for Sc = 1,√Pr  Sc ⎪ ⎪ √ √ ⎪ ⎪ − Ra(M Ra+s)y − Ra(Ra Q+s)y − Ras y ⎪ u ¯ e + u ¯ e + u ¯ e ⎪ 13 14 15 ⎪ ⎩ for Pr  1, Sc = 1

(34)

(35)

where

S¯2 S¯3 RaGr ¯ RaGr ¯ , u¯ 2  − Pr + RaGm − S S + RaGm S¯ , u¯ 3  − RaGm S¯ , u¯ 1  1s + Pr 1 −1 Sc−1 Pr −1 −1 1 Pr −1 3 Sc−1 2     A(s+Ra Q) 1−e−s 1 1 1 S¯1  s 2 (s−a 1 − e−s , S¯2  s−a + (Pr −Sc)s 2((s−a1 )) , s 2) 3   A(s+Ra Q)(1−e−s ) 1 1 RaGr ¯ RaGm ¯ ¯3 , S¯3  (s−a Q ¯  + + − Q Q 4 1 2 2 (s−a ) , u −Sc)s s Pr −1 Pr −1 ) (Pr 2 1   RaGr ¯ 1 u¯ 5  − Pr 1 − e−s , Q 1 + RaGm Q¯ 3 , u¯ 6  − RaGm Q¯ 2 Q¯ 1  s 2 (s−a −1 Pr −1 Pr −1 ) 2     Sr Pr(s+Ra Q)(1−e−s ) Q)(1−e−s ) 1 1 ¯ 3  1 Sr Pr(s+Ra , Q , Q¯ 2  s−a + 2 Ra Q 2 Ra Q s s s−a s 3

2 Gr P¯1 SrSc P¯2 + Ra(Q−M) u¯ 7  1s + Q−M + RaGm P¯3 , Sc−1 Gr P¯1 RaGmSrSc u¯ 8  − Q−M − Ra(Q−M)(Sc−1) P¯3 , u¯ 9  − RaGm P¯ , Sc−1 2

  SrSc(Ra Q+s)(1−e−s ) 1 1 , − P¯1  s12 1 − e−s , P¯2  s−a 2 s s (Sc−1)(s−a¯ 1 ) 3

256

P¯3 

N. Ahmed (Ra Q+s)(1−e−s ) , u¯ 10 s 2 (s−a¯ 1 )



1 s

+

RaGr ¯ S Pr −1 1

− RaGm



Pr Sr S¯5 S¯4 , + (Pr M Ra −1)2 Sr Pr(Ra Q+s)(1−e−s ) 1 + s 2 (Pr −1)(s−d1 ) , s

Pr Sr S¯5 RaGr ¯ u¯ 11  − Pr , u¯ 12  Gm S + RaGm S¯ , S¯4  −1 1 M 4 (Pr −1)2 −s (Ra Q+s)(1−e ) , u¯  1 + Gr T − Gm T − S¯  5

13 s 2 (s−a2 )(s−d1 ) s Q−M 1 M 2 Gr SrGm Gm u¯ 14  − Q−M T1 + Q Ra(Q−M) T3 , u¯ 15  M T2 , T1 Sr(Ra Q+s)(1−e−s ) (Ra Q+s)(1−e−s ) , T3  . T2  1s + Q Ra s 2 s2

SrGm T3 , Q Ra(Q−M)   1 −s  s2 1 − e ,

Solving the ordinary 2nd order linear differential equations subject to the conditions (31) and (32) and taking inverse Laplace Transforms of the solutions, the expressions for temperature θ, concentration φ and velocity field u are obtained as follows θ  ψ 1 − ψ2 ,  φ

φ1 + φ2 − φ3 for Pr  Sc , φ¯ 1 + φ¯ 2 − φ¯ 3 for Pr  Sc

⎧ 7  ⎪ ⎪ G α , Pr  1, Sc  1, Pr  Sc ⎪ ⎪ ⎪ α1 ⎪ ⎪ ⎪ 7  ⎪ ⎪ ⎪ Hα , Pr  1, Sc  1, Pr  Sc ⎪ ⎪ ⎪ ⎪ ⎨ α1 7  ; u Jα , Pr  1, Pr  Sc ⎪ α1 ⎪ ⎪ ⎪ 7  ⎪ ⎪ ⎪ K α , Sc = 1, Pr  Sc ⎪ ⎪ ⎪ α1 ⎪ ⎪ 7 ⎪ ⎪ ⎪ L α , Pr  1, Sc  1 ⎩

(36) (37)

(38)

α1

where A φ1  ψ5 , φ2  Pr −Sc [A1 (ψ3 − ψ4 ) + A2 (ψ5 − ψ6 ) + A3 (ψ7 − ψ8 )], A φ3  Pr −Sc [A1 (ψ9 − ψ10 ) + A2 (ψ11 − ψ12 ) + A3 (ψ1 − ψ2 )], φ¯ 1  φ1 , SrSc SrSc φ¯ 2  Ra [Ra Q(ψ7 − ψ8 ) + ψ5 − ψ6 ], φ¯ 3  Ra [Ra Q(ψ1 − ψ2 ) + ψ11 − ψ12 ], Q Q RaGr G 1  ψ13 , G 2  [A4 (ψ14 − ψ15 ) + A5 (ψ13 − ψ16 ) + A6 (ψ17 − ψ18 )], Pr −1   RaGm A ψ19 + ξ1 , G3  Sc − 1 Pr −Sc ξ1  A7 (ψ20 − ψ21 ) + A8 (ψ22 − ψ23 ) + A9 (ψ13 − ψ16 ) + A10 (ψ17 − ψ18 ), RaGm A ξ2 , G4  − (Pr −1)(Pr −Sc) ξ2  A11 (ψ20 − ψ21 ) + A12 (ψ14 − ψ15 ) + A13 (ψ13 − ψ16 ) + A14 (ψ17 − ψ18 ), RaGr ξ3 , G5  − Pr −1

MHD Mass Transfer Flow Past an Impulsively Started …

257

ξ3  A4 (ψ24 − ψ25 ) + A5 (ψ11 − ψ12 ) + A6 (ψ1 − ψ2 ),   RaGm A ψ26 + ξ4 , G6  − Sc − 1 Pr −Sc ξ4  A7 (ψ3 − ψ4 ) + A8 (ψ27 − ψ28 ) + A9 (ψ5 − ψ6 ) + A10 (ψ7 − ψ8 ), RaGm A ξ5 , G7  (Pr −1)(Pr −Sc) ξ5  A11 (ψ29 − ψ30 ) + A12 (ψ26 − ψ27 ) + A13 (ψ11 − ψ12 ) + A14 (ψ1 − ψ2 ), RaGr ξ6 , H1  ψ13 , H2  Pr −1 ξ6  A4 (ψ14 − ψ15 ) + A5 (ψ13 − ψ16 ) + A6 (ψ17 − ψ18 ),   RaGm Sr Sc ψ19 + ξ7 , H3  Pr −1 Ra Q Gm Sr Sc ξ7  C7 (ψ22 − ψ23 ) + C8 (ψ13 − ψ16 ) + C9 (ψ17 − ψ18 ), H4  − ξ8 , Sc − 1 Gr Ra ξ9 , ξ8  C10 (ψ14 − ψ15 ) + C11 (ψ13 − ψ16 ) + C12 (ψ17 − ψ18 ), H5  − Sc − 1 ξ9  A4 (ψ24 − ψ25 ) + A5 (ψ11 − ψ12 ) + A6 (ψ1 − ψ2 ),   Gm Ra Sr Sc ψ26 + ξ10 , H6  Sc − 1 Ra Q Gm Sr Sc ξ10  C7 (ψ27 − ψ28 ) + C8 (ψ5 − ψ6 ) + C9 (ψ7 − ψ8 ), H7  ξ11 , Q(Sc − 1) ξ11  C10 (ψ24 − ψ25 ) + C11 (ψ11 − ψ12 ) + C12 (ψ1 − ψ2 ), Gr J1  ψ13 , J2  (ψ17 − ψ18 ), Q−M Gm Ra Gm J3  Sc−1 (ψ19 − Sr Sc ξ12 ), J4  (Q−M)(Sc−1) ξ13 , ¯ ¯ ¯ ξ12  A7 (ψ31 − ψ32 ) + A8 (ψ22 − ψ23 ) + A9 (ψ13 − ψ16 ) + A¯ 10 (ψ17 − ψ18 ), Gr J5  − (ψ33 − ψ34 ), Q−M ξ13  B1 (ψ31 − ψ32 ) + B2 (ψ13 − ψ16 ) + B3 (ψ17 − ψ18 ),   RaGm SrSc ψ29 − ψ30 ξ14 − ξ15 , ξ14  , J6  − Sc − 1 Sc − 1 a3 ξ15  A¯ 7 (ψ35 − ψ36 ) + A¯ 8 (ψ27 − ψ28 ) + A¯ 9 (ψ5 − ψ6 ) + A¯ 10 (ψ7 − ψ8 ), SrScGm  − (Q−M)(Sc−1) ξ16 , ξ16  B1 (ψ37 − ψ38 ) + B2 (ψ39 − ψ40 ) + J7   ψ13 + PrSrPr B3 (ψ33 − ψ34 ), K 1  ψ13 , K 2  G 2 , K 3  − RaGm ξ , Ra M −1 17 RaGmPrSr ξ17  B¯ 1 (ψ41 − ψ42 ) + B¯ 2 (ψ13 − ψ16 ) + B¯ 3 (ψ17 − ψ18 ), K 4  − ξ18 , (Pr −1)2 ξ18  B4 (ψ41 − ψ42 ) + B5 (ψ14 − ψ15 ) + B6 (ψ13 − ψ16 ) + B7 (ψ17 − ψ18 ),   RaGr Gm SrPr ξ19 , K 6  ψ45 + ξ20 , K5  − Pr −1 M Pr −1

258

N. Ahmed

ξ19  A4 (ψ24 − ψ25 ) + A5 (ψ11 − ψ12 ) + A6 (ψ1 − ψ2 ), RaGmPrSr ξ20  B¯ 1 (ψ43 − ψ44 ) + B¯ 2 (ψ45 − ψ46 ) + B¯ 3 (ψ48 − ψ49 ), K 7  ξ21 , (Pr −1)2 ξ21  B4 (ψ49 − ψ50 ) + B5 (ψ24 − ψ25 ) + B6 (ψ11 − ψ12 ) + B7 (ψ1 − ψ2 ),   Gr Gm Sr ψ13 + ξ22 , L 1  ψ13 , L 2  − (ψ17 − ψ18 ), L 3  − Q−M M Q Ra ξ22  ψ13 − ψ16 + Ra Q(ψ17 − ψ18 ), GmSr L4  − [ψ13 − ψ16 + Ra Q(ψ17 − ψ18 )], Q Ra(Q − M) Gm L5  − [ψ33 − ψ34 ], Q−M   Gm Sr ψ45 + ξ23 , ξ23  ψ45 − ψ46 + Ra Q(ψ47 − ψ48 ), L6  M Q Ra SrGm L7  [ψ33 − ψ34 + Ra Q(ψ33 − ψ34 )], A  SrScPr, B  Pr Ra Q, Q Ra(Q − M) −B Ra Q + a1 Ra Q + a1 Ra Q , A1  , A2  − , A3  − 2 , a1  2 2 Pr −Sc a1 a1 a1 Ra(M − Pr Q) 1 1 Ra Q + a1 , A4  2 , A5  −A4 , A6  − , A7  2 , a2  Pr −1 a2 a2 a1 (a1 − a3 ) Ra Q + a3 Ra Q Ra Q + a1 A8  2 , A9  −( A7 + A8 ), A10  , , A11  2 a1 a3 a3 (a3 − a1 ) a1 (a1 − a2 ) Ra Q + a2 Ra Q M Ra A12  2 , , a3  , A13  −(A11 + A12 ), A14  a1 a2 Sc − 1 a2 (a2 − a1 ) Ra Q + a3 Ra Q Ra Q + a2 C7  , C8  −C7 , C9  − , C10  , C11  −C10 , a3 a32 a22 Ra Q Ra Q ¯ Ra Q + a¯ 1 ¯ Ra Q + a3 C12  − , A7  2 , a¯ 1  , A8  2 , a2 Sc − 1 a¯ 1 (a1 − a3 ) a3 (a3 − a¯ 1 )   Ra Q Ra Q + a¯ 1 A¯ 9  − A¯ 7 + A¯ 8 , A¯ 10  , B1  , B2  −B1 , a¯ 1 a3 a¯ 12 Ra Q Ra Q Pr ¯ Ra Q + d1 ¯ Ra Q B3  − , B1  , d1  , B2  − B¯ 1 , B¯ 3  − , a¯ 1 1 − Pr d1 d12 d1 + Ra Q a2 + Ra Q Ra Q B4  2 , , B5  2 , B6  −(B4 + B5 ), B7  d1 a2 d1 (d1 − a2 ) a2 (a2 − d1 ) ψ1  f (Pr Ra, Q Ra, y, t), ψ2  f (Pr Ra, Q Ra, y, t − 1)H (t − 1), ψ3  ea1 t ψ(ScRa, a1 , y, t), ψ4  ea1 (t−1) ψ(ScRa, a1 , y, t − 1)H (t − 1), ψ7  λ(RaSc, y, t), ψ8  λ(RaSc, y, t − 1)H (t − 1),

MHD Mass Transfer Flow Past an Impulsively Started …

      y RaSc y RaSc ψ5  er f c , ψ6  er f c H (t − 1), 2 t 2 t −1 ψ9  ea1 t ψ(Pr Ra, Ra Q + a1 , y, t), ψ10  ea1 (t−1) ψ(Pr Ra, Ra Q + a1 , y, t − 1)H (t − 1), ψ11  ψ(Pr Ra, Q Ra, y, t), ψ12  ψ(Pr Ra, Q Ra, y, t − 1)H (t − 1), ψ13  ψ(Ra, M Ra, y, t), ψ16  ψ(Ra, M Ra, y, t − 1)H (t − 1), ψ14  ea2 t ψ(Ra, Ra M + a2 , y, t), ψ15  ea2 (t−1) ψ(Ra, Ra M + a2 , y, t − 1)H (t − 1), ψ17  f (Ra, M Ra, y, t), ψ18  f (Ra, M Ra, y, t − 1)H (t − 1),  1  a3 t ψ19  e ψ(Ra, M Ra + a3 , y, t) − ψ13 , a3 ψ20  ea1 t ψ(Ra, M Ra + a1 , y, t), ψ21  ea1 (t−1) ψ(Ra, M Ra + a1 , y, t − 1)H (t − 1), ψ22  ea3 t ψ(Ra, M Ra + a3 , y, t), ψ23  ea3 (t−1) ψ(Ra, M Ra + a3 , y, t − 1)H (t − 1), ψ24  ea2 t ψ(Pr Ra, Ra Q + a2 , y, t), ψ25  ea2 (t−1) ψ(Pr Ra, Ra Q + a2 , y, t)H (t − 1),  1  a3 t ψ26  e ψ(ScRa, a3 , y, t) − ψ5 , a3 ψ27  ea3 t ψ(ScRa, a3 , y, t), ψ28  ea3 (t−1) ψ(ScRa, a3 , y, t − 1)H (t − 1), ψ29  ea1 t ψ(Pr Ra, Ra Q + a1 , y, t), ψ30  ea1 (t−1) ψ(Pr Ra, Ra Q + a1 , y, t)H (t − 1), ψ31  ea¯ 1 t ψ(Ra, M Ra + a¯ 1 , y, t), ψ32  ea¯ 1 (t−1) ψ(Ra, M Ra + a¯ 1 , y, t − 1)H (t − 1), ψ33  f (Ra, Q Ra, y, t), ψ34  f (Ra, Q Ra, y, t − 1)H (t − 1), ψ35  ea¯ 1 t ψ(RaSc, a¯ 1 , y, t), ψ36  ea¯ 1 (t−1) ψ(RaSc, a¯ 1 , y, t − 1)H (t − 1), ψ37  ea¯ 1 t ψ(Ra, Q Ra + a¯ 1 , y, t), ψ38  ea¯ 1 (t−1) ψ(Ra, Q Ra + a¯ 1 , y, t − 1)H (t − 1), ψ39  ψ(Ra, Q Ra, y, t), ψ40  ψ(Ra, Q Ra, y, t − 1)H (t − 1), ψ41  ed1 t ψ(Ra, M Ra + d1 , y, t), ψ42  ed1 (t−1) ψ(Ra, M Ra + d1 , y, t − 1)H (t − 1), ψ43  ed1 t ψ(Ra, d1 , y, t), ψ44  ed1 (t−1) ψ(Ra, d1 , y, t)H (t − 1), ψ47  λ(Ra, y, t), ψ48  λ(Ra, y, t − 1)H (t − 1),

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ψ45

      y Ra y Ra  er f c , ψ46  er f c H (t − 1), 2 t 2 t −1

ψ49  ed1 t ψ(Pr Ra, Ra Q + d1 , y, t), ψ50  ed1 (t−1) ψ(Pr Ra, Ra Q + d1 , y, t)H (t − 1);        √ y ξ √ y ξ √ 1 √ξη y − ξη y + ηt + e − ηt , er f c er f c ψ(ξ, η, y, t)  e 2 2 t 2 t !      t y ξ √ξη y y ξ √ + + ηt f (ξ, η, y, t)  er f c e 2 4 η 2 t !      t y ξ √ y ξ −√ξη y − − ηt , + er f c e 2 4 η 2 t    √   y ξ y2ξ y ξ t − ξ y2 λ(ξ, y, t)  t + er f c − √ e 4t ; 2 2 t π  0, t < 1 H (t − 1)  is the unit step function. 1, t > 1

8 Rate of Momentum Transfer The viscous drag per unit area on the plate in the direction of the plate velocity is quantified by Newton’s law of viscosity as follows   ∂u  μ ∂u τ   −μ  − . (39) ∂ y y  0 t0 ∂ y y0 The coefficient of skin friction or the coefficient of the rate of momentum transfer at the plate is given by ⎧ 7  ⎪ ⎪ − Mα , Pr  1, Sc  1, Pr  Sc ⎪ ⎪ ⎪ ⎪ ⎪ α1 ⎪ 7  ⎪ ⎪ ⎪− Nα , Pr  1, Sc  1, Pr  Sc ⎪ ⎪ ⎪ α1 ⎪  ⎨ 7  τ ∂u τ ; (40) −  − Pα , Pr  1, Pr  Sc μ/t0 ∂ y y0 ⎪ α1 ⎪ ⎪ ⎪ 7  ⎪ ⎪ ⎪ − Q α , Sc = 1, Pr  Sc ⎪ ⎪ ⎪ α1 ⎪ ⎪ 7 ⎪  ⎪ ⎪ Sα , Pr  1, Sc  1 ⎩− α1

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where RaGr M1  13 , M2  [A4 ( 14 − 15 ) + A5 ( 13 − 16 ) + A6 ( 17 − 18 )], Pr −1   RaGm A RaGm A 19 + η1 , M4  − η2 , M3  Sc − 1 Pr −Sc (Pr −1)(Pr −Sc) η1  A7 ( 20 − 21 ) + A8 ( 22 − 23 ) + A9 ( 13 − 16 ) + A10 ( 17 − 18 ), η2  A11 ( 20 − 21 ) + A12 ( 14 − 15 ) + A13 ( 13 − 16 ) + A14 ( 17 − 18 ), RaGr M5  − [A4 ( 24 − 25 ) + A5 ( 11 − 12 ) + A6 ( 1 − 2 )], Pr −1  RaGm A {A7 ( 3 − 4 ) + A8 ( 27 − 28 ) 26 + M6  − Sc − 1 Pr −Sc +A9 ( 5 − 6 ) + A10 ( 7 − 8 )}], A RaGm M7  − [A11 ( 29 − 30 ) + A12 ( 26 − 27 ) (Pr −1)(Pr −Sc) +A13 ( 11 − 12 ) + A14 ( 1 − 2 )], RaGr N1  13 , N2  [A4 ( 14 − 15 ) + A5 ( 13 − 16 ) + A6 ( 17 − 18 )], Pr −1   RaGm SrSc {C7 ( 22 − 23 ) + C8 ( 13 − 16 ) + C9 ( 17 − 18 )} , 19 + N3  Pr −1 Ra Q GmSrSc N4  − [C10 ( 14 − 15 ) + C11 ( 13 − 16 ) + C12 ( 17 − 18 )], Sc − 1 RaGr N5  − [A4 ( 24 − 25 ) + A5 ( 11 − 12 ) + A6 ( 1 − 2 )], Sc − 1   RaGm SrSc {C7 ( 27 − 28 ) + C8 ( 5 − 6 ) + C9 ( 7 − 8 )} , N6  26 + Sc − 1 Ra Q GmSrSc N7  [C10 ( 24 − 25 ) + C11 ( 11 − 12 ) + C12 ( 1 − 2 )], Q(Sc − 1) Gr P1  13 , P2  ( 17 − 18 ), Q−M  RaGm  19 − ScSr A¯ 7 ( 31 − 32 ) + A¯ 8 ( 22 − 23 ) P3  Sc − 1  + A¯ 9 ( 13 − 16 ) + A¯ 10 ( 17 − 18 ) , Gm P4  [B1 ( 31 − 32 ) + B2 ( 13 − 16 ) + B3 ( 17 − 18 )], (Q − M)(Sc − 1)   Gr RaGm 1 ScSr 51 , P5  − ( 29 − 30 ) − ( 33 − 34 ), P6  − Q−M Sc − 1 a3 Sc − 1 SrScGm P7  − [B1 ( 37 − 38 ) + B2 ( 39 − 40 ) + B3 ( 33 − 34 )], (Q − M)(Sc − 1)

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RaGr Q 1  13 , Q 2  [A4 ( 14 − 15 ) + A5 ( 13 − 16 ) + A6 ( 17 − 18 )], Pr −1    Gm SrPr  ¯ ¯ ¯ B1 ( 41 − 42 ) + B2 ( 13 − 16 ) + B3 ( 17 − 18 ) , Q3  − 13 + M Pr −1 RaGmPrSr Q4  − [B4 ( 41 − 42 ) + B5 ( 14 − 15 ) (Pr −1)2 +B6 ( 13 − 16 ) + B7 ( 17 − 18 )], RaGr Q5  − [A4 ( 24 − 25 ) + A5 ( 11 − 12 ) + A6 ( 1 − 2 )], Pr −1    Gm SrPr  ¯ ¯ ¯ 45 + B1 ( 43 − 44 ) + B2 ( 45 − 46 ) + B3 ( 48 − 49 ) , Q6  M Pr −1 RaGmPrSr Q7  [B4 ( 49 − 50 ) + B5 ( 24 − 25 ) (Pr −1)2 +B6 ( 11 − 12 ) + B7 ( 1 − 2 )], Gr S1  13 , S2  − ( 17 − 18 ), Q−M   Gm Sr { 13 − 16 + Ra Q( 17 − 18 )} , 13 + S3  − M Q Ra SrGm S4  − [ 13 − 16 + Ra Q( 17 − 18 )], Q Ra(Q − M) Gr S5  − ( 33 − 34 ), Q−M   Gm Sr { 45 − 46 + Ra Q( 47 − 48 )} , 45 + S6  M Q Ra SrGm S7  [ 33 − 34 + Ra Q( 33 − 34 )]. Q Ra(Q − M)

9 Rate of Heat Transfer The heat flux q ∗ from the plate to the fluid specified by the Fourier law of conduction is given by   ∂T κ(Tw − T∞ ) ∂θ q ∗  −κ  − . (41) ∂ y y  0 U 0 t0 ∂ y y0 The coefficient of the rate of heat transfer at the plate in terms of Nusselt number Nu is expressed as  U 0 t0 q ∗ ∂θ Nu  −  −( 1 − 2 ). (42) κ(Tw − T∞ ) ∂ y y0

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10 Rate of Mass Transfer The mass flux Mw at the plate determined by Fick’s law of diffusion is given by   ∂C D M (Cw − C∞ ) ∂φ Mw  −D M  − . (43) ∂ y y  0 U 0 t0 ∂ y y0 The coefficient of the rate of mass transfer at the plate in terms of Sherwood number is given by   −(R1 + R2 − R3 ), Pr  Sc M w U 0 t0 ∂φ Sh  −  , (44)   D M (Cw − C∞ ) ∂ y y0 − R¯ 1 + R¯ 2 − R¯ 3 , Pr  Sc where R1  5 , R2 

A [A1 ( 3 − 4 ) + A2 ( 5 − 6 ) + A3 ( 7 − 8 )], Pr −Sc

A [A1 ( 9 − 10 ) + A2 ( 11 − 12 ) + A3 ( 1 − 2 )], R¯ 1  5 , Pr −Sc SrSc R¯ 2  [Ra Q( 7 − 8 ) + 5 − 6 ], Ra Q SrSc R¯ 3  [Ra Q( 1 − 2 ) + 11 − 12 ], Ra Q 1  ϕ(Pr Ra, Ra Q, t), 2  ϕ(Pr Ra, Ra Q, t − 1)H (t − 1), R3 

3  ea1 t (RaSc, a1 , t), 4  ea1 (t−1) (RaSc, a1 , t − 1)H (t − 1), !  RaSc RaSc , 6  − H (t − 1), 5  − πt π (t − 1)   RaSct RaSc(t − 1) , 8  −2 H (t − 1), 7  −2 π π 9  ea1 t (Ra Pr, Ra Q + a1 , t), 10  ea1 (t−1) (Ra Pr, Ra Q + a1 , t − 1)H (t − 1), 11  (Pr Ra, Ra Q, t), 12  (Pr Ra, Ra Q, t − 1)H (t − 1), 13  (Ra, Ra M, t), 16  (Ra, Ra M, t − 1)H (t − 1), 14  ea2 t (Ra, Ra M + a2 , t), 15  ea2 (t−1) (Ra, Ra M + a2 , t − 1)H (t − 1),

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17  ϕ(Ra, Ra M, t), 18  ϕ(Ra, Ra M, t − 1)H (t − 1),  1  a3 t 19  e (Ra, Ra M + a3 , t) − 13 , a3 20  ea1 t (Ra, Ra M + a1 , t), 21  ea1 (t−1) (Ra, Ra M + a1 , t − 1)H (t − 1), 22  ea3 t (Ra, Ra M + a3 , t), 23  ea3 (t−1) (Ra, Ra M + a3 , t − 1)H (t − 1), 24  ea2 t (Pr Ra, Ra Q + a2 , t), 25  ea2 (t−1) (Pr Ra, Ra Q + a2 , t − 1)H (t − 1),  1  a3 t 26  e (ScRa, a3 , t) − 5 , a3 27  ea3 t (ScRa, a3 , t), 28  ea3 (t−1) (ScRa, a3 , t − 1)H (t − 1), 29  ea1 t (Pr Ra, Ra Q + a1 , t), 30  ea1 (t−1) (Pr Ra, Ra Q + a1 , t − 1)H (t − 1), 31  ea¯ 1 t (Ra, Ra M + a¯ 1 , t), 32  ea¯ 1 (t−1) (Ra, Ra M + a¯ 1 , t − 1)H (t − 1), 33  ϕ(Ra, Ra Q, t), 34  ϕ(Ra, Ra Q, t − 1)H (t − 1), 35  ea¯ 1 t (RaSc, a¯ 1 , t), 36  ea¯ 1 (t−1) (RaSc, a¯ 1 , t − 1)H (t − 1), 37  ea¯ 1 t (Ra, Ra Q + a¯ 1 , t), 38  ea¯ 1 (t−1) (Ra, Ra Q + a¯ 1 , t − 1)H (t − 1), 39  (Ra, Ra Q, t), 40  (Ra, Ra Q, t − 1)H (t − 1), 41  ed1 t (Ra, Ra M + d1 , t), 42  ed1 (t−1) (Ra, Ra M + d1 , t − 1)H (t − 1), 43  ed1 t (Ra, d1 , t), 44  ed1 (t−1) (Ra, d1 , t − 1)H (t − 1), !  Ra Ra , 46  − H (t − 1), 45  − πt π(t − 1)   Rat Ra(t − 1) 47  −2 , 48  −2 H (t − 1), π π 49  ed1 t (Ra Pr, Ra Q + d1 , t), 50  ed1 (t−1) (Ra Pr, Ra Q + d1 , t − 1)H (t − 1), 51  A¯ 7 ( 35 − 36 ) + A¯ 8 ( 27 − 28 ) + A¯ 9 ( 5 − 6 ) + A¯ 10 ( 7 − 8 );  √  ξ −η t " e + ξη er f η t , (ξ, η, t)  − πt !  √  " √  ξ ξ t −η t er f η t + t ξηer f η t + e ϕ(ξ, η, t)  − . 4η π

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11 Results and Discussion Numerical computations for the dimensionless velocity, temperature, concentration, skin friction, Nusselt number, and Sherwood number have been carried out by assigning some specific values (arbitrarily chosen) to the physical parameters involved like Soret number Sr, Ramped parameter Ra, radiation parameter Q, magnetic parameter M, solutal Grashof number Gm, thermal Grashof number Gr, Schmidt number Sc and Prandtl number Pr. Thought the investigation, the values of the Pradtl number Pr, thermal Grashof number Gr and Schmidt number Sc have been chosen as 0.71, 30, 0.60 respectively as the numerical calculations are concerned. It is worthwhile to mention that Pr  0.71 corresponds to air at the temperature of 25 °C and 1 atmospheric pressure and Sc  0.60 refers to water vapor diffused in dry air. It is to be noted that in the most of the cases of mass transfer related problems, dry air is considered as an ideal medium of diffusion (solvent). Because, all most all the natural gases diffuse in dry air easily. In the present investigation, the fluid is to be electrically conducting. Dry air is a poor conductor. In contrast, air with moisture is electrically conducting to a reasonable extent. In the present work, a fluid is required which is electrically conducting as well as a good solvent. On the basis of this fact, air is taken as solvent and water vapor as solute. This phenomenon justifies the fact of taking Pr  0.71 and Sc  0.60. Also Gr > 0 presents the externally cool case. Further both the cases of t < 1 and t > 1 are incorporated in performing the computations from the exact solutions of the problem under consideration. It is recalled that t < 1 represents the ramped wall temperature and in contrast t > 1 refers to isothermal plate. The concentration profiles for t < 1 and for t > 1 under the influence of Sr, Ra, Q and normal coordinate y are exhibited in Figs. 1, 2, 3, 4, 5 and 6. These figures show a comprehensive growth in fluid concentration for increasing the Soret number Sr and the radiation parameter Q. An interesting observation on the behavior of φ under the ramped parameter Ra is marked in Figs. 3 and 4. These two figures depict that the concentration level of the fluid goes up marginally in a thin layer adjacent to the plate and thereafter it takes a reverse turn. All the figures uniquely simulate that the concentration distribution φ first increases in a thin layer closed to the wall and after that it falls asymptotically as y → ∞. This is due to the buoyancy force, which is very much effective near the plate and its influence diminishes in fluid region way from the plate. As the effects of Sr, Ra and Q on φ are concerned, it is immaterial whether the plate is isothermal or the temperature of the plate is ramped. However in case of t > 1(isothermal), the asymptotic fall of φ → 0 as y → ∞ is delayed in comparison to the case of t < 1. The influences of Ra and Q on the temperature field θ are displaced in Figs. 7 and 8 for t < 1 and t > 1. These two figures indicate a substantial fall in the temperature field under the effect of Ra as well as the radiation parameter Q. It is worthwhile to mention that, in case of Cogley et al. [11]’s model of radiation, radiation acts like a heat sink. Hence, in the present case, radiation which is similar to heat sink, absorbs heat from the fluid and as a consequence, the fluid temperature falls under

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Fig. 1 Concentration φ versus y for Pr  0.71, Ra  10, Sc  0.6, Q  5, t  0.5

Fig. 2 Concentration φ versus y for Pr  0.71, Ra  10, Sc  0.6, Q  5, t  2

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Fig. 3 Concentration φ versus y for Pr  0.71, Sr  10, Sc  0.6, Q  5, t  0.5

Fig. 4 Concentration φ versus y for Pr  0.71, Sr  10, Sc  0.6, Q  5, t  2

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Fig. 5 Concentration φ versus y for Pr  0.71, Sr  10, Sc  0.6, Ra  10, t  0.5

Fig. 6 Concentration φ versus y for Pr  0.71, Sr  10, Sc  0.6, Ra  10, t  2

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Fig. 7 Temperature θ versus y for Pr  0.71, Q  5, t  0.5, t  2

Fig. 8 Temperature θ versus y for Pr  0.71, Ra  10, t  0.5, t  2

the radiation effect. This phenomenon establishes the fact that the observation on the effect of Q on θ is consistent with the physical reality. Further the role of time t on θ is almost similar to its role on φ.

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Fig. 9 Velocity u versus y for Gr  30, Gm  20, Ra  10, Q  5, Sr  10, Sc  0.6, Pr  0.71, t  0.5

Figures 9, 10, 11, 12, 13 and 14 present the variation of the velocity field u versus y under the effects of M, Q, Sr for the cases t < 1 and t > 1. A comprehensive fall in the fluid velocity for increasing the parameter M is marked in Figs. 9 and 10. It registers the fact that the flow is retarded due to the imposition of the transverse magnetic field. In real situation, a magnetic body force called Lorentz force gets generated on interaction of the magnetic field and electrically conducting fluid in motion. This induced force acts as a resistive force to the flow, which in turn causes the flow to decelerate. It confirms the consistency of our graphical observation with the actual physical situation. It is inferred from Figs. 11 and 12 that the thermal radiation has also some contribution in retarding the fluid motion. Figures 13 and 14 clearly indicate that an increase in the parameter Sr leads the fluid motion to accelerate to a remarkable extent. All the Figs. 9, 10, 11, 12, 13 and 14 show that the fluid velocity first increases in a thin layer adjacent to the plate and thereafter it falls asymptotically as y → ∞. It reveals the fact that the buoyancy force is very much effective near the plate and its influence on the fluid velocity slowly and steadily gets nullified as moved away from the plate. Like the concentration field φ and the temperature field θ, the asymptotic fall of the fluid velocity gets delayed for t > 1 rather than t < 1. Thus the asymptotic fall of temperature θ or concentration field φ or the velocity field u is faster in case of ramped wall temperature than that of the isothermal plate temperature. Figures 15, 16 and 17 demonstrate how the skin friction τ at the plate is affected by the magnetic parameter M, Soret number Sr, solutal Grashof number Gm and time t. It is inferred from the Fig. 15 that the shear stress at the plate is diminished for

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Fig. 10 Velocity u versus y for Gr  30, Gm  20, Ra  10, Q  5, Sr  10, Sc  0.6, Pr  0.71, t  2

Fig. 11 Velocity u versus y for Gr  30, Gm  20, Ra  10, M  4, Sr  10, Sc  0.6, Pr  0.71, t  0.5

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Fig. 12 Velocity u versus y for Gr  30, Gm  20, Ra  10, M  4, Sr  10, Sc  0.6, Pr  0.71, t  2

Fig. 13 Velocity u versus y for Gr  30, Gm  20, Ra  10, M  4, Q  5, Sc  0.6, Pr  0.71, t  0.5

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Fig. 14 Velocity u versus y for Gr  30, Gm  20, Ra  10, M  4, Q  5, Sc  0.6, Pr  0.71, t  2

increasing M. It reflects the fact that the viscous drag at the plate gets suppressed on the imposition of the magnetic field. As such strong magnetic field may be applied in operations for successful inhabitation of the frictional resistance at the plate. The drag force at the plate due to viscosity seems to be enhanced under the thermal diffusion effect or buoyancy force (solutal) effect as visualized in Figs. 16 and 17. These two Figures further predict that the skin friction sharply increases as time progresses in case of ramped wall temperature (t < 1) and it becomes stationary for isothermal plate (t > 1). The effects of the ramped parameter Ra and radiation parameter Q on the Nusselt number Nu are demonstrated in Figs. 18 and 19. It is observed from these two figures that the rate of heat transfer at the plate rises significantly for increasing the ramped parameter Ra as well as radiation parameter Q. It leads to conclude that the low viscosity or high heat absorption raises the heat flux at the plate. The same figures also register the fact that for t < 1, Nu increases linearly at a very fast rate as t increases and it becomes uniform in case of t > 1. Figures 20 and 21 depict that an increase in Sr or Ra results in a substantial rise in the Sherwood number. The two figures uniquely establish that the rate of mass transfer at the plate falls initially and as the time progresses, it rises up linearly for t < 1. Like the Nusselt number, the Sherwood number also becomes stationary for t > 1. It may be concluded from the Figs. 15, 16, 17, 18, 19, 20 and 21 that the rate of momentum transfer, rate of heat transfer and the rate of mass transfer seem to be uniform in case of isothermal plate. In contrast, these transport properties get enhanced almost straight way for ramped wall temperature.

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Fig. 15 Skin friction τ versus t for Gr  30, Gm  20, Ra  10, Sr  10, Sr  10, Q  5, Sc  0.6, Pr  0.71

Fig. 16 Skin friction τ versus t for Gr  30, Gm  20, Ra  10, M  4, Q  5, Sc  0.6, Pr  0.71

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Fig. 17 Skin friction τ versus t for Gr  30, Sr  10, Ra  10, M  4, Q  5, Pr  0.71

Fig. 18 Nusselt number Nu versus t for Q  5, Pr  0.71

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Fig. 19 Nusselt number Nu versus t for Ra  10, Pr  0.71

Fig. 20 Sherwood number Sh versus t for Q  5, Sc  0.6, Ra  10, Pr  0.71

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Fig. 21 Sherwood number Sh versus t for Q  5, Sc  0.6, Sr  10, Pr  0.71

12 Conclusions The present investigation may be summarized to the following conclusions: • Drag force at the plate due to viscosity seems to be enhanced under the thermal diffusion effect or buoyancy force (solutal) effect. • Skin friction sharply increases as time progresses in case of ramped wall temperature (t < 1) and it becomes stationary for isothermal plate (t > 1). • Low viscosity or high heat absorption raises the heat flux at the plate. • For t < 1, Nu increases linearly at a very fast rate as t increases and it becomes uniform in case of t > 1. • An increase in Sr or Ra results in a substantial rise in rate of mass transfer at the plate. • Rate of mass transfer at the plate falls initially and as the time progresses, it rises up linearly for t < 1.

13 Future Scope This chapter deals with the study of an impulsively started convective flow problem. The mathematical model of the problem is idealized to a considerable extent on imposition of some physically realistic constraints. A set of closed form solutions of the resultant governing equations are obtained by adopting the Laplace transform

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technique. There is wide scope of reinvestigation of the same problem numerically by reducing the number of constraints. In this regard, Crank-Nicolson type implicit finite difference scheme may be suggested.

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20. Mankinde, O.D.: Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate. Int. Commun. Heat Mass Transf. 32(10), 1411–1419 (2005) 21. Mbeledogu, I.U., Amakiri, A.R.C., Ogulu, A.: Unsteady MHD free convection flow of a compressible fluid past a moving vertical plate in the presence of radiative heat transfer. Int. J. Heat Mass Transf. 50(9–10), 326–331 (2007) 22. Orhan, A., Ahmet, K.: Radiation effect on MHD mixed convection flow about a permeable vertical plate. Heat Mass Transf. 45, 239–246 (2008) 23. Prasad, N.R., Reddy, N.B., Muthucumaraswamy, R.: Transient Radiative Hydro-Magnetic free convection flow past an impulsively started vertical plate with uniform heat and mass flux. Theor. Appl. Mech. 33(1), 31–63 (2006) 24. Postelnicu, A.: Influence of a magnetic field on heat and mass transfer by natural convection from vertical surface in porous media considering Soret and Dufour effects. Int. J. Heat Mass Transf. 47, 1467–1472 (2004) 25. Raptis, A., Perdikis, C.: Radiation and free convection flow past a moving plate. Int. J. Appl. Mech. Eng. 4(4), 817–821 (1999) 26. Shah, N.A., Zafar, A.A., Akhtar, S.: General solution for MHD-free convection flow over a vertical plate with ramped wall temperature and chemical reaction. Arab. J. Math. 7(1), 49–60 (2018) 27. Samad, M.A., Rahman, M.M.: Thermal radiation interaction with unsteady MHD flow past a vertical porous plate immersed in a porous medium. J. Nav. Archit. Mar. Eng. 3(1), 7–14 (2006) 28. Shercliff, J.A.: A Text Book of Magnetohydrodynamics. Pergamon Press, London (1965) 29. Seth, G.S., Kumbhakar, B., Sharma, R.: Unsteady MHD free convection flow with Hall effect of a radiating and heat absorbing fluid past a moving vertical plate with variable ramped temperature. J. Egypt. Math. Soc. 24(3), 471–478 (2016) 30. Seth, G.S., Sharma, R., Kumbhakar, B.: Heat and mass transfer effects on unsteady MHD natural convection flow of a chemically reactive and radiating fluid through a porous medium past a moving vertical plate with arbitrary ramped temperature. J. Appl. Fluid Mech. 9(1), 103–117 (2016) 31. Seth, G.S., Sharma, R., Sarkar, S.: Natural convection heat and mass flow with Hall current, rotation and heat absorption past an accelerated moving vertical plate with ramped temperature. J. Appl. Fluid Mech. 8(1), 7–20 (2015) 32. Seth, G.S., Sarkar, S., Hussain, S.M., Mahato, G.K.: Effects of Hall current and rotation on hydromagnetic natural convection flow with heat and mass transfer of a heat absorbing fluid past an impulsively moving vertical plate with ramped temperature. J. Appl. Fluid Mech. 8(1), 159–171 (2015) 33. Seth, G.S., Hussain, S.M., Sarkar, S.: Hydromagnetic natural convection flow with heat and mass transfer of a chemically reacting and heat absorbing fluid past an accelerated moving vertical plate with ramped temperature and ramped surface concentration through a porous medium. J. Egypt. Math. Soc. 23, 197–207 (2015) 34. Seth, G.S., Ansari, M.S., Nandkeolyar, R.: MHD Natural convection flow with radiative heat transfer past an impulsively moving plate with ramped temperature. Heat Mass Transf. 47, 551–561 (2011) 35. Takhar, H.S., Gorla, R.S.R., Soundalgekar, V.M.: Radiation effects on MHD free convection flow of a radiating gas past a semi infinite vertical plate. Int. J. Numer. Meth. Heat Fluid Flow 6(2), 77–83 (1996)

Secure Communication Systems Based on the Synchronization of Chaotic Systems Samir Bendoukha, Salem Abdelmalek and Adel Ouannas

Abstract Over the last three decades, chaotic dynamical systems have found many applications in science and engineering particularly in the field of secure communications and data encryption. The vast majority of such applications start from the synchronization control of two chaotic systems where one system’s states are forced to follow the exact same trajectory set out by another system with different initial conditions. The general theme seems to be that a master system is placed at the transmitter and a slave at the receiver. Once the pair is synchronized, the states can be used to secure the communication channel in one of four ways: chaotic modulation schemes, chaotic multi–carrier schemes, chaotic multiple access schemes, and chaos–based encryption schemes. This chapter aims to give an overview of secure communications and chaos and summarize the latest advancements in the field of chaos based communications. In addition, a case study is selected assuming antipodal chaos shift keying (ACSK) modulation and the complete communication system is described. Simulation results are presented to highlight the performance of chaotic modulation systems. Keywords Chaotic dynamical systems · Secure communications · Data encryption · Modulation · Multi–carrier modulation · Multiple access

S. Bendoukha Department of Electrical Engineering, Taibah University, Yanbu, Saudi Arabia e-mail: [email protected] S. Abdelmalek (B) Department of Mathematics, University of Tebessa, 12002 Tebessa, Algeria e-mail: [email protected] A. Ouannas Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Larbi Tebessi, 12002 Tebessa, Algeria e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_9

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1 Introduction Communication systems have become an everyday necessity for humans around the globe whether in the form of mobile phone, computer networks, TV and radio broadcasting... etc. In fact, digital communication has become a tool for commerce, industry, education, social interaction, and the exchange of a vast amount of personal information. For this reason, security has become a necessity for all sorts of communication methods. This gave rise to cryptography, which is concerned with the encoding of the transmitted data such that it may only be accessed by those to whom it belongs. Cryptography is used to ensure: privacy, authentication of sources, and the integrity of data. Substantial research has been carried out on the subject for many decades. In recent years, with the increasing interest in chaotic dynamical systems and their applications in science and engineering, many researchers interested in cryptography and security have turned towards chaos. This shift is due to the many similarities between cryptography and chaos. Chaotic systems are those dynamical systems that are extremely sensitive to small variations in the initial conditions. This property seemed undesirable at first, but since Pecora and Carroll [1] defined and illustrated the plausibility of chaotic synchronization, this became a desirable one. Synchronization refers to the tuning of one chaotic system so that it mimics a totally different one and produces the same outputs. This quickly picked up and attention was paid to it from different fields and disciplines including telecommunications. If a master chaotic system is placed at the transmitter and a slave is mounted at the receiver, then by synchronizing the two one may use the seemingly random behaviour of chaos to ensure no other parties are able to intercept and make sense of the data. In this chapter, we aim to present an overview of chaotic synchronization and its applications in telecommunications. The next section will give a broad description of communication systems and the issues considered therein. Section 3 will summarize the mot common conventional cryptociphers and describe their applicability. Section 4 will introduce dynamical systems and the properties of chaos. This is followed by Sect. 5, which describes the concept of chaotic synchronization and gives a time line of its development including the most recent trends on the subject. Section 6 is the core of this chapter. It classifies the applications of chaotic synchronization in telecommunications into four main classes: conventional cryptography, modulation, multiple access, and multi-carrier schemes. A few examples are selected from the literature concerning each of these classes. Section 7 considers a specific case study and presents computer simulation results that illustrate the applicability and performance of chaos–based communications. Finally, concluding remarks are given in Sect. 8 concerning the conducted literature review and the simulation results obtained for the case study.

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2 Basics of Telecommunications Communication systems have been around for a very long period of time. For thousands of years, smoke signals, drums, horns... etc have been used to notify or inform individuals in remote areas of certain information. As for contemporary telecommunications, perhaps the earliest attempt was made by french scientist Claude Chappe in 1792 through the semaphore line, which is a mechanical encoding scheme that became very popular in the late eighteenth century and early nineteenth century. Some more attempts were made in the decades that followed including signals lamps, the acoustic phonograph, and the telegraph. In 1958, the first transatlantic telegraph cable was implemented. The continued work of specialists culminated in the invention of the telephone, which came about in 1876. The basic idea was to convert the human voice into an analogue electrical signal that is transmitted through a cable and recovered on the other side. Once the idea of telecommunications was noticed, more types of systems appeared such as radio (1896), television (1927), videophones (1936), car phones (1946), satellite communications (1962), fiber optical networks (1964), computer networks (1969), and mobile phone networks (1981). With the rapid increase in telecommunication systems, governing bodies and standardization organizations appeared. A conceptual model was proposed for all networking strategies in order to ensure interoperability. This is referred to as the open system interconnection (OSI) model. This model, depicted in Fig. 1, divides any telecommunication or computer network into 7 layers addressing separate issues and operating different protocols. Below is a brief description of these layers:

Fig. 1 The seven layer open system interconnection (OSI) model for any telecommunication system

The User

Appli tion Layer Presentation Layer Session Layer Transport Layer Network Layer Data-Link Layer Physical Layer

Medium

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1. Physical layer: This layer is the lowest in the model. It specifies the physical nature of the transmission itself and the properties of the electrical signal to be transmitted. In other words, it clearly defines the relationship between the telecommunication device and the transmission medium. Different standards exist depending on the nature of the transmission. For instance, the IEEE802.11 standards (a) through (n) refer to wireless local area networks (WLAN) or wifi. These standards specify the operating frequencies, bandwidth, allowable number of transmission antennas, data rate, encoding schemes, modulation schemes... etc. 2. Data–link layer: This layer moves beyond the dirty nature of the transmission. It is concerned with creating a node–to–node communication link. One of its main objectives is the detection, and potentially correction, of errors in the transmission through specific encoding schemes such as the cyclic redundancy check (CRC), convolutional codes... etc. In the IEEE802 wireless standards, this layer is subdivided into two sublayers: medium access control (MAC) and logical link control (LLC). 3. Network layer: This layer is responsible for dividing the data to be transmitted into variable–length packets, or datagrams, and arranges for the addressing and routing of such packets. Among the most famous protocols is the internet protocol (IP). 4. Transport layer: This layer is responsible for enuring the reliability of the network. It maintains a certain connection and ensures a frame of data is retransmitted if errors exist that cannot be corrected and that affect the integrity of the information. As for the internet, this layer involves the transmission control protocol (TCP) and the user datagram protocol (UDP). 5. Session layer: This layer comes in handy in applications that require some form of remote access. It is responsible for maintaining a certain dialogue between endpoints. 6. Presentation layer: This layer basically translates data between the raw form (lower layers of the protocol stack) and form suitable for a certain application. 7. Application layer: This layer works directly with the software requiring the transmission and reception of data. This is the highest level of the OSI model, and thus simply does not care about the technicalities of the transmission. It deals mainly with synchronization and resource availability for the software with which it interacts. Perhaps the most important of th above mentioned layers of communication systems is the physical layer. As mentioned earlier, it is responsible for the physical nature of the transmission and the encoding/decoding techniques utilized to ensure the transmitted data is properly recovered at the other end. Among the most basic techniques required at this stage is modulation, which is the process of carrying data within steady waveforms that have a constant amplitude and frequency such as the sine and cosine functions. The data can be encoded into the carrier waveform either by changing its frequency characteristics (FM), amplitude (AM), phase (PM), or polarization in time. Modulation can be analog or digital. The latter is preferred

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in contemporary telecommunications due to its many advantages in terms of noise immunity, ease of multiplexing, security, transmission error detection and correction and more. Since the inception of telecommunication systems, they received great interest from governments as well as the public. Whether the data we transmit represents highly classified military secrets or simply a social message, people expect and require a high level of privacy and security. This gave rise to data encryption, which is basically a process for rearranging data in a way that is only readable by the two concerned parties. Hence, if a message is intercepted by a third party, it cannot be understood. In the above mentioned OSI model, encryption takes place in the presentation layer. The next chapter will present a brief history of the conventional cryptography schemes used in communication systems.

3 A Summary of Conventional Cryptography Cryptography is concerned with encoding the transmitted data such that it may only be accessed by those to whom it belongs. Cryptography has been used by humans for thousands of years to protect secrets. As for modern communication systems, cryptography is used to ensure: privacy, authentication of sources, integrity of data, non–repudiation, and the exchange of keys. Numerous methods exist for encluding and decoding data falling mainly under one of three types: secret key cryptography (SKC), public key cryptography (PKC), and Hash functions. For more information, see [2]. In what follows, we will give a brief introduction into existing SKC and PKC methods.

3.1 Secret–Key (Symmetric) Cryptography Historically, cryptography was thought of as two parties encoding data in such a manner that is only fully understood by them. Anyone who tries to make sense of the data is required to know exactly how it was encoded. This is now referred to as symmetric cryptography. The transmitter uses a secret key to encode the data and sends the key to the receiver by means of a secure channel. Data is, then, transmitted publicly and decoded on the receiver side using the special key. The simplest form of a symmetric cipher is the substitution cipher, where a table (key) is established replacing each character in the alphabet by a different one. This, of course, is easily broken by means of an exhaustive search for a meaningful message or through letter frequency analysis. It is important to note that scientists and academics are not only interested in securing communications but also in breaking the key, a process referred to as cryptanalysis.

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Another historical cipher is Caesar cipher, where the characters in an alphabet are shifted by a fixed number of positions (the key). The affine cipher is similar. It encodes characters by multiplying their position by a fixed number and adding another number. While these ciphers are simple and easy to implement using modular arithmetic, they are not very safe as the number of possible combinations of messages are easily evaluated on today’s computers. For more on these historical ciphers, the reader is referred to [3, 4]. As for modern–time telecommunications, perhaps the earliest work where encryption was considered is that of Shanon in 1949 [5]. Symmetric cryptography schemes can be divided into stream ciphers and block ciphers. The first type is serial, i.e. bits are encrypted individually and the key varies with each bit. If the cipher’s key stream depends only on the key itself, the method is said to be synchronous, and if it depends as well on the encrypted text it is said to be asynchronous. On the other hand, block ciphers are those methods that use the same key to encode a complete block of bits. Block ciphers are more practical in computer communications where the computation resources are vast. Usually, in practical stream ciphers, the key is used as a seed for a pseudo–random number generator that produces a key stream. Existing stream ciphers include linear feedback shift registers (LFSR), Trivium [6], RC4 [7], SEAL [3], and many more. As for block ciphers, perhaps the most commonly used are the data encryption standard (DES), which is now considered non–secure, and the advanced encryption standard (AES) [2].

3.2 Public–Key (Asymmetric) Cryptography Asymmetric ciphers were first introduced in 1976 by Whitfield Diffie, Martin Hellman and Ralph Merkle [8] and quickly became more popular than their symmetric counterpart. As mentioned earlier, secret–key encryption requires the ability to share the key between interested parties by means of a secure channel, which is not always plausible. In addition, with today’s trend in telecommunications, a huge number of keys would be required to secure all existing communication sessions. Symmetric Cryptography also cannot support non–repudiation, which refers to preventing both ends of the transmission from cheating. The main idea for public–key encryption is that the encoder may use a public key but the message can only be decrypted by means of a private key that is known at the receiving side. This type of ciphers is more complicated and involves a number of mechanisms including key establishment, non–repudiation, identification, and encryption algorithms such as the RSA cryptosystem, the discrete Logarithm problem, Elgamal, or elliptic curve cryptosystems. Generally speaking, a trade–off exists between the complexity of the encryption process and its vulnerability to interception. Therefore, the main objective of researchers in the field has been the development of a low complexity and high security cryptography algorithm. In recent years, the synchronization of chaotic dynamical systems has presented a very attractive alternative to the previously discussed

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schemes as it offers a high level of security at a minimalistic cost. Throughout the remainder of this chapter, we will present some of the fundamental mathematics related to chaotic systems and examine existing chaos based encryption techniques with a novel case study.

4 Chaotic Dynamical Systems A dynamical system is a system of partial differential equations whose functions describe the time dependence of an arbitrary point in an n–dimensional space. Historically, the first questions concerning dynamic systems where in the field of mechanics back when it was a branch of mathematics. A major question that motivated mathematical research is the problem of the stability of the solar system. The work of Lagrange (1736–1813) on the subject consisted of interpreting the influence bodies other than the Sun have on a certain planet. These works find echoes in the theorem of Kolmogorov, Arnold, and Moser. Perhaps one of the earliest works concerning the application of dynamical systems in engineering is that of Henri Poincaré (1854– 1912) who used mathematical modeling to study the motion of three celestial bodies in space. This was followed by the revolutionary work of Lyapunov concerning the stability of such systems. The subject soon became very popular among engineers and scientists around the globe. A dynamic system is a classical system that evolves over time in a way that is causal, i.e. its future depends only on phenomena of the past or the present, and deterministic, i.e. given a present initial condition one and only one possible future state can arise. Generally speaking, in continuous time, a dynamical system is of the form dX = F (X , t, P) , (1) dt where X ∈ Rn is the vector of states, P ∈ Rr are some parameters, and t ∈ T . Alternatively, in discrete time, a dynamic system can be represented as Xk+1 = f (Xk , P) , k = 1, 2, 3, ...,

(2)

with Xk ∈ Rn and P ∈ Rr . As soon as the dimension n of the system exceeds unity, it becomes quite difficult to represent mentally how the system evolves. The basic tool to overcome this is the phase space, which is a simple and efficient qualitative technique that makes it possible to determine the type of stability of the equilibrium point based on the nature of the eigenvalues of the Jacobian matrix of the linearized function f around the system’s equilibrium point. Due to their endless applications, the general behavior and stability of dynamical systems have been of interest to mathematicians, scientists, and engineers for a very long time. The stability of such systems has been studied extensively. There are different types of stable systems including ones that have an equilibrium point, a

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periodic solution, or a quasi–periodic solution. However, it was noticed that certain linear or nonlinear dynamical systems have a bounded solution that exhibits a seemingly extremely unpredictable behavior subject to very small variations in the initial states, thus the term chaos or chaotic systems. This behavior was later found to be exactly deterministic but the term chaos stuck. Initially, these systems were thought of as being undesirable due to the complexity involved in their analysis and control. However, in recent years, since the emergence of the topic of chaos synchronization, chaotic systems have become an attractive subject for researchers in numerous scientific and engineering disciplines. For instance, chaotic systems have been used in communications for encryption and security purposes. Chaotic dynamical systems have three main characteristics: – Sensitivity to initial conditions: For a chaotic system, a very small error in the initial state X0 in the phase space will almost always be rapidly amplified. From a mathematical point of view, we say that f : I → I shows a sensitive dependence on initial conditions when :   ∃δ > 0 ∀X0 ∈ I , ε > 0 ∃n ∈ N, Y0 ∈ I : |X0 − Y0 | < ε ⇒  f n (X0 ) − f n (Y0 ) > δ.

(3) – The strange attractor: A dissipative chaotic system has at least an attractor of a particular type called strange attractor. Geometrically, such an attractor can be described as the result of operations of stretching and folding a cycle of phase space, repeated an infinite number of times. The length of the attractor is infinite, although it is contained in a finite space. – Power spectrum: A simple way to characterize chaos is to compute the Fourier spectrum of the temporal evolution of one of the system’s variables. When the system is integrable, that is to say that it is possible to completely determine the trajectories of a system in its phase space, the trajectories are the composition of oscillation movements each having a certain pulsation. The existence of wide spectra is an essential characteristic of the chaotic movements of a system. A considerable amount of literature exists concerning different types of chaotic/ hyperchaotic systems suitable for use in scientific and engineering applications including the well–known 3–component model of atmospheric convection commonly referred to as the Lorenz system [9], which is given by ⎧ dx ⎪ ⎨ dt = σ (y − x) , dy = x (ρ − z) − y dt ⎪ ⎩ dz = xy − βz, dt

(4)

and variations thereof. The above mentioned properties of chaos lead to a strong bond between chaos and encryption [10]. Since chaotic systems are very sensitive to small variations

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in the initial conditions, their behavior is said to be unpredictable, which makes them desirable in encryption. Another interesting property is their highly irregular oscillation patterns. Although at first glance these oscillations look like random noise, they may be exactly reproduced given the same initial values. This is quite similar to the idea of a pseudo-random generator, where we have a specific seed used to produce a random sequence as we saw earlier. A more detailed comparison between the properties of chaos and those of an effective cryptosystem can be found in [11].

5 Recent Trends in Chaos Synchronization The application of chaotic dynamical systems in secure telecommunications can be attempted from different perspectives as will be seen in the next section. In this chapter, we are concerned with those methods based on chaos synchronization. The term chaos synchronization refers to the phenomenon where multiple chaotic systems evolve in synchrony and converge towards the exact same trajectory given enough time. It is a multidisciplinary subject that has attracted considerable attention during the last few years. Several concepts of chaotic synchronization were first proposed by Yamada and Fujisaka [12, 13] followed by the work of Afraimovich et al. in [14]. Later, Pecora and Carroll [1] defined the chaotic synchronization known as identical synchronization developed on the basis of chaotic circuits. Other types of synchronization followed including generalized synchronization, phase synchronization, anticipated synchronization, lag synchronization, amplitude envelop synchronization... and many more. Synchronization methods can be classified into two main types: unidirectional and bidirectional. In the bidirectional case, the return loop is applied to both systems at once [15]. On the other hand, in the case of unidirectional synchronization, the feedback loop is only applied to one of the two systems. Since the topic of chaos synchronization remains to this day relatively new, we have made it the purpose of this section to present a brief description of the subject and show a list of the most promising studies related to it from a mathematical perspective. The basic idea behind the use of chaos synchronization to secure communications is that the information signal m (k) is encoded on the transmitter side by means of a chaotic dynamical system with a state vector u (k). The resulting signal s (k) is then transmitted through the channel (medium) to be picked up by the receiver, which uses it to recover the original message. The recovery process is where synchronization becomes handy as a slave chaotic system v (k) with a number of control variables is synchronized to the master u (k) such that ∀u (0) , v (0) :

lim u (k) − v (k) = 0.

k→∞

(5)

Over the last few decades, researchers have considered numerous types of synchronization schemes for a wide range of continuous–time as well as discrete–time

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integer–order differential dynamical systems [1, 16–20] or systems characterized by difference equations [21, 22]. The objective of chaos synchronization is to make the variables of a slave system synchronized in time with the variables of a corresponding chaotic master system [23]. Throughout the years since its inception, numerous synchronization types and methods have been considered in the literature and applied in science and engineering, especially for control purposes. A list of the most prominent synchronization schemes as well as the chaos synchronization based control strategies used in engineering is given in Tables 1 and 2, respectively. It is important to note that most of these works and others have been concentrated on continuous–time rather than discrete–time chaotic systems. Table 1 A list of the most prominent synchronization schemes considered in the literature for chaotic and hyperchaotic systems

Synchronization schemes

Table 2 A list of the most prominent control strategies based on chaos synchronization

Control methods

Complete synchronization [24] Anti–synchronization [25] Phase and antiphase synchronization [26] Projective synchronization [27] Function projective synchronization [28] Function lag projective synchronization [29] Matrix projective synchronization, hybrid synchronization [30] Generalized synchronization [31, 32] Lag synchronization [33] Impulsive synchronization Q–S synchronization [34] – synchronization [35] –ϕ generalized synchronization [36]

PC method [37] OGY method [17] Active control approach [38, 39] Adaptive control [40] Backstepping design [41] Sliding mode control method [42] Generalized Hamiltonian systems approach Feedback control method [43] Linear and nonlinear control methods [44]

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In practice, discrete time chaotic dynamical systems play a more important role than their continuous counterparts. Therefore, it is important to consider chaos or hyperchaos synchronization for discrete time dynamical systems [45–47]. Recently, more and more attention is being paid to the synchronization of chaos in discrete time dynamical systems due to their many applications in secure communications and cryptography [48–50]. This focus has resulted in a variety of approaches that have been proposed for the chaos synchronization of discrete–time systems [51–66]. When examining the literature we observe specific trends being studied heavily by researchers in the subject, the most important of which are: – Synchronization of fractional–order chaotic systems: Traditionally, differentiation and integration were thought of in the integer domain, i.e. one could obtains the first derivative, second derivative, and so on. However, as early as 1695, Leibniz raised the question of whether it is possible to differentiate to a fractional order, which gave rise to the idea of fractional calculus. In the last few decades, fractional calculus found itself a place in dynamical systems and the modelling of natural phenomena. It turned out that with fractional differential equations, one can obtain a more precise model for a specific phenomenon. Efforts have been made to consider the synchronization of fractional–order chaotic systems [67–70]. Most of the work however, relied on numerical solutions based on the stability criteria of linear fractional–order systems including [71–77], or based on the Laplace transform theory such as the works presented in [78–80]. In addition to synchronizing fractional slaves to fractional masters, the idea of cross synchronization between fractional and integer–order ones was also noticed. However, many scientists interested in the field struggled to achieve successful synchronization, which is why the topic remains unexplored for the most part. Interested readers may have a look at [81–85]. – Inverse Synchronization: Studying inverse types of synchronization is important as their reduced complexity can play an important role in many applications. In recent years, various inverse synchronization types have been developed such as inverse hybrid function projective synchronization [86], inverse matrix projective synchronization [87–89], inverse full state function projective synchronization [90, 91], and inverse generalized synchronization [92]. – Coexistence of synchronizations types: A very interesting phenomenon that may sometimes occur is the coexistence of several synchronization types. Some research studies have considered the coexistence problem such as the coexistence of projective synchronization, full state hybrid projective synchronization and generalized synchronization between discrete–time hyperchaotic systems [93]. A general control scheme was proposed in [94], to study the coexistence of inverse projective synchronization, inverse generalized synchronization and Q–S synchronization between arbitrary 3D hyperchaotic maps. Also, in discrete time, the coexistence of full state hybrid projective synchronization, generalized synchronization and antiphase synchronization is shown to be feasible with different dimensions [95]. Again, the coexistence of two synchronization schemes was studied in [96].

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As for fractional–order differential systems, many schemes for the coexistence of various types of synchronization have been investigated [97–101]. It is important to note that the topic of coexistence is important and has many applications in various disciplines. For instance, it can be used to enhance the level of security in wireless communication networks utilizing chaotic encryption.

6 Chaos–Synchronization–Based Communications As mentioned earlier, chaos has a very close relation to encryption and security, and thus has been widely studied by specialists in the field of telecommunications. The main reason for this consideration is the extreme sensitivity of chaotic systems to changes in the initial conditions. Chaos can be used at different stages of a communication system including: compression, encryption [102, 103], and modulation [104, 105]. For more details on the subject, the reader is referred to [106]. The first widely used application of chaotic systems to secure communications is in the encryption of data streams. As discussed in earlier in the chapter, there are two main types of encryption: secret–key and public–key. Numerous efforts have been made to develop block secret–key ciphers including ones that utilize digital chaotic maps as in [107–109]. In [109], the authors observe that many of the chaotic ciphers proposed in the literature do not live up to their claimed level of security versus computational cost. Although many of them seem simple, they involve a vast amount of floating–point multiplications and divisions, which makes for a high computational complexity. The authors consider two classes of chaotic finite-state maps: key-dependent chaotic S-boxes and chaotic mixing transformations. Based on these classes, they propose two distinct block ciphers, uniform and Feistel, and examine, to a high level of scrutiny, their resilience to cryptanalysis (attempts to break them). Another important aspect is random generators, which have a variety of applications including the generation of keys based on a specific seed. Poincaré pointed out a close relationship between chaotic systems and randomness in that he stated that randomness is synonymous with unpredictability. He, then, claimed that unpredictability can be characterized by sensitivity to initial states which is a key property of chaotic systems. This makes chaos highly suitable for the generation of random numbers. There are two main types of chaos–based random generators, pseudo– random generators [110–112] and truly–random ones [113]. Attempts have also been made considering public–key chaos–based cryptosystems. One of the most promising is based on Chebyshev maps [114, 115]. A good description of this type of encryption is given in Chap. 2 of [10] with different floating–point as well as integer implementations and a detailed cryptanalysis. Chaos synchronization has found many applications in secure telecommunications especially in optical transmission. There are different ways of employing synchronization in chaos–based communications. For the most part, these methods may be divided into four main categories: modulation schemes, multi-carrier schemes,

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multiple access schemes, and conventional encryption schemes. Throughout the remainder of this section, we go through these three classes and briefly explain the most common schemes within each class.

6.1 Chaotic Modulation Schemes Digital modulation is considered as an essential component of any modern telecommunication system. Originally, RF communications were carried out using sine and cosine signals as carriers and different types of analog modulation schemes were implemented. These signals are periodic and have a narrow band in the frequency domain, which allows for multiple access to the transmission medium. This narrowband nature turns out to be undesirable as it suffers from substantial attenuation levels when the transmission medium is dynamic, a phenomenon commonly referred to as multipath propagation. For this reason, specialists decided to enter the realm of wideband communications, which require some other form of multiple access as opposed to simply dividing the frequency spectrum into narrow strips. Several studies found in the literature have successfully achieved wideband communications by replacing the sine and cosine carriers with chaotic ones. There are many proposed schemes that use the information message to be transmitted to modulate signals from chaotic systems. The most common of these are additive masking, chaos shift keying (CSK), parametric modulation, and message–embedding. These will be explained in the following.

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Additive Masking

This is perhaps the easiest of the chaos–synchronization based communication strategies. In this approach, the transmitted message is directly added to the output of the master chaotic system [116]. As the name additive masking suggests, the message is buried or masked within the chaotic signal. The master system employed on the transmitter side is of the form  u (k + 1) = f [u (k)] , (6) s (k) = h [u (k)] + m (k) . It is easy to see that the first equation determines the discrete time evolution of the states whereas the second finds the outputs based on current states. At the receiver side, we have a slave system denoted by v (k) and given by 

v (k + 1) = f [v (k)] ,  s (k) = h [v (k)] .

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Note that u (k) ∈ Rn and v (k) ∈ Rn . This system is synchronized with the master by means of an appropriate method. Once synchronization is achieved, we simply subtract the the output of the slave from the received signal (assuming the effect of the transmission medium has been compensated for) yielding m (k) =  s (k) − s (k) m (k) .

(8)

Although this method is easy and straight forward, it suffers from a major deficiency due to the fact that the existence of the message signal hinders the synchronization process thereby reducing the performance of the communication system. 6.1.2

Chaotic Switching: Chaos Shift Keying (CSK)

In CSK–based communications, variations in the wavelength are used to encode and hide the message [117]. In this scenario, instead of a single chaotic system, we have N , where N is the size of our alphabet m (k). Note that by alphabet, we refer to the set of all symbols used to represent our message. The simplest form of CSK is the binary case where N = 2, i.e. the message is encoded using two symbols. In this case the alphabet is simply m (k) = {m (k) ∈ {0, 1}}. The basic idea is to switch between the two systems  u (k + 1) = f 1 [u1 (k)] , u0 (k + 1) = f 0 [u0 (k)] , and 1 s0 (k) = h0 [u0 (k)] , s1 (k) = h1 [u1 (k)] ,



(9)

depending on whether the current symbol is a zero or a one. The transmitted signal s (k) is, thus, a combination of pieces taken from the first and second chaotic systems depending on the sequence of bits in m (k). At the receiver, we also have two different slaves that are synchronized to the received signal. The slaves are of the form  v1 (k + 1) = f 0 [v0 (k)] , f 1 [v1 (k)] , v0 (k + 1) = and h1 [v1 (k)] .  s0 (k) = h0 [v0 (k)] ,  s1 (k) =



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Deciding whether the correct bit transmitted is a zero or a one follows one of two logical steps. The first is by minimizing the synchronization error between the two slave outputs  s0 (k) and  s1 (k) and the received signal. The synchronization errors are given by 

e0 (k) =  s0 (k) − s (k) , s1 (k) − s (k) . e1 (k) = 

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For instance, if the transmitted bit is 1, then e1 (k) will tend to zero but e0 (k) will not. The second logical reasoning is that the correct bit is selected based on the most correlated slave output to the received signal s (k). This is usually preferred over the first method as it is more resilient to noise.

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Usually, in order to achieve the synchronization of the slave and master systems, a reference signal is transmitted every other symbol. This, of course, reduces the data rate of the communication system, which is a major disadvantage of CSK. It is important to note that many variations exist based on this very simple explanation of CSK. For instance, instead of having multiple chaotic systems at the transmitter, we could simply have one but use different initial conditions to differentiate between the carriers. Other variations include antipodal CSK, unipodal CSK, chaos on–off keying (COOK), differential CSK (DCSK), frequency–modulation DCSK (FM–DCSK), and many others. In fact, DCSK–based schemes are the most popular at the moment [118–120]. A good description of DCSK and FM–DCSK along with some enhancements can be found in [121].

6.1.3

Parametric Modulation

In parametric modulation, the information message is not represented using different chaotic signals as is the case with CSK but is rather used to modulate a certain set of parameters within the chaotic system itself [122]. The general form for the master system is described as 



u (k + 1) = f u (k) , p (k) , s (k) = h [u (k)] ,

(12)

where p (k) represents a set of parameters that depend directly on the message m (k). Again, synchronization is used to drive a slave system at the receiver side to coincide with the received signal thereby identifying the states of the master system as well as the modulated parameters, which can in many cases be treated as extensions to the states. The message is, then, recovered from the parameters. In [123], a similar system was proposed with an extended Kalman filter (EKF) used for the synchronization process, which makes the system resilient to the additive noise affecting the transmission.

6.1.4

Message–Embedding

The message embedding approach [124] is quite similar to parametric modulation. The message is directly embedded into the master system as 

u (k + 1) = f [u (k) , m (k)] , s (k) = h [u (k)] .

(13)

In the simplest case, the functionals f and h are assumed to be linear leading to the linear embedding configuration described in [124] by

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u (k + 1) = Au (k) + Bm (k) , s (k) = Cu (k) ,

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where A ∈ Rn×n is called the input matrix, B ∈ Rm×m is called the mixing matrix, and C ∈ Rr×r is called the output matrix. In [124], the reception process as based on an unknown input observer (UIO), which works by selecting appropriate matrices Q, N , and L such that the slave system  s (k + 1) = N s (k) + Ls (k) , v (k + 1) = s (k + 1) + Qs (k + 1) ,

(15)

synchronizes with the master thereby minimizing the error e (k) = u (k) − v (k) .

(16)

An alternative scheme for decoding the original message is by synchronizing a slave system to the master and then using the inverse of the resulting output matrix C.

6.2 Multiple Access Schemes Since the necessary move towards wideband communications, the issue of multiple access to the medium while utilizing the same band of the spectrum became more important. Most multiple access techniques work by dividing the resources (time, frequency, and space) between users. The most common schemes include [125]: – Frequency division multiple access (FDMA): the frequency band allocated for the standard is divided between active users. This scheme has been used in advanced mobile phone systems, the CT2 cordless telephone, and the digital European cordless telephone. – Time division multiple access (TDMA): In TDMA, all users utilize the whole frequency band but alternate in different time slots. TDMA has been the basis for many systems including the initial digital cellular systems in the US and Japan and the global systems for mobile (GSM) communications standard. – Space division multiple access (SDMA). – Spread spectrum multiple access (SSMA): This technique uses a pseudo-noise to convert a narrowband signal to a wideband signal with characteristics similar to noise prior to transmission. SSMA schemes include frequency hopped multiple access (FHMA), direct sequence multiple access (DSMA) and code division multiple access (CDMA). Among the above mentioned multiple access schemes, CDMA is perhaps the most widely used as it is the basis of all 3G cellular telecommunications standards, which is why it has found itself a way into chaos. The basic idea behind CDMA is quite old and, in fact, dates back to the 1940s. However, the actual standards and specifications

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did not come about until the 1990s as a result of the big telecommunications boom that took place in the previous decade. The reason why CDMA was very popular for two decades is its ability to substantially improve the network capacity and the fact that it allows the hand-held device to communicate with more than one base station simultaneously. CDMA uses specific codes to spread the transmitted data from different users and ensure their orthogonality so as to be able to recover the data after it is coupled by the medium. These keys are usually pseudo-random, which means they can be generated by means of a chaotic system [126]. The received signal in a cell containing N users is of the form r (k) =

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(17)

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where ci (k) and di (k) are the orthogonal spreading code and information symbol belonging to the ith user, respectively, gi is the attenuation gain of the channel, τi is the time delay for the ith user, and n (k) is additive white Gaussian noise. The code ci (k) is generated by means of a chaotic system as 

u (k + 1) = f [u (k)] , ci (k) = h { f [u (k)]} .

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(18)

Here, the functional h is used to determine the spreading codes based on the state vector u (k), the chaotic nonlinearity f , and the orthogonality principle. At the receiver side, the slave system is chosen as 

v (k + 1) = f [v (k)] + Ki [r (k) − ci (k)] , ci (k) = h { f [v (k)]} ,

k = 1, 2, ..., N ,

(19)

where ci (k) is the estimated spreading code and Ki is a coupling parameter matrix. Once the spreading code is recovered, its orthogonality to other users’ codes allows for decoupling of the signal of interest. Although CDMA was a very attractive technology for almost 20 years, later cellular standards starting from WiMAX and LTE dropped it and moved towards OFDM and other division multiplexing techniques. However, when examining the literature related to chaotic CDMA, one finds quite recent studies on the subject such as [127, 128]. This, of course, warrants the question: is CDMA coming back? In addition to CDMA, there exist other chaos–based multiple access schemes such as multiple access antipodal chaos shift keying (MA–ACSK) and multiple access generalized correlation delay shift keying (MA–CDSK). These schemes are very promising and have received considerable attention from the research community in recent years. An exact explanation of the concept and implementation issues can be found in [129].

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6.3 Multicarrier Modulation Schemes (MDM) The concept of multicarrier modulation is quite similar to that of multiple access. However, the main idea here is that the data from a single user is split into components and transmitted over separate carriers each with a narrow bandwidth. MDM has many advantages over conventional modulation schemes, the most important of which is that by dividing the bandwidth into smaller portions the effects of multipath fading and inter–symbol interference are minimized. On the other hand, MCM usually suffers from synchronization and high peak to average power ratio (PAPR) issues. Many multicarrier schemes have been studied and deployed including frequency division multiplexing (FDM) and time division multiplexing (TDM), which simply work by dividing the frequency or time, respectively. More advanced techniques include orthogonal FDM (OFDM) and wavelength division multiplexing (WDM). 6.3.1

Orthogonal Frequency Division Multiplexing (OFDM)

Nowadays, the most widely used multicarrier scheme is orthogonal frequency division multiplexing (OFDM). It was first proposed in the 1960s and has been the basis for WiFi transmission ever since the inception of the first IEEE802.11a standard. In addition, it was selected for latest 4G cellular network standards such as WiMAX, LTE, and LTE–A. In fact, many researchers and academics consider it as the strongest contender for 5G networks and generally all future wireless communications. OFDM is a form of multicarrier modulation where data is multiplexed into a number of streams each modulating a different subcarrier. The bandwidth allowed for the transmission is divided equally between these subcarriers. Multicarrier schemes that proceeded OFDM had a strict requirement that the sidebands from these subcarriers do not overlap, which is necessary for reconstruction of the signals. OFDM alleviated this requirement by orthogonalizing the subcarriers. The exact details of OFDM and the way it operates are beyond the scope of this chapter. Interested readers are referred to [130]. As soon as chaos became a serious contender for telecommunications, it found itself a place in OFDM and more specifically OFDM passive optical networks (PON). Existing chaos–based OFDM schemes take advantage of the nice properties of chaos in different ways including signal scrambling [131], a chaotic QAM [132], encoding of the DFT matrix [133]... etc. A good survey of chaotic–OFDM methods can be found in [131]. 6.3.2

Wavelength Division Multiplexing (WDM)

Although chaotic systems have been employed in all types of communication systems, its main field of application is optical communications, where pulses of light are used to carry data through a fiber–optic cable. Since fiber-optics were first introduced in the 1970s, they have been able to revolutionize telecommunications by achiev-

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ing very high data rates, long distances, and a resilience to interference. In the last decade, concerns grew among the industrial and academic communities regarding the security of optical communications. One of the most common multicarrier schemes in optical communications is wavelength division multiplexing (WDM). The basic idea is that the data from a user is demultiplexed and carried on multiple subarriers (lasers) with different wavelengths (colors). In fact, since the wavelength λ of a light beam is directly related to its frequency f through the relation λ = v/ f , where v is the speed of light, many refer to WDM simply as FDM. Similar to OFDM in wireless communications, WDM increases the capacity of the channel and allows for bidirectional transmission over a single fibre. For more on WDM, the reader may look it up in [134–136].

6.4 Conventional Secret–Key Encryption Schemes The previously mentioned chaos–based schemes operate in the physical layer of the communication system. This is different from the conventional cryptography methods we saw in Sect. 3, which mostly operate in the presentation layer (layer 6) of the OSI model. Researchers have also been looking at way to utilize chaos in conventional secret or public–key ciphers. Many studies related to this subject can be found in the literature. For simplicity, we will only describe two of the most straight forward ones. 6.4.1

Mixed Encryption

The basic idea behind the mixed encryption proposed in [137] is fundamentally different from the previous masking and modulation schemes. This scheme is based on the impulsive synchronization of chaotic systems whose stability and feasibility is proven. Yang et al. proposed combining chaotic dynamical systems with conventional secret–key encryption. The information message is divided into frames. Each frame starts with impulses that are required for the synchronization process and then a block of symbols taken from the intended message and scrambled by means of an n–shift cipher according to a secret key extracted from the chaotic system. 6.4.2

Two–Way Transmission

Another encryption scheme was proposed in [138] based on chaotic systems. In this method, two distinct signals are transmitted s (k) and  m (k). These signals are generated by means of the master chaotic system defined by ⎧ ⎨ u (k + 1) = f [u (k)] , s (k) = h [u (k)] , ⎩  m (k) = Et [K (k) , m (k)] ,

(20)

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where K (k) is the secret key generated from the chaotic system and Et is a given encryption scheme. At the receiver, the decryption process follows the slave system ⎧ f [v (k)] , ⎨ v (k + 1) = s (k) = h [v (k)] , ⎩ m (k)] , m (k) = Er [K (k) , 

(21)

m (k) is the recovered with Er being the decryption process corresponding to Et and message. In summary, we have two chaotic systems which are synchronized by means of s (k) (transmitted through the first link) to produce the same key. The key is, then, used to encrypt and decrypt the message, which is transmitted separately in a second communication link.

7 Case Study In this section, we construct a simple chaos–based communication system and assess its performance and characteristics. Recall the chaos shift keying (CSK) modulation scheme we discussed in Sect. 6.1.2. For simplicity, let us consider the antipodal CSK (ACSK) scenario with a binary phase shift keying (BPSK) constellation as shown in Fig. 2, see [129]. BPSK simply encodes the bits {0, 1} as symbols {−1, 1}. These symbols are assumed to be independent and generated by means of a uniform distribution. In ACSK, the symbols are used to modulate a chaotic signal generated by a master system. Let us consider as our master the Lorenz chaotic system we saw earlier in the Chap. [9], which is given by ⎧ dx ⎨ dt = σ (y − x) , dy = x (ρ − z) − y, dt ⎩ dz = xy − βz, dt

(22)

where σ1 = 10, ρ1 = 28, and β1 = 83 are appropriately chosen parameters. It suffices to choose as our carrier one of the states and let that be x (t). At the receiver, the slave is chosen to be a replica of the same system used at the transmitter but with a slight modification that an estimate x (t) of x (t) is used instead of u (t) in the second and third equations. Hence, the slave is formulated as follows ⎧ du ⎨ dt = σ (v − u) , dv = x (ρ − w) − v, ⎩ ddtw = xv − βw, dt

(23)

with the same set of parameters from (22). It can be easily shown that given sufficient time, state u (t) converges towards x (t) and synchronization is achieved. The trajec-

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Fig. 2 Antipodal CSK system model with BPSK modulation: a the transmitter structure, and b the receiver structure

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tories generated by the proposed Lorenz system is well known as shown in Fig. 3. For each symbol in the transmit sequence, N successive symbols are taken from x (t) and multiplied by the symbol (1 or −1) before transmission. Figure 4 shows a portion of the transmitted signal s (t) along with the chaotic signal x (t). Assuming a wireless communication channel, the transmitted signal s (t) undergoes a low pass filtering process due to the signal reflecting and scattering off the many obstacles present in the environment. For simplicity, we consider that the channel is is modelled as an FIR filter with coefficients h = [1, 0.5, 0.2], which is frequency selective, i.e. affects the frequencies present in the transmitted signal s (t) differently.

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The frequency selectiveness of the channel leads to subsequent symbols becoming correlated, an effect that is commonly referred to as inter-symbol-interference. We also assume that the channel is stationary, i.e. does not change over time. Although this assumption might not be realistic in many scenarios, in practice, we usually transmit a training sequence repeatedly to estimate the channel and assume that the estimate remains reasonably accurate for a certain duration. In addition, imperfections in the reception process can be modeled as an additive white Gaussian noise (AWGN) sequence v (t). Hence, the received signal may be formulated as r (t) = h ⊗ s (t) + v (t) ,

(24)

where ⊗ denotes the convolution operation. Normally, the channel coefficients h are identified and accounted for by means of a process referred to as equalization. There are many equalization algorithms in use today. One of the simplest algorithms is the adaptive least mean square (LMS) scheme which uses a training sequence to identify and combat the effect of the channel adaptively. Table 3 describes the steps of the LMS algorithm. The equalized signal corresponding to the portion of s (t) shown in Fig. 4 is depicted in Fig. 5 for a signal to noise ratio (SNR) of 40 dB. Note that the SNR is defined as

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(25)

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Table 3 Steps of the least mean square (LMS) equalization algorithm Initialization: L = 10 μ = 10−5 W0 = [1, 0, 0, ..., 0]T Iteration: Rk = [r (k) , r (k − 1) , ..., r (k − L + 1)]T s (k) = WkH Rk e (k) = s (k) − s (k) Wk+1 = Wk + μe∗ (k) Rk

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Once the equalization and the above mentioned synchronization schemes are achieved, we move into the detection process. We can see that the very simple synchronization scheme we chose is efficient and produces a synchronization error that decays away to zero given sufficient time. The synchronization error is depicted in Fig. 6. As for the detection process, the simplest possible strategy is based on the observation that for a transmitted symbol of 1 the synchronization should be successful unlike when a −1 is transmitted, in which case no adequate synchronization will be achieved given the negation process involved in the modulation. Hence, we may simply make a decision on whether the transmitted symbol was a 1 or −1 based on the magnitude of the error e (t) = x (t) − x (t) .

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However, although this strategy is simple, the decision threshold varies considerably depending on the accuracy of the equalization process and the level of AWGN present. This makes for a problematic detection process. Another common strategy works by calculating the correlation between s (t) and x (t) for symbol period k as  y (kTb ) =

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(27)

where Tb is the symbol period and Ts is the time required for synchronization. Then, if the correlation produces a positive number, the received symbol is deemed to be a 1, otherwise it is a −1. Figure 7 shows the correlation of two arbitrary symbols taken from the coherent receiver at an SNR of 40dB. For more on the correlation based reception, see [129]. In order to assess the error performance of the proposed communication system, computer simulations were carried out where the SNR is varied from 0 to 12dB. The bit error rate (BER) is simply calculated as the number of errors present in a specific frame of transmitted data divided by the frame size. Figure 8 shows the resulting BER.

8 Summary In this chapter, we have seen a general overview of chaos synchronization and its applications in communication systems. Chaotic dynamical systems have a close relationship with encryption due to the many nice properties of chaos including the

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apparent randomness of the trajectories, the high sensitivity to variations in the initial conditions, the strange attractor, and the wide spectra of chaotic signals. Chaotic systems have attracted the interest of researchers in numerous fields including engineering and science. In particular, a vast amount of literature can be found ranging over the last three decades concerning the application of chaos in wireless and optical communications. Chaotic systems have been implemented as electronic circuits or laser diodes. We have made the observation that to the best of our knowledge, most of

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the chaos–based communication schemes found in the literature can be classified into four main categories: modulation schemes, multiuser access schemes, multicarrier schemes, and secret/public key cryptography schemes. In addition to producing a summary of the most promising chaos–based communication schemes, we have also considered a simple case study where we developed a complete physical layer communication system using the antipodal chaos shift keying (ACSK) modulation scheme and a correlation based coherent receiver. Although computer simulations have shown the feasibility and applicability of the proposed scheme, its computational complexity is considerable due to the high number of floating point multiplications and divisions involved. This raises many questions as to the realistic benefits from using chaotic modulation in telecommunications.

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Numerical Techniques for Fractional Competition Dynamics with Power-, Exponential- and Mittag-Leffler Laws Kolade M. Owolabi and Hemen Dutta

Abstract This chapter deals with modelling and analysis fractional competition system with power law, exponential law and the Mittag-leffler law in which the standard derivative in time is replaced with the Caputo, Caputo-Fabrizio and AtanganaBaleanu fractional derivatives. A fractional version of the Adams-Bashforth scheme is formulated for the approximation of these derivatives. To justify the applicability and suitability of these derivatives, we drawn comparison by applying them to solve some problems for specific value of fractional power α. In the simulation framework, we consider a number of fractional competition dynamics arising in applied areas of engineering and science. Keywords Atangana-Baleanu derivative · Fractional Adams-Bashforth-Moulton methods · Fractional order competition dynamics · Numerical simulations · Stability analysis 2010 Mathematics Subject Classification: 34A34 · 35A05 · 35K57 · 65L05 · 65M06 · 93C10

1 Introduction Consideration is given to n-species competing dynamics that is described by the following system of ordinary differential equations K. M. Owolabi (B) Faculty of Natural and Agricultural Sciences, Institute for Groundwater Studies, University of the Free State, Bloemfontein 9300, South Africa e-mail: [email protected]; [email protected] K. M. Owolabi Department of Mathematical Sciences, Federal University of Technology, PMB 704 Akure, Ondo State, Nigeria H. Dutta Department of Mathematics, Gauhati University, Guwahati 781014, India © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_10

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Dtα ui (t) = ui Ni (u1 , u2 , . . . , un ) = Fi (u1 , u2 , . . . , un ), i = 1, 2, . . . , n

(1)

where Dtα is the fractional operator of order α ∈ (0, 1], u = (u1 , u2 , . . . , un ) is a n vector that is situated in the positive cone C = R+ , Ni denote the per capital growth rates which holds for competing condition ∂Ni /∂uj ≤ 0 for i = j. The symbol Fi account for the local kinetics or reaction. When Ni are affine, the above system corresponds to the standard Lotka-Volterra equations. The standard competition models with local derivative is a known to be a predictive system of equations, since solutions are somewhat gained. The fractional order cases are similarly important and useful. The main advantage of adopting fractional order differential operator in mathematical modelling is dues to their nonlocal and nonsingular properties, as in the case of the Atangana-Baleanu fractional derivative [5, 7–9], compare to the classical differential equation which has a local operator. For survey of classical competitive and similar types of dynamical systems, see the book by Murray [22, 23], Allen [2], Kot [19], Petrás [37], Berryman [11] and Owolabi [27]. Some of the recent research papers include Owolabi and Patidar [26], Kandler [18], Berec et al. [10], Patriarca and Heinsalu [36] and Otero-Espinar et al. [35], among many others. Over the years, non-integer order differential equations have become a useful mathematical tool to model real-life problems encountered in applied areas of science and engineering [13, 16, 17, 21, 24, 25, 29, 38, 39]. Till date, and as one of the examples considered in this chapter, we are not aware of any work reported on the language competition dynamics where the classical derivative is modelled as non-integer order case. Hence, we are motivated in this chapter by obtaining a better representation of the competition models consisting of two- and three-species dynamics, using the Caputo and Atangana-Baleanu fractional derivatives. The rest of this work is sectioned as follows: In Sect. 2, we introduce some important fractional operators. We give an alternate representation of the competition dynamics in Sect. 3. Some numerical experiments for various fractional values are presented in Sect. 5. The manuscript is concluded in Sect. 6.

2 Preliminaries on Fractional Operators In the following, we present a review of some important definitions on fractional calculus. The Riemann-Liouville fractional operator of order α > 0 of a function f : (0, ∞) → R is defined by [30] Dα0 f (t) =

1 dn dt n (n − α)



t

(t − ξ)n−α−1 f (ξ)d ξ,

0 0) is given by the definition

Numerical Techniques for Fractional Competition Dynamics … RL α a It f

(t) =

1 (α)



t

a

f (ξ) , (t − ξ)1−α

315

0 0) is given by [31, 32, 34, 39] C α a Dt f

(t) =

1 (n − α)



t a

f (n) d ξ, n − 1 < α ≤ n. (t − ξ)α−n+1

The Caputo-Fabrizio fractional derivative of order α as reported in [12], is defined by    t M (α) −α(t − ξ) CF α  d ξ, 0 < α ≤ 1, (2) f (ξ) exp a Dt = (1 − α) a 1−α where f ∈ H 1 (a, b), b > a and M (α) is a normalization constant, such that M (0) = M (1) = 1. The Laplace transform of Caputo-Fabrizio fractional derivative is given   sn+1 L T [f (t)] − f (0)sn − f  (0)sn−1 · · · − f n (0) α L CF , 0 < α ≤ 1. 0 Dt f (t) = s + (1 − s)α

For f ∈ H 1 (a, b), b > α, α ∈ [0, 1] with f being differentiable function. Then, the Atangana-Baleanu fractional derivative in the sense of Caputo is define as ABC α Dt {f a

(t)} =

B(α) 1−α



t a

  (t − ξ)α d f (ξ)Eα −α d ξ, dt 1−α

(3)

where the function B satisfy B(0) = B(1) = 1, just as in the Caputo-Fabrizio case. The Laplace transform of the above equation is given by L

ABC a

 B(α) sα F(s) − sα−1 f (0) Dαt [f (t)] (s) = . α 1−α sα + 1−α

The term Eα in (3) denotes one-parameter Mittag-Leffler function defined by the series Eα (z) =

∞ n=0

zn , α > 0, z ∈ C, (αn + 1)

 α  t 1 = L Eα − α>0 1−α ξ x + xξ α

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It should be noted that E1 (z) =

∞ n=0



zn zn = = ez . (n + 1) n! n=0

A two-parameter generalization of the Mittag-Leffler function is given as follows: Eα,β =

∞ n=0

zn , (αn + β)

with R(α) > 0 and R(β) > 0,

where R(·) denotes the real part. With β = 1, we recover the classical case because Eα,1 (z) = Eα (z). Let f ∈ H 1 (a, b), b > α, 0 < α ≤ 1 and not necessarily differentiable, the fractional derivative with Mittag-Leffler law in the sense of Riemann-Liouville is defined by Atangana-Baleanu [5] as ABR α b Dt {f

(t)} =

B(α) d 1 − α dt



t b

  (t − ξ)α f (ξ)Eα −α d ξ, 1−α

(4)

where B(·) is the normalization function as given in [12]. The Laplace transform of the above derivative is given by L

ABR 0

   (t − ξ)α B(α) d t d ξ (s) f (ξ)Eα −α 1 − α dt b 1−α B(α) sα {f (t)}(s) = α 1 − α sα + 1−α

 Dαt f (t) (s) = L



3 Competitive System and Its Asymptotic Stability By setting n = 2, and replacing the classical time derivative with the fractional type, the general n−species system (1) reduces to two-component time-fractional dynamics Dtα u1 (t) = F1 (u1 , u2 ),

Dtα u2 (t) = F2 (u1 , u2 ), 0 < α ≤ 1

(5)

subject to initial values of the form u1 (0) = u10 ,

u2 (0) = u20 .

Next, we assume Dtα ui (t) = 0, which implies that Fi (u1∗ , u2∗ ) = 0, for i = 1, 2 in attempt to evaluate the stationary points at the neighbourhood of (u1∗ , u2∗ ).

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For the asymptotic stability, we assume ui (t) = ui∗ + σi (t), in such that Dtα (ui∗ + σi ) = Fi (u1∗ + σ1 , u2∗ + σ2 ) shows that

Dtα σi (t) = Fi (u1∗ + σ1 , u2∗ + σ2 )

bear in mind that Fi (u1∗

+

σ1 , u2∗

+ σ2 )

Fi (u1∗ , u2∗ )



∂Fi + ∂u1





∂Fi σ1 + ∂u2 ∗

 ∗

σ2 + · · ·

(6)

which implies that Fi (u1∗

+

σ1 , u2∗



∂Fi + σ2 ) ∂u1





∂Fi σ1 + ∂u 2 ∗

 ∗

σ2 + · · · .

Note that the we have already assumed Fi (u1∗ , u2∗ ) = 0, with Dtα σi (t)



∂Fi ∂u1

we have system





∂Fi σ1 + ∂u 2 ∗

 ∗

σ2

Dtα σ = Aσ

(7)

subject to initial conditions σ1 (0) = u1 (0) − u1∗ and σ2 (0) = u2 (0) − u2∗ , with Jacobian (denoting the community matrix) defined as A=

σ1 a11 a12 , and σ = . a21 a22 σ2

In a more compact form, we have B−1 AB = C where C=

λ1 0 0 λ2

denotes the diagonal matrix of A. Then, AB = BC and A = BCB−1 , which shows that Dtα σ = (BCB−1 )σ, Dtα (B−1 σ) = C(B−1 σ), then Dtα χ = Cχ, χ = B−1 σ, χ = [χ1 , χ2 ]T

(8)

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that is,

Dtα χ1 = λ1 χ1 ,

Dtα χ2 = λ2 χ2 .

(9)

The solution similar to the above system has been reported in terms of the oneparameter Mittag-leffler functions in [1] χ1 (t) =

∞ (λ1 )n t nα χ1 (0) = Eα (λ1 t α )χ1 (0), (nα + 1) n=0

∞ (λ2 )n t nα χ2 (0) = Eα (λ1 t α )χ2 (0). χ2 (t) = (nα + 1) n=0

(10)

By arguments [30], | arg(λ1 )| > απ/2 and | arg(λ2 )| > απ/2, then χ1 (t) and χ2 (t) are decreasing, so also σ1 (t) and σ2 (t). Hence, we conclude that the stationary point (u1∗ , u2∗ ) is locally asymptotically stable provided the eigenvalues of A have negative real part (| arg(λ1 )| > απ/2), | arg(λ2 )| > απ/2.

4 Numerical Methods of Approximation In a general term, the fractional system described by (1) can be represented as Dtα u(t) = F(t, u(t)), u(t0 ) = u0 , 0 < α ≤ 1,

t0 < t ≤ T .

(11)

Following the numerical idea formulated in [6], we replace the discrete interval [t0 , T ] by set of points tk = t0 + kh, h = (T − t0 )/N , k = 0, 1, 2, . . . , N , so that we can approximate the solution by {uk } for k = 0, 1, 2, . . . , N such that uk ≈ u(tk ). It has been shown that the exact solution of (11) can take the kernel form u(t) = u(t0 ) +

1 (α)



t

(t − τ )α−1 F(τ , u(τ ))d τ ,

(12)

t0

which further transforms into 1 u(ts+1 ) = u(t0 ) + (α) s

k=0



tk+1

(ts+1 − τ )α−1 F(τ , u(τ ))d τ .

(13)

tk

Next, in each of subinterval we consider the product trapezoidal rule to construct a numerical scheme 1 (α) s

k=0

 tk

tk+1

hα ωk,s+1 F(tk , u(tk )), (α + 2) k=0 (14) s+1

(ts+1 − τ )α−1 F(τ , u(τ ))d τ ≈

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319

where ⎧ ⎨

sα+1 − (s − α)(s + 1)α , for k = 0, + (s − k)α+1 − 2(s − k + 1)α+1 , for 1 ≤ k ≤ s, ωk,s+1 = (s − k + 2) ⎩ 1 for k = s + 1. (15) Approximation in (14) results to the implicit scheme us+1

α+1

  s hα = u0 + ωk,s+1 F(tk , uk ) + F(ts+1 , us+1 ) , (α + 2)

(16)

k=0

which is known as the fractional one-step Adams-Moulton scheme [4]. To compute p us+1 in the present chapter, we replace us+1 by us+1 to serve as predictor, so that the iteration becomes   s hα (j−1) (j)s+1 = u0 + ωk,s+1 F(tk , uk ) + F(ts+1 , us+1 ) , u (α + 2) k=0

u

(0)s+1

=

p us+1 ,

j = 1, 2, . . . , m.

(17)

We apply one-step fractional Adams-Bashforth scheme to compute the predictor formula as s hα p γk,s+1 F(tk , uk ). (18) us+1 = u0 + (α + 1) k=0

The schemes (17) and (18) now serve as the fractional predictor-corrector algorithms based on the generalized one-step Adams-Bashforth-Moulton methods. The smoothness assumption results on fractional operator Dtα , and solution u have been reported in [15, 20] as follows. Theorem 1 Let α > 0 and assume Dα u ∈ C 2 [t0 , T ] for some T . Then  max |u(tk ) − uk | =

0≤k≤N

O(h2 ), α ≥ 1, O(hα+1 ), α < 1.

(19)

Theorem 2 Let 0 < α < 1 and assume u ∈ C 2 [t0 , T ] for some T . Then, for 1 ≤ k ≤ N we get ⎧ 1 ⎪ ⎨ hα+1 , 0 < α < , 2 (20) |u(tk ) − uk | ≤ Ctkα−1 × ⎪ ⎩ h2−α , 1 ≤ α < 1 2 where C denotes a constant that is independent of k and h. As a result of some challenges faced by the power law representation, we formulate a viable numerical approximation technique based on the Mittag-Leffler concept as

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suggested by Atangana and Baleanu in [5]. Recall, by definition, we have ABR α 0 Dt {f

where

AB(α) 1−α

G(t) =



t 0

so that ABR α 0 Dt {f

(t)} =

and G(tj ) =

AB(α) 1−α AB(α) 1−α



tj+1

0

 0

(21)

G(t + t) − G(t) d G(t) = , dt t

0ABR Dαt {f (tj )} =

G(tj+1 ) =

d G(t) dt

  α (t − ξ)α d ξ, f (ξ)Eα − 1−α

which means

with

(t)} =

tj+1

G(tj+1 ) − G(tj ) + O(t), t

(22)

  α (tj+1 − ξ)α d ξ f (ξ)Eα − 1−α   α (tj − ξ)α d ξ f (ξ)Eα − 1−α

Next, we evaluate AB(α) G(tj+1 ) = 1−α j



k=0

tk+1

tk

  α f (tk+1 ) − f (tk ) α Eα − (tj+1 − ξ) d ξ + ϒα,j,k 2 1−α

AB(α) f (tk+1 ) − f (tk ) = 1−α 2 j

k=0





tk+1

Eα tk

 α α − (tj+1 − ξ) d ξ + ϒα,j,k 1−α

AB(α) f (tk+1 ) − f (tk ) α j + ϒα,j,k 1−α 2 j

=

k=0

where

  α (tj+1 − ξ)α Eα − 1−α tk   α (tj+1 − tk+1 )α = (tj+1 − tk+1 )Eα,2 − 1−α   α (tj+1 − tk )α + (tj+1 − tk )Eα,2 − 1−α

jα =



tk+1

The remainder term ϒα,j,k is expressed as

(23)

Numerical Techniques for Fractional Competition Dynamics … ϒα,j,k =

321

  j  AB(α) tk+1 α (f (ξ) − f (tk+1 ))Eα − (tj+1 − ξ)α d ξ 1−α 1−α tk k=0

=

  j  α AB(α) tk+1 (f (ξ) − f (tk+1 ))(ξ − tk+1 ) Eα − (tj+1 − ξ)α d ξ 1−α ξ − tk+1 1−α tk k=0

=

  j   AB(α) tk+1   α f (t)(ξ − tk+1 ) Eα − (tj+1 − ξ)α d ξ, ξ < t ≤ tj+1 , 1−α 1−α tk k=0

AB(α)t max {f  (t)} 1 − α 0≤t≤tj+1 j

=



tk+1

k=0 tk

 Eα −

 α (tj+1 − ξ)α d ξ 1−α

  j

AB(α)t α max {f  (t)} (tj+1 − tk+1 )α (tj+1 − tk+1 )Eα,2 − 1 − α 0≤t≤tj+1 1−α k=0   α (tj+1 − tk )Eα,2 − (tj+1 − tk )α , 1−α AB(α)t ≤ max {f  (t)}C a . 1 − α 0≤t≤tj+1

=

(24)

In a similar fashion, we evaluate G(tj ) = =

AB(α) 1−α



tj

0 j

  α (tj − ξ)α d ξ, f (ξ)Eα − 1−α

AB(α) f (tk+1 ) + f (tk ) α j + Hα,j,k 1−α 2

(25)

k=1

where   α (tj − ξ)α Eα − 1−α tk     α α (tj − tk+1 )α + (tj − tk )Eα,2 − (tj − tk )α , = (tj − tk+1 )Eα,2 − 1−α 1−α

αj =



tk+1

and the remainder term AB(α) = 1−α j

Hα,j,k

k=1

AB(α) = 1−α



tk+1 tk

 j

k=1

tk+1 tk

  α α (tj − ξ) d ξ (f (ξ) − f (tk+1 ))Eα − 1−α   α (f (ξ) − f (tk+1 ))(ξ − tk+1 ) α (tj − ξ) d ξ Eα − ξ − tk+1 1−α

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AB(α)t max {f  (t)} 1 − α 0≤t≤tj j

=

k=1



tk+1 tk

  α Eα − (tj − ξ)α d ξ 1−α

 

AB(α)t α (tj − tk+1 )Eα,2 − max {f  (t)} (tj − tk+1 )α 1 − α 0≤t≤tj+1 1−α k=1   α (tj − tk )α , (tj − tk )Eα,2 − 1−α AB(α)t ≤ max {f  (t)}C b . (26) 1 − α 0≤t≤tj j

=

Therefore, the numerical approximation of Atangana-Baleanu time-fractional derivative based on the Mittag-Leffler function in the sense of the Riemann-Liouville is given as ABR α 0 Dt {f (t)}

AB(α) = t(1 − α) ⎫ ⎧ j j−1 ⎨ f (tk+1 ) + f (tk ) α AB(α) f (tk+1 ) + f (tk ) α ⎬ j − j + Ejα , ⎭ ⎩ 2 1−α 2 k=0

k=0

for |Ejα | ≤

AB(α) max {f  (t)}C . 1 − α 0≤t≤tj

5 Numerical Examples and Computational Results All numerical computation results that are presented in this section is obtained using a super-fast computer with an Intel Core i5, 2.6-GHz processor, 16.0 GB RAM (MATLAB R.2013a).

5.1 Fractional Blood Alcohol Model with Exponential Decay Law We consider here the fractional version of the blood alcohol model [3, 40] CF α 0 Dt u(t) CF β 0 Dt v(t)

= −ω1 u(t), u(0) = u0 , = ω1 u(t) − ω2 v(t), 0 < α, β < 1, v(0) = 0,

(27)

where u(t) denotes the alcohol concentration in the stomach, v(t) represents the alcohol concentration in the blood, the initial alcohol consumed is given by u0 , ω1 and ω2 are positive parameters.

Numerical Techniques for Fractional Competition Dynamics …

323

By using the Laplace transform of Caputo-Fabrizio fractional derivative as defined in Sect. 2, and taking the Laplace transform of both sides of Eq. (27), we obtain u0 , αω1 + s(1 + ω1 (1 − α)) ω1 F(s)(α + s(1 − α)) v(s) = , αω2 + s(1 + ω2 (1 − α)) u(s) =

(28)

with inverse Laplace transform   αω1 t u0 exp − , u(t) = 1 + ω1 (1 − α) 1 + ω1 (1 − α)

  β + ω1 (β − α) αω1 t ω1 u0 exp − v(t) = (ω2 β − ω1 α) + ω1 ω2 (β − α) 1 + ω1 (1 − α) 1 + ω1 (1 − α)   β βω2 t − exp − . (29) 1 + ω2 (1 − β) 1 + ω2 (1 − β) In the simulation experiment, we set u0 ∈ [200, 375], ω1 = 0.066 and ω2 = 0.0084. For this case, we employ the three-step Adams-Bashforth scheme earlier developed by Owolabi and Atangana [28] for the Caputo-Fabrizio fractional derivative is given as 1−α 1−α 23αh 16αh + + yn+1 = yn + f (tn , yn ) − f (tn−1 , yn−1 ) M (α) 12M (α) M (α) 12M (α) 5αh + f (tn−2 , yn−2 ) + Rαn (t), (30) 12M (α) where Rαn (t) = Rαn (t) ∞

α M (α)

 0

t

3 (4) f (ξ)h3 d ξ 8

 t   3 (4) α  3  f (ξ)h  d ξ =  M (α) 0 8 ∞  t  (4)  3αh3 f (ξ) d ξ ≤ ∞ 8M (α) 0   3αh3 ≤ .Tmax (χ), χ = max f (4) (ξ)∞ . ξ∈[0,t] 8M (α)

(31)

Experimental results as shown in Fig. 1 revealed that u(t) which denotes the concentration of alcohol in the stomach is decreasing exponentially with time regardless of the value of fractional power. In Fig. 2, the alcohol concentration in the blood is decaying faster with respect to time for different values of α and β for 0 < α, β ≤ 1. Figure 3 depicts the behaviour of both u(t) and v(t) for some instances of fractional powers.

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Fig. 1 The concentration of alcohol in stomach for different values of fractional parameters α = 0.1(0) ∈ (0, 1]. The first and second plots correspond to parameters u0 = 200, ω1 = 0.064, ω2 = 0.008, t = 400 and u0 = 200, ω1 = 0.64, ω2 = 0.08, t = 100 respectively

Fig. 2 The concentration of alcohol in blood for different values of fractional parameters α, β ∈ (0, 1] and simulation time t = 300, u0 = 375, ω1 = 0.064, ω2 = 0.008 Fig. 3 The distribution of u(t) and v(t) for different instances of α, β ∈ (0, 1] and simulation time t = 2

Numerical Techniques for Fractional Competition Dynamics …

325

5.2 Fractional Language Competition Model with Power Law, Exponential Decay Law and Mittag-Leffler Law Representations For the second example, we consider the fractional version of the language competition model govern by the system of equation [18] u(t) , = φu(t)v(t) + ϕu(t) 1 − κ − v(t) v(t) , Dαt v(t) = φu(t)v(t) + ψv(t) 1 − κ − u(t) Dαt u(t)



(32)

where φ > 0 is the switch rate from speakers of language v to that of language u, the carrying capacities for both languages is assumed to be equal and is denoted by κ. The fractional derivative Dαt is given in terms of the power law as in the case of the Caputo derivative (C0 Dαt ), the exponential decay law as in the case of α the Caputo-Fabrizio derivative (CF 0 Dt ), and the Mittag-Leffler law as in the case of α the Atangana-Baleanu fractional derivative (ABR 0 Dt ) in Riemann-Liouville sense. In the simulation, we experiment with u0 = 0.1, v0 = 0.05, φ = and the corresponding numerical results for different values of α are given in Figs. 4, 5 and 6.

5.3 Fractional-Order Dadras Attractor Model In this case, we consider the fractional-order Dadras system described by the Atangana-Baleanu fractional derivative [41]

Fig. 4 The distribution of the language competition model with the Caputo fractional derivative for different instances of α and t = 40

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Fig. 5 Behaviour of the language competition model with the Caputo-Fabrizio fractional derivative for different instances of α and t = 20 ABR α 0 Dt u(t) ABR α 0 Dt v(t) ABR α 0 Dt w(t)

= v(t) − Au(t) + Bv(t)w(t), = Cv(t) − u(t)w(t) + w(t) = Du(t)v(t) − Ew(t)

(33)

with A = 3, B = 2.7, C = 1.7, D = 2 and E = 9. Computational results obtained for α = 0.36, α = 0.55 and α = 0.89 are respectively shown in Figs. 7, 8 and 9. Figure 10 shows the behaviour of the Dadras system with the Caputo fractional derivative obtained at α = 0.89. The results in Fig. 11 correspond to α = 0.89 when the Dadras dynamic system is modelled with the Caputo-Fabrizio fractional derivative. For the same parameter values, we compare the fractional-order system with the integer-order case when α = 1.0, obviously, the behaviour of the attractors as seen in Fig. 12 differ. Above all, only the Atangana-Baleanu depicts the actual behavioural representation of the Dadras system.

Numerical Techniques for Fractional Competition Dynamics …

327

Fig. 6 Behaviour of the language competition model with the Atangana-Baleanu fractional derivative for different instances of α and t = 3

Fig. 7 Chaotic attractor for fractional-order Dadras system (33) at α = 0.36 and simulation time t = 50

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Fig. 8 Chaotic attractor for fractional-order Dadras system (33) at α = 0.55 and simulation time t = 50

Fig. 9 Chaotic attractor for fractional-order Dadras system (33) at α = 0.89 and simulation time t = 50

Numerical Techniques for Fractional Competition Dynamics …

329

Fig. 10 Chaotic attractor for fractional-order Dadras system (33) with the Caputo fractional derivative at α = 0.89 and simulation time t = 50

Fig. 11 Chaotic attractor for fractional-order Dadras system (33) with the Caputo-Fabrizio fractional derivative at α = 0.89 and simulation time t = 50

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Fig. 12 Behaviour of fractional-order Dadras system (33) with α = 1.0 and simulation time t = 50

6 Conclusion In this chapter, modelling and analysis of the competition system with fractionalorder derivatives have been considered. the classical time-derivatives in such dynamics are replaced with the Caputo derivative (with power law concept), the CaputoFabrizio derivative (which takes into account the exponential law), and the AtanganaBaleanu derivative in the Riemann-Liouville sense (which is modelled based on the Mittag-Leffler law that has both the nonlocal and nonsingular properties). Numerical schemes based on the Adams-Bashforth methods are formulated for the approximation of these derivatives. Numerical experiment for different values of α are presented for the fractional blood alcohol model, language competition system, and the Dadras system with promising results. The overall credit is given to the Atangana-Baleanu fractional derivative for its ability to replicate the exact behaviour of noninteger-order dynamics. Extension to time-space fractional system is left for future research.

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Existence of Periodic Solutions for First Order Differential Equations with Applications Smita Pati

Abstract In this chapter, by using a fixed point theorem in cones in a Banach space, we present different sufficient conditions for the existence of at least two positive periodic solutions of first order functional differential equations. The results, presented in this chapter, are then applied to the Nicholson’s Blowflies model and the generalized Michaelis-Menton type single species growth model. Keywords Existence of solutions · Positive solution · Periodic solution · Fixed point theorem · Functional differential equation

1 Introduction Qualitative theory of differential equations deal with the behaviour of solutions without solving the related equation. Dynamical characteristics, such as, existence, uniqueness, oscillation, nonoscillation, periodicity, stability, persistence and global attractivity of solutions are studied under the theory. Ordinary differential equations are often first approximations to real world systems, but inclusion of information about past states of time in the equations, e.g. formulating them as functional differential equations, the existing model can be refined into more accurate one. In evolution theory, the selective forces on ecosystems in a fluctuating environment differ from those in a stable environment. Thus it is reasonable to study the existence and stability behaviour of positive periodic solutions of mathematical models occurring in biology and ecology, which are strongly influenced by periodic environmental variations. From biological point of view, only positive solutions are important.

S. Pati (B) Department of Mathematics, Amity School of Engineering and Technology, Amity University Jharkhand, Ranchi 834002, Jharkhand, India e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_11

333

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The rapid progress in the area of functional differential equations is due to the ability to capture the dynamics in a periodically fluctuating environment. There has also been great interest in finding conditions under which the functional differential equation admit multiple positive periodic solutions, which have wide applications in biology. One characteristic phenomenon of population dynamics is the often observed oscillation behaviour of the population densities. To better understand such a phenomenon, one mechanism is to introduce time delays in the models, which results in models described by functional differential equations. A functional differential equation is a more general type of differential equation, in which the unknown function occurs with various different arguments. A first order delay differential equation is of the form x  (t) = f (t, x(t), x(t − τ (t))) is a first order delay differential equation; x  (t) = f (t, x(t), x(t + τ (t))) represents a first order advanced differential equation, and x  (t) = f (t, x(t), x(t − τ (t)), x  (t − τ (t))) is a first order neutral delay differential equation, where 0 < τ (t) ≤ t. The same definition can be applied to higher order functional differential equations and system of differential equations. One of the simplest and most natural type of functional differential equation is a delay differential equation. The salt brine example in Driver [14] shows the occurrence of a first order delay differential equation in our day to day life. Similarly, the predator-prey model in Driver [14] shows the existence of a first order system of delay differential equation in ecology. One may refer Driver [14] for a detailed elementary study on the existence, nonexistence, stability and asymptotic stability of solutions of first order functional differential equations and of system of functional differential equations. Method of steps [14] is one of the important and elementary method for finding solution of a delay differential equation. There are many examples arising in physical, biological and ecological systems, which shows that the present rate of change of some unknown function(s) depends upon some past values of the same function(s) [14, 32]. In [32], Murray showed how a model using delay differential equation is capable of generating limit cycle periodic solutions. Consider the first order nonlinear functional differential equations of the form

and

x  (t) = −a(t)x(t) + f (t, x(h(t)))

(1)

x  (t) = a(t)x(t) − f (t, x(h(t))),

(2)

Existence of Periodic Solutions for First Order Differential …

335

where a and h ∈ C(R, R + ) are T -periodic, T is a positive constant, R = (−∞, ∞), R + = [0, ∞), a(t) = 0 and f ∈ C(R × R + , R + ) is periodic with period T , with respect to the first variable. Functional differential equations of the form (1) and (2) include many mathematical, ecological and biological models, directly or after some transformations. Here we deal with the following mathematical models: (i) Nicholson’s Blowflies model x  (t) = −a(t)x(t) + b(t)x m (t − τ (t))e−γ(t)x

p

(t−τ (t))

,

(3)

(ii) The generalized Michaelis-Menton type single species growth model  

x (t) = x(t) a(t) −

n  i=1

 bi (t)x(t − τi (t)) , 1 + ci (t)x(t − τi (t))

(4)

where a, b, γ, bi , ci , τ and τi ∈ C(R, R+ ), 1 ≤ i ≤ n are T -periodic functions, m > 1, and p > 0 are reals, T > 0 is a constant. Another important model in ecological dynamics is the model with Allee effect. Any ecological mechanism that can lead to a positive relationship between a component of individual fitness and either the number or density of conspecifies can be termed as a mechanism of Allee effect [4, 25, 44, 48]. Several mechanisms generating Allee effects have been suggested by Berec et al. [4]. Consider the model representing the dynamics of a renewable resources x, subjected to Allee effects: x  (t) = a(t)x(t)(x(t) − b(t))(c(t) − x(t)),

(5)

where a, b and c ∈ C(R, R + ) with a(t) > 0 and 0 < b(t) < c(t). As a particular case of (5), we consider x  (t) = ax(t)(x(t) − b)(c − x(t)), a > 0, 0 < b < c.

(6)

Here a, c and b represent, respectively, the intrinsic growth rate, carrying capacity of the resource and the threshold value below which the growth rate of the resource is negative. In recent years, researchers have been fascinated discovering the nonlinear differential equations of order greater than or equal to one, which, for certain values of their parameter have one of the following characteristics: (i) Every solution of the equation is periodic with the same period. (ii) Every solution of the equation is eventually periodic with a prescribed period, and (iii) Every solution of the equation converges to a periodic solution with the some prescribe period.

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Functional differential equations of the form (2) can be interpreted as the standard Malthus population model x  (t) = a(t)x subject to a perturbation with periodic delay. One of the rapidly growing research problem at the present scenario is whether these periodic functional differential equations (both the general functional differential equations and the mathematical models) can support any positive periodic solution. The functional differential boundary value problems can be considered in the frame of general theory of functional differential equations. This is evident from the book [2]. One may refer to [26] on the formulation of basic concepts for the periodic solutions and original results on the existence, uniqueness and positivity of solutions for linear and nonlinear functional differential equations. For various results on the positivity of solutions of periodic problems for functional differential equations can be found in [9, 19, 20]. One of the first results on positivity of solutions of linear functional differential equation with general type operator x  (t) + (Bx)(t) = f (t), t ∈ [0, ω], where B : C[0,ω] → L [0,ω] is a linear continuous operator, f ∈ L [0,ω] , in the form of results on positivity of Green’s function were studied in [18]. We note that L [0,ω] and C[0,ω] are the spaces of essentially bounded and continuous functions x : [0, ω] → R respectively. Results on existence of periodic solutions to system of delay differential equations xi (t) +

n 

pi j (t)x j (h i j (t)) = f i (t), i = 1, 2, . . . , n,

(7)

j=1

and their positivity in the case of nonpositive off-diagonal coefficients pi j (t) ≤ 0 for i, j = 1, 2, . . . , n, j = i.

(8)

were studied in [10]. In an another work, Domoshnitsky et al. [13] obtained some important developments for the Eq. (7) in the case, where the condition (8) is not fulfilled. The main idea of the approach in [13] was to construct a corresponding scalar functional differential equation of the first order xn (t) + (Bxn )(t) = f (t), t ∈ [0, ω],

(9)

for the n-th component of a solution vector, where B : C[0,ω] → L [0,ω] is a linear continuous operator, f ∈ L [0,ω] and then the results on positivity of solutions of the periodic solution to scalar functional differential equation (9) were used. One may refer to [1, 3, 6, 15, 21, 31, 35, 36, 39, 40, 47] for some recent results on the positivity of solutions of periodic problems, and to [11, 12] for positivity of Green’s functions of functional differential equations. Many authors have studied the existence of at least one or two positive periodic solutions of (1) and (2). One may refer [7, 24, 50, 51] and the references cited therein

Existence of Periodic Solutions for First Order Differential …

337

for (1) and (2). The method used in some of the references given above are mainly the upper lower solution method , Krasnoselskii fixed point theorem, fixed point theorem of cone expansion and cone compression, and fixed point index theory [8]. Our main emphasize, in this chapter, is the use of Leggett-Williams multiple fixed point theorem [28] for the existence of at least two positive periodic solutions of the considered equation. One may refer [24, 27, 45, 46, 51] on the use of Krasnoselskii fixed point theorem [8] to study existence of at least one or two positive periodic solutions of (5) with m = 1 and p = 1. Though existence of one periodic solution of (5) is largely studied in the literature, study on existence of at least two or three periodic solutions of (3) are less studied. Padhi et al. [37, 38] have used Leggett-Williams multiple fixed point theorem [28] to show existence of at least three positive periodic solutions of (5). To study the existence of three periodic solutions, the function f needs to be unimodal, that is, the function f first increases and then decreases eventually, which excludes many mathematical and ecological models, such as, (3) and (4). That motivates to study the existence of two periodic solutions of the mathematical models (3) and (4).

2 Basic Concepts Let X be a real Banach space. A closed convex set K ⊂ X is called a (positive) cone if the following conditions are satisfied: (i) if x ∈ K , then λx ∈ K for λ ≥ 0; (ii) if x ∈ K and −x ∈ K , then x = 0. A completely continuous map means a continuous function which takes bounded sets into relatively compact sets. A continuous map ψ : K → R + is said to be a continuous concave positive functional on K if ψ(ρx + (1 − ρ)y) ≥ ρψ(x) + (1 − ρ)ψ(y), x, y ∈ K , ρ ∈ [0, 1]. One may refer [8, 28] to define different types of positive concave functional. For a > 0, define K a = {x ∈ K ; ||x|| < a}. Then K a = {x ∈ K ; ||x|| ≤ a}. Let b, c > 0 be constants with K and X as defined above. Define K (ψ; b, c) = {x ∈ K ; ψ(x) ≥ b, ||x|| ≤ c}. With this, we now state the following Leggett-Williams fixed point theorem. Theorem 2.1 ([28]): Let X = (X, . ) be a Banach space and K ⊂ X a cone, and c3 > 0 be a constant. Assume that A : K c3 → K is completely continuous, there exists a concave nonnegative functional ψ with ψ(x) ≤ x , x ∈ K and numbers c1 and c2 with 0 < c1 < c2 < c3 satisfying the following conditions:

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S. Pati

(i) {x ∈ K (ψ, c2 , c3 ) : ψ(x) > c2 } = φ and ψ(Ax) > c2 if x ∈ K (ψ, c2 , c3 ); (ii) Ax < c1 if x ∈ K c1 ; and (iii) ψ(Ax) > cc23 Ax for each x ∈ K c3 with Ax > c3 . Then A has at least two fixed points x1 , x2 in K c3 . Furthermore, x1 ≤ c1 < x2 < c3 . Lemma 2.1 ([7]) The set X = {x ∈ C[0, T ]; x(0) = x(T )} endowed with the norm

x = sup x(t), 0≤t≤T

where C[0, T ] is the set of all real valued continuous functions defined on [0, T ] is a Banach space. Lemma 2.2 ([43]) Let X be Banach space and K be a cone in X . Define a concave nonnegative continuous function ψ on K by ψ(x) = min x(t). 0≤t≤T

Then ψ is concave. Consider Eq. (1), which is equivalent to the integral equation 

t+T

x(t) =

G(t, s) f (s, x(h(s))) ds,

(10)

t

where G(t, s) =

e

e

T 0

s t

a(θ) dθ

a(θ) dθ

−1

is the Green’s function satisfying 0 1.

(11)

(12)

Let X = {x(t); x ∈ C(R, R), x(t) = x(t + T )} with the norm x = supt∈[0,T ] |x(t)|, then X is a Banach space with the norm . . Define a cone K in X by K = {x(t); x ∈ X, x(t) ≥

x

, t ∈ [0, T ]} δ

(13)

Existence of Periodic Solutions for First Order Differential …

339

and an operator E on X by 

t+T

(E x)(t) =

G(t, s) f (s, x(h(s))) ds.

t

Lemma 2.3 The operator E satisfies E(K ) ⊂ K and E : K → K is compact and completely continuous. Proof First to show E x ∈ K , we have 

t+2T

(E x)(t + T ) = =

t+T  t+T

G(t + T, s) f (s, x(h(s))) ds

G(t + T, r + T ) f (r + T, x(h(r + T ))) dr

t



t+T

=

G(t, r ) f (r, x(h(r ))) dr

t

= (E x)(t). This shows (E x) ∈ K . For x ∈ K we have 

T

E x ≤ β

f (s, x(h(s))) ds

(14)

0

and  (E x)(t) ≥ α

T

f (s, x(h(s))) ds,

0

we have (E x)(t) ≥

E x

α

E x ≥ . β δ

This shows that (E x) ∈ K , that is, E(K ) ⊂ K . Next, to show that E is a completely continuous operator. We have considered X as a Banach space consisting of all positive T - periodic functions equipped with the sup norm, and K be a positive cone in X . Assume that for any M > 0 and > 0, there exists a δ > 0 such that for u, v ∈ K with ||u|| ≤ M, ||v|| ≤ M, and ||u − v|| < δ, we have sup | f (s, u(h(s))) − f (s, v(h(s)))| < 0≤s≤T

. βT

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S. Pati

This implies 

t+T

|(E x)(t) − (E y)(t)| ≤ t



T

≤β < .

|G(t, s)|| f (s, x(h(s))) − f (s, y(h(s)))| ds | f (s, x(h(s))) − f (s, y(h(s)))| ds

0

Hence E is continuous. Next, we show that f maps bounded sets into bounded sets. Set = 1; then | f (s, x(h(s))) − f (s, y(h(s)))| < 1. Choose a positive integer N , such that for i = 0, 1, ..., N . If x ≤ M, then

xi − xi−1 = sup |x(t) t∈R

M N

< δ. Let x ∈ X and define xi (t) = x(t)( Ni )

i −1

x

M i − x(t) |= ≤ < δ. N N N N

Hence, | f (s, xi (h(s))) − f (s, xi−1 (h(s)))| < 1 for s ∈ [0, T ], which implies | f (s, x(h(s)))| ≤

N 

| f (s, xi (h(s))) − f (s, xi−1 (h(s)))| + | f (s, 0)|

i=1

≤ N + f (s, 0)

= M1 (say). From (14) and using the fact that t ∈ [0, T ], we have 

T

E x ≤ β

f (s, x(h(s))) ds ≤ βT M1 .

(15)

0

Finally, for t ∈ R, d (E x)(t) = [G(t, t + T ) f (t + T, x(h(t + T ))) − G(t, t) f (t, x(h(t)))] − a(t)(E x)(t). dt

From (15) and the choice of M1 , we get |

d (E x)(t)| ≤ [ a βT M1 + M1 ]. dt

Hence {(E x) : x ∈ K , x < M} is a family of uniformly bounded and equicontinuous function on [0, T ]. Thus by Ascoli-Arzela theorem (Royden [43], p.169), E is a completely continuous operator. This completes the proof of the lemma.

Existence of Periodic Solutions for First Order Differential …

341

Lemma 2.4 The existence of a positive periodic solution of (1) is equivalent to the existence of a fixed point of E in K . Proof First, for x ∈ K and E x = x, we have  t+T  d G(t, s) f (s, x(h(s))) ds dt t = G(t, t + T ) f (t + T, x(h(t + T ))) − G(t, t) f (t, x(h(t)))  t+T ∂ G(t, s) f (s, x(h(s))) ds + ∂t t = [G(t, t + T ) − G(t, t)] f (t, x(h(t))) − a(t)(E x)(t) ∂ = f (t, x(h(t))) − a(t)x(t), where G(t, s) = −a(t)G(t, s). ∂t

x  (t) =

Next, if x is a positive T -periodic solution, we have  (E x)(t) =

t+T

G(t, s) f (s, x(h(s))) ds

t

 =

t+T

G(t, s)(x  (s) + a(s)x(s)) ds

t

 =

t+T

G(t, s)x  (s) ds +

t



t+T

G(t, s)a(s)x(s) ds t

 t+T ∂ G(t, s)x(s) ds + G(t, s)a(s)x(s) ds ∂s t t  t+T = [G(t, t + T ) − G(t, t)]x(t) − G(t, s)a(s)x(s) ds 

= [G(t, s)x(s)]t+T − t

t+T

t



t+T

+

G(t, s)a(s)x(s) ds t

= x(t),

where

∂ G(t, s) = a(s)G(t, s). ∂s

Clearly, (E x)(t) ≥ 0, t ∈ [0, T ]. This completes the proof of the lemma.



One may obtain similar results for Eq. (2). In this case, Eq. (2) is equivalent to the integral equation  x(t) =

t+T

G 1 (t, s) f (s, x(h(s))) ds,

t

where G 1 (t, s) =

e−

s t

1 − e−

a(θ) dθ T 0

a(θ) dθ

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S. Pati

represents the Green’s function, and 0 < α1 =

δ1 1 ≤ G 1 (t, s) ≤ = β1 , for all s ∈ [t, t + T ], 1 − δ1 1 − δ1

are the bounds of the Green’s function, where δ1 = e −

T 0

a(θ) dθ

< 1.

In this case, one may define a cone K 1 in X by K 1 = {x(t); x ∈ X, x(t) ≥ δ1 ||x||, for all t ∈ [0, T ]}.

3 Application to Nicholson’s Blowflies Model In this section, we deal with the existence of at least two positive periodic solutions of the nonlinear functional differential equation x  (t) = −a(t)x(t) + p(t)x m (t − τ (t))e−γ(t)x

n

(t−τ (t))

− H (t, x(t)),

(16)

where a, p, τ , γ ∈ C(R, R + ) are T -periodic functions, H ∈ C(R × R + , R + ) and is periodic with respect to the first variable, m > 1 and n > 0 are reals and T > 0 is a constant. By the periodicity of τ (t), we have limt→∞ (t − τ (t)) = ∞. Gurney et al. [17] proposed a mathematical model x  (t) = −ax(t) + px(t − τ )e−γx(t−τ ) to describe the population of Australian sheep-blowfly that agrees well with the experimental data of Nicholson [33, 34] and hence it is referred as the Nicholson’s blowflies model. Later, Berezansky et al. [5] made a review about the Nicholson’s Blowflies differential equation, and presented the model x  (t) = px(t − τ )e−x(t−τ ) − ax(t), where p > 0 is the maximum per capita daily egg production rate, a > 0 is per capita daily adult mortality rate, τ is the generation time (or the time taken from birth to maturity), σ is the time of maturity. One may refer [22, 23, 30, 49] for the existence of at least one positive periodic solution of x  (t) = −a(t)x(t) + p(t)x m (t − τ (t))e−γ(t)x

n

(t−τ (t))

(17)

with m = 1 and n = 1. Krasnoselskii’s fixed point theorem [8] has been used to prove the results. One may refer the monograph due to Padhi et al. [40] for a detailed study

Existence of Periodic Solutions for First Order Differential …

343

on the multiple positive periodic solutions of the model (17). In [16], Guang and Yong studied the existence of two positive periodic solutions for (17) with m = 2, n = 1. Zhao et al. [52] used Krasnoselskii’s fixed point theorem for the existence of at least one positive periodic solution of the generalized Nicholson’s blowflies model with delay and harvesting x  (t) =

m 

p(t)x(t − τi )e−x(t−τi ) − δ(t)x(t) − H (t, x(t)).

i=1

Long et al. [29] used continuation theorem of coincidence degree to study the existence and uniqueness of positive periodic solutions of N  (t) = −δ(t)N (t) + p(t)N (t − τ (t))e−a(t)N (t−τ (t)) − H (t)N (t − τ (t)), where δ, p, a ∈ C(R, R + ) and τ , H ∈ C(R, R + ) are T -periodic functions. Though the existence of positive periodic solutions of (17) has been studied by many authors using different fixed point theorems, the study of existence of positive periodic solutions of (16) have also been given sufficient attention. Leggett-Williams multiple fixed point theorem [28] has been used to obtain several results concerning the existence of three positive periodic solutions of (16). One may observe that different mathematical models exhibit several positive periodic solutions, such as, stable periodic solutions and unstable periodic solutions. This motivates to study on the existence of multiple positive periodic solutions of different models. Set g(t, x(t)) = 1 +

H (t,x(t)) ; a(t)x(t)

then (16) can be rewritten as

x  (t) = −a(t)g(t, x(t))x(t) + p(t)x m (t − τ (t))e−γ(t)x

n

(t−τ (t))

.

(18)

Assume that (H ) : g ∈ C(R, R + ), and g(t, x) is T -periodic with respect to the first variable. The function g(t, x) satisfies 0 < l ≤ g(t, x) ≤ L < ∞ for all x > 0 and t ∈ [0, T ], where l and L are positive constants. Let X be the set of all continuous positive T -periodic functions endowed with sup norm x = supt∈[0,T ] |x(t)|. Then X forms a Banach space. For x ∈ X , Eq. (18) is equivalent to the integral equation  x(t) =

t+T

G x (t, s) p(s)x m (s − τ (s))e−γ(s)x

t

where G x (t, s) =

e

e

T 0

s t

a(θ)g(θ,x(θ)) dθ

a(θ)g(θ,x(θ)) dθ

−1

n

(s−τ (s))

ds,

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S. Pati

is the Green’s kernel. This Green’s Kernel satisfies the integral property 

t+T



e

s t

a(θ)g(θ,x(θ)) dθ

T

a(s)g(s, x(s)) ds t e 0 a(θ)g(θ,x(θ)) dθ − 1  t+T

t e t a(θ)g(θ,x(θ)) dθ − e t a(θ)g(θ,x(θ)) dθ T

= e 0 a(θ)g(θ,x(θ)) dθ − 1 T

e 0 a(θ)g(θ,x(θ)) dθ − 1

= 1. = T e 0 a(θ)g(θ,x(θ)) dθ − 1

G x (t, s)a(s)g(s, x(s)) ds =

t

Set σ = e

t+T

T 0

a(θ) dθ

; then the Green’s kernel G x (t, s) is bounded by α=

σL

1 σL ≤ G x (t, s) ≤ l = β. −1 σ −1

Thus, if we define an operator A on X by 

t+T

(Ax)(t) =

G x (t, s) p(s)x m (s − τ (s))e−γ(s)x

n

(s−τ (s))

ds,

t

then for x ∈ X , we have the following inequalities: σL ||Ax|| ≤ l σ −1



Hence, setting ρ = property

1 L σ −1

σl −1 , σ L (σ L −1)

p(s)x m (s − τ (s))e−γ(s)x

n

(s−τ (s))

ds

(19)

0

and (Ax)(t) ≥

T



T

p(s)x m (s − τ (s))e−γ(s)x

n

(s−τ (s))

ds.

(20)

0

0 < ρ < 1, the inequalities (19) and (20) yields the

(Ax)(t) ≥

σl − 1 ||Ax|| = ρ||Ax||. σ L (σ L − 1)

(21)

Thus, if we define a cone P on X by P = {x ∈ X ; x(t) ≥ ρ||x|| > 0}, then A(P) ⊂ P, and A : P → P is completely continuous. The existence of a positive periodic solution of (18) is equivalent to the existence of a fixed point of the operator A in P.

Existence of Periodic Solutions for First Order Differential …

345

On the cone P, we define a nonnegative concave functional ψ by ψ(x) = min x(t). t∈[0,T ]

Let γ = max γ(t). 0≤t≤T

We shall use the following notations: u∗ = max u(t), u∗ = min u(t). 0≤t≤T

0≤t≤T

Theorem 3.1 Assume that m > 1, n > 0, m−1 (H0 ) a∗lγ n < p ∗ ρm−1 , and m−1 T σn L −1)γ n (H1 ) 0 p(s) ds > e (σρ2m−1 holds. Then (16) admits at least two T -periodic solutions; one nonnegative and one positive. ρ 3 Proof Set c2 = γ 1/n and c3 ∈ ( cρ2 , σρ c2 ): then 0 < c2 < c3 . As φ0 = c2 +c ∈ {x; x ∈ 2 P(ψ, c2 , c3 ), ψ(x) > c2 }, the set {x; x ∈ P(ψ, c2 , c3 ), ψ(x) > c2 } is nonempty. For x ∈ P(ψ, c2 , c3 ), we obtain



t+T

ψ(Ax) = min

0≤t≤T

G x (t, s) p(s)x m (s − τ (s))e−γ(s)x

n

(s−τ (s))

ds

t

 T 1 n ≥ L p(s)x m (s − τ (s))e−γ(s)x (s−τ (s)) ds σ −1 0  T 1 n −n n p(s)ρm c2m e−γc2 ρ σ ds ≥ L σ −1 0 n −n n  T ρm c2m e−γc2 ρ σ p(s) ds ≥ σL − 1 0 n  T ρm c2m−1 e−σ p(s) ds ≥ c2 σL − 1 0  ρm−1 T ρm p(s) ds = c2 σn L e (σ − 1) γ m−1 n 0 > c2 . Hence, assumption (i) of Theorem 2.1 is satisfied. Now, choosing a constant  the 1  m−1 a∗ l , we observe that the inequality c1 < c2 < c3 holds. Hence, for c1 ∈ 0, p∗ x ∈ P c1 , we have

346

S. Pati



t+T

||Ax|| = sup 0≤t≤T



0≤t≤T



n

(s−τ (s))

ds

t t+T

≤ sup c1m

G x (t, s) p(s)x m (s − τ (s))e−γ(s)x

t



G x (t, s) p(s)||x||m ds t+T

sup 0≤t≤T

G x (t, s) p(s)

t

a(s)g(s, x(s)) ds a(s)g(s, x(s))

 t+T p∗ ≤ c1m sup G x (t, s)a(s)g(s, x(s)) ds a∗ l 0≤t≤T t p∗ ≤ c1 c1m−1 < c1 , a∗ l

which implies the assumption (ii) of Theorem 2.1. Finally, for x ∈ P(ψ, c2 , c3 ) with ||Ax|| > c3 , the inequality c3 < ||Ax|| ≤

σL σl − 1



T

p(s)x m (s − τ (s))e−γ(s)x

n

(s−τ (s))

ds

0

implies that 

t+T

ψ(Ax) = min

0≤t≤T

G x (t, s) p(s)x m (s − τ (s))e−γ(s)x

n

(s−τ (s))

ds

t

 t+T 1 n p(s)x m (s − τ (s))e−γ(s)x (s−τ (s)) ds L σ −1 t (σl − 1) α ≥ L

Ax = Ax = ρ Ax

L (σ − 1)σ β c2 > Ax . c3 ≥

This shows that assumption (iii) of Theorem 2.1 is satisfied. By Theorem 2.1, the operator A has at least two positive fixed points in P, and hence (18) admits two positive T -periodic solutions. Consequently, (16) has at least two positive T -periodic solutions. The proof is complete. Theorem 3.2 Let m > 1,

m

(m−1) n (H2 ) (a∗l)n σ n(m−1) γ m−1 < p ∗ ρn(m−1) ln σ Lρ−1 , and m − (m−1) (m−1) T n n σ m−1 (H3 ) 0 p(t) dt > γρm−1 ln σ Lρ−1 hold. Then (16) admits at least two T -periodic solutions; one nonnegative and one positive. m

1/n and a constant c3 satisfying c3 ∈ ( cρ2 , σρ c2 ); then Proof Set c2 = σγρ1/n ln σ Lρ−1 0 < c2 < c3 . Clearly, the set {x; x ∈ P(ψ, c2 , c3 ), ψ(x) > c2 } is nonempty. Now, for

Existence of Periodic Solutions for First Order Differential …

347

x ∈ P(ψ, c2 , c3 ), we obtain 

t+T

ψ(Ax) = min

0≤t≤T

G x (t, s) p(s)x m (s − τ (s))e−γ(s)x

n

(s−τ (s))

ds

t

 T 1 n p(s)x m (s − τ (s))e−γ(s)x (s−τ (s)) ds L σ −1 0  T 1 n p(s)ρm ||x||m e−γ(s)||x|| ds ≥ L σ −1 0  T 1 n .ρm c2m−1 e−γ||x|| p(s) ds ≥ c2 L σ −1 0  m−1    n ρm 1 σL − 1 T ρm−1 m .ρ . ≥ c2 L ln p(s) ds m−1 σ −1 σL − 1 ρm 0 σ m−1 γ n > c2 . ≥

Hence, the assumption (i) of Theorem 2.1 is satisfied. Choosing c1 as in Theorem 3.1, we see that c1 < c2 . Now the proof of the remainder of the theorem is similar to that of Theorem 3.1, and hence is omitted. Theorem 3.3 Let m > 1 and ρm−1 > eσ γ n , and (H0 ) hold. Further, suppose that T L (H4 ) 0 p(t) dt > (σ ρm−1) holds. Then (16) admits at least two T - periodic solutions; one nonnegative and one positive. n

m−1

Remark 3.1 Let (H1 ) holds. If eσ γ n

m−1 n

> ρm−1 ,

then, from (H1 ), we have 

T

eσ (σ L − 1)γ ρ2m−1 n

p(t) dt >

0

m−1 n

σ L − 1 eσ γ n ρm ρm−1 L σ −1 > , ρm n

m−1

=

that is, (H4 ) is satisfied. m−1 n On the other hand, if ρm−1 > eσ γ n holds, then from (H4 ), we have 

T 0

σ L − 1 ρm−1 (σ L − 1)eσ γ > ρm ρm−1 ρ2m−1 n

p(t) dt >

m−1 n

,

348

S. Pati

that is, (H1 ) is satisfied. Thus, the conditions of Theorem 3.3, given above, imply the conditions of Theorem 3.1. Consequently, (16) has at least two positive T -periodic solutions. However, for completeness, the proof of Theorem 3.3 is given below. Further, from the above discussion, it follows that it would be possible to obtain an example, to which Theorem 3.1 can be applied where as, Theorem 3.3 cannot be applied. ρ and c3 ∈ ( cρ2 , σρ c2 ); then 0 < Proof (Proof of Theorem 3.3) Set constants c2 = γ 1/n c2 < c3 holds and the set {x; x ∈ P(ψ, c2 , c3 ), ψ(x) > c2 } is nonempty. Now, for x ∈ P(ψ, c2 , c3 ), we obtain



t+T

ψ(Ax) = min

0≤t≤T

G x (t, s) p(s)x m (s − τ (s))e−γ(s)x

n

(s−τ (s))

ds

t

 T 1 n p(s)x m (s − τ (s))e−γ(s)x (s−τ (s)) ds L σ −1 0  T 1 n p(s)ρm ||x||m e−γ(s)||x|| ds ≥ L σ −1 0  T 1 n .ρm c2m−1 e−γ||x|| p(s) ds ≥ c2 L σ −1 0 > c2 . ≥

This shows that condition (i) of Theorem 2.1 is satisfied. The remainder of the proof is similar to that of Theorem 3.1. This completes the proof of the theorem. The following theorem provide easily verifiable sufficient condition for the existence of two positive T -periodic solutions of (16) on the existence of two positive T -periodic solutions. Theorem 3.4 Let m > 1 and ρ2m−1 > eσ γ n , and (H0 ) hold. Further, suppose that T L 0 p(t) dt > σ − 1 holds. Then (16) admits at least two T - periodic solutions; one nonnegative and one positive. n

m−1

Theorem 3.5 Let m > 1, (H5 ) a∗ lρm < p ∗ , and  n −n(2m−1)/(m−1) T (H6 ) 0 p(t) dt > (σ L − 1)eγσ ρ hold. Then (2) admits at least two T - periodic solutions; one nonnegative and one positive. 1 Proof Set c2 = ρm/(m−1) and c3 ∈ ( cρ2 , σcρ 2 ); then 0 < c2 < c3 , and the set {x; x ∈ P(ψ, c2 , c3 ), ψ(x) > c2 } is nonempty. For x ∈ P(ψ, c2 , c3 ), we obtain

Existence of Periodic Solutions for First Order Differential …



t+T

ψ(Ax) = min

0≤t≤T

t

349

G x (t, s) p(s)x m (s − τ (s))e−γ(s)x

n

(s−τ (s))

ds



T 1 n p(s)x m (s − τ (s))e−γ(s)x (s−τ (s)) ds σL − 1 0  T 1 n p(s)ρm ||x||m e−γ(s)||x|| ds ≥ L σ −1 0  T c2 n n 1 .ρm c2m−1 e−γ( ρ ) σ p(s) ds ≥ c2 L σ −1 0  T 1 n (2m−1)/(m−1) p(s) ds ≥ c2 L e−σ γρ σ −1 0 > c2



This proves the assumption (i) of Theorem 2.1. Choosing c1 =



a∗ l p∗

1

(m−1)

, we observe

that c1 < c2 and proceeding as in Theorem 3.1, we can prove that A : P c1 → Pc1 . Now, the rest of the proof is similar to that of Theorem 3.1 and hence we omit it. The theorem is proved. Theorem 3.6 Let m > 1, 0 < σ L < 2, (H5 ), and T

 − (m−1) (m−1) n (H7 ) 0 p(s) ds > γ n ρ−(2m−1) ln σ L1−1 σ (m−1) hold. Then (16) admits at least two T - periodic solutions; one nonnegative and one positive.

 1/n and c3 ∈ ( cρ2 , σcρ 2 ); then 0 < c2 < c3 , Proof Set constants c2 = σγρ1/n ln σ L1−1 and the set {x; x ∈ P(ψ, c2 , c3 ), ψ(x) > c2 } is nonempty. For x ∈ P(ψ, c2 , c3 ) we obtain  t+T n G x (t, s) p(s)x m (s − τ (s))e−γ(s)x (s−τ (s)) ds ψ(Ax) = min 0≤t≤T

t

 T 1 n ≥ L p(s)x m (s − τ (s))e−γ(s)x (s−τ (s)) ds σ −1 0  T 1 n p(s)ρm ||x||m e−γ(s)||x|| ds ≥ L σ −1 0  T c2 n n 1 .ρm c2m−1 e−γ( ρ ) σ p(s) ds ≥ c2 L σ −1 0 > c2 .  1  m−1 ; then This proves condition (i) of Theorem 2.1. Set a constant c1 ∈ 0, ap∗∗l 0 < c1 < c2 . Now the rest of the proof is similar to that of Theorem 3.1, and hence omitted.

350

S. Pati

4 Application to Michaelis Menton Model Consider the generalized Michaelis-Menton type single species growth model:  x (t) = x(t) a(t) − 

 b(t)x(t − τ (t)) . 1 + c(t)x(t − τ (t))

(22)

The Green’s kernel for this equation is G 1 (t, s) =

e−

s t

1 − e−

a(θ) dθ T 0

a(θ) dθ

,

(23)

which is bounded by 0 < α1 = where δ1 = e−

T 0

δ1 1 ≤ G 1 (t, s) ≤ = β1 , s ∈ [t, t + T ], 1 − δ1 1 − δ1

a(s) ds

(24)

< 1.

Let X be the set of all continuous functions which forms a Banach space with sup norm

x = sup |x(t)|. t∈[0,T ]

Define a cone P1 on X by P1 = {x ∈ X ; x(t) ≥ δ1 x , t ∈ [0, T ]} and an operator A1 on P1 by  (A1 x)(t) = t

t+T

G 1 (t, s)

b(s)x(s)x(s − τ (s)) ds. 1 + c(s)x(s − τ (s))

From the periodicity and the continuity of c(t), it follows that there exists a constant η > 0 such that 0 < c(t) ≤ η, 0 ≤ t ≤ T . In the following we provide two different theorems for the existence of two positive periodic solutions of (22) for the cases η < δ12 and η ≥ δ12 . First suppose that η ≥ δ1 2 . Then there exists a real γ > 0 such that δ12 − γη > 0. We have the following theorem. T 1 , then (22) has at least two positive Theorem 4.1 Let η ≥ δ12 . If 0 b(t) dt > 1−δ δ1 γ periodic solutions.

Existence of Periodic Solutions for First Order Differential …

351

Proof From the definition of the cone P1 , we observe that δ1 x ≤ x(t − τ (t)) ≤

x for t ∈ [0, T ]. Then 1

x



T 0

1 b(s)x(s)x(s − τ (s)) ds ≤ G 1 (t, s) 1 + c(s)x(s − τ (s)) 1 − δ1



T

0

b(s) x

ds. 1 + c(s)δ1 x

x Since lim x→0 1+c(t)δ = 0, t ∈ [0, T ], then the right hand side of the above inequality 1x approaches to zero as x → 0, which proves the condition (ii) in Theorem 2.1 holds for arbitrarily small c1 . 1 . We note Now, we prove (i) in Theorem 2.1. For this, let c3 = cδ21 and c2 = δ2γδ 1 −ηγ that the set {x ∈ P(ψ, c2 , c3 ) : ψ(x) > c2 } is not empty because a constant function c2 +c3 is in the set. For each x in the cone P1 with c2 ≤ ψ(x) and x ≤ c3 , we have 2 c2 ≤ x ≤ c3 and δ1 c2 ≤ x(t − τ (t)) ≤ c2 /δ1 , t ∈ [0, T ]. Then

ψ(A1 x) ≥

δ1 1 − δ1



T

b(s) 0

(δ1 c2 )c2 ds 1 + c(s)c2 /δ1

c22 δ13 (1 − δ1 ) (δ1 + ηc2 ) δ12 c22 > γ(δ1 + ηc2 ) = c2 .



T

>

b(s) ds 0

Hence (i) in Theorem 2.1 is satisfied. It is easy to verify that the condition (iii) in Theorem 2.1 holds. By Theorem 2.1, Eq. (22) has at least two positive T -periodic solutions. This completes the proof of the theorem. T 1 holds, then (22) has at least two Theorem 4.2 Let η < δ12 . If 0 b(s) ds > 1−δ δ1 positive periodic solutions. Proof From the definition of the cone P1 , we observe that δ1 x ≤ x(t − τ (t)) ≤

x for t ∈ [0, T ] holds. Then 1

x



T 0

G 1 (t, s)

1 b(s)x(s)x(s − τ (s)) ds ≤ 1 + c(s)x(s − τ (s)) 1 − δ1

 0

T

b(s) x

ds. 1 + c(s)δ1 x

x Since lim x→0 1+c(t)δ = 0, t ∈ [0, T ], then the right hand side of the inequality 1x approaches to zero as x → 0, which implies that (ii) in Theorem 2.1 holds for arbitrarily small c1 . 1 . Clearly Now, we prove (i) in Theorem 2.1. For this, let c3 = cδ21 and c2 = δ2δ−η c2 +c3 2

1

∈ {x ∈ P(ψ, c2 , c3 ) : ψ(x) > c2 } shows that the set is not empty. For each x

352

S. Pati

in the cone P1 with c2 ≤ ψ(x) and x ≤ c3 , we have c2 ≤ x ≤ c3 and δ1 c2 ≤ x(t − τ (t)) ≤ c2 /δ1 , t ∈ [0, T ]. Then 

b(s)x(s)x(s − τ (s)) ds 1 + c(s)x(s − τ (s)) t  T δ1 b(s)c2 δ1 c2 ≥ ds 1 − δ1 0 1 + c(s)c2 /δ1  T δ12 δ1 c22 > b(s) ds (1 − δ1 ) (δ1 + ηc2 ) 0 δ12 c22 (1 − δ1 ) δ1 > (1 − δ1 ) (δ1 + ηc2 ) δ1 = c2

ψ(A1 x) = min

t+t

0≤t≤T

G 1 (t, s)

holds, that is, (i) in Theorem 2.1 is satisfied. The condition (iii) in Theorem 2.1 is easy to prove. Hence Eq. (22) has at least two positive T -periodic solutions. Thus the theorem is proved. Now, we assume that the population is subject to harvesting under the catch-perunit-effort hypothesis [19]. Then the harvested population’s growth reads  x (t) = x(t) a(t) − 

 b(t)x(t − τ (t)) − q E x(t), 1 + c(t)x(t − τ (t))

(25)

where q and E are positive constants denoting the catch ability coefficient and the harvesting effort, respectively. Since c(t) is continuous and periodic with period T , there exists a constant c > 0 such that 0 < c(t) < c, for 0 ≤ t ≤ T . Theorem 4.3 Suppose that q E T < 1 − δ1 and  δ12 (1 − δ1 )

T

 b(t) dt ≥ c

0

1 (1 − δ1 ) − q E T δ1

 (26)

hold. Then (25) has at least two positive periodic solutions. Proof We define the Banach space X , cone P1 in X and Green’s kernel G 1 (t, s) in the same way as they are defined for Eq. (22). We define an operator A2 on X by 

t+T

(A2 x)(t) = t

 b(s)x(s)x(s − τ (s)) + q E x(s) ds. G 1 (t, s) 1 + c(s)x(s − τ (s)) 

  −q E T ) T1 Choose a constant c1 ∈ 0, (1−δ . Then 0

b(t) dt

  T  1 c1 b(t) dt + q E T < 1. (1 − δ1 ) 0

Existence of Periodic Solutions for First Order Differential …

353

For x ∈ P c1 , we have   T b(s)x(s)x(s − τ (s)) 1

A2 x ≤ + q E x(s) ds 1 − δ1 0 1 + c(s)x(s − τ (s))  T 1 ≤ [b(s)x(s)x(s − τ (s)) + q E x(s)] ds 1 − δ1 0   T  c1 c1 b(s) ds + q E T < c1 . ≤ 1 − δ1 0 This proves (ii) in Theorem 2.1. Next, we prove the condition (i) in Theorem 2.1. Set c2 =

( δ1 (1−δ1 )−q E T ) 1 T δ12 0 b(t) dt

and c3 =

c2 . Then c1 δ1

< c2 < c3 . Now, for each x in the cone

P1 with c2 ≤ ψ(x) and x ≤ c3 , we have c2 ≤ x ≤ c3 and δ1 c2 ≤ x(t − τ (t)) ≤ c2 /δ1 , t ∈ [0, T ]. Then 

 b(s)x(s)x(s − τ (s)) + q E x(s) ds 1 + c(s)x(s − τ (s)) 0   T b(s)c2 δ1 c2 δ1 ≥ + q Ec2 ds 1 − δ1 0 1 + c cδ21    T δ12 c2 c2 δ1 b(t) dt + q E T . ≥ 1 − δ1 (δ1 + cc2 ) 0

ψ(A2 x) ≥

δ1 1 − δ1



T

With the above choice of c2 , the above inequality yields, using (26) that ψ(A2 x) > c2 for c2 ≤ x ≤ c3 . Hence (i) in Theorem 2.1 is satisfied. The proof of (iii) in Theorem 2.1 is easy and hence omitted. Consequently, Eq. (25) has at least two positive T -periodic solutions. This completes the proof of theorem. Ye et al. [50] considered the single species harvested population’s growth model   b(t)x(t) − q E x(t). x (t) = x(t) a(t) − 1 + cx(t) 

(27)

They proved the following result. Theorem 4.4 If 0 < q E


1 − δ1 δ12 T

hold, then (27) has at least one positive T -periodic solution, where bm = min0≤t≤T b(t). A direct application of Leggett-Williams multiple fixed point Theorem 2.1 to Eq. (27), yields the following theorem.

354

S. Pati

Theorem 4.5 If 0 < q E
c(1 − δ1 )

0

hold, then (27) has at least two positive periodic solutions. Proof We define the Banach space X , cone P1 and Green’s kernel G 1 (t, s) in the same way as defined for Eq. (22). We define operator A3 by 

t+T

(A3 x)(t) =

 G 1 (t, s)

t

 b(s)x 2 (s) + q E x(s) ds. 1 + cx(s)

  −q E T ) T1 Choose a constant c1 ∈ 0, (1−δ . Then 0

b(t) dt

  T  1 c1 b(t) dt + q E T < 1. (1 − δ1 ) 0 For x ∈ P c1 , we have   T b(s)x 2 (s) 1

A3 x ≤ + q E x(s) ds 1 − δ1 0 1 + cx(s)  T

 1 ≤ b(s)x 2 (s) + q E x(s) ds 1 − δ1 0   T  c1 c1 b(s) ds + q E T ≤ 1 − δ1 0 < c1 . This proves (ii) in Theorem 2.1. Next, we prove the condition (i) in Theorem 2.1. and c3 = cδ21 . A simple Choose a constant c2 > 0 such that c2 = 2  T δ1 (1−δ1 ) δ1

0

b(s) ds−c(1−δ1 )

calculation shows that c1 < c2 < c3 . Now, for each x in the cone P1 with c2 ≤ ψ(x) and x ≤ c3 , we have that c2 ≤ x ≤ c3 , t ∈ [0, T ]. Then for x ∈ P(ψ, c2 , c3 ), we have   T b(s)x 2 (s) δ1 + q E x(s) ds ψ(A3 x) ≥ 1 − δ1 0 1 + cx(s)   T b(s)c22 δ1 ≥ + q Ec2 ds 1 − δ1 0 1 + c cδ21    T δ1 c2 c2 δ1 b(s) ds + q E T ≥ 1 − δ1 (δ1 + cc2 ) 0 > c2 .

Existence of Periodic Solutions for First Order Differential …

355

The property (iii) of Theorem 2.1 is easy hence omitted. This completes the proof of theorem. A particular case of  x (t) = x(t) a(t) − 

 b(t)x(t) − q E x(t). 1 + c(t)x(t)

(28)

is (27), that is, if c(t) = c, a constant, then (28) reduces to (27). Remark 4.1 The models (22), (25) and (28) exhibit Allee effect if a(t) < b(t) for c(t) b(t) ∗ the Eq. (22) or if a(t) < c(t) and a < q E for the Eqs. (25) and (28). Observe that the condition b∗ > a ∗ c∗ implies that a(t) < b(t) . In the following, we derive some c(t) sufficient conditions different from the earlier results. Theorem 4.6 Let solutions.

b∗ c∗

> a ∗ holds. Then (22) has at least two positive T -periodic

Proof Setting F(t, x) = we observe that lim sup x→∞

b(t)x(t)x(t − τ (t)) , 1 + c(t)x(t − τ (t)) b∗ F(t, x) > ∗ ∗. a(t)x c a

This implies that there exists a positive constant c2 > 0 such that b(t)x(t)x(t − τ (t)) b∗ > ∗ ∗ c2 ≥ c2 a(t)(1 + c(t)x(t − τ (t))) c a holds for c2 ≤ x ≤ c3 and 0 ≤ t ≤ T where c3 = cδ21 . Hence, for x ∈ K (ψ, c2 , c3 ) we have  t+T b(s)x(s)x(s − τ (s)) ds G 1 (t, s) ψ(A3 x) = min 0≤t≤T t 1 + c(s)x(s − τ (s))  t+T ≥ c2 G 1 (t, s)a(s) ds = c2 . t

To complete the proof of the theorem, it is enough to find the existence of a positive constant c1 ∈ (0, c2 ) such that the condition (ii) in Theorem 2.1 holds. = 0 holds, then there exists a 0 < < 1 and ρ ∈ (0, c2 ) such Since lim supx→0 F(t,x) a(t)x that F(t, x) < a(t)x for 0 < x < ρ. Now choosing ρ = c1 , and using the property  t+T G 1 (t, s)a(s) ds ≡ 1, we can prove the required property. The theorem is proved. t

356

S. Pati

Again consider Eq. (28), which is equivalent to the integral equation  x(t) =

t+T

G 1 (t, s)Q(s, x) ds,

t

where G 1 (t, s) is the Green’s kernel given in (23) that satisfy the property (24), and Q(t, x) =

b(t)x 2 (t) + q E x(t). 1 + c(t)x(t)

We assume the Banach space X and the cone P1 as considered for the Eq. (22). Here we define an operator A4 by  (A4 x)(t) =

t+T

G 1 (t, s)Q(s, x) ds.

t

Then A4 (P1 ) ⊂ P1 and A4 is a completely continuous operator. Further, the existence of positive periodic solutions of (28) is equivalent to the existence of fixed points for A4 in P1 . Theorem 4.7 Let a∗ > q E holds. If b∗ > a ∗ c∗ hold. Then (28) has at least two positive T -periodic solutions. Proof Since lim supx→∞

Q(t,x) a(t)x

>

c2 > such that

b∗ , c∗ a ∗

holds, then there exists a positive constant

1 (a∗ − q E) > 0 b∗

b(t)x(t) qE b∗ + > ∗ ∗ a(t)(1 + c(t)x(t)) a(t) c a

holds for c2 ≤ x ≤ c3 and 0 ≤ t ≤ T where c3 = cδ21 . Then for x ∈ P(ψ, c2 , c3 ), we have  t+T G 1 (t, s)Q(s, x) ds ψ(A4 x) = min 0≤t≤T t    t+T b(s)x(s) x(s) + q E ds = min G 1 (t, s)a(s) 0≤t≤T t a(s) 1 + c(s)x(s)  t+T > c2 min G 1 (t, s)a(s) ds 0≤t≤T

= c2 .

t

Existence of Periodic Solutions for First Order Differential …

357

Choose c1 ∈ (0, b1∗ (a∗ − q E)). Then 0 < c1 < c2 holds. If x ∈ P c1 , then 

t+T

A4 x ≤ sup 0≤t≤T

G 1 (t, s)Q(s, x) ds

t

  b(s)x(s) x(s) + q E ds = sup G 1 (t, s)a(s) a(s) 1 + c(s)x(s) 0≤t≤T t  t+T (b∗ c1 + q E) ≤ c1 sup G 1 (t, s)a(s) ds a∗ 0≤t≤T t (b∗ c1 + q E) ≤ c1 ≤ c1 a∗ 

t+T

holds. The reminder of the proof of the theorem is similar to the proof of Theorem 4.3. By Theorem 2.1, (28) has at least two positive T -periodic solutions. The proof is complete. Theorem 4.8 If a ∗ < q E holds, then (28) has no positive T -periodic solutions. Proof Clearly (28) can be expressed in the form x  (t) = −q E x(t) + g(t, x(t)), where

 g(t, x(t)) = x(t) a(t) −

(29)

 b(t)x(t) . 1 + c(t)x(t)

Let x(t) be a positive T -periodic solutions of (29). Here we consider the Banach space X as in previous theorems. Define a cone P2 on X by P2 = {x ∈ X ; x(t) ≥ e−q E T x , t ∈ [0, T ]} and an operator A4 by  (A4 x)(t) =

t+T

H (t, s)g(s, x) ds,

t

where H (t, s) =

eq E(s−t) eq E T − 1

is the Green’s Kernel for the Eq. (29) which satisfies the integral property  t

t+T

q E H (t, s) ds ≡ 1.

358

S. Pati

Then A4 (P2 ) ⊂ P2 and A4 is a completely continuous operator. Further, the existence of solutions of (29) is equivalent to the existence of fixed points for A4 in P2 . Since x(t) is a positive periodic solution of (29), then x(t) = (A4 x)(t) for all x ∈ P2 . Hence  t+T H (t, s)g(s, x) ds

x = A4 x = sup 0≤t≤T



t+T

≤ sup 0≤t≤T



H (t, s)a(s)x(s) ds

t t+T

≤ sup 0≤t≤T

t

t

H (t, s)a ∗ x ds



t+T

< x sup 0≤t≤T

H (t, s)q E ds

t

= x , which is a contradiction. Hence the theorem is proved. Example 4.1 Consider  1 1 (2 + sin2t)x(t) − (ln 1.038)x(t). (30) x (t) = x(t) (1.1 + sin2t) − 8 1 + (2 + cos2t)x(t) π 



Here a(t) = 18 (1.1 + sin2t), b(t) = 2 + sin2t, c(t) = 2 + cos2t and Eq = π1 ln 1.038. Clearly, T = π, a∗ = 0.0125, a ∗ = 0.2625 and c∗ = 3 implies that q E = 0.0118 < 0.0125 = a∗ and a ∗ c∗ = 0.2625 × 3 = 0.7875 < 1 = b∗ hold. Hence Theorem 4.7 can be applied to this example, and so (30) has at least two positive π-periodic solutions. Example 4.2 Consider the equation x  (t) = x(t)



 (2 + sin2t)x(t) 1 1 (1.1 + sin2t) − − (ln 1.038)x(t). 8 1 + 3x(t) π

(31)

Here a(t) = 18 (1.1 + sin2t), b(t) = 2 + sin2t, c = 3 and Eq = π1 ln 1.038. As in Example 4.1, Theorem 4.7 can be applied to (31). The calculations 

π



0

and

π

a(t) dt = 0.43196899,

b(t) dt = 6.283185

0

δ1 = e −

π 0

a(s) ds

= 0.649230,

Existence of Periodic Solutions for First Order Differential …

359

q Eπ = 0.037296 < 0.350770 = 1 − δ1 , implies that  δ12

T

b(t) dt = 2.6483599 > 1.05231 = c(1 − δ1 )

0

implies that Theorem 4.5 can be applied to (31). It is easy to see that Theorem 4.4 can be applied to (31) and it gives the existence of at least one positive T -periodic solutions. Notice that Theorem 4.5 guarantee the existence of at least two positive π-periodic solutions of (31).

5 Conclusion The results on existence of periodic solutions of Eqs. (1), (2) and (22) are from [41]. The results on existence of periodic solutions of Eq. (16) are from [42]. Existence of periodic solutions of Eqs. (1), (2) and (22) can also be found in [40]. Acknowledgements The author is thankful to the referees for their helpful suggestions and constructions in improving the chapter to the present form.

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11. A. Domoshnitsky and M. Drakhlin; On boundary value problems for first order impulse functional differential equations, Boundary Value Problems for Functional Differential Equations, Editor J. Henderson, World Scientific, Singapore-New Jersy-London- Hong Kong, (1995), 107–117 12. Domoshnitsky, A., Drakhlin, M.: Nonoscillation of first order impulse differential equations with delay. J. Math. anal. Appl. 206, 254–269 (1997) 13. A. Domoshnitsky, R. Hakl, and J. Šremr; Component-wise positivity of solutions to periodic boundary value problem for linear functional differential system, J. Ineq. Appl., 2010(2012):112, https://doi.org/10.1186/1029-242X-2012-112. 14. Driver, R.D.: Ordinary and Delay Differential Equations. Springer, New York (1977) 15. Gopalsamy, K., Trofimchuk, S.I.: Almost periodic solutions of Lasota-Wazewska type delay differential equation. J. Math. anal. Appl. 237, 106–127 (1999) 16. Guang, Z.S., Yong, F.Z.: Existence of two positive periodic solutions for Nicholson’s blowflies functional diffrential equations. Natural Science Journal of Xiangtan University 34(1), 11–15 (2012) 17. Gurney, W., Blythe, S., Nisbet, R.: Nicholson’s Blowflies revisited. Nature 287, 17–21 (1980) 18. Gusarenko, S.A., Domoshnitsii, A.I.: Asymptotic and oscillation properties of first order linear scalar functional-differential equations. Differentsial’nye Uravneija 25(12), 2090–2103 (1989) ˇ 19. R. Hakl, A. Lomtatidze and J. Sremr; Some Boundary value Problems for First Order Scalar Functional Differential Equations, FOLIA Facul. Sci. Natur. Univ. Masar. Brun., Mathematica 10, Brno: Masaryk University, 2002 20. R. Hakl, A. Lomtatidze and J. Šremr; On a boundary value problem of periodic type of firstorder linear functional differential equations, Nonlinear Oscillations, 5(2002), 408–425 21. Han, F., Wang, Q.: Existence of multiple positive periodic solutions for differential equation with state-dependent delays. J. Math. anal. Appl. 324, 908–920 (2006) 22. Jhang, G., Zhu, D., Bi, P.: Existence of periodic solutions of a scalar functional diffrential equation via a fixed point theorem. Math. Comput. Model. 46(5–6), 718–729 (2007) 23. Jiang, D., Wei, J.: Existence of positive periodic solutions for nonautonomous deley diffrential equations,(in Chinese) Chinese Ann. Math. Ser. A 20(6), 715–720 (1999) 24. D. Jiang, J. Wei and B. Jhang; Positive periodic solutions for functional differential equations and population models, Electron. J. Differen. Eqns. 2002(71), 1–13 (2002) 25. Kent, A., Doncaster, C.P., Sluckin, T.: Consequences for depredators of rescue and Allee effects on prey. Eco. Modelling 162, 233–245 (2003) 26. Kiguradze, I., Puza, B.: Boundary Value Problems for Systems of Linear Functional Differential Equations. Brno, Czech Republic, FOLIA (2002) 27. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993) 28. Leggett, R.W., Williams, L.R.: Multiple positive fixed points of nonlinear operators on ordered Banach Spaces. Ind. Univ. Math. J. 28, 673–688 (1979) 29. F. Long and M. Yang; Positive periodic solutions of delayed Nicholson’s blowflies model with a linear harvesting term, Electron. J. Qual. Theo. Diff. Eqns., No.41(2011), 1–11 30. Lu, S., Ge, W.: On the existence of positive periodic solutions for neutral functional diffrential equation with multiple deviating arguments. Acta Math. Appl. Sin. Engl. Ser. 19(4), 631–640 (2003) 31. Lu, S., Ge, W.: Existence of positive periodic solutions for neutral population model with multiple delays. Appl. Math. Comput. 153, 885–902 (2004) 32. Murray, J.D.: Mathematical Biology I: An Introduction. Springer, New York (1989) 33. Nicholson, A.: The self adjustment of population to change. Cols. Spring Harb’s Syrup Quant. Bzol. 22, 153–173 (1957) 34. Nicholson, A.: The balance of animal population. J. Animal Ecol. 2, 132–178 (1993) 35. Padhi, S., Pati, S.: Multiple periodic solutions for system of first order differential equations. Appl. Anal. 88(7), 1005–1014 (2009) 36. S. Padhi, C. Qian and S. Srivastava;Multiple periodic solutions for a nonlinear functional differential equation with application to population dynamics, Comm. Appl. Anal., 12(3)(2008), 341–352

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Dynamic Programming Viscosity Solution Approach and Its Applications to Optimal Control Problems Bing Sun, Zhen-Zhen Tao and Yang-Yang Wang

Abstract This chapter is concerned with optimal control problems of dynamical systems described by partial differential equations (PDEs). Firstly, using the Dubovitskii-Milyutin approach, we obtain the necessary condition of optimality, i.e., the Pontryagin maximum principle for optimal control problem of an age-structured population dynamics for spread of universally fatal diseases. Secondly, for an optimal birth control problem of a McKendrick type age-structured population dynamics, we establish the optimal feedback control laws by the dynamic programming viscosity solution (DPVS) approach. Finally, for a well-adapted upwind finite-difference numerical scheme for the HJB equation arising in optimal control, we prove its convergence and show that the solution from this finite-difference scheme converges to the value function of the associated optimal control problem. Keywords Optimal feedback control · Viscosity solution · Dynamic programming approach · Numerical solution · Convergence

1 Introduction Optimal control problem is one of the core issues of modern control theory, while seeking the optimal control in feedback form is of fundamental importance [1]. It can be argued as the Holy Grail of control theory [2]. In 1744, the great mathematician Leonard Euler published a monograph to present a method for finding curved lines This work was supported in part by the National Natural Science Foundation of China under Grant No. 11471036. B. Sun (B) · Z.-Z. Tao · Y.-Y. Wang School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China e-mail: [email protected] B. Sun Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_12

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enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense. In this book, he studied many real-world problems and claimed that nothing in the world can escape the law of extrema. This marks the beginning of calculus of variations as a practical theory. As one of the several generalizations and applications of calculus of variations, optimal control mainly deals with seeking the control function of time or feedback gain to minimize the cost functional constrained by differential equations. The design of control systems can not do without optimization. Furthermore, optimization and feedback are in every corner of control system design [3]. In fact, feedback and optimization are the most fundamental definitions and methods, while numerical solution is the key link to digitalization trend of modern science and technology. Finding the optimal feedback control of investigational problems is one of most exciting projects among control theory and applications [4]. The development of optimal control can be described in two categories: one is the abstract theory and the other is the computational method. The Pontryagin maximum principle and the Bellman dynamic programming principle are two creative contributions in theory and two landmarks in optimal control theory. Nevertheless, the Pontryagin maximum principle only presents the necessary optimality condition. And obtained control is usually in open-loop not feedback form. Moreover, it is upsetting that “those sophisticated necessary conditions rarely give an insight into the structure of the optimal controls” [5]. It is especially prominent in distributed parameter system control investigations. Even so, due to the existence and uniqueness of the optimal control for most of practical problems, the Pontryagin maximum principle provides a possibility of seeking numerical solution for the optimal control at least in open-loop form. Basically, there are two ways of numerical methods solving the optimal control problems through the necessary conditions. It is generally believed that the indirect method that is mainly the multiple shooting method is the most powerful numerical method in seeking the optimal control through solving a two-point boundary-value problem obtained by the Pontryagin maximum principle. However, except for the complexity when the original problem involves inequality constraints of both state variables and controls, the big difficulty for shooting method is the “guess” for the initial data to start the iterative numerical process. It demands that the user understands the essential of the problem well in physics, which is often not a trivial task [6]. For the direct method [7], the simplification for the original problem leads to the fall of the reliability and accuracy, and when the degree of discretization and parameterization is very high, the work of computation stands out and the solving process gives rise to “curse of dimensionality” [6]. Over the last two decades, much progress has been made on the numerical methods for solving optimal control problems. Teo et al. [8] dealt with computational methods for several general classes of optimal control problems subject to constraints. A finite difference scheme was developed for solving a class of hyperbolic partial differential equations in [9]. Teo et al. [10] presented a powerful computational method, where the switching time points can also be considered as decision variables. In addition, many softwares on the optimal control are available now, such as [11], which was

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used to compute the optimal controls in [12] where the performance of the computed optimal feedback control is compared with the open-loop optimal control with the same initial condition. For other developments in computing the optimal control, we refer to [13–17]. In contrast to the Pontryagin maximum principle that deals with one extremal at a time, the dynamic programming method deals with, on the other hand, families of extremals. Once the Hamilton-Jacobi-Bellman (HJB for short) equation satisfied by the value function is established, the feedback control law can be found by solving this first order nonlinear partial differential equation. However, the new problem is raised immediately even from Pontryagin’s time that the HJB equation may have no classical solution no matter how smooth its coefficients are. The fundamental turn comes when the viscosity solution of the HJB equation was introduced in the 1980s [18]. The advantage of viscosity solution is that this kind of weak solution not only exists but also is unique for most of HJB equations. Once this mathematical basis for the dynamic programming approach was rigorously established, the seeking of optimal (feedback) control becomes possible [19]. For infinite-dimensional optimal control problem by viscosity solution approach, we refer to [20–24] and the references therein. Certainly, there are still other problems for seeking optimal control by solving viscosity solution of the HJB equation satisfied by the value function of the optimal control problem. One of the apparent problems is that in order to get the optimal feedback, we need not only the value function but also its gradient. The latter may not exist because the viscosity solution may not be differentiable. But for some big inertial systems, proper difference replacing the gradient of the value function still works for the numerical solution of the optimal feedback control. It was shown in [25–30] that some simple difference scheme may produce numerically the viscosity solution. A recent interesting effort in solving numerically the HJB equation can be found in [31–33] for some finite dimensional control problems. This chapter is a new attempt to numerically solve the optimal feedback control by the DPVS approach. When it is very difficult to solve the optimal control problems by using analytic methods, we turn to numerical solution. Though there are many numerical methods in optimal control, we believe that the viscosity solution approach holding the fundamental superiority. We will show its power in an example of optimal birth feedback control of a McKendrick type age-structured population dynamics. Then we prove the convergence of an effective and well-adapted numerical scheme for the HJB equation arising in optimal control. Before the discussion of DPVS approach, we establish the Pontryagin maximum principle for optimal control of age-structured population dynamics for spread of universally fatal diseases. It finally gives us a clear sense of the defects of the Pontryagin maximum principle. The rest of the paper is organized as follows. In Sect. 2, we consider the optimal control of an age-structured population dynamics for spread of universally fatal diseases in both fixed and free final horizon cases. Next, by the DPVS approach, we investigate the optimal birth feedback control of a McKendrick type age-structured population dynamics in Sect. 3. We formulate a finite difference scheme for the HJB equation of the population system with linear quadratic optimal control. The numerical solution of optimal feedback control is then presented. Section 4 is devoted to the conver-

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gence proof of an upwind finite-difference numerical scheme for the HJB equation. Section 5 concludes the chapter with remarks.

2 Optimal Control of Age-Structured Population Dynamics for Spread of Universally Fatal Diseases 2.1 Model Background and Its Well-Posedness In this section, taking an age-structured population dynamics as an example, we are concerned with its optimal control problem and derive its Pontryagin maximum principle. This population dynamics describes the spread of universally fatal diseases. First of all, the existence and uniqueness of solution of the system, which consists of a group of partial differential equations with nonlocal boundary conditions, is proved. The Dubovitskii and Milyutin functional analytical approach is adopted in investigation of the Pontryagin maximum principle of the system. The necessary conditions are presented for the optimal control problems in both fixed and free final time horizon cases. People can refer to [34, 35] for the details. Among numerous population dynamics presented by the applied mathematical methodology, the Susceptible-Infectious-Recovered (SIR) model is an important epidemiological model that describes the relationships among the number of susceptible people, number of people infected, and number of people who have recovered. Moreover, people prefer the SIR model to study the diseases spread via droplet and aerosol, such as pandemic flu, seasonal flu, SARS, and smallpox [36]. The KermackMcKendrick model is a simple SIR model [37]. Modelling and control of the SIR epidemic has been one of active issues of biomedical engineering and biomedical mathematics. As a result, so far, there are many interesting results available, such as [38–42]. In the survey paper [39], the state of the art for the mathematics of infectious diseases is reviewed and many models for the spread of infectious diseases in populations are analyzed mathematically and applied to specific diseases. The mathematical modelling of epidemic is deeply discussed in [40]. Li and Wang [43] give the global stability results in some SEIR epidemic models. Iannelli and Martcheva [41] present the analysis of an age-structured infinite dimensional homogeneous SIR model with incidence of proportionate mixing type and derives the locally stability result of the system. During the process of modelling and analysis of SIR model, the relation of the epidemic to death is a key problem. In view of this fact, epidemics can be divided into two subclasses. Among them, one class of diseases may obtain the permanent immunity cure, such as measles. The other describes such an epidemic that once infected, the individual can not be cured until death. Rabies belongs to the latter. Such an epidemic can be called the universally fatal disease. The investigational model in this section, a Susceptible-Infectious (SI) model, is just to describe this kind of diseases spread.

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To find a truly effective cure for the epidemic is the ultimate aim of the epidemiology research. To this point, in mathematics, we need formulate an optimal control problem and derive the optimal treatment strategy mathematically. This section is concerned with the optimal control of age-structured population dynamics for spread of universally fatal diseases. Adopting the optimal control theory to study the epidemic dynamics, people has done lots of work. Various deterministic optimal control models for SIR-epidemics are investigated in paper [44] and in all cases the maximum effort control on some initial time interval is obtained. Rowthorn, Laxminarayan and Gilligan [42] consider how best to deploy scarce resources for disease control when epidemics occur in different but interconnected regions by a combination of optimal control methods and epidemiological theory for metapopulations. In paper [45], Iacoviello and Liuzzi introduce the multiple controls in epidemic model and investigate the possible advantages in case of fixed final time. Other relevant literature include [36, 46–50], to name just a few. In this section, we firstly present the investigational model and prove its wellposedness. Suppose the total population is divided into two groups. The first one consists of the class of susceptible individuals who are not infected at present. Denote its density distribution by S(r, t) in which r is age and t is time. The second class is the infected individuals denoted by the density distribution I (r, t). We assume that the epidemic is fatal so that the infected individual can not get any immune therapy until death. Let λ(r, t) be the age-specific transmission coefficient, that is, the average contact rate made by every infected individual per unit time at time t and age r . Therefore the average effective  r contact times of all infected individuals per unit time at the instant of t are 0 m λ(r, t)I (r, t) dr , where rm is the highest age ever attained by individuals  r of the population. At the instant of t, the total size of population is N (t) = 0 m [S(r, t) + I (r, t)] dr . And at time t, the force of infection is η(t) =

1 N (t)



rm

λ(r, t)I (r, t) dr.

0

Throughout this paper, we assume that all newborn individuals are categorized into the first group, i.e., the susceptible individual group, regardless of the inheritance factor. And we assume that the transmission rate contributes nothing to the procreation. So the number of new births only depends on that of the susceptible people. Moreover, the effect of the latency is also omitted. Regularize the age-specific transmission coefficient  r λ(r, t) = β(t)h(r, t). Here h(r, t) is the age-specific mode of transmission and 0 m h(r, t) dr = 1. β(t) is the specific transmission rate at time t, whose practical explanation is the age-specific average fatal contact number of the infected by the infectivity rate at the instant of t. Therefore, the age-structured Kermack-McKendrick control model of fatal epidemics can be written as the partial differential equation that follows [38].

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⎧ ⎪ ⎪ ∂ S(r, t) + ∂ S(r, t) = −μ(r, t)S(r, t) − β(t)χ (t)S(r, t), ⎪ ⎪ ⎪ ∂t ∂r ⎪ ⎪  r2 ⎪ ⎪ ⎪ ⎨ S(0, t) = b(r, t)S(r, t)dr, S(r, 0) = S0 (r ), r1

⎪ ⎪ ⎪ ∂ I (r, t) ∂ I (r, t) ⎪ ⎪ + = −μ(r, t)I (r, t) + β(t)χ (t)S(r, t) − δ I (r, t), ⎪ ⎪ ∂t ∂r ⎪ ⎪ ⎪ ⎩ I (0, t) = 0, I (r, 0) = I0 (r ), (r, t) ∈ Q ∞ = (0, rm ) × (0, +∞).

(1)

Here, [r1 , r2 ] is the fecundity period of females. S0 (r ), I0 (r ) are the initial age distribution. μ(r, t) is the natural relative mortality. b(r, t) is the fertility ratio of females ∞ and vanishes outside [r1 , r2 ]. δ −1 = 0 e−δτ dτ is the average infectious period and e−δτ denotes the rate of the people  r infected by time t live up to t + τ . β(t) is the control variable and χ (t) = N1(t) 0 m h(r, t)I (r, t) dr . Let λ(r, t), b(r, t) are bounded measurable nonnegative functions. The measurable function μ(r, t) satisfies  1 μ ∈ L loc (Q ∞ ), μ(r, t) ≥ 0,

rm

μ(r, t) dr = +∞, (r, t) ∈ Q ∞ .

0

In the context of b(r, t) = b(r ), μ(r, t) = μ(r ), which is the approximate description to the short-term case without mutations, Guo [51] proves the existence and uniqueness of the nonnegative classical solution to (1) for any initial data (S0 (r ), I0 (r )), if mes{r | S0 (r ) = 0, r ∈ [0, rm ]} > 0. In order to address the optimal control problem, we revisit the well-posedness of the system. The following two assumptions are assumed while the conditions of b(r, t), μ(r, t) being constant with respect to time t above are removed. (a) β ∈ U = { ∈ L ∞ (0, +∞) | 0 ≤ β0 ≤ (t) ≤ β1 , ∀ t > 0}, βi , i = 0, 1 are constants. (b) S0 , I0 ∈ L ∞ (0, rm ), S0 (r ) ≥ 0, I0 (r ) ≥ 0, ∀ r ∈ (0, rm ). Mainly making use of the contraction mapping principle, we obtain the following Theorem 1 that characterizes the existence and uniqueness of solution to the system (1), namely, its well-posedness. Theorem 1 For any given β ∈ U , there is a unique solution (S β , I β ) to the system (1), such that (i) S β , I β ∈ C(0, ∞; L 2 (0, rm )); ¯ t), 0 ≤ I β (r, t) ≤ I¯(r, t), ∀ (r, t) ∈ Q ∞ , in which S(r, ¯ t), (ii) 0 ≤ S β (r, t) ≤ S(r, I¯(r, t) is respectively the unique nonnegative and bounded solution of the following systems

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⎧ ∂ S(r, t) ∂ S(r, t) ⎪ ⎪ + = −μ(r, t)S(r, t), ⎪ ⎪ ∂t ∂r ⎪ ⎨  r2 S(0, t) = b(r, t)S(r, t) dr, ⎪ ⎪ ⎪ r1 ⎪ ⎪ ⎩ S(r, 0) = S0 (r ) ⎧ ⎨ ∂ I (r, t) + ∂ I (r, t) = −(μ(r, t) + δ)I (r, t), ∂t ∂r ⎩ I (0, t) = 0, I (r, 0) = I0 (r ).

and

Here (r, t) ∈ Q T = (0, rm ) × (0, T ), T is any given positive constant. (iii) S β , I β depends continuously upon β. 

2.2 Optimal Control Formulation Now consider the optimal control issues of the investigated system. Unless otherwise stated, in what follows when we speak of a solution of (1), we shall always mean the solution in the sense of Theorem 1. In the beginning, the attention is paid to the case of fixed final horizon. For T > 0, consider an optimal control problem for the system (1) with the general cost functional  T  rm L(S(r, t), I (r, t), β(t), r, t) dr dt (2) min J (S, I, β) = min β(·)∈Uad

β(·)∈Uad

0

0

and the control constraint Uad = {β ∈ L ∞ (0, T ) | 0 ≤ β0 ≤ β(t) ≤ β1 , t ∈ [0, T ] a.e.}. The cost function L is quite general in the sense that it contains most practically concerned cost functional like quadratic cost functional of the following form 

T

J (S, I, β) = 0



rm

 α1 [S(r, t) − S † (r, t)]2 + α2 [I (r, t) − I † (r, t)]2 dr dt

0



T

+α3

β 2 (t) dt

0

where αi > 0, i = 1, 2, 3 are constants, and S † , I † are, respectively, the given observations for the density distribution of susceptible individual who are not infected and the already infected. The last quadratic term reflects the cost of the treatment or

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the severity of the side effects of the drugs. One wants to drive the state variables S, I approach the given density distributions by the control function β [52]. Remember that the practical explanation of β(·) is the age-specific average fatal contact number of the infected by the infectivity rate, which is, directly or indirectly, controlled by the vaccines, treatment, or quarantine. In addition, due to the existence of the control constraints, the usual variations in control can not deal with this kind of problems. It is necessary to investigate the problem (2) by other approaches. Take S, I ∈ C(0, T ; L 2 (0, rm )). The control space is L ∞ (0, T ) and the control constraint is β0 ≤ β(t) ≤ β1 in (0, T ). The following assumptions for the cost functional are assumed: (c). L is a functional defined on (L 2 (0, rm ))2 × [β0 , β1 ] × [0, rm ] × [0, T ] and ∂ L(S, I, β, r, t) ∂ L(S, I, β, r, t) ∂ L(S, I, β, r, t) , , ∂S ∂I ∂β exist for every (S, I, β) ∈ (L 2 (0, rm ))2 × [β0 , β1 ] and L is continuous in its variables. (d).



 rm

 rm

∂ L(S, I, β, r, t)

∂ L(S, I, β, r, t)

dr,

dr,







∂S ∂I 0 0

 rm

∂ L(S, I, β, r, t)

dr



∂β 0

are bounded for t ∈ [0, T ]. Define X T = C(0, T ; L 2 (0, rm ; R2 )) × L ∞ (0, T ). Let (S ∗ , I ∗ , β ∗ ) be the optimal solution to the control problem (2) subject to the equation (1). Set Ω1 = {(S, I, β) ∈ X T | β0 ≤ β(t) ≤ β1 , t ∈ [0, T ] a.e.}, Ω2 = {(S, I, β) ∈ X T |

∂ S(r, t) ∂ S(r, t) + = −μ(r, t)S(r, t) ∂t ∂r −β(t)χ (t)S(r, t),  r2 S(0, t) = b(r, t)S(r, t) dr, S(r, 0) = S0 (r ), r1

∂ I (r, t) ∂ I (r, t) + = −μ(r, t)I (r, t) ∂t ∂r +β(t)χ (t)S(r, t) − δ I (r, t), I (0, t) = 0, I (r, 0) = I0 (r ), S(r, T ) = S ∗ (r, T ), I (r, T ) = I ∗ (r, T )}.

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Then the problem (2) is equivalent to seeking for (S ∗ , I ∗ , β ∗ ) ∈ Ω = Ω1 ∩ Ω2 such that (3) J (S ∗ , I ∗ , β ∗ ) = min J (S, I, β). (S,I,β)∈Ω

It is seen that the problem (3) is an extremum problem on the inequality constraint Ω1 and the equality constraint Ω2 . In this situation, the Dubovitskii and Milyutin functional analytical approach has been turned out to be very powerful to solve such kind of extremum problems [53–55]. It is first suggested by Dubovitskii and Milyutin for general extremal problems [56] and then enriched, generalized by their colleagues and many followers. The general Dubovitskii and Milyutin’s theorem for the problem (3) can be stated as the following Theorem 2. Theorem 2 (Dubovitskii-Milyutin, [57]) Suppose the functional J (S, I, β) assumes a minimum at the point (S ∗ , I ∗ , β ∗ ) in Ω. Assume that J (S, I, β) is regularly decreasing at (S ∗ , I ∗ , β ∗ ) with the cone of directions of decrease K 0 and the inequality constraint is regular at (S ∗ , I ∗ , β ∗ ) with the cone of feasible directions K 1 ; and that the equality constraint is also regular at (S ∗ , I ∗ , β ∗ ) with the cone of tangent directions K 2 . Then there exist continuous linear functionals f 0 , f 1 , f 2 , not all identically zero, such that f i ∈ K i∗ , the dual cone of K i , i = 0, 1, 2, which satisfy the condition f 0 + f 1 + f 2 = 0. 

2.3 Fixed Final Horizon Case In order to apply Theorem 2, we have to determine all cones K i , i = 0, 1, 2. First, let us find the cone of directions of decrease K 0 . The cone of directions of decrease of the functional J (S, I, β) at the point (S ∗ , I ∗ , β ∗ ) is determined by



K 0 = (S, I, β) ∈ X T J (S ∗ , I ∗ , β ∗ ; S, I, β) < 0

 T  rm ∂ L(S ∗ , I ∗ , β ∗ , r, t) ∂ L(S ∗ , I ∗ , β ∗ , r, t)

S+ I = (S, I, β) ∈ X T

∂S ∂I 0 0

 ∂ L(S ∗ , I ∗ , β ∗ , r, t) + β dr dt < 0 . ∂β If K 0 = ∅, then for any f 0 ∈ K 0∗ , there exists a κ0 ≥ 0 such that  f 0 (S, I, β) = −κ0 0





∂ L(S ∗ , I ∗ , β ∗ , r, t) ∂ L(S ∗ , I ∗ , β ∗ , r, t) S+ I ∂S ∂I 0

∂ L(S ∗ , I ∗ , β ∗ , r, t) β dr dt. + ∂β

T

rm

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Secondly, we give the cone of feasible directions K 1 . Since Ω1 = C(0, T ; L 2 (0, rm ; R2 )) × Ω˜ 1 , in which Ω˜ 1 = {β ∈ L ∞ (0, T ) | β0 ≤ o

β(t) ≤ β1 , t ∈ [0, T ] a.e.} is a closed convex subset of L ∞ (0, T ), the interior Ω 1 of Ω1 is not empty and at point (S ∗ , I ∗ , β ∗ ), the cone of feasible directions K 1 of Ω1 is determined by  o K 1 = κ(Ω 1 −(S ∗ , I ∗ , β ∗ )) | κ > 0  o = h | h = κ(S − S ∗ , I − I ∗ , β − β ∗ ), (S, I, β) ∈ Ω 1 , κ > 0 . Therefore, for an arbitrary f 1 ∈ K 1∗ , if there is an a¯ ∈ L(0, T ), such that the linear functional defined by  T f 1 (S, I, β) = a(t)β(t) ¯ dt 0

is a support to Ω˜ 1 at point β ∗ , then   a(t) ¯ β(t) − β ∗ (t) ≥ 0, ∀ β ∈ [β0 , β1 ], t ∈ [0, T ] a.e.

(4)

Thirdly, we derive the cone of tangent directions K 2 . Assume that the linearized system ⎧   N (t)β ∗ (t)χ ∗ (t)S ∗ (r, t) ⎪ ∗ ∗ ⎪ S (r, t) + S (r, t) = − μ(r, t) + β (t)χ (t) S(r, t) + ⎪ t r ⎪ N ∗ (t) ⎪ ⎪ ⎪  ⎪ r m ⎪ β ∗ (t)S ∗ (r, t) ⎪ ⎪ ⎪ − h(r, t)I (r, t) dr − β(t)S ∗ (r, t)χ ∗ (t), ⎪ ∗ ⎪ N (t) ⎪ 0 ⎪ ⎪  r2 ⎪ ⎪ ⎪ ⎨ S(0, t) = b(r, t)S(r, t) dr, S(r, 0) = 0, r1

⎪ ⎪ ⎪ N (t)β ∗ (t)χ ∗ (t)S ∗ (r, t) ⎪ ⎪ (r, t) + I (r, t) = −[μ(r, t) + δ]I (r, t) − I ⎪ t r ⎪ N ∗ (t) ⎪ ⎪ ⎪  rm ⎪ ∗ ∗ ⎪ β (t)S (r, t) ⎪ ⎪ ⎪ + h(r, t)I (r, t) dr + β ∗ (t)χ ∗ (t)S(r, t) + β(t)S ∗ (r, t)χ ∗ (t), ⎪ ∗ (t) ⎪ N ⎪ 0 ⎪ ⎪ ⎩ I (0, t) = 0, I (r, 0) = 0 (5) r is controllable, in which χ ∗ (t) = N ∗1(t) 0 m h(x, t)I ∗ (x, t) d x and N ∗ (t) =  rm ∗ ∗ 0 [S (y, t) + I (y, t)] dy. We can obtain that (S, I, β) satisfies the following equation in X T :

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⎧   N (t)β ∗ (t)χ ∗ (t)S ∗ (r, t) ⎪ ⎪ St (r, t) + Sr (r, t) = − μ(r, t) + β ∗ (t)χ ∗ (t) S(r, t) + ⎪ ⎪ N ∗ (t) ⎪ ⎪ ⎪  ⎪ ⎪ β ∗ (t)S ∗ (r, t) rm ⎪ ⎪ ⎪ − h(r, t)I (r, t) dr − β(t)S ∗ (r, t)χ ∗ (t), ⎪ ∗ (t) ⎪ N ⎪ 0 ⎪ ⎪  r2 ⎪ ⎪ ⎪ ⎨ S(0, t) = b(r, t)S(r, t) dr, S(r, 0) = 0, r1

⎪ ⎪ ⎪ N (t)β ∗ (t)χ ∗ (t)S ∗ (r, t) ⎪ ⎪ (r, t) + I (r, t) = −[μ(r, t) + δ]I (r, t) − I ⎪ t r ⎪ N ∗ (t) ⎪ ⎪ ⎪  ⎪ r ∗ ∗ m ⎪ β (t)S (r, t) ⎪ ⎪ ⎪ h(r, t)I (r, t) dr + β ∗ (t)χ ∗ (t)S(r, t) + β(t)S ∗ (r, t)χ ∗ (t), ⎪ + ∗ ⎪ N (t) ⎪ 0 ⎪ ⎪ ⎩ I (0, t) = 0, I (r, 0) = 0 (6) and S(r, T ) = 0, I (r, T ) = 0. (7) Define K 21 = {(S, I, β) ∈ X T | (S, I, β) satisfies (6)}, K 22 = {(S, I, β) ∈ X T | (S, I, β) satisfies (7)}. Then the cone of tangent directions K 2 = K 21



K 22 . Hence

∗ ∗ + K 22 . K 2∗ = K 21

For any f 2 ∈ K 2∗ , decompose f 2 = f 21 + f 22 , f 2i ∈ K 2i∗ , the dual cone of K 2i , i = 1, 2. Then f 21 (S, I, β) = 0 and for all (S, I ) ∈ C(0, T ; L 2 (0, rm ; R2 )) satisfying S(r, T ) = 0, I (r, T ) = 0, there exist φ, ψ ∈ L 2 (0, rm ) such that  f 22 (S, I, β) =

rm

[S(r, T )φ(r ) + I (r, T )ψ(r )] dr.

0

It then follows from Theorem 2 that there exist continuous linear functionals, not all identically zero, such that f 0 + f 1 + f 21 + f 22 = 0. Therefore, when selecting (S, I, β) satisfying (6), f 21 (S, I, β) = 0. Moreover,

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f 1 (S, I, β) = − f 0 (S, I, β) − f 22 (S, I, β)  T  rm ∂ L(S ∗ , I ∗ , β ∗ , r, t) ∂ L(S ∗ , I ∗ , β ∗ , r, t) S(r, t) + I (r, t) = κ0 ∂S ∂I 0 0

 rm ∂ L(S ∗ , I ∗ , β ∗ , r, t) + β(t) dr dt − [S(r, T )φ(r ) + I (r, T )ψ(r )] dr. ∂β 0 Next, we define the adjoint system of (5) as ⎧

N ∗ (t) ⎪ ∗ ∗ ⎪ Φt (r, t) + Φr (r, t) = μ(r, t) − ∗ + β (t)χ (t) Φ(r, t) ⎪ ⎪ ⎪ N (t) ⎪ ⎪ ⎪  ⎪ ⎪ β ∗ (t)χ ∗ (t) rm ∗ ⎪ ⎪ − S (r, t) [Φ(r, t) − Ψ (r, t)] dr − b(r, t)Φ(0, t) ⎪ ⎪ ⎪ N ∗ (t) 0 ⎪ ⎪ ⎪ ⎪ κ0 ∂ L(S ∗ , I ∗ , β ∗ , r, t) ⎪ ⎪ ⎪ − β ∗ (t)χ ∗ (t)Ψ (r, t), + ⎪ ∗ (t) ⎪ N ∂ S ⎪ ⎪ ⎪ ⎪ ⎨ Φ(rm , t) = 0, N ∗ (T )Φ(r, T ) = φ(r ),

⎪ N ∗ (t) ⎪ ⎪ ⎪ ⎪ Ψt (r, t) + Ψr (r, t) = μ(r, t) + δ − N ∗ (t) Ψ (r, t) ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ β ∗ (t)h(r, t) rm ∗ ⎪ ⎪ − S (r, t) [Ψ (r, t) − Φ(r, t)] dr ⎪ ⎪ ⎪ N ∗ (t) 0 ⎪ ⎪  ⎪ ⎪ β ∗ (t)χ ∗ (t) rm ∗ κ0 ∂ L(S ∗ , I ∗ , β ∗ , r, t) ⎪ ⎪ ⎪ − , S (r, t) [Ψ (r, t) + Φ(r, t)] dr + ∗ ⎪ ∗ ⎪ N (t) N (t) ∂I ⎪ 0 ⎪ ⎪ ⎩ Ψ (rm , t) = 0, N ∗ (T )Ψ (r, T ) = ψ(r ). (8) As with (1), the existence and uniqueness of solution to (8) can be obtained similarly. Theorem 3 The solution of system (5) and the solution of its adjoint system (8) have the following relation:  κ0  −  = 0



T

0 rm

0 T

rm

0



∂ L(S ∗ , I ∗ , β ∗ , r, t) ∂ L(S ∗ , I ∗ , β ∗ , r, t) S(r, t) + I (r, t) dr dt ∂S ∂I

[S(r, T )φ(r ) + I (r, T )ψ(r )] dr

 0

rm



S ∗ (r, t) (Φ(r, t) − Ψ (r, t))



rm

h(x, t)I ∗ (x, t) d x β(t) dr dt.

0

Now, by virtue of Theorem 3, we can rewrite f 1 (S, I, β) as



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375



∂ L(S ∗ , I ∗ , β ∗ , r, t) ∂β 0 0

  rm ∗ ∗ +S (r, t) (Φ(r, t) − Ψ (r, t)) h(x, t)I (x, t) d x dr β(t) dt.

f 1 (S, I, β) =

T

rm

κ0

0

Therefore ∂ L(S ∗ , I ∗ , β ∗ , r, t) κ0 ∂β 0

 rm +S ∗ (r, t) (Φ(r, t) − Ψ (r, t)) h(x, t)I ∗ (x, t)d x dr 

a(t) ¯ =

rm

0

and (4) then reads 

rm 0

∂ L(S ∗ , I ∗ , β ∗ , r, t) κ0 ∂β

+S ∗ (r, t) (Φ(r, t) − Ψ (r, t))



rm

  h(x, t)I ∗ (x, t) d x dr β(t) − β ∗ (t) ≥ 0,

0

∀ β(t) ∈ [β0 , β1 ], t ∈ [0, T ] a.e., where κ0 , Φ(r, t) and Ψ (r, t) are not identical to zero simultaneously. Otherwise, there are definitely f 0 = 0, f 1 = 0, f 22 = 0 and f 21 = 0, which contradict with the fact in Theorem 2 that these continuous linear functionals are not all identically zero. For other special cases, such as K 0 being a null set, can also be overcome. Combining the facts above, we have obtained the Pontryagin maximum principle for the problem (2) subject to the system (1). Theorem 4 Suppose (S ∗ , I ∗ , β ∗ ) is a solution to the optimal control problem (2). Then there exist κ0 ≥ 0 and Φ(r, t), Ψ (r, t), not identically zero, such that the following maximum principle holds true: 

rm 0 ∗

κ0

∂ L(S ∗ , I ∗ , β ∗ , r, t) ∂β



+S (r, t) (Φ(r, t) − Ψ (r, t))

rm



h(x, t)I (x, t) d x dr [β(t) − β ∗ (t)] ≥ 0,

0

∀ β ∈ [β0 , β1 ], t ∈ [0, T ] a.e., where the functions Φ(r, t), Ψ (r, t) satisfy the adjoint equation (8). 

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2.4 Free Final Horizon Case In the foregoing derivations, we give the Pontryagin maximum principle for the system in fixed final horizon case. These results were derived under two additional conditions. The first one is that the admissible control set must be convex and contains interior points, and the second requires the cost functional to be differentiable with respect to the control variable. Now we consider optimal control problems of agestructured population dynamics for the spread of universally fatal diseases in free final horizon case without these assumptions. We directly give the obtained results and the detailed deductions can be found in [35]. Consider the following control system defined in the fixed domain [0, rm ] × [0, t1 ]: ⎧ ∂ S(r, t) ∂ S(r, t) ⎪ ⎪ + = −μ(r, t)S(r, t) − β(t)χ (t)S(r, t), ⎪ ⎪ ⎪ ∂t ∂r ⎪ ⎪  r2 ⎪ ⎪ ⎪ ⎪ S(0, t) = b(r, t)S(r, t) dr, S(r, 0) = S0 (r ), S(r, t1 ) = S1 (r ), ⎪ ⎪ ⎨ r1 (9) ∂ I (r, t) ∂ I (r, t) ⎪ ⎪ + = −μ(r, t)I (r, t) + β(t)χ (t)S(r, t) − δ I (r, t), ⎪ ⎪ ⎪ ∂t ∂r ⎪ ⎪ ⎪ ⎪ I (0, t) = 0, I (r, 0) = I0 (r ), I (r, t1 ) = I1 (r ), ⎪ ⎪ ⎪ ⎪ ⎩ (r, t) ∈ Q = (0, r ) × (0, t ), t > 0, β ∈ M ⊂ R+ , t1 m 1 1 and formulate the optimal control problem below. Surely it is worth emphasizing the cancellation of assumptions imposed on the preceding fixed final horizon problem. That is to say, in this subsection the admissible control set M neither needs be convex nor contains interior points as well as the cost functional L(S, I, β) needs not be differentiable with respect to the control variable β. Therefore, M can be any set. An interesting case is that M is allowed to contain only finite many points. In addition, we still adopt the weak solution for the system (9) that has been justified before. Precisely speaking, there exists a unique solution S, I ∈ C(0, t1 ; L 2 (0, rm )) for any β ∈ L ∞ (0, t1 ). The optimal control problem with free final horizon t1 is to minimize 

t1

J (S, I, β) = 0



rm

P(S(r, t), I (r, t), β(t)) dr dt

(10)

0

for S, I ∈ C(0, t1 ; L 2 (0, rm )), β ∈ L ∞ (0, t1 ), under the constraints (9), where the functional P defined on [0, rm ] × R+ satisfies (e) P(S, I, β) is continuous about β.





∂ P(S, I, β) ∂ P(S, I, β)

,

are bounded for every bounded subset of (f)



∂S ∂I + [0, rm ] × R .

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By the Dubovitskii-Milyutin functional analysis approach, we can prove the Pontryagin maximum principle of (10) with free final horizon, which can be stated as the following Theorem 5, which is Theorem 5 Suppose the functions Φ(r, t), Ψ (r, t) satisfy ⎧

N ∗ (t) ⎪ ∗ ∗ ⎪ Φt (r, t) + Φr (r, t) = μ(r, t) − ∗ + β (t)χ (t) Φ(r, t) ⎪ ⎪ ⎪ N (t) ⎪ ⎪ ⎪  ⎪ ⎪ β ∗ (t)χ ∗ (t) rm ∗ ⎪ ⎪ − S (r, t) [Φ(r, t) − Ψ (r, t)] dr − b(r, t)Φ(0, t) ⎪ ⎪ ⎪ N ∗ (t) 0 ⎪ ⎪ ⎪ ⎪ κ0 ∂ P(S ∗ , I ∗ , β ∗ , r, t) ⎪ ⎪ ⎪ − β ∗ (t)χ ∗ (t)Ψ (r, t), + ∗ ⎪ ⎪ N (t) ∂ S ⎪ ⎪ ⎪ ⎪ ⎨ Φ(rm , t) = 0, N ∗ (t1 )Φ(r, t1 ) = φ(r ),

⎪ N ∗ (t) ⎪ ⎪ Ψ (r, t) (r, t) + Ψ (r, t) = μ(r, t) + δ − Ψ ⎪ t r ⎪ ⎪ N ∗ (t) ⎪ ⎪ ⎪  ⎪ ⎪ β ∗ (t)h(r, t) rm ∗ ⎪ ⎪ − S (r, t) [Ψ (r, t) − Φ(r, t)] dr ⎪ ⎪ ⎪ N ∗ (t) 0 ⎪ ⎪  ⎪ ⎪ β ∗ (t)χ ∗ (t) rm ∗ κ0 ∂ P(S ∗ , I ∗ , β ∗ , r, t) ⎪ ⎪ ⎪ − , S (r, t) (r, t) + Φ(r, t)] dr + [Ψ ⎪ ⎪ N ∗ (t) N ∗ (t) ∂I ⎪ 0 ⎪ ⎪ ⎩ Ψ (rm , t) = 0, N ∗ (t1 )Ψ (r, t1 ) = ψ(r ), (r, t) ∈ Q t1 , with 



rm

N (t) = 0





1 S (r, t) + I (r, t) dr, χ (t) = ∗ N (t) ∗







rm

h(r, t)I ∗ (r, t) dr.

0

Furthermore, suppose (S ∗ , I ∗ , β ∗ , t1 ) is a solution to Problem I (10), then there exist κ0 ≥ 0 and Φ(r, t), Ψ (r, t) above, not identically zero, such that: 

rm



κ0 P(S ∗ (r, t), I ∗ (r, t), β ∗ (t))

0

  −N ∗ (t) Sr∗ (r, t) + μ(r, t)S ∗ (r, t) + β ∗ (t)S ∗ (r, t)χ ∗ (t) Φ(r, t) ∗

−N (t)



Ir∗ (r, t)

 + (μ(r, t) + δ)I (r, t) − β (t)S (r, t)χ (t) Ψ (r, t) dr = 0, ∗



∀ t ∈ [0, t1 ] a.e.,





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rm

κ0 P(S ∗ (r, t), I ∗ (r, t), β ∗ (t))

0

  −N ∗ (t) Sr∗ (r, t) + μ(r, t)S ∗ (r, t) + β S ∗ (r, t)χ ∗ (t) Φ(r, t) ∗

−N (t)



Ir∗ (r, t)

 + (μ(r, t) + δ)I (r, t) − β S (r, t)χ (t) Ψ (r, t) dr ≥ 0, ∗

∀ β ∈ M, t ∈ [0, t1 ] a.e.







3 Optimal Birth Feedback Control of a McKendrick Type Age-Structured Population Dynamics In this section, we study the optimal control problem for a McKendrick type agestructured population dynamics by viscosity solution approach. This is an infinite dimensional bilinear control problem. Many works have been done in the setting of partial differential equation models describing the population dynamics. In [53, 54], the optimal birth control problems of an age-structured population system of McKendrick type are investigated and maximum principles for problems are established. The optimal control problem for a Gurtin-MacCamy type system describing the evolution of an age-structured population is studied in [58] and necessary optimality conditions are given. Ani¸ta [59] investigates an optimal harvesting problem for a nonlinear age-dependent population system, and for some approximating problems, optimal controllers in feedback form are given by the dynamic programming method. Ani¸ta et al. [60] consider the optimal harvesting problem for the linear Lokta-McKendrick model with periodic vital rates and a periodic forcing term that sustains oscillations. Necessary optimality conditions are presented and a numerical algorithm is developed to approximate the optimal control and optimal harvest. The model in this paper describes the population evolution of human beings with birth control, which was developed in [61]. The HJB equation satisfied by the value function will be derived. It is shown that the value function is the viscosity solution of the HJB equation. The optimal birth feedback control is thus found through the value function and its gradient. A finite difference scheme is designed to obtain the numerical solution of the optimal birth feedback control. The validity of the optimality of the obtained control is verified numerically by comparing with different controls under the same constraint. All the data utilized in the computation are taken from the census of the population of China in 1989 [62]. This section is organized as follows. In Sect. 3.1, the dynamic programming principle for the value function of the optimal control problem is established. Some other properties of the solution as well as the continuity of the value function are presented. Section 3.2 is devoted to show that the value function is just the viscosity solution of the corresponding HJB equation. In Sect. 3.3, the optimal feedback control is formulated by the value function under the smooth assumption. In Sect. 3.4, we formulate a finite difference scheme for the HJB equation of the population system

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with linear quadratic optimal control. The numerical solution of optimal feedback control is presented. The validity of the optimality of the obtained control is verified numerically.

3.1 Problem Formulation and Dynamic Programming Principle A McKendrick type model of age-structured population dynamics developed in [61] is a first order partial differential equation with nonlocal boundary condition described by ⎧ ∂ p(r, t) ∂ p(r, t) ⎪ ⎪ + = −μ(r ) p(r, t), 0 < r < am , t > 0, ⎪ ⎪ ∂r ⎪ ⎨ ∂t p(r, 0) = p0 (r ), 0 ≤ r ≤ am , ⎪  a2 ⎪ ⎪ ⎪ ⎪ ⎩ p(0, t) = β(t) b(r ) p(r, t)dr, t ≥ 0

(11)

a1

where p(r, t) denotes the age density distribution at time t and age r for a closed population. μ(r ) is the relative mortality of the population, which is a nonnegative measurable function and satisfies  am  r μ(ρ)dρ < ∞ for r < am and μ(ρ)dρ = ∞. 0

0

am is the highest age ever attained by individuals of the population. b(r ) = k(r )h(r ). k(r  a2 ) is the ratio of females and h(r ) is the fertility pattern of females satisfying a1 h(r )dr = 1. It is reasonable to assume that b(r ) is bounded and measurable in [a1 , a2 ], the fecundity period of females, 0 < a1 < a2 < am , and b(r ) = 0 outside [a1 , a2 ]. p0 (r ) is the initial distribution. β(t) is the specific fertility rate of the females at time t, which is considered to be the birth control of the population in macro-level. Let H = L 2 (0, am ) be the state space with the usual inner product ·, · and the inner product induced norm  · . For any t > s ≥ 0, let U[s, t] = {β(τ ) ∈ [β0 , β1 ] ⊂ R+ | τ ∈ [s, t], β(τ ) is measurable on [s, t]}. Given T > 0 and p0 ∈ H, the optimal control problem is to find an optimal control β ∗ (·) ∈ U[0, T ] such that ⎧ J (β ∗ ) = inf J (β), ⎪ ⎪ ⎨ β(·)∈U [0,T ]  T  am  am (12) ⎪ ⎪ L( p(r, t), β(t), r, t)dr dt + f 0 (r, p(r, T ))dr ⎩ J (β) = 0

0

0

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where p(r, t) is the solution to (11) corresponding to β(·), and L, f 0 satisfy

 am



 am







⎪ f 0 (r, p(r ))dr ,

L( p(r ), β, r, t)dr

≤ C1 + C2  p, ⎪

⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ∀ (t, β) ∈ [0, T ] × [β0 , β1 ], p ∈ H, ⎪ ⎪ ⎪

 a m

⎪ ⎪



⎪ ⎪

, ⎪ [ f (r, p (r )) − f (r, p (r ))]dr 0 1 0 2 ⎨

0

 a m



⎪ ⎪

[L( p1 (r ), β, r, t) − L( p2 (r ), β, r, t)]dr

≤ C3  p1 − p2 , ⎪ ⎪

⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ∀ (t, β) ∈ [0, T ] × [β0 , β1 ], p1 , p2 ∈ H, ⎪ ⎪ ⎪ ⎪  am ⎪ ⎪ ⎪ ⎩ L( p(r ), β, r, t)dr is continuous in (t, p, β) ∈ R+ × H × [β0 , β1 ]

(13)

0

for some constants Ci , i = 1, 2, 3. Define the time-dependent operators Aβ (t) as follows: ⎧ ⎪ ⎨ Aβ (t)φ(r ) = −φ (r ) − μ(r )φ(r ), ∀ φ ∈ D(Aβ (t)), 



⎪ ⎩ D(Aβ (t)) = φ(r ) φ, Aβ (t)φ ∈ H, φ(0) = β(t)

a2

 b(r )φ(r )dr .

a1

Then the (11) can be written as a first order evolution equation in H: ⎧ ⎨ ∂ p(r, t) = Aβ (t) p(r, t), ∂t ⎩ p(r, 0) = p0 (r ).

(14)

From now on, we also use Aβ to denote the operator Aβ (t) when β(t) ≡ β is independent of time t, and the semigroup generated by Aβ will be denoted as Tβ (t). It is seen that Aβ (t) = Aβ(t) . Define a family of evolution operators T(t, s; β), 0 ≤ s ≤ t < ∞ by T(t, s; β)φ(r ) r ⎧ − μ(ρ)dρ ⎪ , r ≥ t − s, ⎪ φ(r − t + s) e r −t+s ⎪ ⎪  ⎨ a2 τ r = β(t − r ) b(τ )φ(τ − t + s + r ) e− τ −t+s+r μ(ρ)dρ dτ e− 0 μ(ρ)dρ , r < t − s, ⎪ a1 ⎪ ⎪ ⎪ ⎩ ∀ φ ∈ H, 0 ≤ t − s ≤ a1 .

Dynamic Programming Viscosity Solution Approach …



381

[a ]

1  t −s T(t, s; β) = T(t, s + T(s + na1 , s + (n − 1)a1 ; β), t − s > a1 a1 ; β) a1 t−s

n=1

(15) where [x] denotes the maximal integer not exceeding x. T(t, s; β) is uniquely determined by {β(τ ) | τ ∈ [s, t]}. The following Theorem 6 comes from [63], which can be verified directly by a straightforward computation. Theorem 6 (i) For each β(·) ∈ U[s, t], T(t, s; β) is a bounded linear operator on H with T(s, s; β) = I, T(t, s; β) = T(t, t0 ; β)T(t0 , s; β), ∀ 0 ≤ s ≤ t0 ≤ t < ∞ where I stands for the identity operator on H. (ii) T(·, ·; β)φ is (strongly) continuous for each φ ∈ H uniformly in β(·) ∈ U[s, t]. (iii) Let Tβ1 (t) be the C0 -semigroup generated by Aβ 1 on H . Then ([61]) T(t, s; β) ≤ Tβ1 (t) ≤ Meω(t−s) for some M > 1 and ω ∈ R. (iv) The family of evolution operators {T(t, s; β)} defined by (15) has the iterative relation r ⎧ − μ(ρ)dρ ⎪ , r ≥ t − s, ⎨ φ(r − t + s) e r −t+s  a2 r T(t, s; β)φ(r ) = ⎪ b(τ )T(t − r, s; β)φ(τ )dτ e− 0 μ(ρ)dρ , r < t − s. ⎩ β(t − r ) a1

(16)

(v) If β(t) ∈ C 1 (s, ∞), then T(t, s; β) is an evolution system, i.e., d T(t, s; β) p0 = Aβ (t)T(t, s; β) p0 , ∀ p0 ∈ D(Aβ (0)). dt (vi) For any t > s ≥ 0 and any βi (·) ∈ L 2 (s, t) (i = 1, 2), T(t, s; β1 ) p0 − T(t, s; β2 ) p0  ≤ b p0 β1 (·) − β2 (·) L 2 (s,t) . Theorem 7 Let A∗β be the adjoint operator of Aβ on H. Then there exists an operator C on H that is bounded, linear, self-adjoint and positive definite such that for each β ∈ [β0 , β1 ], A∗β C is a bounded linear operator on H and √ sup A∗β C ≤ 1 + β1 b am .

β∈[β0 ,β1 ]

Furthermore, the set D∗ defined by

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D∗ = D(A∗β ) = {φ(r ) φ(r ), φ (r ) − μ(r )φ(r ) ∈ H, φ(am ) = 0}

(17)

is independent of β ∈ [β0 , β1 ] and dense in H. Proof Let the operator D be defined as 

Dφ(r ) = φ (r ) − μ(r )φ(r ),

D(D) = {φ(r ) φ, Dφ ∈ H, φ(am ) = 0}. 

Then

am

−φ(0) = 0



am

=



r d [φ(r )e− 0 μ(ρ)dρ ]dr dr

[φ (r ) − μ(r )φ(r )]e−

r 0

μ(ρ)dρ

dr = Dφ, e−

r 0

μ(ρ)dρ

.

0

It is easy to check that A∗β φ(r ) = φ (r ) − μ(r )φ(r ) + βb(r )φ(0) = Dφ(r ) − βb(r )Dφ, e−

r

μ(ρ)dρ

. (18) 1 So D(A∗β ) = D(D). Furthermore, let C = (I + D∗ D)− 2 . Then it follows from Theorem 13.13 of [64] that C is a linear bounded self-adjoint positive operator and DC ≤ 1. By (18), √ A∗β C ≤ 1 + β1 b am . 0



This is the desired result.

Theorem 8 Let D∗ be defined as (17) and let q(·) ∈ D∗ . Then q(·), p(·, t) is differentiable almost everywhere on [0, T ] and d q(·), p(·, t) = A∗β (t)q(·), p(·, t), t ∈ [0, T ] a.e. dt

(19)

where p(r, t) = T(t, 0; β) p0 (r ) for any p0 ∈ H and β(·) ∈ U[0, T ]. A∗β (t) = A∗β(t) is the adjoint operator of Aβ (t). Proof Let t ∈ (0, T ). For simplicity, we only consider the right derivative. Let δ > 0, t + δ < a1 . Then from (16), it has p(r, t + δ) r ⎧ − μ(ρ)dρ ⎪ , r ≥ δ, ⎨ p(r − δ, t) e r −δ  a2 τ r = ⎪ b(τ ) p(τ − r + δ, t) e− τ −r +δ μ(ρ)dρ dτ e− 0 μ(ρ)dρ , r < δ. ⎩ β(δ + t − r ) a1

So

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q(·), p(·, t + δ) − p(·, t)  am r = q(r )[ p(r − δ, t) e− r −δ μ(ρ)dρ − p(r, t)]dr δ



δ

+

 q(r ) β(δ + t − r )

0

·e 



r

μ(ρ)dρ

0

b(τ ) p(τ − r + δ, t) e−



τ −r +δ

0

am

am −δ

 −

t δ+t

 ·

q(r ) p(r, t)dr

b(τ ) p(τ + r − t, t) e





τ +r −t

μ(ρ)dρ

dτ dr

q(r ) p(r, t)dr

0 am −δ 0



t+δ

+

    r +δ p(r, t) q(r + δ) e− r μ(ρ)dρ − q(r ) dr −

a2

b(τ ) p(τ + r − t, t) e





τ +r −t

μ(ρ)dρ

am

am −δ

 δ+t−r q(δ + t − r ) e− 0 μ(ρ)dρ β(r )

t

 ·

p(r, t)q(r )dr

 δ+t−r q(δ + t − r ) e− 0 μ(ρ)dρ β(r )

a2

δ

+ 

δ

a1



=



0





μ(ρ)dρ

a1

    r +δ − r μ(ρ)dρ p(r, t) q(r + δ) e − q(r ) dr −

am −δ

=

− p(r, t) dr

a2

q(r ) p(r, t)dr

dτ dr

a1

and hence   p(·, t + δ) − p(·, t) lim q(·), δ↓0 δ   am p(r, t)[q (r ) − μ(r )q(r )]dr + q(0)β(t) = 0

a2

b(τ ) p(τ, t)dτ

a1

= A∗β (t)q(·), p(·, t), for all such t that the above limit exists, which are dense in [0, T ].



By virtue of Theorem 8, we will consider p(r, t) = T(t, 0; β) p0 (r ) as the (weak) solution of (11) in the sense of (19), which is obtained by integrating (11) along the characteristics. The value function V (t, p0 ) for the optimal control problem is defined by

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 V (t, p0 ) =

T



am

inf

β(·)∈U [t,T ]

t



am

L( p(r, τ ), β(τ ), r, τ )dr dτ +

0

 f 0 (r, p(r, T ))dr

0

where p(r, τ ) = T(τ, t; β) p0 (r ) is the solution of (11) corresponding to β(·) ∈ U[t, T ]. Theorem 9 (Dynamic Programming Principle). For any 0 < δ < T − t, ⎧ V (t, p0 ) ⎪ ⎪ ⎪ ⎪  t+δ  am  ⎪ ⎪ ⎨ L( p(r, τ ), β(τ ), r, τ )dr dτ + V (t + δ, p(·, t + δ)) , = inf β(·)∈U [t,t+δ] 0 t ⎪ ⎪  am ⎪ ⎪ ⎪ ⎪ f 0 (r, p0 (r ))dr ⎩ V (T, p0 ) = 0

where p(r, τ ) = T(τ, t; β) p0 (r ). Proof Let W (t) =



t+δ



am

inf

β(·)∈U [t,t+δ]

t

 L( p(r, τ ), β(τ ), r, τ )dr dτ + V (t + δ, p(·, t + δ)) .

0

ˆ ∈ U[t, T ] It is obvious that V (t, p0 ) ≤ W (t). For any given ε > 0, there exists a β(·) such that  am  T  am ˆ ), r, τ )dr dτ + L( p(r, ˆ τ ), β(τ f 0 (r, p(r, ˆ T ))dr ≤ V (t, p0 ) + ε 0

t

0

ˆ p0 (r ). Since where p(r, ˆ τ ) = T(τ, t; β)  V (t + δ, p(·, ˆ t + δ)) ≤

T

t+δ

it follows that  t+δ  W (t) ≤  ≤ t

T

t am

 0

am



am

ˆ ), r, τ )dr dτ + L( p(r, ˆ τ ), β(τ

0



am

f 0 (r, p(r, ˆ T ))dr,

0

ˆ ), r, τ )dr dτ + V (t + δ, p(·, L( p(r, ˆ τ ), β(τ ˆ t + δ))

0

ˆ ), r, τ )dr dτ + L( p(r, ˆ τ ), β(τ



am

f 0 (r, p(r, ˆ T ))dr ≤ V (t, p0 ) + ε.

0

By the arbitrariness of ε, we obtain that W (t) ≤ V (t, p0 ). Therefore W (t) =  V (t, p0 ).

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385

Theorem 10 With M and ω as in Theorem 6 and constants Ci , i = 1, 2, 3 in (13), the following assertions are true: (i) V (t, p0 ) ≤ (1 + T )C1 + MC2 (1 + ω−1 ) eωT  p0 , for all t ∈ [0, T ], p0 ∈ H. (ii) |V (t, p1 ) − V (t, p2 )| ≤ (1 + ω−1 )MC3 eωT  p1 − p2 , for all t ∈ [0, T ], p1 , p2 ∈ H. (iii) For any fixed p0 , V (t, p0 ) is continuous about t. Proof For any β(·) ∈ U[t, T ], 

T

|V (t, p0 )| ≤

(C1 + C2 T(τ, t; β) p0 ) dτ + C1 + C2 T(T, t; β) p0 

t

M C2 eωT  p0  + C1 + MC2 eωT  p0  ω 1 = (1 + T )C1 + (1 + )MC2 eωT  p0 . ω ≤ C1 T +

ˆ ∈ U[t, T ] such that This is (i). For any ε > 0, take β(·) 

T

V (t, p2 ) ≥



t

am

ˆ ), r, τ )dr dτ + L( pˆ 2 (r, τ ), β(τ



0

am

f 0 (r, pˆ 2 (r, T ))dr − ε

0

ˆ p2 (r ). Let pˆ 1 (r, τ ) = T(τ, t; β) ˆ p1 (r ). Then where pˆ 2 (r, τ ) = T(τ, t; β) V (t, p1 ) − V (t, p2 )  T  am  ˆ ≤ L( pˆ 1 (r, τ ), β(τ ), r, τ )dr dτ + t

0



T

− t T

 ≤ t



+









0



f 0 (r, pˆ 1 (r, T ))dr

0 am

0 am

ˆ ), r, τ )dr dτ − L( pˆ 2 (r, τ ), β(τ



am

f 0 (r, pˆ 2 (r, T ))dr + ε

0



ˆ ˆ L( pˆ 1 (r, τ ), β(τ ), r, τ ) − L( pˆ 2 (r, τ ), β(τ ), r, τ ) dr

dτ 0

am  

f 0 (r, pˆ 1 (r, T )) − f 0 (r, pˆ 2 (r, T )) dr

+ ε T

≤ C3

am



   pˆ 1 (·, τ ) − pˆ 2 (·, τ ) dτ + C3  pˆ 1 (·, T ) − pˆ 2 (·, T ) + ε

t

M ≤ C3 eωT  p1 − p2  + C3 MeωT  p1 − p2  + ε ω = (1 + ω−1 )MC3 eωT  p1 − p2  + ε. (ii) is thus proved. Notice that in the last line above, we used the fact from (iii) of Theorem 6 that

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ˆ  pˆ 2 (·, τ ) − pˆ 2 (·, τ ) ≤ T(τ, t; β) p1 − p2  ≤ Meω(τ −t)  p1 − p2 , ∀ τ ∈ [t, T ].

Finally, for given p0 ∈ H and any t1 > t2 ∈ [0, T ], there exists a β(·) ∈ U[t2 , T ] such that  T  am  am L( p 2 (r, t), β(t), r, t)dr dt + f 0 (r, p 2 (r, T ))dr − ε V (t2 , p0 ) ≥ t2

0

0

where p 2 (r, t) = T(t, t2 ; β) p0 (r ). Let p 1 (r, t) = T(t, t1 ; β) p0 (r ). Then V (t1 , p0 ) − V (t2 , p0 )  T  am  ≤ L( p 1 (r, t), β(t), r, t)dr dt + t1

0



T

− 

t1

=− t2

T

+ 

am



am

L( p 2 (r, t), β(t), r, t)dr dt −

f 0 (r, p 2 (r, T ))dr + ε

0



am

L( p 2 (r, t), β(t), r, t)dr dt

0 am



 L( p 1 (r, t), β(t), r, t) − L( p 2 (r, t), β(t), r, t) dr dt

0

t1 am

+





f 0 (r, p 1 (r, T ))dr

0

0

t2



am



 f 0 (r, p 1 (r, T )) − f 0 (r, p 2 (r, T )) dr + ε

0

≤ (C1 + C2 M eωT  p0 )(t1 − t2 ) + C3



T

T(t, t1 ; β) p0 − T(t, t2 ; β) p0 dt

t1

+C3 T(T, t1 ; β) p0 − T(T, t2 ; β) p0  + ε ≤ (C1 + C2 M eωT  p0 )(t1 − t2 ) + C3 (1 + T )M eωT T(t1 , t2 ; β) p0 − p0  + ε. Hence as t1 − t2 → 0, V (t1 , p0 ) − V (t2 , p0 ) → 0, provided that T(t1 , t2 ; β) p0 − p0  → 0. Now assume t1 − t2 < a1 . We have

Dynamic Programming Viscosity Solution Approach …

387

T(t1 , t2 ; β) p0 − p0 2  am  2 r − μ(ρ)dρ p0 (r − t1 + t2 ) e r −t1 +t2 = − p0 (r ) dr t1 −t2



t1 −t2

+

 β(t1 − r )

0

·e  ≤2

a2

b(τ ) p0 (τ − t1 + t2 + r ) e





τ −t1 +t2 +r

μ(ρ)dρ



a1



r 0

am

t1 −t2



+2

μ(ρ)dρ

2

− p0 (r ) dr

[ p0 (r − t1 + t2 ) − p0 (r )]2 dr am

t1 −t2

 r 2 − μ(ρ)dρ (e r −t1 +t2 − 1) p0 (r − t1 + t2 ) dr 

1

+2β12 b(a2 − a1 ) 2  p0 |t1 − t2 | + 2

t1 −t2

| p0 (r )|2 dr → 0

0

uniformly for β as t1 − t2 → 0. So (iii) is valid. The proof is complete.



Remark 1 The proof of (iii) of Theorem 10 shows that for any fixed p0 , T(t1 , t2 ; β) p0 − p0  tends to zero uniformly for β(·) ∈ U[t2 , t1 ] as t1 − t2 → 0. We will use this fact frequently in the sequel.

3.2 HJB Equation and Viscosity Solution For brevity in notation, we rewrite (14) as ⎧ ⎨ dP(t) = Aβ (t)P(t), dt ⎩ P(0) = P where P(t) = p(·, t), P = p0 (·). The value function is rewritten as  V (t, P) = where

inf

β(·)∈U [t,T ] t

T

f (τ, P(τ ), β(τ ))dτ + ψ(P(T ))

⎧ P(τ ) = T(τ, t; β)P, ⎪ ⎪ ⎪  am ⎪ ⎪ ⎨ f (τ, P(τ ), β(τ )) = L( p(r, τ ), β(τ ), r, τ )dr, 0 ⎪ ⎪  am ⎪ ⎪ ⎪ ⎩ ψ(P(T )) = f 0 (r, p(r, T ))dr. 0

(20)

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B. Sun et al.

Theorem 11 If the value function V ∈ C 1 ([0, T ] × H), then V satisfies the following HJB equation ⎧ ⎪ ⎪ ⎪ ⎨ Vt (t, P) +

inf

β∈[β0 ,β1 ]

 V p (t, P), Aβ P + f (t, P, β) = 0,

⎪ ⎪ V (T, P) = ψ(P), ∀ t ∈ [0, T ], P ∈ ⎪ ⎩



(21)

D(Aβ ).

β∈[β0 ,β1 ]

Proof For any 0 < δ < T − t and β ∈ [β0 , β1 ], the dynamic programming principle implies that  V (t, P) ≤

t+δ

f (τ, Tβ (τ − t)P, β)dτ + V (t + δ, Tβ (t + δ)P).

(22)

t

If P ∈ D(Aβ ), since Tβ (t)P = P, it follows that P − Tβ (t + δ)P = P − [P + δAβ P + ◦(δ)] = −δAβ P + ◦(δ). Hence V (t, P) − V (t + δ, Tβ (t + δ)P) = −Vt (t, P)δ − V p (t, P), Aβ Pδ + ◦(δ), from which (22) can be rewritten as 1 −Vt (t, P) − V p (t, P), Aβ P ≤ δ



t+δ

f (τ, Tβ (τ − t)P, β)dτ +

t

◦(δ) . δ

Letting δ → 0, we obtain that −Vt (t, P) − V p (t, P), Aβ P ≤ f (t, P, β). So Vt (t, P) +

inf

β∈[β0 ,β1 ]

 V p (t, P), Aβ P + f (t, P, β) ≥ 0.

ˆ ∈ U[t, t + δ] such that On the other hand, for any given ε > 0, there exists a β(·)  V (t, P) ≥

t+δ

ˆ ), β(τ ˆ + δ)) − ε ˆ ))dτ + V (t + δ, P(t f (τ, P(τ

(23)

t

ˆ ) = T(τ, t; β)P. ˆ where P(τ By (vi) of Theorem 6, we may assume without loss ˆ = β for any given β ∈ ˆ ∈ C 1 [t, t + δ] ∩ U[t, t + δ] and β(t) of generality thatβ(·) ˆ + δ) is differentiable with respect to δ [β1 , β2 ]. If P ∈ β∈[β0 ,β1 ] D(Aβ ), then P(t and

Dynamic Programming Viscosity Solution Approach …

389

d ˆ

P(t + δ)

= Aβˆ (t)P. δ=0 dδ This together with (23) gives ˆ −Vt (t, P) − V p (t, P), Aβˆ (t)P ≥ f (t, P, β(t)) − ε, i.e.,

ˆ Vt (t, P) + V p (t, P), Aβ(t) ˆ P + f (t, P, β(t)) ≤ ε.

inf

 V p (t, P), Aβ P + f (t, P, β) ≤ ε

inf

 V p (t, P), Aβ P + f (t, P, β) ≤ 0

So Vt (t, P) +

β∈[β0 ,β1 ]

and hence Vt (t, P) +

β∈[β0 ,β1 ]



proving the theorem. Remark 2 We shall denote  



D= D(Aβ ) ⊃ φ(r ), φ (r ) + μ(r )φ(r ) ∈ H φ(0) =

a2

 b(r )φ(r )dr = 0 .

a1

β∈[β0 ,β1 ]

From now on, we always assume that Aβ (t) is dissipative for all β ∈ [β0 , β1 ]. The assumption is reasonable since instead of (20), we may consider ⎧ ⎨ dQ(t) = [Aβ (t) − C]Q(t), dt ⎩ Q(0) = P and then Aβ (t) − C with C = 21 b2 is dissipative for each β ∈ [β0 , β1 ] because ReAβ (t)P, P ≤

1 b2 P, P. 2

Since Q(t) = P(t) e−Ct , the value function becomes  V (t, P) =

inf

β(·)∈U [t,T ]

T

 f (τ, Q(τ ) eCτ , β(τ ))dτ + ψ(eC T Q(T ))

t

where Q(τ ) = e−Ct T(τ, t; β)P. First, we give a definition for the solution of the HJB equation (21) in the “viscosity sense” [65].

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Let Ω be an open set of H and set USC([0, T ] × Ω) = {upper-semicontinuous mappings u : [0, T ] × Ω → R}, LSC([0, T ] × Ω) = {lower-semicontinuous mappings u : [0, T ] × Ω → R}. Definition 1 U (t, P) ∈ USC([0, T ] × Ω) (respectively U (t, P) ∈ LSC([0, T ] × Ω)) is a subsolution (respectively, supersolution) of (21) on [0, T ] × Ω if for every test function Φ = ϕ + g, ϕ, ϕ p ∈ C([0, T ] × Ω; R), g ∈ C 1 (Ω; R) satisfying the condition (24) and the condition (25) below: ⎧ Range(ϕ p ) ⊂ D∗ , ⎪ ⎪ ⎨ the mapping (t, P) → A∗β ϕ p (t, P), P (24) ⎪ ⎪ ⎩ from [0, T ] × Ω to R is equicontinuous in β ∈ [β0 , β1 ]; ⎧ there exists a mapping h˜ : [0, ∞) → R such that ⎪ ⎪ ⎨ h˜ is nondecreasing, h˜ (0) = 0 and ⎪ ⎪ ⎩ ˜ g(P) = h(P), ∀ P∈H

(25)

and for the local maximum point (respectively, the minimum point) (t, P) of U − Φ (respectively U + Φ), we have ϕt (t, P) +

inf

β∈[β0 ,β1 ]

 ∗ Aβ ϕ p (t, P), P + f (t, P, β) ≥ 0.

(26)

(respectively, −ϕt (t, P) +

inf

β∈[β0 ,β1 ]



A∗β ϕ p (t,

 P), P + f (t, P, β) ≤ 0.)

U (t, P) ∈ C([0, T ] × Ω) is a viscosity solution of (21) if it is both a subsolution and a supersolution on [0, T ] × Ω. Theorem 12 The value function V is a viscosity solution of the HJB equation (21). Proof Assume that (t, P) is a local maximum point of V − Φ for the test function Φ. Then there exists a real number δ > 0 such that V (τ, Q) − Φ(τ, Q) ≤ V (t, P) − Φ(t, P), for all |τ − t|, Q − P < δ, i.e., Φ(τ, Q) − Φ(t, P) ≥ V (τ, Q) − V (t, P), for all |τ − t|, Q − P < δ.

Dynamic Programming Viscosity Solution Approach …

391

By Remark 1, there exists a real number ε > 0 such that for all β ∈ [β0 , β1 ], Φ(t + δ0 , Tβ (δ0 )P) − Φ(t, P) ≥ V (t + δ0 , Tβ (δ0 )P) − V (t, P)  t+δ0 f (τ, Tβ (τ − t)P, β)dτ as 0 < δ0 < ε. ≥− t

The last inequality comes from Theorem 9. Notice that under conditions (24) and (25), we have  ϕ(τ + δ0 , Tβ (δ0 )Q) − ϕ(τ, Q) = 

δ0

=

0 δ0

 ϕt (τ + ρ, Tβ (ρ)Q)dρ +

0

0

δ0

d ϕ(τ + ρ, Tβ (ρ)Q)dρ dρ

A∗β ϕ p (τ

(27)

+ ρ, Tβ (ρ)Q), Tβ (ρ)Qdρ

that holds for all τ, δ0 and all Q by first considering those Q ∈ D(Aβ ) and then for Q ∈ H via the density argument. Also (25) implies that g(Tβ (δ0 )Q) − g(Q) ≤ 0

(28)

for all Q ∈ H, and all β ∈ [β0 , β1 ] by first for Q ∈ D(Aβ ) and then via the density argument. Putting things all together, we have 

t+δ0



f (τ, Tβ (τ − t)P, β)dτ ≤ Φ(t + δ0 , Tβ (δ0 )P) − Φ(t, P)

t



δ0

≤ ϕ(t + δ0 , Tβ (δ0 )P) − ϕ(t, P) =

ϕt (t + τ, Tβ (τ )P)dτ

0



δ0

+ 0

A∗β ϕ p (t + τ, Tβ (τ )P), Tβ (τ )Pdτ

for all 0 < δ0 < ε. Dividing δ0 on both sides above and letting δ0 go to zero, we obtain ϕt (t, P) + A∗β ϕ p (t, P), P + f (t, P, β) ≥ 0. And hence (25) holds by the arbitrariness of β. Next, assume that (t, P) is a local minimum point of V + Φ for the test function Φ. Then there exists a real number ε > 0 such that −Φ(τ, P(τ )) + Φ(t, P) ≤ V (τ, P(τ )) − V (t, P) for all |τ − t| < ε and all admissible control β(·) ∈ U[t, t + ε], in which P(τ ) = T(τ, t; β)P. Let 0 < δ0 < ε. By Theorem 9,

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 V (t, P) = ≥

t+δ0

inf

β(·)∈U [t,t+δ0 ]

f (τ, P(τ ), β(τ ))dτ + V (t + δ0 , P(t + δ0 ))

t



t+δ0

inf

β(·)∈U [t,t+δ0 ]



f (τ, P(τ ), β(τ ))dτ + V (t, P)

t

 −Φ(t + δ0 , P(t + δ0 )) + Φ(t, P) .

So 



t+δ0

f (τ, P(τ ), β(τ ))dτ − Φ(t + δ0 , P(t + δ0 )) + Φ(t, P) ≤ 0

inf

β(·)∈U [t,t+δ0 ]

t

(29)

holds for all 0 < δ0 < ε. As (27) and (28), we have 

δ0

ϕ(t + δ0 , P(t + δ0 )) − ϕ(t, P) =

ϕt (t + τ, P(t + τ ))dτ

0



δ0

+ 0

A∗β (t)ϕ p (t + τ, P(t + τ )), P(t + τ )dτ

and g(P(t + δ0 )) − g(P) ≤ 0 for all β(·) ∈ U[t, t + ε]. (29) then implies that 

t+δ0

inf

β(·)∈U [t,t+δ0 ]

 f (τ, P(τ ), β(τ ))dτ − ϕ(t + δ0 , P(t + δ0 )) + ϕ(t, P) ≤ 0

t

or 

t+δ0

inf

β(·)∈U [t,t+δ0 ]



t



δ0

− 0

δ0

f (τ, P(τ ), β(τ ))dτ −

ϕt (t + τ, P(t + τ ))dτ

0

A∗β (t)ϕ p (t



+ τ, P(t + τ )), P(t + τ )dτ

(30) ≤ 0.

Suppose on the contrary that −ϕt (t, P) +

inf {−A∗β ϕ p (t, P), P + f (t, P, β)} > 2ϑ

β∈[β0 ,β1 ]

for some ϑ > 0. Then for all β ∈ [β0 , β1 ], −ϕt (t, P) − A∗β ϕ p (t, P), P + f (t, P, β) > 2ϑ. By Remark 1, there exists a real number δ ∈ (0, ε), such that

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−ϕt (t, P(t + τ )) − A∗β (t)ϕ p (t, P(t + τ )), P(t + τ ) + f (t + τ, P(t + τ ), β(τ )) > ϑ

for all τ < δ and all β(·) ∈ U[t, t + δ]. Contradicts to (30). So V is a supersolution. The proof is complete. 

3.3 Optimal Birth Feedback Control A necessary condition as well as a characterization of optimal trajectory-control pairs can be formulated by using the value function V (t, P). Theorem 13 Let V be the value function. Then for any trajectory-control pair (P∗ , β ∗ ), P∗ (t) = T(t, 0; β ∗ )P, the function t → V (t, P∗ (t)) −



T

f (τ, P∗ (τ ), β ∗ (τ ))dτ

(31)

t

is nondecreasing on [0, T ]. Moreover, (P∗ (·), β ∗ (·)) is an optimal pair if and only if the above function is constant on [0, T ]. Consequently, if V ∈ C 1 ([0, T ] × H), β ∗ (·) ∈ C 1 [0, T ], P∗ (0) ∈ D(Aβ ∗ (0) ), then (P∗ (·), β ∗ (·)) is an optimal pair if and only if 

Vt (t, P∗ (t)) + V p (t, P∗ (t)), Aβ ∗ (t) P∗ (t) + f (t, P∗ (t), β ∗ (t)) = 0, V (T, P∗ (T )) = ψ(P∗ (T )), ∀ t ∈ [0, T ].

(32)

Proof The first assertion is a direct consequence of the dynamic programming principle. The last assertion comes from the differentiation of (31) with respect to time t and (v) of Theorem 6. As for the second assertion, if (P∗ , β ∗ ) is not optimal then there exists another trajectory-control pair (P, β), P(t) = T(t, 0; β)P, such that 

T

 f (τ, P(τ ), β(τ ))dτ + ψ(P(T ))


T

f (τ, P(τ ), β(τ ))dτ + ψ(P(T )) − ε.

0

Hence  T



T

f (τ, P(τ ), β(τ ))dτ + ψ(P(T ))
0, we use the following approximation:

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  Aβ p Aβ p V p (t, p), Aβ p = V p (t, p), ε Aβ p ε 

 Aβ p Aβ p − V (t, p) . ≈ V t, p + ε Aβ p ε For the initial state p0 let pi = pi−1 + (35) by the difference equation:

Aβ pi−1 ε, Aβ pi−1 

i = 1, 2, . . . , M. Approximate

⎧ j j j+1 j ⎪ Vi − Vi−1 Vi − Vi am j ⎪ ⎪ + Aβ j pi  + [β − β¯ j ]2 = 0, ⎪ ⎨ i Δt ε 2 i  j+1  j+1 ⎪ Vi − Vi−1 am ⎪ j+1 j+1 2 ⎪ Aβ pi  + [β − β¯ ] inf ⎪ ⎩ βi ∈ arg β∈[β ε 2 0 ,β1 ]

(36)

j ¯ j ). for i = 1, 2, . . . , M and j = 0, 1, 2, . . . , N , where Vi ≈ V (t j , pi ), β¯ j = β(t Referring to [68], the following necessary condition for the stability of the difference scheme will be assumed:

|Δt| max Aβ pi  ≤ 1. ε 1≤i≤M j

Set αi = follows.

Δt Aβ j ε i

pi . We give the algorithm for the numerical solution of (35) as

Algorithm of solving the HJB equation. Step 1: initialization. Set ⎧ Aβ pi−1 0 ⎪ ⎪ ⎪ Vi = V (T, pi ) = ψ( pi ), pi = pi−1 + A p  ε, ⎪ β i−1 ⎪ ⎪ ⎨  ψ( p ) − ψ( p ) am i i−1 0 0 2 ¯ A [β − β β ∈ arg inf p  + ] β i i ⎪ ⎪ β∈[β0 ,β1 ] ε 2 ⎪ ⎪ ⎪ ⎪ ⎩ := arg inf {H (β, p)} , i = 1, 2, . . . , M.

(37)

β∈[β0 ,β1 ]

Here the formula for βi0 comes from β(t, p(·)) ∈ arg

 am ¯ 2 . V p (t, p), Aβ p + [β − β(t)] β∈[β0 ,β1 ] 2 inf

Step 2: iteration. By (36),

(38)

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⎧ am j j+1 j j j j ⎪ V (β − β¯ j )2 Δt, = (1 − αi )Vi + αi Vi−1 − ⎪ ⎪ ⎨ i 2 i  j+1  j+1 Vi − Vi−1 am ⎪ j+1 j+1 2 ⎪ inf Aβ pi  + [β − β¯ ] ⎪ ⎩ βi ∈ arg β∈[β ε 2 0 ,β1 ]

(39)

for i = 1, 2, . . . , M and j = 0, 1, 2, . . . , N − 1. It is seen that ⎧ β , if H (β1 , p) = min{H (β1 , p), H (β0 , p), H (β¯ 0 , p)}, ⎪ ⎪ ⎨ 1 βi0 = β0 , if H (β0 , p) = min{H (β1 , p), H (β0 , p), H (β¯ 0 , p)}, ⎪ ⎪ ⎩ ¯0 β , if H (β¯ 0 , p) = min{H (β1 , p), H (β0 , p), H (β¯ 0 , p)} j

for i = 1, 2, . . . , M. The similar formula can be also obtained for βi , i = 1, 2, . . . , M, j = 1, 2, . . . , N − 1. From (38), the optimal feedback control is  am ¯ 2 V p (t, p ∗ (·, t)), Aβ p ∗ (·, t) + [β − β(t)] β∈[β0 ,β1 ] 2 (40) in which p ∗ (·, t) is the optimal trajectory of the system. Because it involves the optimal trajectory in (40), we need to find the solution of (11). Here we utilize the difference scheme to compute the approximate solution of the initial value problem (11) [61]. Adopting the equally spaced grid method, we have β ∗p0 (t, p ∗ (·, t)) ∈ arg

inf

⎧p −p pi, j − pi, j−1 i, j i−1, j ⎪ + = −μi pi, j , 1 ≤ j ≤ J, 1 ≤ i ≤ 2K , ⎪ ⎪ ⎪ Δs Δs ⎪ ⎪ ⎨ pi,0 = p0 (iΔs), 0 ≤ i ≤ 2K , ⎪ a2 ⎪  ⎪ ⎪ ⎪ bi pi, j , 1 ≤ j ≤ J ⎪ p0, j = β j Δs ⎩ i=a1

in which bi = b(iΔs), β j = β( jΔs), i = a1 , . . . , a2 , 2K Δs = am , J Δs = T . It follows that ⎧ pi−1, j + pi, j−1 ⎪ , 1 ≤ j ≤ J, 1 ≤ i ≤ 2K , ⎪ pi, j = ⎪ 2 + μi Δs ⎪ ⎪ ⎪ ⎨ pi,0 = p0 (iΔs), 0 ≤ i ≤ 2K , ⎪ a2 ⎪  ⎪ ⎪ ⎪ ⎪ bi pi, j Δs, 1 ≤ j ≤ J. ⎩ p0, j = β j

(41)

i=a1

Next we give steps of finding the optimal feedback control in detail. It focuses on the solving the difference scheme (41) to obtain the optimal trajectory. Moreover, in

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the process, the algorithm of solving the HJB equation will be called frequently to get the corresponding feedback control function (Figs. 1, 2 and 3). Steps of finding the optimal feedback control. Step 1: Call the algorithm of solving the HJB equation to get the feedback control function β(t, p0 ). Substitute β(0, p0 ) into (41) to get the optimal trajectory p ∗ (·, s1 ), s1 = Δs. Step 2: Take p ∗ (·, s1 ) as the initial data p0 as in the first step, and call the algorithm of solving the HJB equation again to get the feedback control β(s1 , p ∗ (·, s1 )). Substitute β(s1 , p ∗ (·, s1 )) into (41) to get the optimal trajectory p ∗ (·, s2 ), s2 = 2Δs. Step 3: Repeating the processes above, we get all feedback control functions β(s j , p ∗ (·, s j )), s j = jΔs, j = 0, 1, . . . , J , that is to say, β ∗p0 (t, p ∗ (·, t))  = β(0, p0 (·)), β(s1 , p ∗ (·, s1 )), β(s2 , p ∗ (·, s2 )), · · · , β(T, p ∗ (·, T )) , which is the optimal feedback control. Now we are in a position to find the numerical solution of linear quadratic optimal control for the Chinese population based on the scheme (37), (39) and (41). The initial data are extracted from the Chinese population sampling census in 1989 [62], which are plotted by MATLAB 6.1 as Fig. 4 (age-structure of females); Fig. 5 (the relative mortality); Fig. 6 (the age-structure of the total population). The agestructure of an ideal society taken from [61] are listed in Table 1, which is used to get p(r ¯ ) by multiplying the proportion in the table with the total ideal population Nsum = 1,400,000. The fertility pattern h(r ) is approximated by the Gamma density

Fig. 1 The value function V (t, p ∗ (·, t))

*

22

value function : v(t,p (⋅,t))

x 10

5

v

4

3

2

1

0 0

5

10

15

t

20

25

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399

optimal feedback control : β* (t,p*(⋅,t)) p

0

3 2.8 2.6 2.4

β

2.2 2

1.8 1.6 1.4 1.2 1 0

5

10

15

20

25

t

Fig. 2 The optimal feedback control β ∗p0 (t, p ∗ (·, t)) Table 1 The age-structure of an ideal society Age r Proportion 0 1–5 6–10 11–15 16–20 21–25 26–30 31–35 36–40 41–45

0.013 0.065 0.065 0.065 0.065 0.065 0.065 0.065 0.064 0.064

Age r

Proportion

46–50 51–55 56–60 61–65 66–70 71–75 76–80 81–85 >85

0.063 0.062 0.061 0.057 0.053 0.047 0.037 0.024 0.003

distribution curve in statistics [61] in which the peak value of the fertility age is assumed to be 24. Other parameters are listed as follows: 

¯ = 2, T = 25, β0 = 1, β1 = 3, β(t) a1 = 15, a2 = 49, am = 99, ε = 0.01.

In addition, due to the limitation of computer memory, the data in Figs. 4 and 6 and Nsum are divided by 10,000. All computations are performed in Visual C++ 6.0 and numerical results are plotted by MATLAB 6.1. Figure 1 is the value function V (t, p ∗ (·, t)) and Fig. 2 is the optimal feedback control β ∗p0 (t, p ∗ (·, t)).

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B. Sun et al. β (t)

β (t)

2

1

3

3

2

2 1

1 0

5

5

0

25

20

15

10

15

10

β (t)

20

25

20

25

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25

20

25

β (t)

3

4

3

3

2

2 1

1 0

5

5

0

25

20

15

10

15

10

β (t)

β (t)

6

5

3

3

2

2

1

1 0

5

10

15

20

0

25

5

10

15

β* (t,p*(⋅,t))

β (t)

p

7

0

3

3

2

2

1

1 0

5

10

15

20

0

25

5

10

15

Fig. 3 Seven different arbitrarily chosen control βi and the optimal feedback control β ∗p0 (t, p ∗ (·, t)) Fig. 4 The age-structure of females of Chinese population in 1989

female number

15000

10000

5000

0 0

10

20

30

40

50

age

60

70

80

90

100

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Fig. 5 The relative mortality of Chinese population in 1989

relative mortality

0.5

μ(r)

0.4 0.3 0.2 0.1 0 0

10

20

30

40

50

60

70

80

90

100

80

90

100

r

Fig. 6 The age-structure of Chinese population in 1989

Initial density distribution

4

3

x 10

2.5

0

p (r)

2

1.5

1

0.5

0

0

10

20

30

40

50

60

70

r

Now, we check the optimality of the numerical solution of optimal feedback control. This is done by comparing the cost functional J (β ∗p0 (t, p ∗ (·, t))) of obtained optimal control-trajectory pair with that of arbitrarily chosen control β p0 (t), J (β p0 (t)) under the same initial condition, that is, J (β ∗p0 (t, p ∗ (·, t))) ≤ J (β p0 (t)). Referring to Fig. 3, we compute these costs J (βi ) (i = 1, 2, . . . , 7) for seven different controls βi ∈ U[0, T ] respectively. The cost value J (β ∗p0 (t, p ∗ (·, t))) is also computed. These results are listed in Table 2. It is seen from Table 2 that for the optimal feedback control β ∗p0 (t, p ∗ (·, t)), J (β ∗p0 (t, p ∗ (·, t))) = 11641607717.814667,

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Table 2 Different β and its corresponding cost J (β) β J (β) β β1 (t) β2 (t) β3 (t) β4 (t)

11641608005.975365 11641608005.975365 11641607901.862444 11641608172.639233

β5 (t) β6 (t) β7 (t) β ∗p0 (t, p ∗ (·, t))

J (β) 11641608172.639233 11641608172.639233 11641608172.639233 11641607717.814667

which is evidently less than other costs J (βi ), i = 1, 2, . . . , 7. In other words, we do get the numerical solution of optimal birth feedback control for the Chinese population from 1989 to 2014.

4 Convergence of an Upwind Finite-Difference Scheme for HJB Equation This section considers convergence of an upwind finite-difference numerical scheme for the HJB equation arising in optimal control. This effective scheme has been welladapted and successfully applied to many examples. Now, we show that the solution from this finite-difference scheme converges to the value function of the associated optimal control problem. People can refer to [26] for the details.

4.1 Background The Bellman dynamic programming principle is commonly acknowledged as one of milestones in modern control theory. Simply speaking, it was established that the optimal feedback control law can be synthesized by solving the HJB equation, a first-order nonlinear PDE. There is one fundamental issue though in this design, that is, the HJB equation usually has no classical solution regardless of the smoothness of its coefficients. Fortunately, this problem can be circumvented with considering the viscosity solution to the HJB equation, a notion introduced by Crandall and Lions in the 1980s [18, 69]. As a type of weak solution, the viscosity solution is shown to be unique and stable. More importantly, the value function itself often turns out to be the unique viscosity solution to the HJB equation associated with the optimal control problem [25]. Accordingly, the viscosity solution can be used to synthesize the optimal feedback law by the DPVS approach. On the other hand, for almost all nonlinear optimal control problems (with the exception of LQ problems and a few particular examples), it is infeasible to obtain the viscosity solution for the HJB equation in closed form. This makes obtaining numerical solutions the only practically significant way of dealing with the HJB equation and hence optimal control. In the last two decades, several finite difference schemes has

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been developed for this purpose. These include the upwind finite-difference scheme [33], the method of vanishing viscosity [70], and the parallel algorithm based on the domain decomposition technique [71]. Most of these algorithms suffer from the socalled “curse of dimensionality” [72] and are thus confined to only “toy” problems of low dimension. One exception is the max-plus method [73] for nonlinear system control and estimation. It exploits semiconvex dual operators and max-plus linearity by considering the dual-space semigroup corresponding to the HJB equation. This semigroup is particularly useful owing to its max-plus integral operator form with kernel obtained from the originating semigroup. Repeated application of the dual-space semigroup can lead to a numerical solution [72, 73]. Another interesting work presented in [74] combines the recent results on the structure of the HJB equation and its reduction to a linear PDE. The curse of dimensionality is avoided with separated low rank tensor representations. The core idea is to approximate the solution and its associated operators by low rank tensors. If the components of the problem can be adequately modeled in the region, the complexity grows in terms of the rank of the approximation rather than the dimensionality. The curse of dimensionality can also be circumvented via effective model reduction, for which many techniques, such as proper orthogonal decomposition (POD) [75], are available. With the POD technique and the Galerkin projection approach, the state equation (even if it is a PDE) can be approximated by a low order ordinary differential equation. Then the corresponding HJB equation of reduced problem can be solved numerically by the difference scheme. Among these numerical schemes, the upwind finite-difference scheme is welladapted and has been successfully applied to many examples [33, 75]. Its easy implementation to low-dimensional systems allows a numerical solution suitable for feedback control. Moreover, with the aid of model reduction technique, the upwind finite-difference scheme is applicable to both finite-dimensional and infinitedimensional systems. In addition, not only the max-plus method but also its alternative given in [74] could potentially take advantage of the scheme. Actually, the upwind finite-difference scheme has been potentially applied to many numerical solutions of the HJB equations and the optimal feedback controls, see, for instance [76–79]. Despite the effectiveness of the upwind finite-difference scheme, however, the important issue of convergence remains to be addressed [80–82]. The objective of this section is thus to establish a convergence result for this scheme. Using the methods presented in [25, 83], we show that the solution obtained from the upwind finitedifference scheme converges to the value function of the optimal control problem. The remainder of the paper is organized as follows. In Sect. 4.2, we formulate an upwind finite-difference scheme for the HJB equation for a general finite-dimensional optimal control problem. The convergence of this numerical scheme is established in Sect. 4.3.

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4.2 Upwind Finite-Difference Scheme Based Algorithm for the HJB Equation Consider the following control system: ⎧ ⎨ d y(t) = f (t, y(t), u(t)), t ∈ [0, T ], dt ⎩ y(0) = z,

(42)

where y ∈ Rn is the state, u ∈ U[0, T ] = L ∞ ([0, T ]; U ) is the admissible control with U ⊂ Rm being a compact set, z is the initial value, and f : [0, T ] × Rn × U → Rn is continuous. Let a cost functional associated with system (42) be given by 

T

J (u) =

L(τ, y(τ ), u(τ )) dτ + ψ(y(T )),

(43)

0

where L(t, y, u) represents a running cost and ψ(y) a terminal cost. The optimal control problem is to minimize (43) over U[0, T ], that is, to seek an optimal control u ∗ ∈ U[0, T ] such that inf J (u(·)). J (u ∗ ) = u(·)∈U [0,T ]

Following the dynamic programming principle, we consider a family of optimal control problems: for each (t, x) ∈ ΣT = [0, T ) × Rn , minimizing the cost functional  T

Jt,x (u) =

L(τ, yt,x (τ ), u(τ )) dτ + ψ(yt,x (T )),

t

subject to

⎧ ⎨ d yt,x (s) = f (s, yt,x (s), u(s)), s ∈ [t, T ], ds ⎩ yt,x (t) = x.

We define the value function V (t, x) =

inf

u(·)∈U [t,T ]

Jt,x (u), ∀ (t, x) ∈ ΣT ,

(44)

with the terminal value V (T, x) = ψ(x), ∀ x ∈ Rn .

(45)

It is well known that if the value function V is smooth enough, e.g., V ∈ C 1 ([0, T ] × Rn ), then V satisfies the following HJB equation [19]:

Dynamic Programming Viscosity Solution Approach …

⎧ ⎨ − ∂ V (t, x) + H(t, x, ∇x V (t, x)) = 0, (t, x) ∈ ΣT , ∂t ⎩ V (T, x) = ψ(x), x ∈ Rn ,

405

(46)

where H(t, x, ∇x V (t, x)) = sup{− f (t, x, u) · ∇x V (t, x) − L(t, x, u)}. u∈U

From (46), we see that in order to get the feedback control law, we need to know not only the value function V itself but also its gradient ∇x V . According to the viscosity solution theory, we know that the value function V becomes the viscosity solution to (46) under the following three assumptions on the coefficients f and L [19]. Assumption A1: The functions f , L, L t , and L x are bounded on ΣT × U . Assumption A2: The function f (·, ·, u) is of class C 1 (ΣT ), where ΣT is the closure of ΣT . In addition, there exists a constant C > 0 such that | f t | + | f x | ≤ C, | f (t, x, u)| ≤ C(1 + |x| + |u|). Here, the subscripts t and x denote, respectively, the partial derivatives with respect to t and x. Note that the first inequality above is equivalent to the Lipschitz condition | f (s, y, u) − f (t, x, u)| ≤ C[|s − t| + |y − x|] for all (s, y), (t, x) ∈ ΣT . Assumption A3: L and ψ are continuous and satisfy the following polynomial growth conditions for all (t, x) ∈ ΣT , |L(t, x, u)| ≤ C(1 + |x| + |u| ), |ψ(x)| ≤ C(1 + |x| ) with a suitable constant  > 0, where the notation | · | denotes the operator norm. Definition 2 (Viscosity solution) Let V ∈ C([0, T ] × Rn ). Then (i) V is a viscosity subsolution to (46) in ΣT if for every test function φ ∈ C 1 (ΣT ), −

! ∂ φ(t¯, x) ¯ + sup − f (t¯, x, ¯ u) · ∇x φ(t¯, x) ¯ − L(t¯, x, ¯ u) ≤ 0, ∂t u∈U

at every (t¯, x) ¯ ∈ ΣT that is a maximizer of V − φ on ΣT with V (t¯, x) ¯ = φ(t¯, x); ¯ (ii) V is a viscosity supersolution to (46) in ΣT if for every test function φ ∈ C 1 (ΣT ), −

! ∂ φ(t¯, x) ¯ + sup − f (t¯, x, ¯ u) · ∇x φ(t¯, x) ¯ − L(t¯, x, ¯ u) ≥ 0, ∂t u∈U

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at every (t¯, x) ¯ ∈ ΣT that is a minimizer of V − φ on ΣT with V (t¯, x) ¯ = φ(t¯, x); ¯ (iii) V is a viscosity solution to (46) in [0, T ] × Rn if it is both a viscosity subsolution and a viscosity supersolution to (46) in ΣT . We now study the unique viscosity solution of the HJB equation (46). Under Assumptions A1–A3, the value function V in (44) and (45) is the unique viscosity solution to (46) according to [19, 25]. Since a closed-form solution is not feasible, we propose an upwind finite-difference scheme to obtain a " numerical solution. For practical consideration, we consider a finite region nk=1 [ak , bk ] rather than Rn for the HJB equation (46). Let V(t, x) = V (T − t, x). Equation (46) becomes an initial value problem: ⎧  n ⎪ ∂ ⎪ ⎪ V(t, x) + sup f k (T − t, x, u) · ∂k V(t, x) − ⎪ ⎪ ⎪ ∂t u∈U ⎪ k=1 ⎨  ⎪ −L(T − t, x, u) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V(0, x) = ψ(x),

(47)

" where (t, x) ∈ (0, T ) × nk=1 [ak , bk ] ⊂ Rn+1 , x = (x(1) , x(2) , . . . , x(n) ), f = ( f 1 , f 2 , . . . , f n ) ∈ Rn and ∂k V(t, x) = ∂xk V(t, x). Set i = (i 1 , i 2 , . . . , i n ), ik + = (i 1 , i 2 , . #. . , i k−1 , i k + 1, $i k+1 , . . . , i n ), ik − = (i 1 , i 2 , . . . , i k−1 , i k − 1, i k+1 , . . . , i n ), xi = xi1 , xi2 , . . . , xin , xik = ak + i k Δxk , i k = 0, 1, . . . , Mk , with Δxk = (bk − ak )/Mk , t j = ( j − 1)Δt, j = 1, 2, . . . , N + 1, with Δt = T /N . Here Mk , k = 1, 2, . . . , n, and N are given positive inte# $ j− # $ j+ gers. Let f k,i (u) = max f k (T − t j , xi , u), 0 , f k,i (u) = min f k (T − t j , xi , u), 0 , j j j j V + − Vi V − Vik − + j − j Dik V = ik , Dik V = i . Δxk Δxk Let us consider the following upwind finite-difference scheme for the HJB equation (47), n  j+1 j  Vi − Vi j+ j + j = f k,i (u i ) · Dik V Δt k=1  j− j j − j + f k,i (u i ) · Dik V + L(T − t j , xi , u i ), (48)  n  j j+ + j f k,i (u) · Dik V u i = arg sup − u∈U

+

j− f k,i (u)

k=1

·

− j Dik V



 − L(T − t j , xi , u) ,

with Vi1 = ψ(xi )

(49)

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407

for j = 1, 2, . . . , N , i k = j to Mk − j, k = 1, 2, . . . , n. Note that the computational region is a hyper-pyramidal region that contains the original orthotope as a subset due to explicit nature of the discrete scheme. Hence index i k should vary from j to Mk − j not 1 to Mk . This has been explained in more detail in [33] through a simple case. In addition, for the stability of difference scheme, the Courant-Friedrichs-Lewy (CFL) condition n   f k ∞ ≤1 (50) Δt · Δxk k=1 should be assumed [75], where  f k ∞ = max(t,x,u)∈Ξ | f k (t, x, u)| and Ξ  [0, T ] × "n [a k=1 k , bk ] × U . Condition (50) is a necessary condition for stability in solving certain PDEs (usually hyperbolic PDEs) numerically by the finite-difference method. The above finite-difference approximation scheme is firstly proposed in [33] for one-dimensional problems. The scheme is subsequently generalized in [75] to the ndimensional case. Based on this finite-difference scheme, we propose the following algorithm to find a numerical solution to the HJB equation (47). M

M

k k Algorithm 1 Initialize the value function {Vi1 }i =0 and the control {u i1 }i =0 , k = 1, 2, . . . , n. k k For j = 1 to N do For i k = j to Mk − j, k = 1, 2, . . . , n do 

n # j+ j j− + j − j$ V + f k,i (u) · Dik V − L(T − t j , xi , u) , f k,i (u) · Dik Hi (u) = −

k=1 j

j

u i = arg sup Hi (u), j+1

u∈U j

j

j

Vi = Vi − Δt Hi (u i ), end do i k ; end do j.

It is worth pointing out that the efficiency and effectiveness of the above method have been verified by numerical experiments presented in [33]. Example 1 Consider the following HJB equation ⎧ ∂ ⎪ ⎨ V(t, x) + sup {−xuVx (t, x)} = 0, (t, x) ∈ [0, 1] × [−1, 1], ∂t u∈[0,1] ⎪ ⎩ V(0, x) = −x, which associates with the optimal control problem that minimizes −y(1) subject to ⎧ ⎨ d y(s) = y(s)u(s), s ∈ [t, 1], ds ⎩ y(t) = x, with control constraint u ∈ [0, 1]. This HJB equation has the analytical solution

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 V(t, x) =

−xet , x > 0, −x,

x ≤ 0.

The problem has been solved by Algorithm 1 and the numerical solution is shown to be convergent to the analytic solution at the rate of order one [33]. Example 2 Consider the following HJB equation ⎧ ∂ ⎪ ⎪ V(t, x) + sup {−uVx (t, x)} = 0, ⎪ ⎪ ⎨ ∂t u∈[−1,1] ⎪ ⎪ ⎪ ⎪ ⎩

(t, x) ∈ [0, 1] × [−1, 1], V(0, x) = −x 2 ,

which is deduced from the optimal control problem which minimizes −y 2 (1) subject to ⎧ ⎨ d y(s) = u(s), s ∈ [t, 1], ds ⎩ y(t) = x, with the control constraint u ∈ [−1, 1]. This HJB equation also has an analytical solution given by V(t, x) = − (|x| + t)2 . In the same way, the problem is also solved by Algorithm 1 and the numerical solution is shown to be convergent to the analytic solution [33].

4.3 Convergence of Numerical Approximation By virtue of [82, 83], to show the convergence of the upwind finite-difference scheme, we need to show its monotonicity, stability, and consistency. The first two properties have been proven in [75]. For the one-dimensional case, the stability has also been addressed in [33]. Both are listed here for completeness of the proof and for convenience in reference. The consistency has not been available in the literature. j+1 in Firstly we present the monotonicity of the difference scheme. Writing Vi j j j function of Vi−1 , Vi , and Vi+1 , we obtain

Dynamic Programming Viscosity Solution Approach …

Vi

j+1

j

= Vi − Δt



n  

j+

409

+ j f k,i (u i ) · Dik V j

k=1

 j j j − j + f k,i (u i ) · Dik V − L(T − t j , xi , u i ) −

 %n = 1 − Δt k=1

j+

j−

j

j

f k,i (u i )− f k,i (u i ) Δxk

 Vi

+

+Δt

j



j j j j j j n  f k,i (u i )Vik + − f k,i (u i )Vik −

Δxk

k=1

j

+ Δt L(T − t j , xi , u i ). (51)

By condition (50), it follows that +

1 − Δt



j j j j n  f k,i (u i ) − f k,i (u i )

Δxk

k=1

= 1 − Δt

j j n  | f k,i (u i )| k=1

j+

Δxk j−

j

≥ 0.

(52)

j

In addition, since Δt > 0, Δxk > 0, f k,i (u i ) ≥ 0 and − f k,i (u i ) ≥ 0, substituting j+1 j (52) into (51), we conclude that Vi is a non-decreasing monotone function of Vi−1 , j j Vi , and Vi+1 . Next, we need to prove the stability of the scheme. As shown in [83], the stability of the scheme is implied by the CFL condition. Moreover, it suffices to verify the stability condition for L = 0 according to [84]. j Taking L(T − t j , xi , u i ) = 0 in (51), we obtain ⎛ ⎞ j+ j j− j n



 f k,i (u i ) − f k,i (u i )

j+1

⎝ ⎠Vj

Vi = 1 − Δt i Δx k

k=1

j j j− j j

n  f k,i (u i )Vik + − f k,i (u i )Vik −



Δx j+

+Δt

k=1

k





j+ j j− j

n 

f k,i (u i ) − f k,i (u i )

· |V j | ≤

1 − Δt i

Δxk



k=1



n j+ j j j− j j

 f k,i (u i )Vik + − f k,i (u i )Vik −



+Δt

Δxk

k=1







j j

n f (u )



k,i i

sup |V j | ≤

1 − Δt i Δxk

i

k=1



+ n

f j (u j )

+

f j (u j )

 i i k,i k,i

j

+Δt sup Vi

Δxk i k=1









n

f j (u j )



n

f j (u j )





 

k,i i

k,i i

j

+ Δt =

1 − Δt sup Vi .

Δxk

Δxk i

k=1

k=1

(53)

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By condition (50), it follows that









n

f j (u j )



n

f j (u j )

 



k,i i k,i i

= 1 − Δt

1 − Δt .



Δx Δxk k



k=1 k=1 Thus (53) becomes





n

f j (u j )

n

f j (u j )









  k,i i k,i i

j+1

j

j

sup Vi = sup Vi , + Δt

Vi ≤ 1 − Δt Δx Δx k k i i k=1 k=1 i.e.,

 j+1  V 



    ≤ V j ∞ ≤ V 0 ∞ .

This proves the stability of the difference scheme. Now it is the turn of consistency. Let j+1 φi

=

j φi

u∈U



− + f k,i (u) · Dik φ j



− Δt sup

n  #

j+

+ j f k,i (u) · Dik φ

k=1

$ j

 − L(T − t j , xi , u) ,

# $ − and # define f k+ (T − $ s, y, u) = max f k (T − s, y, u), 0 , f k (T − s, y, u) = min f k (T − s, y, u), 0 . Denote by δk an n-dimensional vector with the k-th element being 1 and others zero, namely, δk = (0, . . . , 0, 1, 0, . . . , 0), k = 1, 2, . . . , n. Let δxk = Δxk · δk . Deφ(s, y + δxk ) − φ(s, y) − φ(s, y) − φ(s, y − δxk ) , Dk φ(s, y) = . fine Dk+ φ(s, y) = Δxk Δxk Then φ(s + Δt, y)  n # + = φ(s, y) − Δt sup − f k (T − s, y, u) · Dk+ φ(s, y) u∈U

+

f k− (T

− s, y, u) ·

The consistency means that

k=1

Dk− φ(s,

$



y) − L(T − s, y, u) .

Dynamic Programming Viscosity Solution Approach …

lim

(s,y)→(t,x) Δt+Δx→0+



u∈U

+

φ(s + Δt, y) − φ(s, y) Δt



+ sup

411

n  

f k+ (T − s, y, u) · Dk+ φ(s, y)

k=1

f k− (T

− s, y, u) ·

= φt (t, x) + sup



u∈U

Dk− φ(s,

n 







y) − L(T − s, y, u)

(54)

f k (T − t, x, u) · ∂k φ(t, x)

k=1

−L(T − t, x, u) for every test function φ(·, ·) ∈ C 1,1 (Rn+1 ), Δx  max{Δx1 , Δx2 , . . . , Δxn }. To verify (54), we need verify only the approximation of the gradient of the value function. To this end, note that for any positive integer k between 1 and n, f k+ (T − s, y, u) · Dk+ φ(s, y) + f k− (T − s, y, u) · Dk− φ(s, y)   o(Δxk ) . = f k (T − s, y, u) · ∂k φ(s, y) + Δxk

(55)

By assumption for f , (55) means that f k+ (T − s, y, u) · Dk+ φ(s, y) + f k− (T − s, y, u) · Dk− φ(s, y) ≈ f k (T − s, y, u) · ∂k φ(s, y) for sufficiently small Δxk . This shows the consistency of (54). We come to prove the convergence of the upwind finite-difference numerical scheme. Set V ∗ (t, x) = lim sup VΔx,Δt (s, y), (s,y)→(t,x) Δt+Δx→0+

V∗ (t, x) = lim inf VΔx,Δt (s, y),

(56)

(s,y)→(t,x) Δt+Δx→0+

for any (t, x) ∈ Q T := (0, T ] × Rn , in which VΔx,Δt satisfies the upwind scheme for "n the HJB equation in Algorithm 1 with initial value VΔx,Δt (0, x) = ψ(x), x ∈ k=1 [ak , bk ]. Therefore, the upper semicontinuous envelope V ∗ of VΔx,Δt is the smallest upper semicontinuous function on Q T , which is greater than or equal to VΔx,Δt . The dual fact holds for V∗ .

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Lemma 1 Let V ∗ and V∗ be defined by (56). Then V ∗ is a viscosity subsolution of (47), and V∗ is a viscosity supersolution of (47). Proof Define

V ∗ (t, x) = lim sup VΔx,Δt (s, y), (s,y)→(t,x) Δt+Δx→0+

V∗ (t, x) = lim inf VΔx,Δt (s, y), (s,y)→(t,x) Δt+Δx→0+

for any (t, x) ∈ ΣT , where VΔx,Δt satisfies the following upwind finite-difference scheme for the HJB equation (46), similarly to scheme (48) and (49) for (47). Vi

 n  j − Vi j+ + j = sup − f k,i (u) · Dik V Δt u∈U k=1

j+1

+

j− f k,i (u)

·

− j Dik V





− L(t j , xi , u)

with Vi N = ψ(xi ) for j = 1, 2, . . . , N , i k = j to Mk − j, k = 1, 2, . . . , n. On the one hand, suppose that φ is a test function such that V ∗ − φ has a maximum at (t¯, x) ¯ ∈ ΣT . We show that V ∗ is a viscosity subsolution. We may suppose without loss of generality that this maximum is strict [25]. Then there is a sequence converging to zero, denoted by Δx + Δt, such that VΔx,Δt −"φ has a maximum at (sδ , yδ ) which $ tends to (t¯, x) ¯ as Δx + Δt → 0+ . For any y ∈ nk=1 x(i) − 21 Δxk , x(i) + 21 Δxk , it holds VΔx,Δt (sδ , yδ ) − φ(sδ , yδ ) ≥ VΔx,Δt (sδ + Δt, y) − φ(sδ + Δt, y). Notice that for any (t, x) ∈ ΣT , the value function satisfies the dynamic programming principle:  V (t, x) =

inf

u(·)∈U [t,t+Δt]

t+Δt

L(τ, yt,x (τ ), u(τ )) dτ

t

 +V (t + Δt, yt,x (t + Δt)) , where yt,x (·) is the trajectory corresponding to the control u(·). Therefore,

Dynamic Programming Viscosity Solution Approach …

413

0 ≤ VΔx,Δt (sδ , yδ ) − φ(sδ , yδ ) − VΔx,Δt (sδ + Δt, y) + φ(sδ + Δt, y)  sδ +Δt L(τ, y(τ ), u(τ )) dτ − φ(sδ , yδ ) + φ(sδ + Δt, y). ≤ sδ

Dividing by Δt on both sides above, letting Δx + Δt → 0+ , and using the consistency property of the finite-difference scheme gives −

! ∂ ¯ u) · ∇x φ(t¯, x) ¯ − L(t¯, x, ¯ u) ≤ 0. φ(t¯, x) ¯ + sup − f (t¯, x, ∂t u∈U

Since V(t, x) = V (T − t, x), we have that V ∗ is a viscosity subsolution of (47). On the other hand, suppose that V∗ − φ attains a minimum at (sδ , yδ ) ∈ ΣT . By the definition, for any ε > 0, we can find u(·) = u ε,sδ +Δt (·) ∈ U[sδ , T ] satisfying 0 ≥ VΔx,Δt (sδ , yδ ) − φ(sδ , yδ ) − VΔx,Δt (sδ + Δt, y(sδ + Δt)) +φ(sδ + Δt, y(sδ + Δt))  sδ +Δt L(τ, y(τ ), u(τ )) dτ ≥ −εΔt + sδ

−φ(sδ , yδ ) + φ(sδ + Δt, y(sδ + Δt)). Dividing by Δt on both side above, letting Δx + Δt → 0+ , and again applying the consistency property gives −ε ≤

 sδ +Δt 1 − L(τ, y(τ ), u(τ )) dτ Δt sδ

−φ(sδ + Δt, y(sδ + Δt)) + φ(sδ , yδ ) 

1 = Δt

sδ +Δt

− L(τ, y(τ ), u(τ )) −



∂ φ(τ, y(τ )) ∂t

− f (τ, y(τ ), u(τ )) · ∇x φ(τ, y(τ )) dτ

1 ≤ Δt



sδ +Δt





∂ φ(τ, y(τ )) ∂t

+ sup {− f (τ, y(τ ), u) · ∇x φ(τ, y(τ )) − L(τ, y(τ ), u)}

 dτ

u∈U

→−

! ∂ φ(t¯, x) ¯ + sup − f (t¯, x, ¯ u) · ∇x φ(t¯, x) ¯ − L(t¯, x, ¯ u) . ∂t u∈U

Therefore, V∗ is a viscosity supersolution.



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B. Sun et al.

The following assumption ensures that VΔx,Δt takes the initial data in a uniform way: (57) lim VΔx,Δt (s, y) = ψ(x) (s,y)→(0,x) Δt+Δx→0+

uniformly for x in any compact subset of Rn . Theorem 14 Let VΔx,Δt (t, x) be a solution of the upwind finite-difference scheme. Assume that the monotonicity, stability, consistency, and (57) hold. Then lim

(s,y)→(t,x) Δt+Δx→0+

VΔx,Δt (s, y) = V(t, x)

uniformly on any compact subset of Q T , the closure of Q T . Proof By assumptions A1–A3, a slight modification of results in Sect. 3, Chapter V of [25] shows that V is a bounded, uniformly continuous viscosity solution for HJB equation (47) with the given initial value. By Lemma 1, V ∗ is a bounded upper semicontinuous subsolution of (47) and V ∗ (0, x) = ψ(x) for all x ∈ Rn . By a comparison result, V ∗ ≤ V. Similarly V∗ ≥ V. Since V∗ ≤ V ∗ , the result follows.  Next, we introduce sub- and supersolutions to the upwind finite-difference scheme. A bounded function W is a supersolution of (48) and (49) if W(t, x) ≥ W(t + Δt, x)  n # + +Δt sup − f k (T − t, x, u) · Dk+ W(t, x) u∈U

(58)

k=1



$ + f k− (T − t, x, u) · Dk− W(t, x) − L(T − t, x, u) and W(0, x) ≥ ψ(x)

(59)

for t = t j , j = 1, 2, . . . , N , x = xi , i k = j to Mk − j, k = 1, 2, . . . , n. In the same way, W is a subsolution of (48) and (49) if the inequalities are reversed in (58) and (59). Lemma 2 VΔx,Δt ≤ W for any supersolution W of (48) and (49), and Z ≤ VΔx,Δt for any subsolution Z of (48) and (49). Proof Since VΔx,Δt (t, x) is the solution to the upwind finite-difference scheme (48), we have

Dynamic Programming Viscosity Solution Approach …

415

VΔx,Δt (t, x) = VΔx,Δt (t + Δt, x)  n # + f k (T − t, x, u) · Dk+ VΔx,Δt (t, x) +Δt sup − u∈U

+

f k− (T

k=1

− t, x, u) ·

Dk− VΔx,Δt (t, x)

$

 − L(T − t, x, u)

and VΔx,Δt (0, x) = ψ(x) for t = t j , j = 1, 2, . . . , N , x = xi , i k = j to Mk − j, k = 1, 2, . . . , n. By (58), (59), and the monotonicity property of the upwind finite-difference scheme, VΔx,Δt ≤ W. Similarly, the reversed inequality holds for Z.  The following Theorem 15 shows that the value function VΔx,Δt of the upwind finite-difference scheme converges to the value function V as Δx + Δt → 0+ . Theorem 15 Let VΔx,Δt (t, x) be a solution of (47). In addition to assumptions A1 and A2, we suppose that ψ is bounded and uniformly continuous. Then lim

(s,y)→(t,x) Δt+Δx→0+

VΔx,Δt (s, y) = V(t, x)

(60)

uniformly on any compact subset of Q T . Proof First, suppose that ψ, ψx(k) , k = 1, 2, . . . , n are bounded and uniformly con+ − ψ and Dik ψ are also tinuous. By Taylor’s formula, the difference quotients Dik bounded. Let W(t, x) = −K t + ψ(x), where the constant K is chosen to be large enough so that ˜ x, D + ψ, D − ψ)| K > |H(t, ik ik for all x ∈

"n

k=1 [ak , bk ],

where

˜ x, D + ψ, D − ψ) H(t, ik ik  n  + f k+ (T − t, x, u) · Dik ψ(x) = sup − u∈U

+ It then follows that

k=1

f k− (T

− t, x, u) ·

− Dik ψ(x)



 − L(T − t, x, u) .

416

B. Sun et al.

˜ x, D + ψ, D − ψ) W(t + Δt, x) + Δt H(t, ik ik   + ˜ x, D ψ, D − ψ) = W(t, x) − Δt K − H(t, ik ik ≤ W(t, x). This shows that W is a supersolution. In view of Lemma 2, VΔx,Δt ≤ W. Similarly, let Z(t, x) = K t + ψ(x) be a subsolution. Then Z ≤ VΔx,Δt . Since



VΔx,Δt (s, y) − ψ(x)



≤ VΔx,Δt (s, y) − ψ(y) + |ψ(y) − ψ(x)| ≤ −K s + Dψ|y − x|, we see that (57) holds uniformly for x ∈ Rn . In addition, since the finite-difference scheme satisfies the monotonicity, stability, and consistency conditions, Theorem 15 follows from Theorem 14. When ψ is bounded and uniformly continuous, a standard smoothing technique + ˜ − ˜ ψ and Dik ψ are bounded concludes that for any a > 0, and ψ˜ with ψ˜ − ψ ≤ a, Dik and uniformly continuous. Let V˜ and V˜ Δx,Δt be the corresponding value functions with initial cost function ψ˜ instead of ψ. From the definition, ˜ ≤ a, VΔx,Δt − V˜ Δx,Δt  ≤ a. V − V We have thus shown that lim

(s,y)→(t,x) Δt+Δx→0+

˜ x) V˜ Δx,Δt (s, y) = V(t,

uniformly on compact subsets of Q¯ T . Since a is arbitrary, the same is true for VΔx,Δt and V. The convergence (60) then follows. 

5 Conclusion The control system design can not do without the optimization and feedback. Moreover, to synthesize the optimal feedback control is always one of the most exciting topics in control theory. And it is the Holy Grail of control theory. However, whether by the Pontryagin maximum principle or the Bellman classical dynamic programming approach, it is definitely not possible to find the analytical solutions of the optimal feedback law for the general nonlinear system, especially the infinite dimensional

Dynamic Programming Viscosity Solution Approach …

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systems. The fundamental way out for optimal control lies in numerical solutions. The numerical solution is the only and practically effective approach. The currently corresponding numerical method mainly adopts the multiple shooting method, while it has happened the difficulty in guess of the initial data and the obtained control is of open-loop. The definition of the viscosity solution builds the rigorous mathematical foundations for the classical dynamic programming approach; meanwhile, makes it possible to numerically solve the optimal feedback control law. And the numerical solution to the HJB equation helps us construct the optimal feedback control. To sum up, the DPVS approach does predestinate to be a vivid research area with the broad application prospect.

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A Simple Model of Periodic Reproduction: Selection of Prime Periods Raul Abreu de Assis and Mazílio Coronel Malavazi

Abstract A discrete-time model of periodic reproduction with inter and intra-specific competition is proposed as a tool to investigate the selection of prime reproduction cycle lengths, observed in certain species of cicadas. An approximation for the average populations is proposed and analyzed for the case of 2 and 13 populations. Results indicate that prime periods are in advantage when compared with composite ones, suggesting that the prime periods displayed by cicada species in nature might arise by the process of natural selection of adaptive values (and not as a random result of evolutionary constraints). Keywords Difference equations · Periodic reproduction · Prime numbers · Cicadas · Magicicada

1 Introduction Many species present periodic life cycles. Annual, biennial and perennial plants, [4, 18], salmon and aphids [20] are all examples of such species. In particular, the biological example that inspires the models presented in this work comes from cicada species [1]. Some species of cicadas present a life cycle in which the adults last for only a few weeks [22], while the larval stages, which usually live underground feeding on root xylem fluids, may live from 1 to 17 years [10, 11, 22]. Diverse species of the Magicicada genus (Magicicada septendecim, M. cassini, M. septendecula, M. tredecim, M. tredecassini, M. tredecula), present cycles of 13 or

R. A. de Assis (B) Departamento de Matemática, UNEMAT, Sinop, MT 78550-000, Brazil e-mail: [email protected]; [email protected] M. C. Malavazi Instituto de Ciências Naturais, Humanas e Sociais (ICNHS)—UFMT, Sinop, MT 78557-257, Brazil e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_13

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17 years. Of special interest is the fact that those numbers are prime and that species with intermediate composite life cycles, like 10, 12, 14, 15 or 16 are absent in nature. Gould, Loyd and Dibas [3, 5, 6] were among the first to present possible explanations for the selection of prime periodic cycles. Part of the explanation would be that the evolution of such cycles would be a strategy to avoid periodic parasites or predators. Species with prime reproduction cycles tend to coincide with periodic predators or parasites in a smaller number of years than species with a cycle of composite length. If a predator has a periodic life cycle of 4 years, then it will coincide in all years in which a prey of life cycle of 16 years is present, while if the prey has a life cycle of 13 years, they will coincide only once each 52 years. Some mathematical models were elaborated following this line of reasoning and confirmed that it is possible that the selection of prime reproduction cycles emerges from the interactions of host and parasite [19]. Other models worked with hypotheses on hybridization (adaptive disvantage for offspring of parents with different life cycle length) to obtain selection of prime life cycles [23]. Biological arguments [14, 15] suggested factors of “acceleration and deceleration” of 4 years, with offspring taking minus or plus 4 years to emerge as adults. The discovery of this type of observation led researchers to suggest [8] that a “special explanation” for the periods of 13 and 17 years was unnecessary, since some species [17] present variable cycles that may reach up to 9 years. From a 9-year cycle, two “decelerations” of 4 years might produce cycles of 13 and 17 years. In that case it would not be necessary to explain the fact that species with 11-year cycles, which is also prime, were never found. This is a very interesting example that serves to illustrate the contrast between an adaptive and an explanation due to evolutionary constraint [7]. In the first, the “accelerations” and “decelerations” of 4 years would have been a result of adaptation, shaped by the advantadge given by prime cycles while in the second these would have a genetic explanation, generating, by accident, the prime number cycles that are related with a restriction on the possible phenotypes, that is, they arise due to evolutionary constraints. (of course, the maintenance of the phenotypes depends on selection). To the present date, it is not clear if one of the alternative explanations is correct or if the evolution of prime cycles is a combination of both. In this work we present an extremely simple model of periodic reproduction that indicates that there is an adaptive advantadge to prime life cycles. In this context, the model contributes in favor of the adaptive explanation. As the model is fairly general, in principle, it suggests that it would be possible to set up experiments in which the selection of prime reproduction cycles would be observed. If such experiments were indeed conducted and confirmed the results of the models, we would have strong evidence in favor of the adaptive explanation, and the “accelerations and decelerations” would be a subproduct of the adaptive process that led to the generation of prime life cycles. Although our model does indicate adaptative advantadge of prime periodic cycles, it does not discuss the emergence of those cycles, that is, we take for granted that a single brood of each population with cycles of different periods exist and proceed to show that prime cycles are advantageous. Lehmann-Ziebarth et al. [13] presents

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an evolutionary model that discusses many possible explanations for the emergence and selection of cycles of different lengths, although they failed in finding selection of prime numbers for the periods. One important result presented by Lehman et al. is that nymph competition plays an important role in the evolutionary process and is sufficient to create selective forces to establish periodic life cycles and single brood selection. In our model, nymph competition is also the main component that drives the selection for prime numbers and can be thought as a minimalistic approach that provides some evidence for the selection of prime periods. In a sense, both models provide an incomplete explanation for the emergence of prime periodic life cycles in cicadas. The fusion of both factors (i.e. the emergence of periodic cycles and selection of prime ones) in one model is reserved for future work.

2 The Model In our model, the variable k ∈ N represents the number of time intervals passed between the update of the values of the dependent variables. For the specific example of the cicada species, k is measured in years. We denote by Ni (k) the population of adults with cycles of period i ∈ I = {1, 2, 3, . . . , M} in time k (year, for cicadas). M is an arbitrary limited interval for the possible periods in the population, which aims to compare the population dynamics of populations with different life cycle lengths. In fact, the set of possible period lengths should be the product of the evolution of the species, but such approach is reserved for future works. The present model has the limited scope of analyzing the role of different life cycle lengths in the selection of the populations, as such, we use hypotheses that aim to create homogeneous population in relation to the other aspects of population dynamics. Below, we list the hypotheses used in the model: 1. In the absence of other populations with different life cycles lengths, populations grow logistically, with the same carrying capacity for all populations. 2. Reproduction rates in the logistic growth for each life cycle are identical for all populations. 3. Adults of different populations exert a negative impact in the variation of the number of adults in the populations of different life cycles. The negative impact is proportional to the product of adults of each type. Hypotheses 1 and 2 are an “economic” way to model the competition of individuals inside each population, generating identical population dynamics so that any difference would come from the different life cycle lenghts. In the specific case of hypothesis 2, it is worth to mention some observations. Individuals from population that possess longer life cycles would have, in principle, to spend a longer time underground until emerging as adults, which might affect the rate of survival, resulting in lower effective reproduction rates for populations with longer cycles. However,

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there is evidence [12] that populations with longer life cycles deposit more eggs per adult, acting in the opposit direction to the effect of the longer underground period that the larvae must face before emerging as adults. In fact, some authors consider this trade-off in their models [23]: The clutch size C of a copulated adult female with interval y is assumed to be an increasing function of interval (growth period) y, such that C(y) = Ry, where R is a constant. This assumption is supported by Karban’s [12] demonstration that 17-year cicadas have higher fecundity than 13-year cicadas living nearby. Note that there is a trade-off between intervals with respect to overall juvenile survival rate and clutch size C per cycle. Emergence failure in periodical cicadas is frequently seen in nature and appears to be related to crowding rather than nutrition [21]. Therefore, successful emergence rate E (0 < E < 1) is kept constant, assuming emergence failure is independent of life-cycle length.

Again, since the objective of the model is to analyze the role of the cycle lengths, we assume that the effects cancel out, resulting in logistic growth rates that are identical for all the different period lengths. Hypothesis 3 is a simple way to model competition between population of different periods, assuming that the number of encounter between individuals is proportional to the number of adults in each population and that each encounter reduces the reproductive capacity of the populations in a fixed amount. It is a direct application of the law of mass action and was chosen by parsimony. It is also worth to mention that there is competition between larvae of the same age [2]: “Competition between young nymphs also plays a role, but the density is balanced after the second year.” Therefore, the main source of competition seems to occur in the first year, so it is reasonable to assert that the number of adults that live in the same year (which defines the number of larvae that are going to compete in the following year) is an important variable for the reproduction and mortality rates of the cicadas. In other words, it is possible to model indirectly the competition between larvae without recurring to an age-structured model, because competition is concentrated in the first year. We define parameter K as the carrying capacity of the environment in relation to the cicada populations, r the logistic growth rate per cycle and c as a coefficient that regulates the degree of interference between populations of different periods. With these hypotheses and parameters, we can write the equations for the populations: Ni (k + i) = Ni (k) ((1 − r )Ni (k)/K ) + r ) − c Ni (k)



N j (k)

(1)

j=i

note that, if we write r = 1 + s, the dynamics may be rewritten as Ni (k + i) = Ni (k) + s Ni (k)(1 − Ni (k)/K )) − c Ni (k)



N j (k),

(2)

j=i

or also: i Ni (k) = s Ni (k)(1 − Ni (k)/K )) − c Ni (k)

 j=i

N j (k).

(3)

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Fig. 1 Populations are updated in discrete time steps. The population with cycle length equal to two us present only in years with even indexes, those with cycle of length three only its multiples and son on

In Fig. 1 we present a scheme for the atualization process of the populations with periods of 1, 2 and 3 time intervals (years, for cicadas). Naturally, we can re-scale the variables Ni to Ni∗ = Ni /K , obtaining the non-dimensional model (dropping the stars for convenience of notation): ⎛ ⎞  i Ni (k) = s Ni (k) ⎝1 − Ni (k) − θ N j (k)⎠ (4) j=i

where θ = c/s. The parameters for the model are s, which regulates the maximum growth rate per period for each population, and θ , the competition coefficient between the populations. It is a well-known result that the discrete logistic equation written in the form xk+1 = r xk (1 − xk ) presents chaotic behavior when parameter r crosses a threshold r ∗ ≈ 3.56. If we write the logistic equation in the alternative form yk+1 = yk + syk (1 − xk /K ), and make the trasfornation xk = syk /((1 + s)K ), we obtain the equation xk+1 = (1 + s)xk (1 − xk ), such that the relation between parameters r and s is given by r = s + 1. To avoid chaotic behavior, we will work with values for s always below 1. Parameter θ controls the degree of interference between the populations of different periods. If θ > 1, then competition between populations of different cycle lengths is stronger than those of the same cycle length, while if θ < 1 interspecific competition is weaker than intraspecific competition. In Fig. 2a–d we present some simulations of the model with M = 11 (number of different populations). A fact that is clearly observed in all simulations is that the populations in which the cycle length is a prime number present higher average population numbers than those of composite periods. The population with the shortest period, 1, has average population superior to all others. The analysis of model (4) is made more difficult by the fact that in various instants of time many of the populations are absent. For instance, the population with period 2 is present only when the time elapsed is an even number of time intervals. Even the existence of equilibrium points is compromised in such formulation, as can be readly observed in the simulations of Fig. 2a–d. On the other hand, the solutions appear to oscillate around average in a consistent way in a way the arises naturally the question whether it is possible to formulate a model that is able to describe the behavior of such averages.

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Fig. 2 Simulations of model (4) with 11 populations. Note that in all simulations the populations with prime cycles display a higher average population than those of composite cycles. a s = 0.8, θ = 0.8. b s = 0.05, θ = 0.8. c s = 0.8, θ = 0.2. d s = 0.05, θ = 0.2

One way to obtain a simplified dynamics for the population Ni , which has period i, is to take a period of length p and observe that, in this period the population Ni contributes to its increase, approximately, only p/i times (because the population is updated only each cycle of i years). Also, another population N j will influence the dynamics of Ni only p GCD(i, j)/j times (GCD(x, y) = greatest common divisor of x and y). For instance, in a period of 12 time intervals (let’s say for k from 0 to 11), in the dynamics for N2 , population N1 will exert an influence a total of 6 times (years 0, 2, 4, 6, 8, 10), because it is present in every cycle of two years, N2 is also present 6 times, N3 will affect the dynamics 2 times (years 0 and 6), N4 3 times (years 0, 4 and 8) and so on. For short periods p, the formula will display significative rounding errors, as in this case for N5 , which will contribute 2 times while the formula presents a result of 6/5 (for a period p = 20 the formula provides an exact result). The relative error of the approximation formula tends to zero as p increases (for p = 256, the approximation is 25.6, and the exact result is 26). If we ¯ p N¯ i , the dynamics for the averages is given denote the approximate dynamics by  by:

A Simple Model of Periodic Reproduction …

⎛ ⎞  GCD(i, j) N¯ j sp ⎠  p N¯ i = N¯ i ⎝1 − N¯ i − θ i j j=i

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(5)

In the following section we present a deduction and a detailed analysis for the approximation in its simplest case, that is, when M = p = 2.

3 The Two-Dimensional Case 3.1 Deduction for the Dynamics of the Averages In this case we will adopt the notation xk = N1 (k) and yk = N2 (k). The model is then described by the following operators: D(z, k) = sz(1 − z − θ yk ) F(z, k) = sz(1 − z − θ xk )

(6)

so that the recurrence equations of the system are given by: xk+1 = xk + D(xk , k) yk+2 = yk + F(yk , k)

(7)

with initial conditions x0 , y0 and y1 = 0. Note that, since the recurrence equation for yk is of second order, the value of y1 is necessary for the complete determination of the solutions. Defined as zero, it generates a population that is present only in even years. For this case, we analyze an approximation for the dynamics of the averages: x¯k = (xk+1 + xk )/2 . y¯k = (yk+1 + yk )/2

(8)

We begin our analysis when k is even. For convenience, we write Dxk in place of D(xk , k) and F yk in place of F(yk , k). In first place, we observe that the average may be written as x¯k =

xk + Dxk + xk xk+1 + xk Dxk = = xk + 2 2 2

and also that a step of length 2 in xk may be approximated by: xk+2 = xk+1 + Dxk+1 = xk + Dxk + sxk+1 (1 − xk+1 ) for k is even and yk+1 = 0. Replacing xk+1 = xk + Dxk and regrouping the term of order s 2 , we obtain:

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xk+2 = xk + Dxk + sxk (1 − xk ) + O(s 2 ) finally, writing Dxk explicitly: xk+2 = xk + 2sxk (1 − xk − θ yk /2) + O(s 2 ) = xk + D2 xk + O(s 2 )

(9)

where D2 xk = 2sxk (1 − xk − θ yk /2). If we write the average value for instant k + 1, we have: x¯k+1 =

xk + D2 xk + O(s 2 ) + xk + Dxk xk+2 + xk = 2 2

which can be written also as:   Dxk D2 x k + O(s 2 ) x¯k+1 = xk + + 2 2 where x¯k+1 = x¯k + sxk (1 − xk − θ yk /2) + O(s 2 ). If we recall that xk = x¯k − Dxk /2 and collect the terms of order s 2 , we arrive at a recurrence relation for x¯k : x¯k = x¯k+1 − x¯k = s x¯k (1 − x¯k − θ yk /2) + O(s 2 ).

(10)

This recurrence equation is valid k is even. Below, we present the deduction for the case k + 1 (whic is odd if k is even) . Just as we made for x¯k , x¯k+1 can be written as: x¯k+1 = xk+1 +

Dxk+1 . 2

(11)

Now, x¯k+2 depends on xk+3 , obtaining: xk+3 = xk+2 + Dxk+2 = xk+1 + Dxk+1 + sxk+2 (1 − xk+2 − θ yk+2 ) since xk+2 = xk+1 + Dxk+1 and yk+2 = yk + O(s), xk+3 = xk+1 + sxk+1 (1 − xk+1 ) + s(xk+1 + Dxk+1 )(1 − xk+1 − Dxk+1 − θ (yk + O(s)))

collecting the terms of order 2 in s, we arrive at: xk+3 = xk+1 + D2 xk+1 + O(s 2 ). With this relation, we can establish a recurrence equation for x¯k+2 :

(12)

A Simple Model of Periodic Reproduction …

x¯k+2 =

429

xk+3 + xk+2 = xk+1 + D2 xk+1 + xk+1 + Dxk+1 + O(s 2 ) 2

using Eq. (11) we have x¯k+2 − x¯k+1 = sxk+1 (1 − xk+1 − θ yk /2) + O(s 2 ). Replacing xk+1 = x¯ + k + 1 − Dxk+1 and collecting the second order terms in s, we obtain: x¯k+1 = x¯k+2 − x¯k+1 = s x¯k+1 (1 − x¯k+1 − θ yn /2) + O(s 2 ).

(13)

Note that the recurrence equation for yk+2 can be written as: yk+2 = yk + syk (1 − yk − x¯k ) + O(s 2 )

(14)

The system composed of recurrence Eqs. (10), (13) and (14) contains two different update intervals, with a step of length 1 for variable x¯k and of length 2 for the variable yk . This also prevents us from using the usual tools of equilibrium analysis and stability for discrete-time systems. A way to solve this issue is simply to use relation (10) in (13), obtaining: x¯k+2 = x¯k + 2s x¯k (1 − x¯k − θ yk /2) + O(s 2 ) yk+2 = yk + syk (1 − yk − x¯k ) + O(s 2 )

(15)

which is a system wher both variables are updated in steps of length 2. Note that Eq. (15) is a particular case of Eq. (5) and that the same type of recurrence equation might be obtained if we used only the value of xk+2 , as in Eq. (9).

3.2 Analysis of the Two-Dimensional Model of the Averages For convenience of notation, we adopt x¯k+2 = u t+1 , x¯k = u t , yk+2 = z t+1 and yk = z t . The system obtained, dropping the terms of order s 2 , is given by: u t+1 = u t + 2su t (1 − u t − z t /2) z t+1 = z t + sz t (1 − z t − u t )

(16)

System (16) thus obtained has four equilibrium points: P0 = (0, 0), P1 = (1, 0), P2 = (0, 1) and P3 = (u ∗ , z ∗ ). The eigenvalues for the Jacobian in equilibrium points P0 , P1 and P2 are given in Table 1. The coordinates of point P3 are given by u ∗ = (2 − θ )/(2 − θ 2 ) and z ∗ = 2(1 − θ )/(2 − θ 2 ), so that it is biologically feasible (non-negative) if θ < 1 or θ > 2. The stability of P3 was analyzed numerically, and its results we incorporated in Fig. 3 that

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Table 1 Stability for points P0 , P1 and P2 Ponto Eigenvalues Stability P0 P1 P2

λ1 λ2 λ1 λ2 λ1 λ2

= 1 + 2s =1+s = 1 − 2s = 1 + s(1 − θ) = 1 + s(2 − θ) =1−s

s > 0, λ1 , λ2 > 1, P0 unstable s > 0, so P1 is unstable if θ < 1 and stable if θ > 1 s > 0, so P2 is unstable if θ < 2 and stable if θ > 2

Fig. 3 Stability of points P1 , P2 and P3 for model (15) in relation with parameters s and θ. If 0 < θ < 1 the model converges to the coexistence equilibrium (P3 ), while if 1 < θ < 2 we have the dominance of the population with period of length 1 (P1 ) and for θ > 2 we have bistability and the determination of which population survives is dependent on the initial conditions

represents the regions of stability of the equilibrium points in relation to parameters s and θ . The behavior of the model can be mapped using parameter θ . If 0 < θ < 1 the model converges to the coexistence equilibrium (P3 ), while if 1 < θ < 2 we have the dominance of the population with period of length 1 (P1 ) and for θ > 2 we have bistability and the determination of which population survives is dependent on the initial conditions. If interference (in the sense of competition or other forms) between individuals of different reproductive cycle lengths is greater than the interference between individuals with identical cycles, then the expected outcome is the survival of only one type of population with a greater chance for for individuals with

A Simple Model of Periodic Reproduction …

431

period 1. When the interference is smaller between individuals of different cycles, both populations survive. With this simple analysis, the behavior of model (15) is clear. However, model (15) seeks to approximate the dynamics of the averages of model (7), therefore it is natural to inquire about the quality of the approximations as we vary the parameters. Such point is approached in the next section.

3.3 Comparison Between the Model for the Averages and the Averages of the Original Model To analyze the quality of the approximations we conducted simulations for different values of parameters s and θ in the rectangle [0.01, 0.99] × [0.05, 2.5], using a total number of discrete-time interactions of TF = 3500 and initial conditions x0 = y0 = 1/2. In the end of each simulation the average for the last 100 (from k = 3400 to k = 3500) values of xk and yk is calculated (excluding the zero values for yk ). With ¯ y¯ ) to be compared with this procedure, we obtain estimates for the averages v E = (x, the values of the equilibrium vT = (u ∗ , z ∗ ), of model (15). The error is evaluated by making E(s, θ ) = v E − vT 2 /v E 2 , the relative deviation in the euclidean norm. In Fig. 4a–b we present an example of comparison ¯ y¯ ) and the equlibrium vT = (u ∗ , z ∗ ), also a between the simulated average v E = (x, mapping of the error in function of the parameters es presented. Fundamentally, the results indicate that inside the region of stability of P3 , the equilibrium of model for

(a)

(b) 2.5 1 2

Populations

0.9 1.5

0.8

1

0.7

0.5

0.6 0

100

200

Years

300

0.2

0.4

0.6

0.8

Fig. 4 To evaluate the relative error, we compare the average of simulations of model (7), (x, ¯ y¯ ), with the coordinates of equilibrium P3 , (u ∗ , z ∗ ), of model (15). a Trajectories of populations xk , yk , averages x, ¯ y¯ and equilibria u ∗ e z ∗ . b Relative error in euclidean norm (x, ¯ y¯ ) − (u ∗ , z ∗ )2 /(x, ¯ y¯ )2 . R1 represents a region of “small error” (between 0.01 and 5%), R2 we have “large” errors (up to 100%) and R3 is a region where the original model results in negative population values, losing its biological significance

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the dynamics of the averages (model (15)) is a good approximation for the simulated average of the original model.

4 Observations for the Case M = 13 For the case M = 13, it is possible to show, using the same ideas of the twodimensional case, that the dynamics for the average of the original model (Eq. (4)) can be approximated by: ⎛ ⎞  GCD(i, j) N¯ j sp ⎠ + O(s 2 ); i = 1, . . . , M  p N¯ i = N¯ i ⎝1 − N¯ i − θ i j j=i

(17)

where p is the period used to calculate the average (ideally, p = LCM(1, . . . , M)). Equation (17), above could easily result in chaotic behavior or negative populations, since sp/i can be greater than one. However, it is worth to observe that the coordinates of the equilbirum points are independent of p. Just as in the twodimensional case we can use the equilbrium points of the model for the dynamic of the averages (N ∗ fixed point of Eq. (17) with all coordinates non-zero) and compare it with the average of the last p periods of simulation of model (4) (which we denote by N¯ ). If we calculate the relative error in the euclidean norm,  N¯ − N ∗ 2 / N¯ 2 , in relation to parameters s and θ , we obtain results very similar to those of Fig. 4b, made for the two-dimensional case. This shows that the behavior of the original model can, for many combinations of parameters, be analyzed through the equilibrium of the approximate dynamics of the averages. Essentially, our objective is to evaluate if populations with prime cycles do better than those of composite cycles. To investigate this, we conducted simulations for different values of s and θ using both the original model (4) and the dynamics for the averages (17). The period p for the averages was taken as 100 interactions and the total number of interactions was T = 10,000. For each simulation we calculated the value of N¯ and N ∗ , and for each population N1 , N2 ,…, N13 we established its rank in comparison to the others when the equilibirum vector N ∗ is re-arranged in crescent order. In this manner, the largest population has rank 13, while the smallest one has rank 1. When populations displayed a population inferior to 10−5 the rank 0 was attributed to these. Through such procedure it is possible to analyze, in terms of parameters θ and s, which populations are the most abundant and when they go extinct. In Fig. 5a–b we present the results for both models with s = 0.1 and θ ∈ [0.05, 1.4]. In Fig. 6a–b we have analogous resuts for s = 0.7. It is easy to note that in both models populations with pime cycles are superior (in the sense of population size) to those woth composite cycles. Besides that, we observe also a very similar qualitative behavior in both models, with populations changing rank or going exctinct for very similar threshold values of θ .

A Simple Model of Periodic Reproduction …

433

(a) 14 Rank (when organized in crescent order)

13 12 11 10 9 8 7 6 5 4 3 2 1 0 0

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Rank (when organized in crescent order)

(b) 14 12

10

8

6

4

2

0 0.4

0.6

0.8

1

1.2

1.4

Fig. 5 Comparison of the behaviors of models (4) and (17) for s = 0.1. Populations are organized in crescent order and their rank is presented in function of parameter θ. A rank value of “0” means that the population has values below 10−5 . a Results for the values of N ∗ of model (17). Prime cycles have advantadge over composite cycles. b Results for the average values of model (4). Prime cycles have advantadge over composite cycles

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Rank (when organized in crescent order)

(a) 14 12

10

8

6

4

2

0 0

0.2

0.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Rank (when organized in crescent order)

(b) 14 12

10

8

6

4

2

0 0.6

0.8

1

1.2

1.4

Fig. 6 Comparison of the behaviors of models (4) and (17) for s = 0.7. Populations are organized in crescent order and their rank is presented in function of parameter θ. A rank value of “0” means that the population has values below 10−5 . a Results for the values of N ∗ of model (17). Prime cycles have advantadge over composite cycles. b Results for the average values of model (4). Prime cycles have advantadge over composite cycles

A Simple Model of Periodic Reproduction …

435

In the following section we present a proof that, for the case M = 13, the equilibrium populations of N ∗ with prime cycles are larger than those with composite cycles.

4.1 Proof N ∗p > Nc∗ for the Case M = 13 Proposition 1 Assume system (5) is at equilibrium with Ni∗ > 0, ∀ i ∈ I M = {1, . . . , M}, with 0 < θ < 1. Then for q, k ∈ I M , with k divisible by q, p = k, we have Nq∗ > Nk∗ . Proof By hypothesis Ni∗ > 0, ∀ i ∈ I M , are equilibrium populations for (5). Therefore, 1 − Ni∗ − θ

 MCD(i, j)N ∗j j

j=i j∈I M

= 0, ∀ i ∈ I M .

(18)

Given q, k ∈ I M , we can re-write its equations from (18), as follows: ⎞

⎛ ⎜ GCD(k, q)Nq∗ ⎜

1 − Nk∗ − θ ⎝

q

+

 GCD(k, j=k j=q

j

j)N ∗j ⎟ ⎟

⎠=0

(19)





⎜ GCD(q, k)Nk∗  GCD(q, j)N ∗j ⎟ ⎟ = 0. 1 − Nq∗ − θ ⎜ + ⎠ ⎝ k j j=q

(20)

j=k

Taking the difference between (20) and (19), recalling that k is divisible by q, so that, GCD(k, q) = q and GCD(k, j) ≥ GCD(q, j), ∀ j ∈ I M , we obtain Nq∗



Nk∗

 − θq

Nq∗

N∗ − k q k

 =θ

Portanto, Nq∗ − Nk∗ − θq ou seja,



(GCD(k, j) − GCD(q, j))

j=q j=k



Nq∗ q



Nk∗ N∗ N∗ + k − k q q k

N ∗j j

≥ 0. (21)

 ≥ 0,

(22)

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(1 − θ ) Nq∗ − Nk∗ ≥ θq Nk∗



1 1 − q k

 > 0,

(23) 

from which the result follows directly.

Proposition 2 Assuming the same hypotheses as Proposition 1, with M = 13. So, for every p ∈ {1, 2, 3, 5, 7, 11, 13} and k ∈ {4, 6, 8, 9, 10, 12}, we have N p∗ > Nk∗ . Proof The proof consists in the comparison between the populations of the first set of indexes and the populations in the second set. To reduce the number of cases, we begin by proving that N4∗ > Ni∗ , ∀ i ∈ {6, 8, 10, 12}. This allows us to focus our comparison in the populations N4∗ and N9∗ of the second set of indexes. Considering the equations of the system in equilibrium and calculating the difference between the appropriate equations, we obtain: N4∗ − N6∗ = θ N4∗



∗ N10



2 ∗ 1 ∗ 1 ∗ 1 ∗ 2 ∗ 1 ∗ N + N4 − N6 − N8 + N9 + N12 3 3 2 3 4 9 6 





 1 ∗ 4 ∗ 1 ∗ 1 ∗ 1 ∗ N + N5 − N8 − N10 − N12 . 2 4 5 4 5 6

Therefore N4∗ − N6∗ = θ 13 N3∗ + 14 N4∗ + ( 13 N3∗ − 13 N6∗ ) + ( 41 N4∗ − 41 N8∗ )+ ∗ + 29 N9∗ + 16 N12 1 3 ∗ ∗ ∗ ∗ ∗ )+ N4 − N10 = θ 12 N4 + 5 N5 + ( 41 N4∗ − 14 N8∗ ) + ( 51 N5∗ − 15 N10 1 1 ∗ ) . +( 12 N4∗ − 6 N12 Proposition 1 assures that the differences of the right-side of the above equation ∗ ∗ . The cases N4∗ > N8∗ and N4∗ > N12 are are positive so that N4∗ > N6∗ and N4∗ > N10 an immediate consequence of Proposition 1. We proceed with the proof of the main result. For p = 1 the result follows from Proposition 1. In the case p = 2, we have from Proposition 1 that N2∗ > N4∗ , so it is enough to show that N2∗ > N9∗ . In analogous manner from which we did before, we obtain: N2∗ − N9∗ = θ 21 N2∗ + 23 N3∗ − 41 N4∗ + 16 N6∗ − 18 N8∗ − 19 N9∗ + 1 1 ∗ ∗ N10 + 12 N12 − 10 ∗ 1 5 ∗ 1 ∗ 1 1 1 1 = θ 4 + 8 + 10 + 40 N2 + 9 + 9 N3 − 4 N4 + 1 1 ∗ ∗ N10 + 12 N12 + 16 N6∗ − 18 N8∗ − 19 N9∗ − 10 > 0. Note that for p = 3, we just have to make the comparison with k = 4 and, for the remaining values of p, we make two comparisons, listed below:

A Simple Model of Periodic Reproduction …

N3∗ − N4∗ = θ 21 N2∗ + 1 ∗ N12 + 12 N5∗ − N4∗ = θ 21 N2∗ + N7∗ − N4∗ = θ 21 N2∗ − ∗ N11 − N4∗ = θ 21 N2∗ − ∗ N13 − N4∗ = θ 21 N2∗ − and

N5∗ − N9∗ N7∗ − N9∗ ∗ − N9∗ N11 ∗ N13 − N9∗

= = = =

437

1 ∗ N 3 3

− 14 N4∗ − 16 N6∗ + 38 N8∗ − 29 N9∗ +

3 ∗ N 4 4 1 ∗ N 4 4 1 ∗ N 4 4 1 ∗ N 4 4

+ + + +

θ 23 N3∗ + θ 23 N3∗ + θ 23 N3∗ + θ 23 N3∗ +

1 ∗ N 5 5 1 ∗ N 6 6 1 ∗ N 6 6 1 ∗ N 6 6

1 ∗ N 5 5 1 ∗ N 3 6 1 ∗ N 3 6 1 ∗ N 3 6

+ + − −

+ + + +

1 ∗ N 6 6 1 ∗ N 7 7 3 ∗ N 8 8 3 ∗ N 8 8

1 ∗ N 3 6 1 ∗ N 7 7 1 ∗ N 9 9 1 ∗ N 9 9

1 N∗ + 10 10

3 ∗ ∗ + 38 N8∗ − 10 N10 + 41 N12 1 ∗ ∗ + 38 N8∗ + 10 N10 + 41 N12 1 1 ∗ ∗ ∗ + 10 N10 + 11 N11 + 14 N12 1 1 ∗ ∗ ∗ + 10 N10 + 41 N12 + 13 N13

1 ∗ ∗ − 19 N9∗ − 25 N10 + 6 N12 1 ∗ 1 ∗ − 9 N9 + 6 N12 1 ∗ ∗ + 11 N11 + 16 N12 1 ∗ 1 ∗ . + 6 N12 + 13 N13

Re-arranging the terms and using Proposition 1 the result is obtained.





5 Conclusions Our results indicate that, indeed, in the simple case of competition between populations of different life cycle lengths, those with prime periods would have advantadge over others with composite ones. That advantadge would be expressed through a smaller chance of extinction due to higher average population levels. This hypothesis is interesting in the sense that it does not rely on negative effects produced by hybridization nor the assumption of the existence of periodic predators to explain the emergence of prime life cycles for the species of cicadas. The fact that the population with period 1 does even better than the ones with prime cycles does not result in any contradiction, since the majority of the cicada species has a life cycle of one year. Sanborn [9] reviews the phenomena of periodical life cycles in insects, providing only seven examples of periodical cicada species, while in [16] more than 300 species can be found only in the United States while more than 3000 can be found worldwide. Naturally, the model presented is extremely simple and at least two major lines of improvement can be traced. In first place, it is expected that the populations that possess longer life cycles descend from populations with shorter life cycles. In this context, it would be interesting to propose a model which included mutations between populations of different life cycle lengths that started with only one population of period 1. Second, real populations are divided in “strains” that emerge periodically, but every year. That is, in every year we have the presence of populations with period 13, whose larvae were conceived 13 before. The model used in this artivle considers as 0 the population in the years that are not a multiple of the period of the population. Finally, the results of the model indicate that if we could manipulate some particular species with a short, discrete, periodic life cycle (days, hours, minutes) it would be possible to observe if indeed the varieties with prime life cycle lenghts would be present in larger abundance. In such case, we would have experimental confirmation

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of the effect and strong evidence that the prime cycles presented by the cicada species are the product of selection through adaptive value.

References 1. Alexander, R.D., Moore, T.E.: The evolutionary relationships of 17-year and 13-year cicadas, and three new species. Miscellaneous Publications Museum of Zoology, pp. 1–59. University of Michigan, Ann Arbor (1962) 2. Behncke, H.: Periodical cicadas. Math. Biol. 40, 413–431 (2000) 3. Dybas, H.S., Lloyd, M.: The periodical cicada problem ii. evolution. Evolution 20, 466–505 (1966) 4. Edelstein-Keshet, L.: Mathematical Models in Biology. McGraw-Hill Inc., New York (1988) 5. Edelstein-Keshet, L.: A mathematical approach to cytoskeletal assembly. Eur. Biophys. J. 24, 521–531 (1998) 6. Gould, S.J.: Of bamboos, cicadas, and the economy of Adam Smith. Ever Since Darwin, 97–102 (1977) 7. Gould, S.J., Lewontin, R.C.: The spandrels of San Marco and the Panglossian paradigm: a critique of the adaptationist programme. Proc. R. Soc. Lond. B 205, 581–598 (1979) 8. Grant, P.R.: The priming of periodical cicada life cycles. Trends Ecol. Evol. 100, 433–438 (2007) 9. Heliövaara, K., Väisänen, R., Simon, C.: Evolutionary ecology of periodical insects. Trends Ecol. Evol. 9(12), 475–480 (1994) 10. Karban, R.: Increased reproductive success at high densities and predator satiation for periodical cicadas. Ecology 63, 321–328 (1982) 11. Karban, R.: Opposite density effects of nymphal and adult mortality for periodical cicadas. Ecology 65, 1656–1661 (1984) 12. Karban, R.: Evolution of prolonged development: a life table analysis for periodical cicadas. Am. Nat. 150(4), 446–461 (1997) 13. Lehmann-Ziebarth, N., Heideman, P.P., Shapiro, R.A., Stoddart, S.L., Hsiao, C.C.L., Stephenson, G.R., Milewski, P.A., Ives, A.R.: Evolution of periodicity in periodical cicadas. Ecology 86(12), 3200–3211 (2005) 14. Lloyd, M., e White, J. A., : Sympatry of periodical cicada broods and the hypothetical four-year acceleration. Evolution 30, 786–801 (1976) 15. Martin, A.P., e Simon, C., : Anomalous distribution of nuclear and mitochondrial DNA markers in periodical cicadas. Nature 336, 237–239 (1988) 16. Sanborn, A.F.: Catalogue of the Cicadoidea (Hemiptera: Auchenorrhyncha). Elsevier, New York (1988) 17. Soper, R.S., Delyzer, A.J., e Smith, L. F. R., : The genus massospora entomopathogenic for cicadas. part. ii. biology of massospora levispora and its host okanagana rimosa, with notes on massospora cicadina on the periodical cicadas. Ann. Entomol. Soc. Am. 29, 89–95 (1976) 18. Tielbörger, K., e Petrù, M., : An experimental test for effects of the maternal environment on delayed germination. J. Ecol. 98, 1216–1223 (2010) 19. Webb, G.F.: The prime number periodical cicada problem. Discrete Continuous Dyn. Syst.— Ser. B 1, 387–399 (2001) 20. Whitham, T.G.: The theory of habitat selection: examined and extended using Pemphigus aphids. Am. Nat. 115, 449–466 (1980) 21. White, J., Lloyd, M., Zar, J.H.: Faulty eclosion in crowded suburban periodical cicadas: populations out of control. Ecology 60(2), 305–315 (1979), Wiley Online Library 22. Williams, K.S., e Simon, C.: The ecology, behavior and evolution of periodical cicadas. Annu. Rev. Entomol. 40, 269–295 (1995) 23. Yoshimura, J., Hayashi, T., Tanaka, Y., Tainaka, K., e Simon, C.: Selection for prime-number intervals in a numerical model of periodical cicada evolution. Evolution 63, 288–294 (2008)

Transport on Networks—A Playground of Continuous and Discrete Mathematics in Population Dynamics Jacek Banasiak and Aleksandra Puchalska

Abstract We consider structured population models in which the population is subdivided into states according to certain feature of the individuals. We consider various rules allowing individuals to move between the states; it may be physical migration between geographical patches, or the change of the genotype by mutations during mitosis. We shall see that, depending on the type of the migration rule, the models can vary from a system of coupled McKendrick equations to a system of transport equations on a graph. We address the well-posedness of such problems but the main interest is the asymptotic state aggregation that, in the presence of different time scales, allows for a significant simplification of the equations. Interestingly enough, the aggregated equations vary widely, from scalar transport equations to systems of ordinary differential equations. Keywords Transport problem on network · Asymptotic state lumping · Convergence of semigroups · Singularly perturbed dynamical systems · Structured population dynamics · Reducible matrices · Long term behaviour.

Research supported by National Science Centre, Poland, grants 2017/25/B/ST1/00051 and 2017/25/N/ST1/00787. J. Banasiak (B) Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa e-mail: [email protected] J. Banasiak Institute of Mathematics, Łód´z University of Technology, Łód´z, Poland A. Puchalska Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_14

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J. Banasiak and A. Puchalska

1 Introduction The world we live in is an evolving system of interacting subsystems that may widely vary in their complexity and its exact description practically is impossible. However, usually we do not need the full information about the world; even if we had, it would not be manageable. Thus, in our description of the world we tend to focus on the subsystems that, in our view, are relevant to the questions we ask. The choice of these subsystems is highly subjective in the sense that researchers can arrive at widely different models of the same phenomenon. This is not something inherently wrong, as the parsimony principle, [23], dictates building models that should offer good value for money. If, for instance, we are interested in a crude description of a process, but we need it quickly and cheaply, then a simple model can do. At other times we need to understand fine details of the process, while the cost of obtaining them is of a secondary importance, and then we opt for more complex models. Thus an ideal situation would be to have a complex model that could be easily restricted according to our needs by ‘switching off’ irrelevant subsystems and, indeed, recent advances in experimental and observational techniques, as well as in the available computational power, have made it possible to develop such complex and realistic models by taking into account fine structures of the subsystems. For instance, in the population theory a crude description of a population is its total size. More complexity could be added by differentiating the individuals in the population with respect to their position in space, or their age, but also by accounting for the interactions with other communities and entities of the food web, see e.g. [34]. Also, the individuals themselves are not homogeneous but differ with respect to the metabolism, physiology and behaviour. If we assume that the additional structure is discrete; that is, the modelled entities can occur in one of the denumerable states, we create an expanded compartmental model of high complexity. Otherwise, if the attribute defining the structure is continuous, we arrive at partial differential equations, or integro-differential equations, the analysis of which brings yet other challenges and is not considered here.

1.1 From Macro to Micro Models Example 1.1 In this chapter we shall consider metapopulations that are networks of many interacting subpopulations. A crude model of such a metapopulation typically consists of a system of ordinary differential equations, with the states of each subpopulation as unknowns. The state of the subpopulation occupying a particular node of the network is given by a number, for instance its size, that changes in time. Since we only will consider transport problems, the network can be represented as a directed graph, such as the one given in Fig. 1. As explained above, the state of the structured population on G can be described by a vector v = (v1 , . . . , v7 ), whose evolution in time is given by (1) v = Lv,

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Fig. 1 The directed graph G representing the physical network considered in the example

e1





e3 •

e4

e2 e5





e6

e7

where L is a matrix describing redistribution of the flow in the nodes (in fact, as we shall see later, it is the adjacency matrix of the line graph of G). ♦ Models of this type are too crude for many purposes. They correspond to what is often referred to as the macroscopic description of the system that usually is a resultant of many dynamical processes taking place in the subpopulations at the nodes and along the edges of such a network. Thus, to get a more realistic description of the system, one should recognize that each subpopulation has its own structure that should be modelled. If, for example, we are interested in the age structure of the subpopulations, then each node would become a one dimensional manifold parametrized by the age of individuals and the model would become a system of McKendrick equations on these manifolds, coupled by an appropriate rule of exchange of individuals between the subpopulations at different nodes, either by migrations, or by the gene exchange during mitosis, [7, 32, 39]. Also, the communication between the subpopulations may be modelled by some dynamical process such as transport or diffusion. Such detailed versions of evolutions on a network are suggested on, respectively, Figs. 2 and 3. Incorporating more details in models is necessary to provide a more realistic description of the processes occurring in nature and to better understand the interdependence of their drivers. The drawback is that adding more and more details makes the model less clear and more difficult to analyse. Indiscriminately increasing the size of the model often results in incorporating redundant information and thus obscuring its essential features. It is thus important to find the details that are important and must be incorporated in the model, while still keeping the model tractable.

vp

∂t uk = ck ∂x uk



vq

uk (0, t) = Φpk (wp )



wp



• ek

∂t wq = Ψq (wq , uk )

• el



Fig. 2 A hypothetical micro-model of a network flow obeying first order equations along the edges with the boundary equation at the entry point determined by the state at the adjacent node. Inside the node the substance may undergo some chemical reaction

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J. Banasiak and A. Puchalska ∂t uk = ∂xx uk

vp





wp

∂x uk (1, t) = Φp (uk , wp )

vq

∂x uk (0, t) = Φq (uk , wq )



• ek

∂t wq =−∂a wq−μwq+Ψq (uk , ul ) wq (0, t) =

∞  βq (a)wq (a, t)da 0

• el



Fig. 3 A hypothetical micro-model related to a network in which the population in the node vq evolves according to the McKendrick model and the exchange between the nodes occurs through diffusion along the edges

1.2 Aggregation of Micro-models The other problem we address in this chapter can be thus phrased as follows. Let us assume that we have a detailed model of some process, called a micro-model, that involves a large quantity of variables allowing for an accurate description of minute interactions between subpopulations. If we only are interested in certain coarser features of the process, called macro-variables, we can find a solution of the micromodel, giving the micro-variables, and then aggregate them in an appropriate way to obtain the macro ones. This, however, is of little interest, as to obtain the macrovariables in this way we need to first solve the detailed model and that, typically, is costly. To avoid dealing with the detailed model, we first may try to aggregate the micro-model to derive a system of equations whose solution are the required macrovariables, avoiding thus solving the micro-model. Of course, the macro-variables obtained in both ways should be at least approximately equal. In other words, the diagram in Fig. 4 should be commutative.

Fig. 4 Commutativity of the aggregation diagram

Solution Micro-model

Microsolution aggregation

Micromodel aggregation

Macro-model

Micro solution

Solution

Macro solution

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Thus, the next question we face is whether there is a systematic way, called the aggregation, of finding a reduced macro-model whose solutions have dynamics that is close to the dynamics of the aggregated micro-variables. Finding in what way the coefficients of the macro-model depend on the micro-model is crucial to understand which micro-scale properties are important for the large scale dynamics; that is, what the macro-model remembers from the micro one. There are various ways of aggregation. Sometimes the structure of the system is such that it can be decomposed into blocks in such a way that the collective dynamics of each block is fully consistent with the dynamics of the original system; that is, the diagram in Fig. 4 commutes. Then we say that the aggregation is perfect. Otherwise, we say that the aggregation is approximate, see e.g. [2, 26, 27]. The perfect aggregation occurs in very particular situations, see e.g. [1], and here we only are concerned with an approximate aggregation, where the agreement between the micro and macro dynamics is required to be only approximate, so that the diagram in Fig. 4 commutes only approximately.

1.3 Multiscale Models Approximate aggregation often can be achieved in multiscale systems; that is, systems that are driven by mechanisms having widely different typical time (or size) units. One example of such a system, that will be discussed later, consists of a population living in different patches (with the typical time unit of demographical processes being one year), that could daily migrate between the patches. Hence, the typical time units of these two processes differ by two orders of magnitude. If such a system is correctly nondimensionalized, then a existence of the different time scales in the model is revealed by the presence of a small (or large) scaling parameter representing the ratio of these typical time units expressed in a common time scale. In most cases the terms of the equations that are multiplied by the small parameters represent fast processes and thus setting a particular scaling parameter to zero ‘switches them off’ and seemingly leads to an aggregate problem. However, such a procedure may dramatically change the type of the involved equations and render the problem illposed. Thus, the aggregation; that is, removing fast processes from the description, if possible at all, cannot be accomplished by simply setting the appropriate scaling parameter to 0 but requires complicated limit procedures which lead to the so-called singular limits of the involved equations. Thus, in this chapter we shall consider two models on graphs that can be thought of as micro-models for the macro-model (1) and that are related to Figs. 2 and 3, and we follow the following strategy, based on the ideas presented in [15]. A. Building a micro-model. Given a network of interconnected collections of agents (e.g. particles or individuals), called subpopulations, we endow the subpopulation at each node with an internal structure and dynamics, which is appropriate for a particular application. We identify each node with a one-dimensional domain in which the agents of each subpopulation evolve, driven either by transport (e.g. ageing in age structured population models), or by diffusion (which could be the

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J. Banasiak and A. Puchalska

effect of the Brownian motion in a physical space [17], or of the size changes in size structured population models [24]). Such models provide a more detailed description of the system under investigation and thus could be considered as micro-models. B. Identifying time scales. Typically processes at the micro-scale occur much faster than at the macro-scale – one can think about timescales of an individual and of the whole society. Thus, we associate appropriate time scales to subsystems evolving at different rates by using a universal reference time unit for the rate coefficients. This results in the occurrence of small parameter(s) in the fast subsystems. C. Checking the consistency of the micro-model. The procedure of enhancing a macro-model to a micro-model should be consistent in the sense that both should give (at least approximately) the same dynamics at the macro-scale, as shown in Fig. 4. This amounts to requiring that the processes occurring at the fast time scale have little or no impact on the macro-dynamics. In mathematical terms, we need to prove that, in the singular limit (as the ratio of the time scales tends to zero), the solutions to the micro-model tend, in an appropriate sense, to the solution of the macro-model. As we shall see, this is not always the case, as the chosen time scales may group together subsystems in a way that is different from how they appeared in the original model. While the above strategy can be applied to both linear and nonlinear problems, here we confine ourselves to the linear ones. Also, we shall focus on transport processes – the reader interested in aggregation of diffusion processes on networks can consult [7, 17].

2 Models We begin with a simplified version of the model in Fig. 2 with transport occurring along the edges but only with algebraic exchange conditions at the nodes. For this we need to recall some facts from graph theory, see e.g. [14, 19].

2.1 Graph-Theoretic Notions and Notations In general, we consider a network represented by a directed, connected graph with no multiple edges, G = (V (G), E(G)) = ({v1 , . . . , vn , . . .}, {e1 , . . . , em , . . .}), where v1 , . . . , vn , . . . , are the vertices and e1 , . . . , em , . . . , are the edges of G. In the directed graph each edge has a prescribed direction and the beginning of an edge is

Transport on Networks—A Playground of Continuous … Fig. 5 The line graph of the directed graph G in Fig. 1. Note that any edge outgoing from the source in G becomes a source in L(G)

445

e1

e2





e3

e4

e5

e6









e7 •

referred to as its tail, while the end is called the head. We allow the number of vertices and edges to be infinite but in such a case we assume that the graph is locally finite; that is, for each vertex the number of edges incident to it is finite. This makes all matrices associated with this graph well-defined. We denote by M = {1, . . . , m, . . . } the set of the indices of the edges of G and by |M| its cardinality. As our aim is to model a flow on the graph, we place a weight wij > 0 on each edge ej outgoing from vi . The weights give the proportions of the flow from vi directed to the edges and thus, in a conservative scenario,  wij = 1; (2) ej

outgoing

from

vi

that is, all material from the vertex is redistributed among the outgoing edges and none is created. A vertex v is called a source if there are no incoming edges and a sink if there are no outgoing edges. The line graph Q of G is defined by Q = (V (Q), E(Q)) = (E(G), E(Q)), where E(Q) = {uv; u, v ∈ E(G), head of u coincides with tail of v}, see Fig. 5. By A we denote the weighted adjacency matrix for the line graph whose coefficients are given by  vk wki if ∃k ej → ei , (3) aij = 0 otherwise. Note that A only involves the weights defined on the edges that are not outgoing from a source in the original graph – by the definition of aij there must be an edge ej feeding into the vertex from which ei is outgoing. If there is an outgoing edge at each vertex of G, then A is column stochastic. Indeed, in this case each column of A consists of the weights placed on the edges of G that are fed by the edge of G corresponding to this column, and these weights sum up to 1 by (2). Otherwise, the columns of A corresponding to edges incoming to a sink of G have only zero entries.

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2.2 Models with Dynamics on the Edges 2.2.1

Flow on a Network

We parameterise the edges of G so that we treat it as a metric graph: • each edge ej is identified with the interval [0, 1]; 0 is the tail and 1 is the head. • any function u on G is written as u(x) = (uj (x))j∈M , x ∈ [0, 1]. We consider the following transport problem. We have a substance moving along the edges of G, described by u(x, t) = (uj (x, t))j∈M , where uj (x, t) is the density of particles at position x ∈ [0, 1] of the edge ej and at time t ≥ 0. At each edge ej the flow occurs from the tail 0 to the head 1 with a continuous velocity cj (x) > 0. A standard assumption is that the flow satisfies the Kirchhoff law at each vertex vk with incoming and outgoing edges, 

cj (1)uj (1, t) =

incoming edges ej



ci (0)ui (0, t), t > 0,

(4)

outgoing edges ei

which, in this context, is the mass conservation law at each such a vertex. In other words, whenever ei is an outgoing edge from a vertex vk , ci (0)ui (0, t) = wki



cj (1)uj (1, t)

incoming edges ej

= aij



cj (1)uj (1, t).

incoming edges ej

Note that the notation is correct as in this case aij is independent of j. Let C(x) = diag(cj (x))j∈M and K = (kij )i,j∈M = C −1 (0)AC(1). Then the above transport problem can be written as ∂t u(x, t) + ∂x (C(x)u(x, t)) = 0, x ∈ (0, 1), t ≥ 0, u(0, t) = Ku(1, t), ˚ u(x, 0) = u(x),

(5)

if (and only if) G does not contain a sink. Moreover, if G contains a sink, then the transport problem is not solvable, [12, Theorem 2.1]. The problem can be converted into a constant speed problem by the change of variables  x ds 0 cj (s)

u¯ j (yj ) = cj (x)uj (x), yj =  1

ds 0 cj (s)

∈ (0, 1), j ∈ M.

Indeed, multiplying the differential equations in (5) by cj for any j ∈ M and using the chain rule, we have

Transport on Networks—A Playground of Continuous … Fig. 6 Graphical representation of the Kimmel–Stivers model

447

d2 v 0 (t)

d1

an

a2

a1

a0

v 1 (t)

···

v 2 (t) b1

dn+1

dn

d3 b2

···

v n (t) bn−1

bn

0 = ∂t u¯ j (yj ) + cj (x)∂x u¯ j (yj ) = ∂t u¯ j (yj ) + c¯ j ∂yj u¯ j (yj ), where the new speed c¯ j on ej is given by 1 = c¯ j

 0

1

ds . cj (s)

Hence, from now on we assume that cj does not depend on x for j ∈ M.

2.2.2

The Kimmel–Stivers Model of Gene Proliferation and the Lebovitz–Rubinow–Rotenberg Model

In [28, 29] the authors considered a gene amplification process; that is, the increase or decrease of the number of copies of a drug resistant gene, observed in tumor cells and associated with their resistance to certain drugs. In this model the population is divided into a denumerable quantity of types according to the number n of the drug resistant genes, (Vn (t))n∈N0 , where N0 = N ∪ {0}, and its evolution is modelled by a branching random walk, see Fig. 6. We assume that at death any cell (apart from class 0) produces two offspring that only can belong to the classes j + 1, j, or j − 1. Then the expectations of the random variables (Vn (t))n∈N0 , denoted by (vn (t))n∈N0 , satisfy the birth-and-death system with proliferation, v0 = a0 v0 + d1 v1 , v1 = a1 v1 + d2 v2 , vn = an vn + bn−1 vn−1 + dn+1 vn+1 , n ≥ 2,

(6)

where dn+1 , bn−1 are the rates of recruitment from the populations n + 1 and n − 1 into the population n, and an is the net growth rate of the population n that incorporates birth, death and loss to other populations of cells of type n. We assume that the sequences (an )n∈N0 , (dn )n∈N , (bn )n∈N are bounded.

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However, the cells have their own vital dynamics. In particular, • they age till they die or divide; • they could have different life-spans depending on n and denoted by τn . Furthermore, the cells can die at any time for various reasons (at the rate μn , n ∈ N) and the mutations can occur: • due to the replication errors occurring during mitosis the number of the drug resistant genes can change in an arbitrary way (at the rates (kij )i,j∈N ); • due to external factors (mutagenes) that may happen at any moment of the cell’s life cycle (at the rates (rij )i,j∈N ). The dynamics presented in Figs. 7 and 8 can be expressed by  ∂t un + ∂x un = −μn un + rnj uj , x ∈ (0, τn ), j  knj uj (τj , t), un (0, t) =

(7)

j

n ≥ 1, where the summation is extended over all states j that, by mutation, can change the number of genes to n. In (7) we recognize the Lebowitz–Rubinow–Rotenberg model of cell maturation, [31, 39].

an

un−1 (τn−1 , t)

un+1 (0, t)

bn−1

bn 0

τn

un (x, t)

dn+1

dn

un+1 (τn+1 , t)

un−1 (0, t)

Fig. 7 Kimmel–Stivers model with vital dynamics

knn

ul (τl , t)

uj (0, t) knj

kln 0 kmn

τn

un (x, t) μn

um (τm , t) Fig. 8 Discrete Lebowitz–Rubinow–Rotenberg type model

rnp

kni

up (x, t) ui (0, t)

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If for each n we rescale x according to τn−1 x → x, then cn = τn−1 can be considered as the speed of maturation in the nth class and (7) can be written in the form analogous to (5), ∂t u + C∂x u = −Mu + Ru, x ∈ (0, 1), u(0, t) = Ku(1, t), (8) ˚ u(x, 0) = u(x), where C = diag(cj )j∈N , M = diag(μj )j∈N , R = (rij )i,j∈N and K = (kij )i,j∈N . We assume (9) 0 < cmin ≤ cn ≤ cmax < ∞, n ∈ N.

2.2.3

Graph Realizability

We observed that the two different models considered above led to essentially the same system ut = Au, (10) ˚ u(0) = Ku(1), u(0) = u, where A is a first order differential operator and K is a nonnegative matrix. As we shall see, the mathematical treatment of both models is the same. However, as seen from the following example, (10) not always defines a flow problem of a network. Example 2.2 Let us consider two problems (10) with the boundary conditions given by, respectively, ⎛ ⎞ ⎛ ⎞ 100 100 K1 = ⎝ 1 0 0 ⎠ , K2 = ⎝ 1 0 0 ⎠ . 010 110 p

We will try to build a directed graph G = (V, E) for which K p = (kij )1≤i,j≤3 , p = 1, 2, is the adjacency matrix of its line graph L(G). Let us denote the edges by 1 = 1 indicates that e1 , e2 , e3 . Let us start the considerations with K 1 . Then entry k11 e1 is loop incident to a vertex, denoted by w1 . Further, since b121 = 1, the edge e1 is connected with e2 and hence w1 is the tail of e2 . Let us denote its head by w2 ; e2 = (w1 , w2 ). Finally, by b123 = 1 we see that w2 is the tail of e3 ; e3 = (w1 , w3 ). Taking V = {w1 , w2 , w3 } , we see that K 1 is the adjacency matrix of a directed graph G = ({w1 , w2 , w3 } , {(w1 , w1 ), (w1 , w2 ), (w1 , w3 )}), given in Fig. 9 and thus in this case (10) can be represented as a network flow problem. Consider now K 2 . The entry, 2 = 1 shows that there is a connection between e1 and e3 but then the vertices w1 k31 and w2 should coincide, in which case e2 becomes a loop and the graph takes the form presented on right-hand side of Fig. 9. On the other hand, this graph allows 2 should be nonzero. for the transport of mass from e2 to e1 and hence the entry k12 This contradiction shows that (10) with the boundary condition given by K 2 does not describe a flow on a network. ♦

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e1



e2 e3

e1



e2



e3





Fig. 9 On the left, the graph representing the flow of (10) with K = K 1 . On the right, an attempt to build a graph in the case K = K 2

As we have seen, the problem is equivalent to the question under what conditions a nonnegative matrix K is a weighted incidence matrix of the line graph L(G) of some graph G. We have already observed that in the adjacency matrix A, defined in (3), the nonzero entries in each row must be equal. It is, in fact, almost full characterization of such matrices. However, to formulate a rigourous result, we must introduce some notation. First we notice that for the problem at hand the actual values of the entries of K are irrelevant. Thus we introduce K = {κij }i,j∈M , where  κij =

1 for kij = 0 0 for kij = 0.

Definition 2.3 We say that problem (10) is graph realizable if there exists a graph G such that the matrix K is the adjacency matrix of its line graph L(G) and A = CKC −1 is column stochastic for some diagonal matric C. Then we have Theorem 2.4 [5] Problem (10) is graph realizable by a graph G without loops and multiple edges if and only if the following conditions are satisfied: 1. each two rows of matrix (columns) of K are either identical or orthogonal; 2. κii = 0 for i ∈ M and if there are two indices i, j such that the i-th and j-th rows are non-zero and identical, then the i-th and j-th columns are orthogonal; 3. there is a diagonal matrix C such that CKC −1 is column stochastic. The extension of this result to graphs with loops is immediate upon noticing that a loop in G creates a loop in L(G) and hence a diagonal entry in K.

2.2.4 |M|

Solvability

By l1 we understand Rn with if |M| = n < ∞, or the space l1 , in both cases normed with the standard l1 norm. Then the abstract Cauchy problem (10) is considered in |M| X = L1 ([0, 1], l1 ). We define A (resp. A0 ) to be the realization of the differential expression Au = A0 u + Qu := ( − cj ∂x uj )j∈M − Mu + Ru

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|M|

on the domain D(A) = {u ∈ W11 ([0, 1], l1 ); u(0) = Ku(1)}. We assume that the matrices C, K, Q, introduced in Sect. 2.2.2, generate bounded operators on X (denoted by the same symbols) and that (9) holds. Theorem 2.5 [8] Under the above assumptions on C and K, (A0 , D(A)) generates a C0 −semigroup (etA0 )t≥0 . This semigroup is positive if and only if K is nonnegative and it is stochastic if and only if K = C −1 AC, where A is a stochastic matrix. The Bounded Perturbation Theorem, [37], yields the following Corollary 2.6 Under assumptions of Theorem 2.5, (A, D(A)) generates a C0 semigroup (etA )t≥0 . This semigroup is positive if K is positive and Q is either positive or diagonal. Remark 2.7 By [10], we can use the integrating factor to relate (10) to its unperturbed version. Since C = diag(cj )j∈M > 0 is diagonal, we can write u(x, t) = exC

−1

Q

v(x, t),

(11)

for some v ∈ X. Inserting this expression into the system (10), we obtain the equivalent problem −1

−1

∂t v(x, t) = −e−xC Q CexC Q ∂x v(x, t), x ∈ (0, 1), t > 0, −1 ˚ =: v˚ (x), x ∈ [0, 1], v(x, 0) = e−xC Q u(x) −1 v(0, t) = KeC Q v(1, t), t ≥ 0.

(12)

In general, (12) is of little use as it is a coupled system of differential equations. If, −1 −1 however, C = cI or if Q is a diagonal matrix, then e−xC Q CexC Q is also diagonal and (12) has the same structure as (10); that is, it is a system of equations only coupled by the boundary conditions. Remark 2.8 The formula for the resolvent of A will be useful in further considerations. Following the argument leading to [8, Eq. (41)] for the resolvent of A0 , we solve the resolvent equation, (λI − Q)u + C∂x u = f,

u(0) = Ku(1).

Using the integrating factor, as when deriving (12), to find the general solution of the differential equation and then taking into account the boundary conditions, we arrive at the resolvent formula x −1 −1 u(x) = e−xC (λI−Q) y + 0 e(s−x)C (λI−Q) C −1 f(s)ds, −1 1 (13) −1 −1 K 0 e(s−1)C (λI−Q) C −1 f(s)ds. y = I − Ke−C (λI−Q)

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2.2.5

A Representation Formula

Theorem 2.9 If C = cI, then the semigroup (etA )t≥0 solving problem (8) is given by 1 n (x−ct+n) x [etA u](x) ˚ = e c Q Ke c Q e− c Q u(x ˚ − ct + n),

(14)

for −n ≤ x − ct ≤ −n + 1, n ∈ N0 .

Proof To show (14), first let us provide a more explicit version of (13) for the case C = cI. Using the Neumann series for the expression for y in (13), we find 1 u(x) = c



x

e c (s−x)(λI−Q) f(s)ds 1

0 ∞

1 + c n=0



1

1 n+1 s λ x e− c (x+n+1−s) e c Q Ke c Q e− c Q f(s)ds.

(15)

0

Let us denote by [(t, f)](x) the right hand side of (14). We see that (t, f) is a piecewise continuous function with respect to t in the L1 -norm, and thus Bochner measurable. It also is exponentially bounded, hence we can evaluate its Laplace transform 



e

−λt

 (t, f)dt = 0

0

+ =

1 c

x c

e−λt esQ f(x − vs)ds

∞   n=1 x



x+n−1 c λ

1 n (x−ct+n) x e−λt e c Q Ke c Q e− c Q f(x − ct + n)dt

e− c (x−t) e

0 ∞

+

x+n c

1 c n=1



1

(x−t) c Q

f(t)dt

1 n t λ x e− c (x+n−t) e c Q Ke c Q e− c Q f(t)dt,

(16)

0

where we used the substitution t = x − cs in the first integral and t = x − cs + n in the second. Changing the parameter of a summation from n into n = n − 1 we see that (16) equals (15). By the uniqueness of the Laplace transform, (t, ·) = etA .  Many a result can be proven using (14) but the additional assumption cj = c for j ∈ M is restrictive. The problem with Q = 0, ut = −C∂x u,

u(0) = Ku(1),

˚ u(0) = u,

(17)

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can be transformed to a one satisfying this assumption, provided ∃c∈R ∀j=1,...,m

c = lj ∈ N, cj

(18)

see [30]. Since τj = 1/cj is the time taken to traverse ej , the above condition can be expressed by saying that all traverse times are natural multiples of a reference time τ0 = 1/c. Note that if M is infinite, then (9) and (18) imply that there can only be a finite number of different values of cj . Though the transformation described below can be carried out for problems with an arbitrary matrix K, it has a clear geometric interpretation if there is an underlying (locally) finite connected digraph G = ({vi }i∈N , {ej }j∈M ) such that K = C −1 AC, where A is the weighted adjacency matrix of L(G). We explain the construction in such a case before formulating the main result in full generality. We convert (17) on G into a similar problem on a ‘blown-up’ G by: • rescaling time as τ = ct to convert the problem to an analogous problem with velocities 1/lj on ej ; • rescaling ej by y = lj x to convert the problem into like problem with unit speed on each edge but parametrised as ej = (0, lj ); • splitting each domain ej into lj domains of unit length and re-parametrizing them to (0, 1) to produce an analogous problem on a set of χ(M) = {j ∈ M, k = 1, . . . , lj } = {1, 2, . . . , |χ(M)|} of unit domains ej,k , see Fig. 10. Theorem 2.10 [12, Sect. 3] Let us consider transport problem (17), where C sat |χ(M)| isfies (18). Let |χ(M)| = i∈M li , X = L1 ([0, 1], l1 ).  The mapping  S : X → X, defined for u = (ui )i∈M , U = (Ui )i∈χ(M) , j ∈ M and s ∈ 1, 2, . . . , lj by  [Su]χ(j)+s = Uχ(j)+s (y) = uj

| l −s j

lj

,

 l −s+1 j

lj

 lj − s + y , for y ∈ [0, 1] , lj

(19)

is an isomorphism with the inverse S−1 : X → X given by [S−1 U]j = uj (x) = Uχ(j)+s (lj x + s − lj ),

 for x ∈

 lj − s l j − s + 1 , lj lj

(20)

Furthermore, etA0 u = S−1 eτ A0 Su,

(21) |χ(M)|

where (eτ A0 )τ ≥0 , τ = ct, is the semigroup on X = L1 ([0, 1], l1

) solving

∂τ U(x, τ ) + ∂x U(x, τ ) = 0, x ∈ (0, 1), τ ≥ 0, ˚ U(x, 0) = U(x), U(0, τ ) = KU(1, τ ),

(22)

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where K incorporates the continuity condition at the new vertices of G and is given by ⎛ ⎞ ⎛ ⎞ T1 0 0 . . . 0 . . . K11 . . . K1j . . . ⎜ .. . . .. .. .. .. ⎟ ⎜ .. .. . . .. ⎟ ⎜ . . . . . . ⎟ ⎜ . . . . ⎟ ⎟ ⎜ ⎟ K=⎜ (23) ⎜ 0 . . . Tj . . . 0 . . . ⎟ + ⎜ Kj1 . . . Kjj . . . ⎟ , ⎝ ⎠ ⎝ ⎠ .. .. .. . . .. .. .. .. .. . . . . . ... . . . . where Tj is either the scalar 1, or a cyclic matrix, and Kij a matrix, and they are given by ⎛ ⎞ ⎛ ⎞ 0 0 ... 0 1 0 0 . . . 0 kij ⎜1 0 ... 0 0⎟ ⎜0 0 ... 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ Kij = ⎜ . . . . ⎟ , (24) Tj = ⎜ . . . . ⎟ , ⎝ .. .. .. .. 0 ⎠ ⎝ .. .. .. .. 0 ⎠ 0 0 ... 1 0

0 0 ... 0 0

where kij are defined under (8). The motivation behind (21) is that, specifying (14) to the case Q = 0, for any U ∈ X we have [etA0 U](x) = [Kn U](t + x − r), n ∈ N0 , 0 ≤ t + x − r ≤ 1.

(25)

Thus the representation (25) reduces the analysis of (etA0 )t≥0 , and thus of (etA0 )t≥0 , to that of the iterates of K. It is worthwhile to observe that if Q = 0 is not a diagonal matrix, then the above procedure leads to a delay problem, see [38, Example 1].

Fig. 10 Blowing-up of G

G ek ej

vi

G ek,2 ej,1

ej,2

ej,3

ek,1

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2.3 Dynamics at the Nodes—Population Equation with Age and Space Structure In the next example we illustrate the micromodel shown in Fig. 3 where, however, we ignore the dynamics occurring along the pathways connecting the states. Following e.g. [10, 33, 40], we consider a population of individuals structured by age a, divided into N patches. We assume that no patch is isolated. We denote by ni the population density in patch i. By rij we denote the migration rate from patch j to patch i, j = i. The transition matrix R(a) = (rij (a))1≤i,j≤N for each a ≥ 0 is a Kolmogorov transition matrix (of a time-continuous process); that is, the off-diagonal entries are positive and columns sum up to 0. In other words, rii (a) = −

N 

rji (a), a ≥ 0.

j=1

j =i

We also introduce a matrix M(a) = (μij (a))1≤i,j≤N that describes the mortality and possible additional intercohort phenomena. We assume that −M is a subKolmogorov matrix; that is, its columns sum up to non-positive numbers, −

N 

μji (a) =: −μi (a) ≤ 0,

(26)

j=1

where μi (a) is the age specific death rate in the patch i. Thus, the vector n(a, t) = (n1 (a, t), . . . , nN (a, t)) satisfies ∂t n1 (a, t) = −∂a n1 (a, t) −

N 

μ1i (a)ni (a, t)

i=1

+(r11 (a)n1 (a, t) + . . . + r1N (a)nN (a, t)),

.. .. .. . . .,

(27)

∂t nN (a, t) = −∂a nN (a, t) −

N 

μNi (a)ni (a, t)

i=1

+(rN 1 (a)n1 (a, t) + · · · + rNN (a)nN (a, t)). Further, we introduce the McKendrick–Von Foerster boundary conditions: nj (t, 0) =

∞  N 0

i=1

βji (a)ni (a, t)da, 1 ≤ j ≤ N ,

(28)

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where B(a) := (βij (a))1≤i,j≤N describes the age and patch specific fertility rates, and the initial distribution of the population n1 (0, a) = n˚ 1 (a), . . . , nN (0, a) = n˚ N (a).

(29)

We observe that (27)–(29) can be written in a form similar to (8), ∂t n + ∂a n = −Mn + Rn, x ∈ (0, ∞), n(0, t) = Bn(·, t), ˚ n(a, 0) = n(a), where Bn :=

2.3.1

∞ 0

(30)

B(a)n(a)da.

Solvability

Similarly to the previous model, the basic space is X := L1 ([0, ∞), l1N ) and we let Y = W11 ([0, ∞), l1N ). Further, let | · | be the norm in the l1N . By · we denote the norm on X. For any other norm in a Banach space U we shall write · U . By B(Z, U) we denote the space of linear bounded operators from Z to U equipped with the standard operator norm ||| · |||B(Z,U) (with short notation ||| · ||| if it does not create any misunderstanding). We will keep the notation for the operators in X generated by the pointwise multiplication of elements of X by the matrices M(a), R(a), . . .; these operators are bounded in X. Solvability of (27)–(29) in the space X is standard and can be derived from the general integral equation formulation, or by the classical semigroup approach. We recall the basic facts. First we note that (27)–(29) involves, as in the previous model, two operators: a differential operator on (0, ∞) and a boundary operator. Thus it is advantageous to separately consider the differential expression A := S − M + R = (−∂a nj )1≤j≤N − M + R,

(31)

defined on Y, and the operator A, given by the restriction of A to the domain D(A) := {u ∈ Y; u(0) = Bu}. We assume that the entries of M(a), R(a) and B(a) satisfy μij , rij and βij ∈ L∞ ([0, ∞)), βij (a) ≥ 0. Denote b¯ := ess sup |||B(a)|||B(l1N ) . Further, the matrix −M a∈[0,∞)

can be decomposed into a Kolmogorov part and the diagonal part diag(−μj )1≤j≤N , N  where μj (a) = μjj (a) + μij (a); these are the ‘true’ death rates, see (26). We denote i=1

m :=

i =j

inf

1≤j≤N ,a∈[0,∞)

μj (a). Then we have

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Theorem 2.11 [10, 25]. The operator A generates in X a strongly continuous semi¯ group (etA )t≥0 which satisfies |||etA ||| ≤ e( b−m )t . We note that, in particular, |||etA ||| is independent of R. This observation will become important below.

3 Asymptotic State Lumping According to the discussion in Introduction, models such as (8) or (30) can be considered as micro-models for (1). It is not clear, however, whether they are consistent with (6), as defined in paragraph C of Introduction. Even more, at this moment we do not know in what way we should aggregate the solutions of, respectively, (8) and (30) to make them comparable with solutions of an appropriate version of (1). Also, as discussed in Introduction, such micro-models, be they built from first principles, or considered as enhancements of macro-model (1), often are exceedingly complex, to the extent of not allowing for any robust analysis. Moreover, the full information they yield is often redundant for many applications. Thus, there is a tendency to use in practice macro-models that are, however, backed up by more detailed micromodels should a need for a more precise description arise. Therefore, as mentioned earlier, we should find a systematic way of checking whether a given micro-model is consistent with the original macro-model in the sense that they both give the same macroscopic description of the process they model. We also can ask a more general question: how to construct a consistent macro-model from a given micro-model, making thus the diagram in Fig. 4 commutative (at least approximately).

3.1 Some Mathematical Tidbits We noted in Introduction that one of the most efficient ways of aggregating complex systems is to exploit their multiscale structure. In many cases, when there are just two characteristic time scales, the micro-models discussed above can be cast into the form of a singularly perturbed Cauchy problems ∂t u = K u ,

t > 0, u (0) = u˚ ∈ X ,

(32)

where {K }>0 is a family of generators of C0 -semigroups (etK )t≥0 in a Banach space X , that corresponds the micro-models. The problem is to determine the limit equation, corresponding to the macro-model, satisfied by u = lim→0 u . We shall briefly recall the theory that can be used to approach it. Let X be a Banach space and (etK )t≥0 ,  > 0, be a family of strongly continuous semigroups in X . We assume these semigroups to be equibounded: there exists an M > 0 such that |||tK ||| ≤ M for all t ≥ 0 and  > 0. For λ > 0 and  > 0, let

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Rλ, f := (λI − K )−1 f , f ∈ X , be the resolvent of K . Before we proceed further, we mention that most convergence results in the theory are formulated for sequences hence, strictly speaking, we should operate with sequences (n )n∈N satisfying n → 0 as n → ∞. Since, however, in the applications discussed here the limits mostly do not depend on the choice of a particular sequence (n )n∈N , we will write  → 0 for simplicity. Let X1 be the space of those f ∈ X such that the limit lim→0+ Rλ, f = R(λ)f exists for all λ > 0; equiboundedness implies that X1 is closed. Next, let X0 be the space of f such that lim→0+ etK f exists and is almost uniform in [0, ∞). The space X0 , often referred to as the regularity space, is also closed and satisfies X0 ⊂ X1 , [16]. The classical Trotter–Kato theorem, [37], states that X0 = X if and only if (a) X1 = X and (b) for some λ > 0 (and then for all λ > 0) the range of the operator R(λ) is dense in X . Then there exists an operator K generating a strongly continuous semigroup (etK )t≥0 such that etK f = lim→0+ etK f , f ∈ X . It often happens that condition (a) is satisfied but (b) fails to hold. An easy modification of the proof of the Trotter–Kato theorem, [16, Theorem 8.4.3], shows that then X0 is the closure of the range of R(λ), X0 = RangeR(λ) and there is an operator K in X0 that generates a strongly continuous limit semigroup (etK )t≥0 in X0 with |||etK || ≤ M and R(λ)|X0 = (λI − K)−1 . For f ∈ / X0 , the limit lim→0+ etK f may not exist [16, 18]. As mentioned before, the convergence for f ∈ X0 is called regular and, by definition, is almost uniform in [0, ∞). The convergence outside of X0 is referred to as irregular, or degenerate; this convergence is almost uniform on (0, ∞), [4, 18]. Though the convergence of resolvents usually is easier to handle than that of semigroups, proving it is still a non-trivial task. The Sova-Kurtz version of the TrotterKato theorem [16, 41] allows us to establish the convergence of resolvents without actually calculating them, by using the notion of the extended limit Kex of generators K ,  > 0. By definition, the extended limit Kex of (K )>0 is a relation in X × X : a pair (f , g) belongs to Kex if and only if there exist f ∈ D(K ) such that lim→0 f = f , and lim→0 K f = g. The domain D(Kex ) of Kex is defined to be the set of all f ∈ X such that there exists a g ∈ X with (f , g) ∈ Kex ; by Kex f = g we mean that (f , g) ∈ Kex . It follows, [16, Theorem 8.4.9], that X1 = X iff the range of λI − Kex is dense in X for some (hence, all) λ > 0 and, if either condition is met, then X0 = D(Kex ), the part of Kex in X0 is single valued and it is the generator of the limit semigroup. The above result will be referred to as the Sova-Kurtz theorem. The Sova-Kurtz theorem, if applicable, gives the regular convergence; that is, the convergence of the family (etK )>0 to a semigroup in X0 . Irregular convergence for a

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large class of problems can be handled by an extension of the above result provided by Kurtz, see [21, Theorem 7.6] or [18, Theorem 33.1]. As before, we consider a family of equibounded semigroups (etK )t≥0 and the generator C of a C0 −semigroup (etC )t≥0 which is assumed to satisfy lim etC f =: Pf ,

t→∞

f ∈ X.

(33)

Then it follows that P is a projection onto the kernel Ker C of C along Range C. Furthermore, if f ∈ X, (34) lim etK f = etC f , →0

then lim etK f = 0,

→0

f ∈ Ker P, t > 0,

(35)

providing the degenerate convergence on Ker P. The regular convergence that occurs on Ker C = Range P must be handled separately and the relevant result is given below. Denote X  = Ker C = Range P; let K be an operator in X  and D ⊂ D(K). Theorem 3.12 Assume that: (a) for f ∈ D, (f , Kf ) ∈ Kex ; (b) for g in a core D of C we have (g, Cg) ∈ Lex , where Lex is the extended limit of K ; (c) the operator PK with domain D ∩ X  is closable and its closure PK generates a strongly continuous semigroup in X  . Then lim etK f = etPK Pf , f ∈ X ,

→0

(36)

almost uniformly in t ∈ (0, ∞). For the proof we refer the reader to [18, Theorem 33.1]. It follows that X  = X0 so that X  coincides with the regularity space. Therefore the convergence in (36) is almost uniform on [0, ∞), provided f ∈ X0 . Example 3.13 In many applications we have 1 K = S + C,  where S and C are operators on X . Then K = S + C. Formally, if lim→0 S = 0 and lim etC f = Pf , t→∞

then X0 = RangeP is the space of the regular convergence and there exists a semigroup (etK )t≥0 on X0 such that

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lim etK f = etK Pf , f ∈ X ,

→0

almost uniformly on each (0, ∞); the convergence is almost uniform on [0, ∞) if f ∈ X0 . In most relevant cases, P is the spectral projection onto the null space of C. Thus, in particular, for a nontrivial limit dynamics, the equation Cf = 0 must have non-trivial solutions.

3.1.1

Asymptotic Expansion

The Kurtz theory is closely intertwined with the asymptotic expansion method of the Chapman–Enskog type, see [4]. To explain, let us assume, at a purely formal level, that K = C + S + O(2 ) so that the assumptions of the Kurtz theory can be assumed to hold. Then, making the formal expansion u (t) = u0 (t) + u1 (t) + · · ·

(37)

in (32), we obtain ∂t u0 = C(u0 + u1 ) + Su0 + O(2 ). Comparing terms multiplying like powers of , at the 0th level we obtain Cu0 = 0.

(38)

As noted above, we assume that (38) has nontrivial solutions, the space of which we denote by V (this space is sometimes called the hydrodynamic space due to the origin of the method, [36]). We note that if λ = 0 is an isolated eigenvalue of C, then V = PX , where P is the spectral projection of C corresponding to λ = 0, in accordance with the Kurtz theory. Now, contrary to standard asymptotic methods, first we split X = V ⊕ W , where W is the complementary space and write u (t) = (v(t), w (t)) = (v(t), w0 (t) + w1 (t) + · · · ), where v = Pu and w = Qu = (I − P)u . Then, operating with P and Q on (32) and using the fact that PC = CP = 0, we obtain ∂t v = PSv + PSw0 + O(2 ), ∂t w0 = QCw0 + QSv + QSw0 + O(2 ).

(39)

Now, comparing the terms multiplying the same powers in  in the second equation, we find w0 = 0 and hence at the 0th level we obtain the limit equation ∂t v = PSv.

(40)

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If we recognize that by taking u = v in the definition of Kex we obtain K v = Sv, so that (v, Sv) ∈ Kex , then S is a good candidate for the limit operator K of Theorem 3.12, with (40) corresponding to (36). Certainly, as stated above, this result is purely formal but it can be used as a guideline for constructing Kex and K of Theorem 3.12, see Lemma 3.24, and in many cases it can be proved with full mathematical rigour, as in Theorem 3.25. It also should be noted that the approximation, constructed above, often is insufficient to provide a good approximation of the solution to (32) especially close to t = 0 and close to spatial boundaries if such are involved. Thus there is a need to augment the expansion with various correctors such as the initial or boundary layers, see [10, 11, 36]. We shall briefly return to this topic in Example 3.26.

3.2 Back to the Models for the Spread of a Genotype in a Population Let us recall the Kimmel-Stivers macro-model without vital dynamics, v = Kv,

(41)

where v = (vi )i∈N gives the numbers of cells with the genotype i and K is the matrix of coefficients of (6), and a simplified Lebovitz–Rubinow–Rotenberg micro-model (7) with the cells’ vital dynamics, ∂t u(x, t) + C∂x u(x, t) = 0, ˚ u(x, 0) = u(x), u(0, t) = Ku(1, t),

(42)

where u(x) = (ui (x))i∈M is the density of cells with genotype i at age x. Since (42) and (41) describe the same phenomenon, the solutions should give the same answer at the level at which they are comparable. Here, v(t) gives the total amount of cells in each class, hence we compare it with the aggregated, or lumped, solution to (42) 1 [PM u](t) = 0

⎛ 1 ⎞  1 u(x, t)dx = ⎝ u1 (x, t)dx, . . . , un (x, t)dx, . . .⎠ . 0

(43)

0

Since (41) does not explicitly contain the ageing process, it is natural to assume that it could provide an approximate solution to (42) if the maturation is practically instantaneous. In other words, we have to assume that the cells divide many times in the chosen reference unit of time. As we shall see in the example below, this assumption is insufficient.

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Example 3.14 For simplicity we consider a finite dimensional problem 1 ˚ ∂t u + ∂x u = 0, u (0, t) = Ku (1, t), u (x, 0) = u(x), 

(44)

where K is assumed to be an irreducible and column stochastic m × m matrix. Following Sect. 3.1.1, we see that the hydrodynamic space V consists of solutions to ∂x v = 0, v(0, t) = Kv(1, t); that is, v(x, t) = v(t) = v(t)N, where N is the normalised Perron eigenvector of K and v is a scalar coefficient. Then the spectral 1 projection onto V is given by v = Pu = 1 · 0 u(x)dx N, where 1 = (1, . . . , 1) is the left eigenvector of K so that 1 · N = 1. It satisfies ∂t v =

1 1 1 P∂x u = ((u (1, t) − u (0, t)) · 1) N = u (1, t) · (I − K T )1 = 0.   

Hence, the projection of (44) onto V is exactly the limit equation and v = ρN, m  1 ˚ i (x)dx is the initial mass. Thus the possible limit solution where ρ = i=1 0 u is the stationary distribution of the initial total mass on the network, weighted by the Perron eigenvector. Hence, it provides neither a simplification of the original problem, nor any interesting approximate dynamics. ♦ It turns out that in the Lebovitz–Rubinow–Rotenberg model an asymptotic state lumping can be done if we assume that, in addition to fast maturation of cells, we assume that the daughter cells have a tendency to be of the same genotype as the mother. Thus, we shall consider the problem ∂t u (x, t) + −1 C∂x u (x, t) = 0, ˚ u (0, t) = (I + B)u (1, t). u (x, 0) = u(x),

(45)

As before, we try the formal asymptotic expansion. Formal steps are similar to that in Example 3.14. The hydrodynamic space V consists of solutions to ∂x u = 0,

u(0, t) = u(1, t),

|M|

(46) |M|

so that V = l1 is the subspace of functions of X = L1 ([0, 1], l1 ) that are independent of x. Clearly, the formally adjoint problem to (46) is given by ∂x φ = 0, φ(0, t) = φ(1, t), and thus the projection onto V is given by (43). Then we have u = PM u + QM u = 1 v + w , where QM = I − PM so that w ∈ W = {w ∈ X; 0 w(x)dx = 0}. Projecting (45) onto V and W, we get ∂t v (t) = CBu(1, t) = CB(v (t) + w (1, t)), v (0) = PM u˚

(47)

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and 1 ∂t w (x, t) = − C∂x w (x, t) − CBv (t) − CBw (1, t),  w (0, t) = w (1, t) + Bw (1, t) + Bv (t), ˚ ˚ w (x, 0) = u(x) − PM u.

(48)

Substituting the expansion w (x, t) = w0 (x, t) + w1 (x, t) + O(2 ) into (48), as in (39) we find that w0 = 0. Thus the limit problem for v is given by ˚ ∂t v(t) = CBv(t), v(0) = PM u;

(49)

˚ Hence in this case we can have a nontrivial limit dynamics. that is, v(t) = etCB PM u. It turns out, however, that the question of the convergence is quite complex and what we can prove is only the regular convergence. Precisely, we have Theorem 3.15 [6] (i) The semigroup (etA )t≥0 solving (45) is equibounded; (ii) if λ > cmin |||B|||, then lim R(λ, A ) = R(λ, CB)PM ;

→0+ |M|

(iii) If u˚ ∈ l1 (that is, the initial condition is independent of x), then |M|

lim etA u˚ = etCB u˚ in L1 ([0, 1], l1 ).

→0+

If, however, the average lifespans τn are integer multiples of some τ (= 1), then we can prove a stronger version of this result. By ‘blowing-up’ the network, see (22), we convert (45) into ∂t υ  (x, t) + −1 ∂x υ  (x, t) = 0, x ∈ (0, 1), t ≥ 0, υ  (x, 0) = υ(x), ˚

(50)

υ  (0, t) = (T + C)υ  (1, t). |χ(M)|

There is a semigroup (etA )t≥0 on X = L1 ([0, 1], l1 for any υ˚ ∈ X it is given by

) solving (50) and, as in (25),

  t t , n ∈ N, 0 ≤ n + x − < 1. ˚ = (T + C)n υ˚ n + x − [etA υ](x)   The matrix T + C has been constructed as in (23), (24), hence T is periodic and thus

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1 = l −1

l−1 

Ti

i=0 |χ(M)|

defines the ‘return’ projection from l1 to the eigenvalue 1. Then

|M|

to the original space l1 , corresponding

1 PM u =

1 u(x)dx = 1 Pχ(M) υ = 1

0

υ(s)ds 0

and we have |M|

Theorem 3.16 [6] For any u˚ ∈ L1 ([0, 1], l1 ) we have ˚ lim PM etA u˚ = etCB PM u.

→0+

(51)

In general, (etA )t≥0 do not converge on x−dependent initial values as demonstrated in the example below. Example 3.17 Consider the scalar problem ∂t u (x, t) + −1 ∂x u (x, t) = 0, x ∈ (0, 1), t ≥ 0, u (x, 0) = u˚ (x), u (0, t) = (1 + b)u (1, t), where b ∈ R. It is easy to see that if u (x, t) = [etA u˚ ](x), then for t satisfying n ≤ t < (n + 1), so that n = t/, we have  u (x, t) = Then

  t (1 + b)  +1u˚ x +  t  +1 − t for 0 ≤ x ≤ t −  t , t (1 + b)   u˚ x +  t  − t for t −  t  ≤ x ≤ 1.

(52)

t/b 1 = ebt , lim+ (1 + b)n = lim+ (1 + b) b

→0

→0

where we used lim+

→0

  t  = t. 

The above is obvious for t = 0 and for t > 0 it follows from   t  n ≤ ≤1 n+1  t and n → ∞ with  → 0.

(53)

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Let us consider t = 1. Then for  = 1/k we obtain   1 k u k1 (x, 1) = 1 + b u˚ (x), 0 ≤ x ≤ 1, k while for  =

2 2k+1

we have 

2 (x, 1) = u 2k+1

(1 +  1+

  x + 21 for 0 ≤ x < 21 ,  − 21 for 21 ≤ x ≤ 1.

2 b)k+1 u˚ 2k+1  k  2 b u˚ x 2k+1

From this it follows that lim e

k→∞

and lim e

k→∞

A

A1 k

u˚ = eb u˚

2 2k+1

u˚ = eb v˚

in L1 ([0, 1], where v(x) ˚ = u˚ (x + 1/2) for x ∈ [0, 1/2) and v(x) ˚ = u˚ (x − 1/2) for x ∈ [1/2, 1]. Thus (etA )t≥0 does not converge as  → 0+ for all initial conditions. However, it is easy to see that lim etA u˚ = ebt u˚ ,

→0+

provided u˚ is a constant, in accordance with Theorem 3.15. We apply Theorem 3.16 to link the Kimmel–Stivers model with the Lebovitz– Rubinow–Rotenberg model. If u (x, t) = (u0, (x, t), . . . , un, (x, t), . . .) is the solution to ∂t un + −1 ∂x un = 0, x ∈ (0, 1), t > 0, un (x, 0) = u˚ n (x), with the boundary conditions u0 (0, t) = u0 (1, t) + (a0 u0 (1, t) + d1 u1 (1, t)), u1 (0, t) = u1 (1, t) + (a1 u1 (1, t) + d2 u2 (1, t)), un (0, t) = un (1, t) + (an un (1, t) + bn−1 un−1 (1, t) + dn+1 un+1 (1, t)), 1 n ≥ 2, then 0 u (x, t)dx converges to the solution of (6) (with the initial condition 1 ˚ v˚ = 0 u(x)dx). The interpretation of the boundary operator I + B is that at the mitosis one daughter cell always has the genotype of its mother, while the probability of mutations described in the Kimmel–Stivers model is very low.

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3.3 Extension to Boundary Matrices of the Form K + B. Comparing the (negative) result of Example 3.14 and the main results of Theorems 3.15 and 3.16, we may notice that the rich asymptotic dynamics comes from the fact that the boundary matrix K = I + B at  = 0 was totally reducible – the identity matrix does not describe any communication between the states. Because of this the results of Theorems 3.15 and 3.16 cannot be used for network flows. Further, dealing with a more involved problem (50) we noted an essential role played by the spectral projection 1 corresponding to the eigenvalue 1. Following these observations, we consider the extension of (50) to more general boundary conditions, ⎧ ⎨ ∂t u(x, t) = − 1 ∂x u(x, t) + Qu(x, t), x ∈ (0, 1), t > 0, ˚ u(x, 0) = u(x), x ∈ [0, 1], ⎩ u(0, t) = (K + B()) u(1, t), t ≥ 0,

(54)

|M|

in X = L1 ([0, 1], l1 ). By re-writing (14) for the case at hand, we obtain that the solution to (54) is given by ˚ =e u (x, t) = [e u](x) tA

xQ

    t Q n −(n+x− t )Q , (55) (K + B())e e u˚ n + x − 

! where n = t − x + 1. We assume that  → B() is continuous and satisfies B(0) = B. Further, we assume K is contractive and 1 is an isolated and semisimple eigenvalue of K.

(56)

This assumption in particular covers the case of general stochastic matrix describing the redistribution of material at the nodes that was considered in the network transport problem (5); B may describe e.g. the loss of material during the redistribution at the nodes and does not have to be stochastic. Let 1 be the spectral projection onto the eigenspace of K belonging to 1. Further, for any operator O, let O11 = 1 O1 . To provide some insight into this result and, in particular, to understand the meaning of the projection 1 , let us consider a simplified model ⎧ ⎨ ∂t u(x, t) = − 1 ∂x u(x, t), x ∈ (0, 1), t > 0, ˚ u(x, 0) = u(x), x ∈ [0, 1], ⎩ u(0, t) = (K + B) u(1, t), t ≥ 0. Then (55) gives   t t ˚ , n ∈ N, 0 ≤ n + x − < 1. = (K + B) u˚ n + x − [e u](x)   tA

n

(57)

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If we wanted to use the Kurtz approach, then formula (34) would give ˚ ˚ = K n u˚ (n + x − t) = [etA u](x), lim [etA u](x)

→0

for n =

t 

! − x + 1; that is, we would get the semigroup (etA )t≥0 solving ⎧ ⎨ ∂t u(x, t) = −∂x u(x, t), x ∈ (0, 1), t > 0, ˚ u(x, 0) = u(x), x ∈ [0, 1], ⎩ u(0, t) = Ku(1, t), t ≥ 0.

(58)

Then the projection P of (34) would be given by lim etA = P,

t→∞

(59)

provided it existed. Let us consider this question.

3.3.1

Interlude—The Long Time Behaviour of (etA )t≥0

The long term behaviour of (58), in both finite and infinite dimensional cases, has been analysed in a series of papers [3, 12, 20, 30]. The approach of [3, 12] is particularly useful for our purpose as it gives us the representation 

˚ = [etA u](x)

˚ + x − t))Vj + O(αt ) λnj (Wj · u(n

j,λj ∈σper (K)

=



j,λj =1

+

˚ + x − t))Vj (Wj · u(n 

˚ + x − t))Vj + O(αt ) λnj (Wj · u(n

j,λj ∈σper (K)\{1}

˚ = [1 etA u](x)  ˚ + x − t))Vj + O(αt ), + λnj (Wj · u(n

(60)

j,λj ∈σper (K)\{1}

where 0 ≤ n + x − t < 1, n ∈ N0 , α < 1, Wj and Vj are, respectively, the left and right eigenvectors belonging to the eigenvalues from the peripheral spectrum of K, denoted here by σper (K). We shall provide a more explicit form of this expansion that also plays an important role in the analysis of (57), even if the assumptions of the Kurtz theorem are not satisfied. As we noted above, in most applications the matrix K describes the redistribution of the material at the nodes of the network and thus it is column stochastic. Henceforth we adopt this assumption and, to simplify the considerations, we assume M = {1, . . . , m}. Otherwise, we would have to introduce additional compactness

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assumptions, see [20]. Then, [22, 35], by a simultaneous permutation of rows and columns K can be put in the so-called normal form ⎛

0 ⎜ K 2,1 ⎜ ⎜ .. ⎜. ⎜ ⎜ K g−1,1 K=⎜ ⎜ K g,1 ⎜ ⎜ K g+1,1 ⎜ ⎜. ⎝ .. K s,1

0 K2 .. .

··· ··· ··· .. .

0 0 .. .

... ... .. .

0 0 .. .

0 0 .. .

K g−1,g−2 K g,g−2 K g+1,g−2 .. .

K g−1 K g,g−1 K g+1,g−1 .. .

0 Kg 0 .. .

0 0 K g+1 .. .

K s,g−1

0 0

· · · K s,g−2

⎞ ... 0 ... 0 ⎟ ⎟ ⎟ .. ⎟ . ⎟ ... 0 ⎟ ⎟. ... 0 ⎟ ⎟ ... 0 ⎟ ⎟ . . .. ⎟ .. ⎠

(61)

· · · Ks

If K is an adjacency matrix of a graph (or a line graph), then this structure corresponds to the so-called topological sorting of the vertices of the graph in which the lowest indices correspond to its acyclic part, then to the transient part consisting of strong components with outgoing pathes and, finally, the highest indices correspond to the ergodic part; that is, the strong components with no outgoing pathes, [14]. Then, [3, 12], • the zeros, occupying the first block row, correspond to the sources in the graph; • The square nonzero irreducible matrices K 2 , . . . , K g−1 correspond to the transient strong components of the graph; • The square irreducible matrices K r with r ≥ g correspond to the ergodic part. Since K is column stochastic, we can use the whole theory of reducible Markov chains, see e.g. [35, Sect. 8.4]. In particular, 1 = (1, . . . , 1) is a (positive) left Perron eigenvector, the spectral radii satisfy ρ(K l ) < 1, 2 ≤ l ≤ g − 1 and ρ(K l ) = 1, g ≤ l ≤ s. Further, λ = 1 is always a semisimple eigenvalue of K. To proceed, first we simplify the notation by writing ⎛

K0 ⎜ Kg ⎜ K = ⎜. ⎝ ..

0 Kg .. .

... ··· .. .

0 0 .. .

⎞ ⎟ ⎟ ⎟, ⎠

(62)

Ks 0 · · · K s

where Kr = (K r,1 , . . . K r,g−1 ) for r = g, . . . , s. Due to the block structure of K, we find it useful to represent a vector x ∈ Rm as x = (x0 , xg , . . . , xs ). Then the eigenvalue problem for the Perron eigenvectors of K can be written as K0 v0 = v0 , Kr v0 + K r vr = vr , r = g, . . . , s,

Transport on Networks—A Playground of Continuous …

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and w0 K0 + wg Kg + · · · + ws Ks = w0 , wr K r = wr , r = g, . . . , s. Since ρ(K0 ) < 1, v0 = 0 and hence the right Perron eigenspace is spanned by the eigenvectors of the form Vr = (0, . . . , vr , . . . , 0), r = g, . . . , s,

(63)

where vr is the right Perron eigenvector of K r . The left Perron eigenspace is spanned by Wr = ((1r Kr )(I − K0 )−1 , 0, . . . , 1r , . . . , 0), r = g, . . . , s, where wr = 1r = (1, . . . , 1). The spectral projection 1 is thus given by 1 x =

s s   (Wr · x)Vr = ((1r Kr )(I − K0 )−1 · x0 + 1r · xr )Vr . r=g

(64)

r=g

From the above, we see that the terms corresponding to different ergodic components K r , r = g, . . . , s, can be calculated independently of each other. Hence, to give a more explicit and ‘applied’ interpretation of this formula, it is sufficient to consider K of the form   K0 0 , (65) K= Kr K r r = g, . . . , s. Then, as we know, the semigroup solving (58) is given by ˚ = K n u˚ (n + x − t) , for n ∈ N, 0 ≤ n + x − t < 1 [etA u](x) and thus, as also observed in (60), the long time behaviour of (etA )t≥0 depends on whether 1 is the only eigenvalue on the unit circle, or not; that is, whether K r is primitive or imprimitive. If K r is primitive, then 1 is a strictly dominant eigenvalue with the corresponding strictly positive eigenvector v normalized so that 1 · v = 1. Then, as above, V = (0, v) is an eigenvector of K corresponding to the (dominant) eigenvalue 1 of K and the corresponding left Perron eigenvector is given by W = ((1r Kr )(I − K0 )−1 , 1r ). The situation changes when K r is imprimitive. Then, by the Perron-Frobenius theory, apart from the eigenvalue 1, for some d ≥ 2 there are d − 1 other eigenvalues on the unit circle, given by 2πi λj = e d j , j = 1, . . . , d − 1.

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Good news is that they are all simple eigenvalues. Denote, respectively, by vj and wj the right and left eigenvectors of K r belonging to λj and by Wj = (w1 , w2 ) = ((wj Kr )(λj I − K0 )−1 , wj ) j

Vj = (0, vj ),

j

the corresponding eigenvectors of K. We assume that the eigenvectors are normalized so that Wj · Vj = 1. Clearly, again W0 = W = ((1r Kr )(I − K0 )−1 , 1r ). Then K n x = (W · x)V + O(αn ) if K r is primitive and K n x = (W · X)V + e + O(αn ),

2πi d n

(W1 · X)V1 + · · · + e

2πi d (d −1)n

(Wd −1 · X)Vd −1

if K r is imprimitive, where 0 < α < 1. Then the expansion (60) for large t can be fine-tuned as follows: ˚ ˚ + x − t))V ≈ (W · u(n [etA u](x) for K r primitive and ˚ ˚ + x − t))V + e ≈ (W · u(n [etA u](x) +e

2πi d (d −1)n

2πi d n

˚ + x − t))V1 + · · · (W1 · u(n

˚ + x − t))Vd −1 , (Wd −1 · u(n

0 ≤ n + x − t < 1, n ∈ N0 , if K r is imprimitive. The expansion in the general case (62) is then obtained by superposition. Example 3.18 Consider (58) with ⎛

0 ⎜0 ⎜ ⎜1 ⎜ ⎜0 K=⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0

0 0 0 1 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 0.5 0 0.5

0 0 0 0 0 0 1 0

0 0 0 0 1 0 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟. 1⎟ ⎟ 0⎟ ⎟ 0⎠ 0

(66)

Transport on Networks—A Playground of Continuous …

471

The characteristic equation of the 4 × 4 ergodic part K 1 is λ(λ3 − 0.5λ − 0.5) = 0 with eigenvalues 1, 0, −0.5 ± 0.5i and thus it is primitive. We have ⎛

(I − K0 )−1

1 ⎜0 =⎜ ⎝1 1

0 1 0 1

0 0 1 1

⎞ 0 0⎟ ⎟, 0⎠ 1

(11 K1 )(I − K0 )−1 = 1.

The Perron eigenvector of K 1 is given by v=

1 (2, 1, 1, 1) 5

and the flow for large t is asymptotically given by ˚ [etA∞ u](x) = u˚ ∞ (n + x − t)(0, v) =

8 

u˚ r (n + x − t)(0, v),

r=1

where 0 ≤ n + x − t < 1, n ∈ N0 .



Hence, even if 1 is the dominating eigenvalue of K, the long time behaviour of (etA )t≥0 is not independent of t and thus it is not a projection, as required by (33). Thus, while we still can have a nontrivial regular convergence, we do not expect the degenerate one in the sense of (35). The theorem below makes this observation precise. Theorem 3.19 [13] (a) Let (56) be satisfied. Then lim etA u˚ = et(B11 +Q11 ) u˚

→0+

(67)

for any u˚ ∈ 1 PM X. Moreover lim 1 PM etA u˚ = et(B11 +Q11 ) 1 PM u˚

→0+

(68)

for any u˚ ∈ X. In both cases the convergence is almost uniform on [0, ∞). (b) If, additionally, sup{|λ| : λ ∈ σ(K), λ = 1} < 1, then lim etA u˚ = et(B11 +Q11 ) 1 u˚

→0+

(69)

for any u˚ ∈ PM X and t > 0 and the convergence is almost uniform on (0, ∞). We observe that in the theorem above we use two projections. The projection PM has a clear physical meaning – it aggregates the mass over the edges giving the vector

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describing how much of the material occupies each edge. The spectral projection 1 does not have an obvious physical significance and our aim now is to explain its meaning and to provide an interpretation of the convergence results (67) and (68). We shall focus on the simplified case given by (65) with primitive K r . We imagine individuals moving along the edges of a graph, that either belong to the transient component indexed by 0, or to the ergodic component indexed by r. The individuals could migrate from states 0 to states r with the rates of migration modelled by K r . Then, as in (64), for u˚ = (u˚ 0 , u˚ r ) ˚ = ((1r Kr )(I − K0 )−1 · u˚ 0 + 1r · u˚ r )V 1 u˚ = (W · u)V = (1r · Kr (I − K0 )−1 u˚ 0 + 1r · u˚ r )V.

(70)

˚ The interpretation of the right hand side is that whatever is the initial distribution u, due to the structure of V, see (63), 1 u˚ is nonzero only at the ergodic component of the underlying graph. However, the magnitude of 1 u˚ is determined also by the initial population from the transient parts. Clearly, the term 1r · u˚ r is the standard term containing the contribution of the initial population in the ergodic component. The first term can be understood by noting that K n u˚ solves the recurrence x0 (n + 1) = K0 x0 (n), x0 (0) = u˚ 0 , xr (n + 1) = K r xr (n) + Kr x0 (n), xr (0) = u˚ r . First, since x0 (n) = K0n u˚ 0 , ∞  n=0

x0 (n) =

∞ 

K n0 u˚ 0 = (I − K 0 )−1 u˚ 0 ,

n=0

is the structured total number of ‘man-days’ in the transient states resulting from the original cohort u˚ 0 born there. To make it more clear, we note that if K0 = k0 is a scalar, it can be interpreted as the survival probability over the unit time. Then μ = 1 − k0 is the death probability and hence (1 − K0 )−1 = 1/μ is the average lifespan of an individual. Thus u˚ /μ is the number of ‘man-days’ expected from a cohort of the initial size u˚ . Thus xr (n) satisfies xr (n + 1) = K r xr (n) + Kr K0n u˚ 0 , xr (0) = u˚ r , whose solution is xr (n) = K nr u˚ r +

n−1 

K lr Kr K0n−l−1 u˚ 0 .

(71)

l=0

The first part describes the redistribution of the native inhabitants of the ergodic component r. If we look at the second part term by term, we see that it describes

Transport on Networks—A Playground of Continuous …

473

˚ 0 ; that is, those who were born the immigrants to r: for l = n − 1 we have K n−1 r Kr u in the state 0, in the first cycle got through the ‘border control’ Kr and then joined the native population of the ergodic states, evolving like it for the n − 1 subsequent periods. On the other hand, for l = 0 we have Kr K0n−l u˚ 0 ; that is, those born in the state 0, who survived there for n − 1 cycles, but who eventually got through Kr and joined the natives of the ergodic state in the last cycle. So, if K r is primitive, 1r · Kr (I − K0 )−1 u˚ 0 = lim

n−1 

n→∞

K lr Kr K0n−l−1 u˚ 0

(72)

l=0

and thus 1 u˚ can be interpreted as the effective initial condition in the ergodic state(s) consisting of all individuals originally present in those states as well as all that reach the ergodic states at some moment in time. In other words, in the effective initial condition the initial transient states are instantaneously lumped together with the ergodic states weighted, however, by the chances of reaching the later. Thus, in the limit (67) ˚ lim+ etA u˚ = etB11 1 u, →0

for u˚ that is constant on each edge, we see that for the approximation we need to consider the ODE v = B11 v ˚ with the effective initial condition v˚ = 1 u. Consider now the case of imprimitive Kr . While again 1 is given by (64), 1r · Kr (I − K0 )−1 u˚ 0 cannot be obtained and interpreted as in (72) since neither K nr nor n−1  l K r Kr K0n−l−1 has a limit as n → ∞. However, [3], for each r = 0, 1, . . . , d − 1 l=0

there exists pd +r

lim

p→∞

 l=0

pd +r−l

K lr Kr K0

u˚ 0 = K r+1 r

d −1 

d −m−1 Km (I − K0d )−1 u˚ 0 =: Fr u˚ 0 , p,r Kr K0

m=0

where Kp,r denotes the periodic part Kr (that also may be interpreted as the limit matrix of K nr as n → ∞), and 1r · Kr (I − K0 )−1 u˚ 0 =

d −1 1 Fr u˚ 0 . d r=0

This formula can be interpreted in a similar way as (72); that is, as the contribution of the transient states to each periodic state of the ergodic component and then averaged over all of them.

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Example 3.20 Consider the flow on the graph from Fig. 1, given by (57) with ⎛

0 ⎜1 ⎜ ⎜0 ⎜ K=⎜ ⎜0 ⎜0 ⎜ ⎝0 0

0 1 0 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 2 1 2

0 0 0 0 1 0 0

⎛ ⎞ 0 0 ⎜1 0⎟ ⎜ ⎟ ⎜0 0⎟ ⎜ ⎟ ⎜0 1⎟ , B = ⎜ ⎟ ⎜0 ⎟ 0⎟ ⎜ ⎝0 0⎠ 0 0

0 1 0 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 1

0 0 0 0 1 0 0

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟. 0⎟ ⎟ 0⎠ 0

The eigenvalues of K are λ1 = 1, λ2 = 0, λ3 = −1+i , λ4 = −1−i , where λ1 is a dou2 2 ble and λ2 is a triple eigenvalue. The normalized left and right eigenvectors of K associated with 1, respectively eli and eri , i = 1, 2, are     el1 = 1 1 0 0 0 0 0 , er1 = 0 1 0 0 0 0 0 ,    1 0001211 , el2 = 0 0 1 1 1 1 1 , er2 = 5 and the associated spectral projections 1 = 11 ⊕ 21 are ⎛

0 ⎜1 ⎜ ⎜0 ⎜ 1 1 = ⎜ ⎜0 ⎜0 ⎜ ⎝0 0

0 1 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

⎞ ⎛ 0 00 ⎜0 0 0⎟ ⎟ ⎜ ⎜0 0 0⎟ ⎟ ⎜ 1 2 ⎟ 0 ⎟ , 1 = ⎜ 00 5⎜ ⎜0 0 0⎟ ⎟ ⎜ ⎝0 0 0⎠ 0 00

0 0 0 0 0 0 0

0 0 0 1 2 1 1

0 0 0 1 2 1 1

0 0 0 1 2 1 1

⎞ 0 0⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟. 2⎟ ⎟ 1⎠ 1

Since |λi | < 1 for i = 2, 3, 4, the assumptions of Theorem 3.19 (a) are satisfied. The dimension of the eigenspace belonging to the semisimple eigenvalue 1 of K equals the number of terminal strong components of G; that is, 2. Hence the limiting system of ordinary differential equations consists of 2 differential equations describing the evolution of the material trapped in each terminal component: " lim+ e u˚ = e tA

→0

tB11

1 u˚ = e

t

2  i=1

# u˚ i



0100000

" 7 #   1 7t  + e5 u˚ i 0 0 0 1 2 1 1 . 5 i=3



Transport on Networks—A Playground of Continuous …

475

for any t > 0 and u˚ ∈ R7 . Analogously, if for an arbitrary u˚ = (˚ui (x))i=1,2,...,7 ∈ ˚ then L1 ([0, 1], R7 ) we write (u1, (x, t), . . . , u7, (x, t)) = etA u, ⎛ ⎞ 2 1    1 PM etA u˚ = ⎝ ui, (x, t)⎠ 0 1 0 0 0 0 0 i=1 0

⎞ ⎛ 7 1   1 ⎝ + ui, (x, t)⎠ 0 0 0 1 2 1 1 5 i=3 0

and we have lim+ 1 PM e u˚ = e tA

t

" 2  

→0

i=1

7 5t

e + 5

u˚ i (x)dx

0

" 7   i=3

#

1

1



#

u˚ i (x)dx

0100000





 0001211 .

0

We observe that the dynamics of the macro-model, obtained as the limit solution, is given by two equations v1 (t) = v1 (t), v2 (t) =

7 v2 (t), 5

(73)

instead of the postulated system (1) of the same dimension as the adjacency matrix of the line graph of G. Moreover, the solution of the above system approximates the total mass of the system at terminal strong components instead of the masses on each edge of G. Note that K is column stochastic and hence there is mass conservation, so that the whole initial mass is eventually distributed according to the Perron eigenvectors of K. However, B is not conservative and thus generates the growth observed in (73).

3.4 Singularly Perturbed Problem (27)–(29) In the structured population theory, the inter-patch migration often happens on a much faster time scale then the demographic processes such as ageing and death. This leads to the scaling 1 ∂t n = Sn − Mn + Rn ,  ∞ ◦ n (0, t) = B[n (·, t)] := B(a)n (a, t)da, n (a, 0) = n(a). 0

(74)

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J. Banasiak and A. Puchalska

The operator A , which is the expression A = S − M + 1 R restricted to Y(W11 ([0, ∞), l1N )) functions satisfying the boundary conditions, is the generator of an equibounded semigroup (etA )t≥0 , see Theorem 2.11 and the comment below it. By the Kurtz theorem, see Example 3.13, we need to understand the long term behaviour of solutions to the problem ∂t u = Ru,

u(0) = u0 ,

(75)

where u = (u1 , . . . , uN ) ∈ l1N , which describes the so-called fast dynamics of (74). This requires the spectral analysis of R. As in the network transport, we are interested in general, possibly reducible, transition matrices. Though here we deal with Kolmogorov; that is, positive off-diagonal matrices, their theory is to large extent the same as for positive matrices – in fact, adding a sufficiently large multiple of identity to a Kolmogorov matrix makes it a positive matrix. Thus, an arbitrary Kolmogorov matrix can be permuted to the form ⎛

R1 ⎜ .. ⎜. ⎜ ⎜ Ar,1 R=⎜ ⎜ Ar+1,1 ⎜ ⎜ .. ⎝.

0 .. .

··· ··· .. .

... .. .

0 .. .

Rr Ar+1,r .. .

0 Rr+1 .. .

Ar+m,1 · · · Ar+m,r 0

... .. .

... ... .. .

0 .. . 0 0 .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

(76)

· · · Rr+m

where the diagonal matrices are irreducible. The interpretation of the decomposition is the same as for (61), though here we do not have the block row of zeros that resulted from the fact that (61) was the adjacency matrix of the line graph of a graph that could contain sources. Here, R can be regarded as an adjacency matrix of a network of connections in the described population and we assumed that there are no isolated states. The analysis of eigenvectors is the same as in Sect. 3.3.1 (though obviously here λ = 0 is the dominant eigenvalue). The kernel Ker R is m dimensional, spanned by eigenvectors er+1 = (0, . . . , 0, eˆ r+1 , 0, . . . , 0) .. . er+m = (0 . . . , 0, . . . , 0, eˆ r+m ), where eˆ i is the normalized strictly positive Perron eigenvector of Ri , i = r + 1, . . . , r + m.

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The left eigenspace is spanned by , . . . , y(r+1) , yr+1 , 0, . . . , 0), xr+1 = ( y(r+1) r 1 .. .

xr+m = ( y(r+m) , . . . , y(r+m) , 0, . . . , 0, yr+m ), r 1 where yr+1 , . . . , yr+m are the vectors 1 of appropriate dimension and the other components will be determined later. Thus we obtain the spectral decomposition, see Sect. 3.1.1, as l1N = Ker R ⊕ W = Span{ e1 , . . . , em } ⊕ W,

(77)

with the spectral projections onto the null-spaces defined by   r  yi(k) · uk ek =: unk ek , Pk u = (xk · u) ek = 1 · uk +

(78)

i=1

for r + 1 ≤ k ≤ r + m. Then v = Pu =

r+m 

Pk u =

k=r+1

r+m 

(xk · u) ek =

k=r+1

r+m 

unk ek .

(79)

k=r+1

We set nk = k − r and let υ = (υ1 , . . . , υm ) = (unr+1 , . . . , unr+m ). The complementary space W = {w ∈ l1N ; Pw = 0} is the range of Q, defined as Q u := [ I − P ]u =: w. Note that KerR, and thus W, are a-dependent. Hence, as for v, we introduce ω = (ωm+1 , . . . , ωN ) as the coordinate representation of w in the a-dependent basis of W. Example 3.21 As in the network model, to understand the meaning of the spectral decomposition, it suffices to restrict our attention to  R=

R1 0 Ai Ri

 ,

g ≤ i ≤ r + m. Then, on one hand, the spectral projection onto the ergodic state is given by Pi u = 1 · (Ai (−R1 )−1 u˚ 0 + u˚ i )ei ,

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where ei is the Perron eigenvector of R and u = (u0 , ui ), and on the other by ⎛ lim ui (t) = lim ⎝etRi u˚ i +

t→∞

∞

t→∞

⎞ e(t−s)Ri Ai esR1 u˚ 0 ds⎠ .

0

In other words, 1 · Ai (−R1 )−1 u˚ 0 is the total number of the individuals from the cohort originally present in the transient states that at any point in time will reach the ergodic state. Thus it is clear that in the considered meta-population the states 1 ≤ j ≤ r eventually will become depleted with the population moving towards states r + 1 ≤ j ≤ r + m. However, according to (78), the population in any final state will contain a contribution from those transient states that communicate with this state (but not from any other ergodic state). ♦ Next, we assume that the structure of R in (76) does not depend on a. Further, 2 a → R(a) ∈ Cb2 ([0, ∞), l1N ) and max

λ∈ σ(R(a)|W )

 λ =: ζ < 0.

(80)

2 N 2 N Under the above assumptions, R|−1 W ∈ Cb ([0, ∞), l1 ) and ej ∈ Cb ([0, ∞), l1 ) for r + 1 ≤ j ≤ r + m, see [9, Lemma 4.1]. Using the spectral coordinates as in (39), we can write (74) in the form 2

∂t υ  = −∂a υ  − M∗ υ  + Fω  , 1 ∂t ω  = −∂a ω  + Gυ  + Hω  + R|W ω  ,  γυ  (t) = B ∗ [υ  (·, t)] + J [ω  (·, t)],

(81)

γω  (t) = BQ,υ [υ  (·, t)] + BQ,ω ω  (·, t), ◦



υ  (a, 0) = υ(a), ω  (a, 0) = ω(a), where γυ = υ|a=0 , F, G, H, J , BQ,υ and BQ,ω are linear bounded operators on respective subspaces of X, whose particular form does not have any bearing on the calculations. Further, −M∗ is a sub-Kolmogorov matrix, given by M∗ = (μ∗jk )r+1≤j,k≤r+m with μ∗jk := xj · Mek 

and ∗

B = 0 ◦





B∗ da, B∗ = (βjk∗ )r+1≤j,k≤r+m with βjk∗ := xj · Bek . ◦







Finally, P u = v = (υ 1 , . . . , υ m ) and Qu = ω. It follows, [9, Sect. 5], that (81) is equivalent to (74). More precisely, the semigroup (etA )t≥0 solving (81) is similar to (etA )t≥0 with the similarity operator being the coordinate transformation from the canonical basis in l1N to the spectral coordinates.

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Example 3.22 To better understand these formulae, let us consider a simplified situation, where K is irreducible with 0 being a simple eigenvalue and M = diag(μj )1≤j≤N . Then the Perron eigenvector e gives the stable population distribution, x = 1 and 1 · Me = μ1 e1 + · · · + μn en is the weighted average mortality of the population in the equilibrium. If we have a reducible part, its contribution is not visible in the projection. ♦ To find the formal limit equation, we follow the Chapman-Enskog procedure as when deriving (49) from (47) and (48). Thus we leave υ = (υ1 , . . . , υm ) unexpanded and write (82) u = (υ1 , . . . , υm , ω) = (υ1 , · · · , υm , ω 0 + ω 1 + · · · ). Inserting this representation into the projected system (81) we get

ω 0,t + 2 ω 1,t

∂t υ = −∂a υ − M∗ υ + F[ω 0 + ω 1 + · · · ], + · · · = −ω 0,a − 2 ω 1,a + · · · + Gυ

(83)

+H[ω 0 + ω 1 + · · · ] + R|W [ω 0 + ω 1 + · · · ] and γυ(t) = B ∗ υ(t) + J [ω 0 (t) + ω 1 (t) + · · · ], γ[ω 0 (t) + ω 1 (t) + · · · ] = BQ,υ [υ(t)] + BQ,ω [ω 0 (t) + ω 1 (t) + · · · ] (84) ◦



υ(0) = υ, ω 0 (0) + ω 1 (0) + · · · = ω. Comparing the terms multiplying like powers of  and using the invertibility of R|W on W, we obtain ω 0 = 0 and thus the formal limit system, whose solution will be denoted by υ, ¯ takes the form ¯ ∂t υ¯ = −∂a υ¯ − M∗ υ, γ υ(t) ¯ = B ∗ υ(t), ¯

(85)



υ(0) ¯ = υ. We observe that, in contrast to the network problems, here the limit equation is a partial differential equation, hence (74) cannot be considered as a micro-model consistent with any ODE macro-model (1). To provide the rigorous result on the convergence of solutions to (74) to the solution of (85), let us introduce the relevant notation. Let X = L1 ([0, ∞), l1m ) × {0}, Aυ = −∂a υ − M∗ υ on W11 ([0, ∞), l1m ) and let A be A restricted to the set of functions satisfying the boundary conditions in (85). Then, similarly to Theorem 2.11, we have

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Theorem 3.23 A generates a strongly continuous semigroup (etA )t≥0 such that ¯ ∗ −m∗ )t

|||etA ||| ≤ e( b

where b¯ ∗ := ess sup |||B∗ (a)|||B(l1m ) and m∗ := a∈[0,∞)

, inf

1≤j≤m,a∈[0,∞)

μ∗j (a).

Let Aex be the extended limit of (A )>0 . A crucial result allowing for the application of Theorem 3.12 is the following lemma. Lemma 3.24 [9] D(A) × {0} ⊂ D(Aex ) ⊂ X and (Aυ, ω) = Aex (υ, 0) for any υ ∈ D(A) and ω ∈ W11 ([0, ∞), l1N −m ). As noted in Sect. 3.1, to prove the statement such as (Aυ, ω) = Aex (υ, 0), we need to find u = (υ  , ω  ) ∈ D(A ) such that u → (υ, 0) and A u → (Aυ, ω). Construction of such a sequence usually is not easy. Here it can be accomplished by a higher order asymptotic expansion and it is given by $),  > 0, (υ  , ω  ) = (υ, (ω 1 + R|−1 W ω) + ω where υ ∈ D(A), ω 1 is obtained from (83) and(84), ω is arbitrary satisfying the assumption of the lemma and ω $ is the zeroth order term of the boundary layer determined from $ = R|W (0)$ ω, γ$ ω = BQ,υ υ, (86) ∂α ω where α = a/, see also [10, 11]. Then we have Theorem 3.25 We have lim etA

→0

   tA  υ e υ = 0 0

(87)

almost uniformly in [0, ∞) for υ ∈ L1 ([0, ∞), l1m ) and lim etA

→0

   tA  υ e υ = ω 0

(88)

almost uniformly in (0, ∞) for (υ, ω) ∈ L1 ([0, ∞), l1N ). Sketch of the proof. Formula (87) follows directly from the Sova-Kurtz theorem. Indeed, by Lemma 3.24, we obtain that for any υ ∈ D(A) and ω ∈ W11 ([0, ∞), l1N −m ), (λυ − Aυ, ω) belongs to the range of λ − Aex and, since A is a generator, such ∗ a set of (υ, ω) is dense in X for λ > b − m∗ . Further, again from the first part of Lemma 3.24, we find that the space of regular convergence, X0 , coincides with X . Since the generator Aˆ of the limit semigroup is the part of Aex in X0 whence, again ˆ D(A)) ˆ and since, by Theorem 3.23, A by Lemma 3.24, ((A, 0), D(A) × {0}) ⊂ (A, is the generator, (87) follows.

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To consistently use Theorem 3.12, we observe that (a) is satisfied by Lemma 3.24 with K = (A, 0), D = D(A) × {0} and ω = 0. Consider now the assumption (b). Let Lex denote the extended limit of A . Since R is bounded, we can write A (υ  , ω  ) = (−∂a υ  − M∗ υ  + Fω  , −∂a ω  + Gυ  + Hω  ) + (0, R|W ω  ) and taking (υ  , ω  ) = (υ, ω) ∈ D(A ), we see that ((υ, ω), (0, R|W ω)) ∈ Lex for any (υ, ω) ∈ D(A ). Hence, we can take the operator (0, R|W ) as the operator C of Theorem 3.12. Since this operator is bounded, by density, D(A ) is its core. It is also clear, by construction, that (33) is satisfied and the projection P is just taking the first m coordinates and replacing the remaining N − m coordinates by zeroes. Hence, the assumption c) of Theorem 3.12 is automatically satisfied by Theorem 3.23.  Since the spectral decomposition (77) is reversible, given the approximate υ(t) ¯ and ω(t), we can reconstruct the explicit approximation to n (t), as will be demonstrated in Example 3.26. ◦ Let us interpret this result. Let n be an initial distribution of the population and n (t) the corresponding solution of the full equation (74). Then we consider v (t) = Pn (t) and w (t) = Qn (t). In particular, by Example 3.21, v (t) contains the potential contributions from all transient states over the whole time ‘glued’ together. Remember, ¯ of the transitions occur instantaneously! Then v (t) converges to the solution υ(t) the aggregated system (85) and w (t) converges to zero but only if t > 0. If we want ◦ the convergence to be uniform in t up to zero, we need to ensure that Qn = 0; that is, that we only begin after all transitions have already occurred. The question is whether the lack of uniform approximation at t = 0 is a technical, or a real issue. To illustrate this question, we present numerical simulations for one component of the solution to (74), taken from [11]. In Fig. 11 we show the error of approximation for arbitrary initial data. We see that, indeed, the approximation is bad close to t = 0 (the initial layer effect), close to a = 0 (the boundary layer effect) and at a = t = 0 (the corner layer effect). It turns out that it is possible to construct corrections in a systematic and algorithmic way by introducing the initial and boundary layers and obtain the convergence uniform in t up to t = 0. The errors of such an approximation are presented in Fig. 12.

Fig. 11 The error of bulk approximation for arbitrary initial data with visible bad approximation close to t = 0 or a = 0

n2 − nk,2

0.7 0.4 0.1 −0.2 −0.5 1

0.8

0.6 t

0.4

0.2

00

0.2

0.4

0.6 a

0.8

482 0.7

J. Banasiak and A. Puchalska n2 − nk,2 − ni,2

n2 − nk,2 − ni,2 − nb,2

0.4

−0.1

0.1 −0.2 1

−0.4 0.8

0.6 t

0.4

0.2

00

0.2

0.4

0.6 a

0.8

−0.7 1

0.8

0.6 0.4 0.2 t

00

0.2

0.4

a

0.6

0.8

Fig. 12 Reducing the error of approximation. From the left: bulk and initial layer approximation and bulk, initial and boundary layer approximation

Example 3.26 In this section we will illustrate the asymptotic procedure as applied to the McKendrick models with 5 patches. To make calculations simpler, we consider age independent matrices M, B and R with ⎛

−2 ⎜ 0 ⎜ R := ⎜ ⎜ 1 ⎝ 0 1

0 −1 1 0 0

0 1 −1 0 0

0 0 0 −1 1

⎞ 0 0 ⎟ ⎟ 0 ⎟ ⎟. 1 ⎠ −1

(89)

The matrix R is reducible, with σ(R) = {0, −2}. The eigenvalue λ = 0 is dominant, semisimple with multiplicity 2, while λ = −2 is of algebraic multiplicity 3 and geometric multiplicity 2. Accordingly, Ker R is also two dimensional. A nonnegative basis of (normalized) vectors for Ker R is given by   1 1 e4 = 0, , , 0, 0 , 2 2   1 1 , e5 = 0, 0, 0, , 2 2 while in W we can select the basis e1 = (0, 1, −1, 0, 0) , e2 = (0, 0, 0, 1, −1) ,   1 1 , e3 = 1, 0, − , 0, − 2 2 where e3 is an associated eigenvector. Similarly, the bi-orthogonal basis of the left null space is

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 1 , 1, 1, 0, 0 , 2   1 , 0, 0, 1, 1 , x5 = 2 

x4 =

while in W we can select the basis   1 1 1 x1 = − , , − , 0, 0 , 4 2 2   1 1 1 , x2 = − , 0, 0, , − 4 2 2 x3 = (1, 0, 0, 0, 0) . Let u := (u1 , u2 , u3 , u4 , u5 ). Accordingly, for u = v + w ∈ Ker R ⊕ W we have v = u4 e4 + u5 e5 with υ1 = u 4 =

1 u1 + u2 + u3 2

and

υ2 = u5 :=

1 u1 + u4 + u5 , 2

while w = w1 e1 + w 2 e2 + w 3 e3 , where 1 1 1 1 1 1 w 1 = − u1 + u2 − u3 , w 2 = − u1 + u4 − u5 , and w 3 = u1 . (90) 4 2 2 4 2 2 In the notation of the theoretical part ω = (ω3 , ω4 , ω5 ) = (w 1 , w 2 , w 3 ). It is easy to see that this transformation is invertible. We take general constant mortality and birth matrices, respectively, M := (μij )1 ≤ i, j ≤ 5 and B := (βij )1 ≤ i, j ≤ 5 . Applying projections Pj , 4 ≤ j ≤ 5, and Q, we get the hydrodynamic part of (81) as ∂t u4 = −∂a u4 − μ∗44 u4 + μ∗45 u5 − μ∗41 w 1 + μ∗42 w 2 + μ∗43 w 3 , ∂t u5 = −∂a u5 − μ∗54 u4 + μ∗55 u5 − μ∗51 w 1 + μ∗52 w 2 + μ∗53 w 3 , where the explicit values of the coefficients μ∗ij are found as follows. First,    P4 M (P4 u+P5 u) = x4 · M u4 e4 + u5 e5 e4

  = μ∗44 u4 + μ∗45 u5 e4 ,

so that μ∗44 μ∗45

 1 μ22 + μ23 + μ32 + μ33 + = 2  1 μ24 + μ25 + μ34 + μ35 + = 2

1 μ12 + 2 1 μ14 + 2

 1 μ13 , 2  1 μ15 . 2

(91)

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Similarly μ∗54 μ∗55

 1 μ42 + μ43 + μ52 + μ53 + = 2  1 μ44 + μ45 + μ54 + μ55 + = 2

1 μ12 + 2 1 μ14 + 2

 1 μ13 , 2  1 μ15 . 2

The other essential term in the expansion is R|W = QRQ, given by     1 1 R|W w = −2w 1 − w 3 e1 + −2w 2 − w 3 e2 − 2w 3 e3 . 2 2

(92)

Using projections Pj , 1 ≤ j ≤ 2, and Q, we can also derive the corresponding expressions of the boundary equations. Hence, system (85) takes the following form, where υ¯ = (¯u4 , u¯ 5 ), ∂t u¯ t4 = −∂a u¯ 4 − μ∗44 u¯ 4 − μ∗45 u¯ 5 , ∂t u¯ t5 = −∂a u¯ 5 − μ∗54 u¯ 4 − μ∗55 u¯ 5 , ∞ 4 ∗ 4 ∗ 5 u¯ (0, t) = (β44 u¯ (a, t) + β45 u¯ (a, t))da, 0

∞ 5 ∗ 4 ∗ 5 u¯ (0, t) = (β54 u¯ (a, t) + β55 u¯ (a, t))da, 0

1◦ ◦ ◦ u1 (a) + u2 (a) + u3 (a), 2 1◦ ◦ ◦ u¯ 5 (a, 0) = u1 (a) + u4 (a) + u5 (a), 2

u¯ (a, 0) = 4

(93)

where (βij∗ )1≤i≤2,1≤j≤2 is given by formulae analogous to (μ∗ij )1≤i≤2,1≤j≤2 . According to [11], to write down the approximation with the error of order  that is uniformly valid up to t = 0, we need the initial layer; that is, the solution to ˜ 0 = R|W w ˜ 0, ∂τ w where τ = t/. Using the coordinate form (92), we find the solution as  τ2 3 τ 2 1 w˚ − w˚ + w˚ , =e 8 2 τ 2 −2τ 3 w˜ 0 (τ ) = e − w˚ + w˚ 2 , 2 w˜ 03 (τ ) = e−2τw˚ 3 , w˜ 01 (τ )

−2τ



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where, by (90), 1 1 1 1 1 1 w˚ 1 = − u˚ 1 + u˚ 2 − u˚ 3 , w˚ 2 = − u˚ 1 + u˚ 4 − u˚ 5 and w˚ 3 = u˚ 1 . 4 2 2 4 2 2

(94)

Thus we can write the combined estimates as u (t) = u(t) + u(τ ) + O() = u¯ 4 (t)e4 + u¯ 5 (t)e5 " # τ τ2 3 τ 2 w˚ − w˚ + w˚ 1 e1 + e−2τ − w˚ 3 + w˚ 2 e2 + e−2τ w˚ 3 e3 + O() + e−2τ 8 2 2

in X = L1 ([0, ∞), l15 ) norm, uniformly in t on [0, T ]. Using the expressions for ej , 1 ≤ j ≤ 5, we can write the approximation in the components as % % %u1 (t) − e−2τ w˚ 3 % L1 ([0,∞)) %  2 % % % τ 1 τ %u2 (t) − u¯ 4 (t) − e−2τ w˚ 3 − w˚ 2 + w˚ 1 % % % 2 8 2 L ([0,∞)) % %1  2  % % τ 1 1 τ %u3 (t) − u¯ 4 (t) + e−2τ w˚ 3 − w˚ 2 + w˚ 1 + e−2τw˚ 3 % % % 2 8 2 2 L1 ([0,∞)) % % τ % % 1 %u4 (t) − u¯ 5 (t) − e−2τ − w˚ 3 + w˚ 2 % % % 2 2 L ([0,∞)) % %1 τ 1 % % 1 %u5 (t) − u¯ 5 (t) + e−2τ − w˚ 3 + w˚ 2 + e−2τw˚ 3 % % % 2 2 2 L1 ([0,∞))

= O(), = O(), = O(), = O(), = O(),

where u¯ i , i = 1, 2, solve (93) and w˚ j , j = 3, 4, 5, are defined by (94).

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Augmenting and Decreasing Systems J. M. Bilbao, A. Jiménez-Losada and M. Ordóñez

Abstract This paper deals with cooperative games in which there exist a feasible coalition structure. Augmenting and decreasing systems are set systems specially introduced for analyzing certain situations of partial cooperation. They are dual structures in the sense that a coalition is feasible in one of them if and only if the complement is feasible in the another one, so if an augmenting system represents the feasible options of the players to cooperate in a game then its dual decreasing system analyze the options of the players out each coalition. Augmenting systems are generalizations of antimatroids and systems of connected subgraphs in a graph. This fact means to relate two interesting structures in games. We study the core and the Weber set for games on augmenting systems. Later, two very known classical solutions for games are defined on augmenting systems: the Shapley value and the Banzhaf one. Finally we leverage the duality relationship to analyze these values for decreasing systems. Keywords Augmenting system · Decreasing system · Core · Weber set · Shapley value · Banzahf value 1991 Mathematics Subject Classification 90C27 · 90D12

J. M. Bilbao · A. Jiménez-Losada · M. Ordóñez (B) Department of Applied Mathematics II, University of Seville, Escuela Superior de Ingenieros, Camino de los Descubrimientos, 41092 Sevilla, Spain e-mail: [email protected] J. M. Bilbao e-mail: [email protected] A. Jiménez-Losada e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_15

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1 Introduction A cooperative game (with transferable utility) is defined by a finite set of agents (players) and a characteristic function given a worth for each coalition (cooperation group) of the players. This paper analyzes a special framework for cooperation. We study a structure of feasible coalitions using the theory of augmenting systems, a notion developed from the combinatorial abstract theory for cooperative games. Classical cooperative games suppose that all the coalitions are feasible because the characteristic function of the game defines a worth for each of them. But in real life there exist situations where this fact is not true. A feasible coalition system in a cooperative game represents the family of possible cooperations among the agents in the game. Aumann and Dreze [3] introduced coalition structures in 1974. This one was the first case of feasible coalition system as the family of subsets of a partition of the set of players. Later, numerous set systems have been used interpreted as feasible coalition systems. Myerson [20] defined the feasible coalitions as the connected subgraphs in a graph representing the bilateral communications among the agents, Gilles et al. [17] used hierarchical functions to determine those coalitions with autonomy to cooperate (these families of coalitions form particular case of antimatroids), Faigle and Kern [13] took as feasible coalitions the ideals in a poset, Bilbao and Edelman [5] considered convex geometries and Algaba et al. [1] antimatroids. All these works revealed the interest of game theory in combinatorial structures and set systems. Recently, Bilbao [6] or Gallardo et al. [15] have showed that also it is possible to define new interesting set systems from game theory. So, they introduced augmenting systems and authorization structures. Given a game with a feasible coalition system there are two different models to undertake the additional information to the game. The model of Auman and Dreze [3] (also Faigle and Kern [13] and Bilbao and Edelman follow this option) consider that it is not possible to estimate the worths of the non-feasible coalitions and then only the characteristic function restricted to the system can be used. But, Myerson [20] (and also Gilles et al. [17] and Algaba et al. [2]) proposed to introduce a way to evaluate the worth of the non-feasible coalitions from the feasible ones. So, a new classical game (called restricted game) is obtained involving the information of the coalition system. Another different model was introduced by Owen [22] using a coalition structure as a priori union system but not as the definitive feasible coalitions. A solution for a cooperative game is a payoff vector giving a payment for each player. A value is a function obtaining a solution for each game. The solution concepts and values for classical cooperative games use the characteristic function and then the information of a feasible coalition system influences in them. In the first model, new solutions concepts must be introduced in order to use only the feasible coalitions. But, in the second model classical solutions can be used over the new game involving the system. This paper develops the analysis of games on augmenting systems following the first model, and also their dual structures, decreasing systems. Section 2 introduces augmenting systems following Bilbao and Ordóñez [7]. Core [16] and Weber set [27] restrict the set of reasonable solutions for a game. The aim of Sect. 3 is the introduction

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of adapted concepts of core and Weber set for games on augmenting systems. The definitions of these sets coincide with those given by a determined restricted game, thus it is connected with second model. Sections 4 and 5 study the extension of several properties of games to these situations: monotonicity, superadditivity and convexity. In Sects. 6 and 7 a Shapley value for games on augmenting systems is defined and it is provided with two axiomatizations. Shapley [24] introduced in 1953 perhaps the most known value for cooperative games. Nearest to the Shapley value, the Banzhaf value [23] is another value with interesting applications in social sciences. Section 8 defines a Banzhaf value for games on augmenting systems. Decreasing systems are the focus of Sect. 9 with special emphasis on the duality relation with the augmenting systems. The duality of the structures is extended to the games in Sect. 10. Sections 11 and 12 analyze the Shapley and the Banzhaf value on decreasing systems using the duality. Finally, in the last section we give an application to greedoids [19], they are structures very useful and used in graph theory.

2 Augmenting Systems Antimatroids were introduced by Dilworth [10] as particular examples of semimodular lattices. Since then, several authors have obtained the same concept by abstracting various combinatorial situations (see Korte, Lovász, and Schrader [19]). Let N be a finite set. A set system over N is a pair (N , F) where F ⊆ 2N is a family of subsets. The sets belonging to F are called feasible. We will write S ∪ i and S \ i instead of S ∪ {i} and S \ {i} respectively. Definition 1 A set system (N , A) is an antimatroid if A1. ∅ ∈ A, A2. for S, T ∈ A we have S ∪ T ∈ A, A3. for S ∈ A with S = ∅, there exists i ∈ S such that S \ i ∈ A. Let (N , A) be an antimatroid and let S, T ∈ A such  that |S| < |T |. Property A3 implies an ordering T = {i1 , . . . , it } with i1 , . . . , i j ∈ A for j = 1, . . . , t. Let / S. Then S ∪ ik = S ∪ {i1 , . . . , ik } ∈ k ∈ {1, . . . , t} be the minimum index with ik ∈ A by Property A2. Therefore, the definition of antimatroid implies the following augmentation property: If S, T ∈ A with |S| < |T | then there exists i ∈ T \ S such that S ∪ i ∈ A. Convex geometries are a combinatorial abstraction of convex sets introduced by Edelman and Jamison [11]. Definition 2 A set system (N , G) is a convex geometry if it satisfies the following properties: G1. ∅ ∈ G, G2. for S, T ∈ G we have S ∩ T ∈ G, G3. for S ∈ G with S = N , there exists i ∈ N \ S such that S ∪ i ∈ G.

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We will introduce a new combinatorial structure as follows. Definition 3 An augmenting system is a set system (N , F) with the following properties: P1. ∅ ∈ F, P2. for S, T ∈ F with S ∩ T = ∅, we have S ∪ T ∈ F, P3. for S, T ∈ F with S ⊂ T , there exists i ∈ T \ S such that S ∪ i ∈ F. The relationship between the combinatorial structures above mentioned is given by Bilbao [6] in the next proposition. Proposition 4 (i) An augmenting system (N , F) is an antimatroid if and only if F is closed under union. (ii) An augmenting system (N , F) is a convex geometry if and only if F is closed under intersection and N ∈ F. A direct consequence of the above proposition is the following Corollary 5 If (N , A) or (N , G) are an antimatroid or a convex geometry respectively, then they are augmenting systems. Example The following collections of subsets of N = {1, . . . , n}, given by F = 2N , F = {∅, {i}} where i ∈ N , and F = {∅, {1}, . . . , {n}}, are augmenting systems over N. Example Let us consider a communication graph G = (N , E), where N is the set of players and E is the set of edges which represents the bilateral communication between some players. Given a coalition S ⊆ N , the set of edges between players in S is denoted by E(S) = {i j ∈ E : i, j ∈ S}. Thus, the set system (N , F) given by F = {S ⊆ N : (S, E(S)) is a connected subgraph of G} , is an augmenting system. Example Gilles et al. [17] showed that the feasible coalition system (N , F) derived from the conjunctive or disjunctive approach contains the empty set, the ground set N , and that it is closed under union. Algaba et al. [1] showed that the coalition systems derived from the conjunctive and disjunctive approach were identified to poset antimatroids and antimatroids with the path property respectively. Thus, these coalition systems are augmenting systems. The next characterization of the augmenting systems derived from the connected subgraphs of a graph is proved by Algaba et al. [2] (Fig. 1). Theorem 6 An augmenting system (N , F) is the system of connected subgraphs of the graph G = (N , E), where E = {S ∈ F : |S| = 2} if and only if {i} ∈ F for all i ∈ N.

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Fig. 1 Augmenting system

1234

123

124

134

234

12

13

24

34

1

4

Let N = {1, . . . , n} be a set of players with n > 2 and we consider a subset S of starting players. If i ∈ S then the coalition {i} is feasible. Each starting player i looks for a player k ∈ / S to generate a new feasible coalition {i, k}. These coalitions with cardinality 2 searching for new players which agree to join one by one. If we assume that common elements of two feasible coalitions are intermediaries between the two coalitions in order to establish the feasibility of its union, we obtain an augmenting system (N , F). Since the individual players k ∈ / S are not feasible coalitions, the family F is not generated by the connected subgraphs of a graph. Moreover, if players i, j ∈ S then {i} , { j} ∈ S and {i, j} ∈ / S and hence (N , F) is not an antimatroid. Example Let N = {1, 2, 3, 4} and we consider S1 = {1, 2, 4} and S2 = {1, 4}. By using the above coalition formation model we can obtain the following augmenting systems: F1 = {∅, {1} , {2} , {4} , {2, 3}} , and F2 = {∅, {1} , {4} , {1, 2} , {3, 4}} . Example Let us consider N = {1, 2, 3, 4} and F = {∅, {1} , {4} , {1, 2} , {3, 4} , {1, 2, 3} , {2, 3, 4} , N } . Since {1, 2, 3} and {2, 3, 4} are feasible coalitions, property P2 implies that the grand coalition N is also feasible. Example The set system given by N = {1, 2, 3, 4} and F = {∅, {1} , {4} , {1, 2} , {1, 3} , {2, 4} , {3, 4} , {1, 2, 3} , {1, 2, 4} , {1, 3, 4} , {2, 3, 4} , N } .

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is an augmenting system. Since {1, 4} ∈ / F the system (N , F) is not an antimatroid. Moreover, {1, 2} ∩ {2, 4} = {2} ∈ / F and hence (N , F) is not a convex geometry. Let (N , F) be a set system and let S ⊆ N be a subset. A feasible subset C ∈ F with C ⊆ S is called a basis of S if C ∪ i ∈ / F for all i ∈ S \ C. The maximal nonempty feasible subsets of S are called components of S. Clearly, every component of S is a basis of S. However, the converse is not true, as the following example shows: Example If N = {1, 2, 3} and F = {∅, {1} , {2} , {2, 3} , N } then C = {1} is a basis of N , but the only component of N is the ground set N . Since {1} ⊂ N and {1} ∪ i ∈ /F for i ∈ {2, 3}, the set system (N , F) is not an augmenting system. Proposition 7 Let (N , F) be an augmenting system and let S ⊆ N be a subset. Then a non-empty feasible subset C ⊆ S is a basis of S if and only if C is a component of S. Proof See [6]. We denote by CF (S) the set of the components of a subset S ⊆ N . Observe that the set CF (S) may be the empty set. This set will play a role in the concept of a game restricted by an augmenting system. Proposition 8 A set system (N , F) satisfies Property P2 if and only if for any S ⊆ N with CF (S) = ∅, the components of S form a partition of a subset of S. Proof See [6].

3 The Core and the Weber Set of (N, v, F ) We consider in this section only non-negative games. That is, a non-negative game, v : F → R+ is a function with non-negative real values such that v(∅) = 0. Given the game (N , v, F) we define a standard cooperative game v F : 2N → R+ , named the extension of (N , v, F), as v F (S) =



v(T ) for all S ⊆ N .

T ∈CF (S)

Note that the family CF (S) forms a partition of a subset of S, and v F (S) = v(S) for all S ∈ F. Now, we define the core of the games (N , v, F) and (N , v F ). Definition 9 Let (N , v, F) be a game on the augmenting system (N , F). Then   Core (N , v, F) = x ∈ Rn+ : x(N ) = v F (N ), x(S) ≥ v(S) for all S ∈ F

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and     Core N , v F = x ∈ Rn+ : x(N ) = v F (N ), x(S) ≥ v F (S) for all S ⊆ N , where we denote x(S) =

 i∈S

xi , and x(∅) = 0.

Proposition 10 The core of (N , v, F) coincides with the core of its extension (N , v F ).   Proof Note that Core N , v F ⊆ Core(N , v, F) and so Core (N , v, F) = ∅ implies  Core N , v F = ∅. Then we can suppose that Core(N , v, F) = ∅. Next, we prove that for all x ∈ Core(N , v, F) we have x ∈ Core(N , v F ). Since x ≥ 0, we obtain 

x(S) =

  x(T ) + x S \T ∈CF (S) T

T ∈CF (S)







x(T ) ≥

T ∈CF (S)

v(T ) = v F (S)

T ∈CF (S)

for all S ⊆ N .



Proposition 11 The core of (N , v, F) is a polyhedron contained in the convex cone Rn+ = {x ∈ Rn : xi ≥ 0 for all i ∈ N }. Moreover, Core(N , v, F) is a bounded polyhedron or polytope. Proof Since Core(N , v, F) is given by a finite set of inequalities and |xi | = xi ≤



xi = v F (N )

i∈N

for every component of each x ∈ Core(N , v, F), we obtain the property.



Let us consider a standard cooperative game v : 2N → R with v(∅) = 0, and a total ordering of the elements of N , given by i1 < i2 < · · · < in . Then we obtain the following chain of coalitions ∅ ⊂ C1 ⊂ · · · ⊂ Cn−1 ⊂ Cn = N where Ck = {i1 , . . . , ik } ⊆ N , for all k = 1, . . . , n. The marginal worth vector aC ∈ Rn with respect to the above total ordering in the game (N , v) is given by aiCk = v(Ck ) − v(Ck−1 ), for all k = 1, . . . , n. The Weber set of the game (N , v) is the convex hull of the marginal worth vectors, W eber(N , v) = conv{aC : C is a total ordering of N }. It is easy to show that k  j=1

aiCj = v(Ck ) for all k = 1, . . . , n.

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Weber [27] showed that any game satisfies Core(N , v) ⊆ W eber(N , v) and Ichiisi [18] obtained the following characterization of convex games: The game (N , v) is convex if and only if Core(N , v) = W eber(N , v). In our model, we can consider the compatible orderings of an augmenting system (N ,  N ∈ F, as the total orderings of N , i1 < i2 < · · · < in such that the set  F) with i1 , . . . , i j ∈ F for all j = 1, . . . , n. A compatible ordering of (N , F) corresponds exactly to a maximal chain in F and we denote by Ch(F) the set of all the maximal chains of F. Definition 12 The Weber set of a game (N , v, F) is given by W eber(N , v, F) = conv{aC : C ∈ Ch(F)} Proposition 13 Let (N , F) be an augmenting system with N ∈ F. If (N , v, F) is a game and C ∈ Ch(F) then 

aiC = v(S) for all S ∈ C.

i∈S

Proof It follows from the definition of aC .



Example Let N = {1, 2, 3} and let F = {∅, {1}, {2}, {1, 2}, {1, 3}, N }. We define the game v : F → R+ by v(∅) = v(1) = v(2) = 0, v(1, 2) = 1 and v(1, 3) = v(N ) = 2. Notice that   Core(N , v, F) = x ∈ R3+ : x (N ) = 2, x1 + x2 ≥ 1, x1 + x3 ≥ 2   = x ∈ R3+ : x (N ) = 2, x1 ≥ 1, x2 = 0, x3 ≤ 1 = conv {(1, 0, 1), (2, 0, 0)} . There are three maximal chains in F given by C1 : ∅ ⊂ {1} ⊂ {1, 2} ⊂ {1, 2, 3} , C2 : ∅ ⊂ {1} ⊂ {1, 3} ⊂ {1, 2, 3} , C3 : ∅ ⊂ {2} ⊂ {1, 2} ⊂ {1, 2, 3} . The marginal worth vectors are aC1 = (v({1}) − v(∅), v({1, 2}) − v({1}), v(N ) − v({1, 2})) = (0, 1, 1) , aC2 = (v({1}) − v(∅), v(N ) − v ({1, 3}) , v({1, 3}) − v({1}) = (0, 0, 2) , aC3 = (v({1, 2}) − v({2}), v({2}) − v(∅), v(N ) − v({1, 2})) = (1, 0, 1) , and hence we obtain W eber(N , v, F) = conv {(1, 0, 1), (0, 1, 1) , (0, 0, 2)} .

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Then we have that Core(N , v, F)  W eber(N , v, F) and W eber(N , v, F)  Core(N , v, F).

4 Monotone Convex Games Faigle and Peis [14] introduced the concept of monotone convex game for a weakly submodular lattice (F, ≤). In our context, we work with augmenting systems (N , F) such that N ∈ F and it is the maximal element of our structure. Proposition 2.1 by Faigle and Peis [14] implies that the dominance relation ≤ among feasible coalitions of an augmenting system coincides with the set inclusion (F, ⊆). Since in an augmenting system the union of any feasible coalitions is not feasible, Axiom (C2) of the weakly submodular lattices is not necessarily true in our model and we obtain: Proposition 14 Let (N , F) be an augmenting system such that N ∈ F. Then (F, ⊆) is not necessarily weakly submodular. A cooperative game (N , υ, F) is monotone  if for  all S, T ∈ F with S ⊆ T , we have v(S) ≤ v(T ). In general, the extension N , v F of a monotone game (N , v, F) is not necessarily monotone as the next example shows. Example Let N = {1, 2, 3, 4} and we consider the augmenting system F = {∅, {1} , {4} , {1, 2} , {3, 4} , {1, 2, 3} , {2, 3, 4} , N } . The game v : F → R+ defined by v(S) = 1 for every non-empty S ∈ F, and v(∅) = 0 is monotone. The extension υ F yields v F ({1, 4}) = v ({1}) + v ({4}) = 2 > 1 = v F (N ). Shapley [25] introduces the notion of convexity for cooperative games v : 2N → R such that v(S) + υ(T ) ≤ v(S ∩ T ) + v(S ∪ T ), for all S, T ⊆ N . Next, we introduce the following concept. Definition 15 A cooperative game (N , υ, F) is said to be convex if for all S, T ∈ F with S ∪ T ∈ F, we have  v(C) + v(S ∪ T ). v(S) + v(T ) ≤ C∈CF (S∩T )

Note that if F = 2N then the above inequality is the classical convexity. Theorem 16 Let (N , F) be an augmenting system such that N ∈ F. If the nonnegative game v : F → R+ is monotone convex then Core(N , v, F) = ∅.

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Proof Since ∅, N ∈ F Property P3 of the augmenting systems implies the existence of a maximal chain of feasible coalitions ∅ ⊂ C1 ⊂ · · · ⊂ Cn−1 ⊂ Cn = N where C j = {i1 , . . . , i j } ∈ F for all j = 1, . . . , n. We define the corresponding marginal worth vector x ∈ Rn+ as follows: xi1 = v(C1 ), xi j = v(C j ) − v(C j−1 ), j = 2, . . . n. Note that x(C j ) =

j 

xik = v(C j )

k=1

for all j = 1, . . . , n and x(N ) = v(N ). Since v is monotone we obtain xi j ≥ 0 for all j = 1, . . . , n. We will show that x(S) ≥ v(S) for all S ∈ F. For this, we consider S ∈ F such that S = C j for all j = 1, . . . , n and suppose that S is a minimal, with respect to the inclusion, feasible coalition such that x(S) < v(S). Since ∅ ⊂ C1 ⊂ · · · ⊂ Cn−1 ⊂ Cn = N is a maximal chain, there exists j ∈ {1, . . . , n} such that S ⊂ C j and S  C j−1 . Then i j ∈ S and hence S ∪ C j−1 = C j ∈ F. By applying the convexity of v to the feasible coalitions S and C j−1 we obtain v(S) + v(C j−1 ) ≤



v(T ) + v(C j ).

T ∈CF (S∩C j−1 )

Note that S is minimal feasible coalition such that x(S) < v(S) and for each T ∈ CF (S ∩ C j−1 ) we have T ⊂ S. Thus x(T ) ≥ v(T ) for all T ∈ CF (S ∩ C j−1 ). Since x ≥ 0, we deduce x(S) + x(C j−1 ) = x(S ∪ C j−1 ) + x(S ∩ C j−1 ) = v(C j ) + x(S ∩ C j−1 )  ≥ v(C j ) + x(T ) T ∈CF (S∩C j−1 )

≥ v(C j ) +



v(T )

T ∈CF (S∩C j−1 )

≥ v(S) + v(C j−1 ) By using x(C j−1 ) = υ(C j−1 ) we conclude that x(S) ≥ v(S), which is a contradiction. 

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Corollary 17 Let (N , v, F) be a non-negative game in (N , F). If v is a monotone convex game in (N , F), then W eber(N , v, F) ⊆ Core(N , v, F). Moreover, any marginal worth vector is a vertex of Core(N , v, F). Proof For all C ∈ Ch(F) the marginal worth vector aC ∈ Core(N , v, F) which is a convex set. Thus conv{aC : C ∈ Ch(F)} ⊆ Core(N , v, F). Given aC ∈ Core (N , v, F), we have aC (C j ) = v(C j ) for all feasible coalition belonging to chain C. Then we obtain n equations which are linear independent and its solution aC is a vertex of the polytope Core(N , v, F).  Definition 18 A game (N , v, F) is superadditive if for all S, T ∈ F with S ∩ T = ∅ and S ∪ T ∈ F, we have v(S) + v(T ) ≤ v(S ∪ T ). We observe that every convex game is superadditive. Proposition 19 If W eber(N , v, F) ⊆ Core(N , v, F) then the game (N , v, F) is superadditive. Proof Let S, T ∈ F with S ∩ T = ∅ and S ∪ T ∈ F. Let C be a chain contains S ∪ T and we take x = aC ∈ Core(N , υ, F). Thus we conclude that υ(S ∪ T ) = x(S ∪ T ) = x(S) + x(T ) ≥ υ(S) + υ(T ). 

5 Hereditary Properties of (N, v, F ) In this section, we show that the superadditivity and the convexity of a monotone  game (N , v, F) imply the corresponding property of the extended game N , v F .   Theorem 20 If (N , v, F) is a monotone superadditive game then the game N , v F is superadditive and monotone.   Proof Suppose that N , v F is not superadditive. Then there exist coalitions S, T ⊆ N with S ∩ T = ∅ and components CF (S) = {S1 , . . . , Sn } , CF (T ) = {T1 , . . . , Tm } , such that v F (S) + v F (T ) > v F (S ∪ T ) and that n is as small as possible. If n ≥ 2 we consider the coalitions S \ Sn and Sn . Since |CF (Sn )| = 1 and |CF (S \ Sn )| = n − 1, we have v F (S \ Sn ) + v F (Sn ∪ T ) ≤ v F (S ∪ T ), v F (Sn ) + v F (T ) ≤ v F (Sn ∪ T ).

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Thus, we obtain v F (S ∪ T ) ≥ v F (S \ Sn ) + v F (Sn ) + v F (T ) = v F (S) + v F (T ), which is a contradiction. Hence we conclude that n = 1. Now CF (S) = {S1 }, CF (T ) = {T1 , . . . , Tm }, such that v F (S) + v F (T ) > v F (S ∪ T ) and we assume that m is as small as possible among all such counterexamples. We next consider the following  three cases.  (i) The set CF (S ∪ T ) = S1 , T1 , . . . , Tm where S1 ⊆ S1 and Ti ⊆ Ti for all i = 1, . . . , m. Since v is monotone, v F (S) + v F (T ) = v (S1 ) + v (T1 ) + · · · + v (Tm )       ≤ v S1 + v T1 + · · · + v Tm = v F (S ∪ T ), and this gives a contradiction. , . . . , Tm where 1 ≤ p < m, Ti ⊆ Ti for all (ii) The set CF (S ∪ T ) = C, Tp+1 i = p + 1, . . . , m and S1 ∪ T1 ∪ · · · ∪ Tp ⊆ C. Then ⎛ v F (S) + v F (T ) = v F (S) + v F ⎝T \

m





m

Ti ⎠ + v F ⎝

i=p+1

⎞ Ti ⎠ .

i=p+1

      Since |CF (S)| = 1 and CF T \ m T i=p+1 i  = p < m, we obtain ⎛ v F (S) + v F ⎝T \

m





Ti ⎠ ≤ v F ⎝(S ∪ T ) \

i=p+1

m

⎞ Ti ⎠ .

i=p+1

           m C = 1 and Moreover, using CF (S ∪ T ) \ m T T   =m−p< i F i i=p+1 i=p+1 m, we deduce that ⎛ ⎞ ⎛ ⎞ m m v F ⎝(S ∪ T ) \ Ti ⎠ + v F ⎝ Ti ⎠ ≤ v F (S ∪ T ). i=p+1

i=p+1

Therefore, v F (S) + v F (T ) ≤ v F (S ∪ T ) and this is a contradiction. (iii) The set CF (S ∪ T ) = {au(C)} where S1 ∪ T1 ∪ · · · ∪ Tm ⊆ au(C). Since S1 ⊂ au(C) and S1 is a maximal feasible subset of S there exists a chain of feasible coalitions S1 = S10 ⊂ S11 ⊂ · · · ⊂ S1k ⊂ · · · ⊂ au(C),

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  such that S1k \ S1k−1 = t k ⊆ T for all k = 1, . . . , |au(C) \ S1 |. Moreover, T1 ∪ p · · · ∪ Tm ⊆ au(C) and we may select the first coalition S1 in the chain such p p that S1 ∩ T j = ∅ for some j ∈ {1, . . . , m}. It follows that S1 ∪ T j ∈ F. Note that p p S1 ∩ T j = {t ∗ } and hence S1 \ {t ∗ } ∈ F satisfying 

 p   p   p S1 \ t ∗ ∪ T j = S1 ∪ T j and S1 \ t ∗ ∩ T j = ∅.

By using the superadditivity of v, we have    p   p   p   v S1 \ t ∗ + v T j ≤ v S1 ∪ T j = v F S1 ∪ T j . Therefore,     v F (S) + v F (T ) = v (S1 ) + v T j + v F T \ T j  p       ≤ v S1 \ t ∗ + v T j + v F T \ T j  p    ≤ v F S1 ∪ T j + v F T \ T j .      p Since CF (S1 ∪ T j ) = 1 and CF T \ T j  = m − 1 < m, we obtain  p     p  v F S1 ∪ T j + v F T \ T j ≤ v F S1 ∪ T = v F (S1 ∪ T ) = v F (S ∪ T ) , and this contradiction completes the proof of the superadditivity of v F . Finally, we will show that v F is monotone. Let S, T ⊆ N with S ⊆ T . Since v F is superadditive and v F (T \ S) ≥ 0, we conclude that v F (S) ≤ v F (S) + v F (T \ S)  ≤ v F (T ) .   Theorem 21 If (N , v, F) is a monotone convex game then the game N , v F is convex.   Proof We remark first that N , v F is superadditive and monotone. Next, suppose  that N , v F is not convex. Then there exist coalitions S, T ⊆ N with components CF (S) = {S1 , . . . , Sn } , CF (T ) = {T1 , . . . , Tm } , such that v F (S) + v F (T ) > v F (S ∩ T ) + v F (S ∪ T ) and that n is as small as possible among all pairs of coalitions satisfying the above inequality. Suppose that n ≥ 2 and let S \ Sn and Sn , where |CF (Sn )| = 1 and |CF (S \ Sn )| = n − 1. Then we have v F (S \ Sn ) + v F (Sn ∪ T ) ≤ v F ((S \ Sn ) ∩ T ) + v F (S ∪ T ), v F (Sn ) + v F (T ) ≤ v F (Sn ∩ T ) + v F (Sn ∪ T ).

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We note that ((S \ Sn ) ∩ T ) ∪ (Sn ∩ T ) = ((S \ Sn ) ∪ Sn ) ∩ T = S ∩ T , ((S \ Sn ) ∩ T ) ∩ (Sn ∩ T ) = ∅. The superadditivity of v F implies v F (S ∩ T ) ≥ v F ((S \ Sn ) ∩ T ) + v F (Sn ∩ T ). Therefore, v F (S ∩ T ) ≥ v F (S \ Sn ) + v F (Sn ∪ T ) − v F (S ∪ T ) +v F (Sn ) + v F (T ) − v F (Sn ∪ T ) = v F (S) + v F (T ) − v F (S ∪ T ), and this contradicts the hypothesis that n ≥ 2. We thus have CF (S) = {S1 } and we assume that m is as small as possible. There are two possible cases:   m = ∅ so that S1 ∩ T j = ∅ for all j = 1, . . . , m. (i) The intersection S1 ∩ T j j=1 Since v F is superadditive and monotone, we obtain ⎛ v F (S) + v F (T ) = v F (S1 ) + v F ⎝

m

⎞ Tj⎠

j=1 F

≤ v (S1 ∪ T1 ∪ · · · ∪ Tm ) ≤ v F (S ∪ T ), which is a contradiction. (ii) There exists at least p ∈ {1, . . . , m} such that S1 ∩ Tp = ∅. Since S1 ∩ Tp = ∅, Property P2 implies that S1 ∪ Tp ∈ F and applying the convexity of v,     v F (S) + v F (T ) = v (S1 ) + v Tp + v F T \ Tp      ≤ v (C) + v S1 ∪ Tp + v F T \ Tp C∈CF (S1 ∩Tp )

    = v F (S1 ∩ Tp ) + v F S1 ∪ Tp + v F T \ Tp . By using the monotonicity of v F , we obtain     v F (S) + v F (T ) ≤ v F (S ∩ Tp ) + v F S ∪ Tp + v F T \ Tp .     Since CF (S ∪ Tp ) = 1 and CF (T \ Tp ) = m − 1 < m, we have

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        v F S ∪ Tp + v F T \ Tp ≤ v F S ∪ Tp ∩ T \ Tp + v F (S ∪ T )    = v F S ∩ T \ Tp + v F (S ∪ T ). From the superadditivity of v F , we deduce that    v F (S ∩ Tp ) + v F S ∩ T \ Tp ≤ v F (S ∩ T ). Therefore,

v F (S) + v F (T ) ≤ v F (S ∩ T ) + v F (S ∪ T ), 

and this contradiction completes the proof.

Corollary  22 Let (N , v, F) be a monotone game. Then (N , v, F) is convex if and only if N , v F is convex.   Proof It sufficient to show that the convexity of N , v F implies that (N , v, F) is convex. Let us consider S, T ∈ F with S ∪ T ∈ F. Then v F (S) + v F (T ) ≤ v F (S ∩ T ) + v F (S ∪ T ), which implies v(S) + v(T ) ≤



v(C) + v(S ∪ T ),

C∈CF (S∩T )

and we obtain the result.



By using Proposition 10 and the classical characterization of convex games, we obtain the following characterization of the convexity for games on augmenting systems. Theorem 23 Let (N , v, F) be a monotone game. Then (N , v, F) is convex if and only if Core(N , v, F) = W eber(N , v F ).

6 Axioms for the Shapley Value In this section an axiomatization of the Shapley value for games defined on augmenting systems will be showed. To this end we recall the following concepts and properties. Definition 24 Let (N , F) be an augmenting system. For a feasible coalition S ∈ F, we define the set au(S) = {i ∈ N \ S : S ∪ i ∈ F} of augmentations of S and the set S + = S ∪ au(S) = {i ∈ N : S ∪ i ∈ F}.   Proposition 25 Let (N , F) be an augmenting system. Then the interval S, S + F =   C ∈ F : S ⊆ C ⊆ S + is a Boolean algebra for every nonempty S ∈ F.

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Definition 26 Let (N , F) be an augmenting system. An element i in S ∈ F is an extreme point of S if S \ i ∈ F. The set of extreme points of S is denoted by ex(S). Note that Property P3 implies A3 and hence |ex (S)| ≥ 1 for any nonempty S ∈ F. Definition 27 A cooperative game on the augmenting system (N , F) is a triple (N , v, F), where v : F → R is a real-valued function such that v(∅) = 0. The coalitions are the feasible sets belonging to F and the players are the elements of N . Let (F) be the real vector space of the games on the the augmenting system F ⊆ 2N . We will follow the work of Weber [27] to obtain an axiomatic development of the Shapley value for games on augmenting system. This way to extend the Shapley value is the logical path to obtain the adaptation of the classical axioms (linearity, dummy, efficiency, and symmetry) to cooperative games on combinatorial structures. For this, we consider the following game on F. For any T ∈ F, T = ∅, the identity game δT : F → R is defined by  δT (S) :=

1 if S = T , 0 if S = T .

Let  : (F) → Rn a map such that  (v) = (1 (v), . . . , n (v)). The meaning of this function is to give the expected payoffs to the players of a game. We introduce several axioms that give rise to a unique function for games on augmenting systems. If F = 2N then this function is equal to the classical Shapley value. First, we consider the linearity property. Theorem 28 Let i : (F) → R be a value for i, a player in N , which satisfies the linearity axiom. Then there exists an unique set of coefficients aSi : S ∈ F, S = ∅ such that  aSi v (S) , i (v) = {S∈F : S=∅}

for every v ∈ (F). Proof The collection {δS : S ∈ F, S = ∅} is a basis of the vector space (F). Then v = {S∈F : S=∅} v(S)δS for every game v ∈ (F). We denote aSi = i (δS ) for every i ∈ N , and every nonempty S ∈ F . Applying the linearity axiom i (v) =  i  {S∈F : S=∅} aS v(S), for every v ∈ (F). We will now introduce the concept of dummy player. Definition 29 The player i ∈ N is a dummy player in the game v ∈ (F) if for all S ∈ F such that i ∈ au(S), we have  v (S ∪ i) − v(S) =

v ({i}) if {i} ∈ F, 0 otherwise.

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This definition derives from the observation that a dummy player has no strategic role in the game, because of such a player contributes precisely v ({i}) or zero. We need a preparatory lemma about some properties of the dummy player in the identity game. Lemma 30 Let (N , F) be an augmenting system and consider a nonempty S ∈ F. Then: (i) If i ∈ S \ ex (S) then player i is dummy in the identity game δS . (ii) If i ∈ / S + then player i is dummy in the identity game δS . (iii) If i ∈ au(S) then player i is dummy in the game δS + δS∪i . Proof 1. Note that if {i} ∈ F then i ∈ ex ({i}), and hence S = {i}. This implies δS ({i}) = 0. Now let C ∈ F be such that i ∈ au(C), and it is sufficient to prove that δS (C ∪ i) − δS (C) = 0. If S = C ∪ i then C = S \ i ∈ F, so that i ∈ ex (S), a contradiction. If S = C then i ∈ au(S) = S + \ S, which is a contradiction. Thus, δS (C ∪ i) = 0 and δS (C) = 0. 2. If S = {i} ∈ F then i ∈ S + , contradicting the hypothesis. Then δS ({i}) = 0. Consider C ∈ F such that i ∈ au(C). Since i ∈ / S + = S ∪ au(S) we have S = C ∪ i and S = C, because otherwise i ∈ S or i ∈ au(S). Thus, δS (C ∪ i) − δS (C) = 0. 3. If {i} ∈ F where i ∈ au(S) then i ∈ / S, and hence S = {i}. Since S = ∅ we also know that S ∪ i = {i}. Then δS ({i}) + δS∪i ({i}) = 0. We now take C ∈ F such that i ∈ au(C). Since i ∈ au(S) = S + \ S we obtain S = C ∪ i and S ∪ i = C, because otherwise i ∈ S or i ∈ C. Thus, (δS + δS∪i ) (C ∪ i) − (δS + δS∪i ) (C) = δS∪i (C ∪ i) − δS (C). Finally, S ∪ i = C ∪ i ⇔ S = C implies δS∪i (C ∪ i) − δS (C) = 0, for all C ∈ F such that i ∈ au(C).  The following axiom gives the payoff received for a dummy player. Dummy axiom If the player i ∈ N is a dummy in v ∈ (F), then  i (v) =

v ({i}) if {i} ∈ F, 0 otherwise.

Theorem 31 Let i : (F) → R be a value for player i ∈ N that satisfies linearity and dummy axioms. Then, for every game v ∈ (F), i (v) =



aTi ∪i [v(T ∪ i) − v(T )] .

{T ∈F : i∈au(T )}

Moreover, if {i} ∈ F then

 {T ∈F : i∈au(T )}

aTi ∪i = 1.

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Proof We know from Theorem 28 that for a fix player i ∈ N , 

i (v) =

aSi v (S)

{S∈F : S=∅}



=



aSi v (S) +

{S∈F : i∈ex(S)}



aSi v (S) +

{S∈F : i∈S} /

aSi v (S) .

{S∈F : i∈S\ex(S)}

Lemma 30 (i) implies that if i ∈ S \ ex (S) then player i is dummy in the identity we obtain aSi =  game δS . Applying dummy axiom  i (δS ) = 0, for all i ∈ S \ ex (S).  Moreover, N \ S = au(S) ∪ N \ S + and au(S) ∩ N \ S + = ∅, and then we have 

i (v) =

{S∈F : i∈ex(S)}



=



aSi v (S) +

aSi v (S)

{S∈F : i∈S} /

aSi v (S)



+

{S∈F : i∈ex(S)}

aSi v (S) +

{S∈F : i∈au(S)}



aSi v (S) .

{S∈F : i∈S / +}

If i ∈ / S + then player i is dummy in the identity game δS by Lemma 30 (ii). Hence / S + . This shows that dummy axiom implies aSi = i (δS ) = 0, for each i ∈ 

i (v) =



aSi v (S) +

{S∈F : i∈ex(S)}

aSi v (S) .

{S∈F : i∈au(S)}

Since i ∈ ex (S) ⇔ S \ i ∈ F ⇔ S = T ∪ i, where T ∈ F and i ∈ au(T ), we have 



aSi v (S) =

{S∈F : i∈ex(S)}

aTi ∪i v (T ∪ i) .

{T ∈F : i∈au(T )}

If i ∈ au(S) then player i is dummy in the game δS + δS∪i by Lemma 30 (iii). By linearity and dummy axioms i = i (δS ) + i (δS∪i ) = i (δS + δS∪i ) = 0, aSi + aS∪i i for all i ∈ au(S). Then the above properties yield which implies that aSi = −aS∪i

i (v) =



aTi ∪i [v(T ∪ i) − v(T )] .

{T ∈F : i∈au(T )}

Now we suppose that {i} ∈ F and compute  {T ∈F : i∈au(T )}

aTi ∪i =

 {T ∈F : i∈au(T )}

⎛ i (δT ∪i ) = i ⎝



{T ∈F : i∈au(T )}

⎞ δT ∪i ⎠ .

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 We claim that player i is dummy in the game w = {T ∈F : i∈au(T )} δT ∪i . Observing that for any C ∈ F such that i ∈ au(C) the feasible set T ∪ i = C, we obtain  δT ∪i (C ∪ i) − δT ∪i (C) =

1 if C = T , 0 if C = T .

This implies that w (C ∪ i) − w (C) = 1. Observe that {i} ∈ F implies i ∈ ∅+ , and hence w ({i}) = δ∅∪i ({i}) = 1. This proves the claim. Finally, by using dummy ax iom, we get i (w) = w ({i}) = 1. If the vector (v) = (1 (v), . . . , n (v)) is a distribution of the available resources to the grand coalition N ∈ F, then  satisfies the following axiom: Efficiency axiom If (N , F) is an augmenting system such that N ∈ F and v ∈   (F) then i∈N i (v) = v(N ). The efficiency axiom implies the following properties for the coefficients of the values that satisfy linearity and dummy axioms. We will assume throughout that (N , F) is an augmenting system such that N ∈ F. Theorem 32 Let  : (F) → Rn be a value defined for every game v ∈ (F) and every player i ∈ N by 

i (v) =

i aS∪i [v(S ∪ i) − v(S)] .

{S∈F : i∈au(S)}

Then  satisfies the efficiency axiom if and only if 

aNi = 1, and

i∈ex(N )





aSi =

i∈ex(S)

i aS∪i

i∈au(S)

for every nonempty S ∈ F such that S = N . Proof First, we compute the sum 

i (v) =





i aS∪i [v(S ∪ i) − v(S)]

i∈N {S∈F : i∈au(S)}

i∈N

=





v(S) ⎝

S∈F

⎛ =⎝



i∈ex(N )



aSi −

i∈ex(S)



aNi ⎠ v(N ) +



⎞ i ⎠ aS∪i

i∈au(S)

 {S∈F : S=N }







aSi −

i∈ex(S)

By considering v(S) as variables, we conclude that if the relations are true.

 i∈N



⎞ i ⎠ v(S). aS∪i

i∈au(S)

i (v) = v(N ) if and only 

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Let v : 2N → R be a standard cooperative game and let π a total ordering of the elements of N , given by i1 < i2 < · · · < in . The classical Shapley value for the player i ∈ N is given by Shi (N , v) =

   1   i v π ∪ {i} − v π i , n! π∈ n

where n is the set of all permutations of N and π i is the set of the predecessors of player i in the order π. Let us consider a compatible ordering of an augmenting system (N , F) such that  as the total ordering of N , given by i1 < i2 < · · · < in such that the set  N ∈ F, i1 , . . . , i j ∈ F for all j = 1, . . . , n. A compatible ordering of (N , F) corresponds exactly to a maximal chain in F and we denote by Ch(F) the set of all the maximal chains in F. Given an element i ∈ N and a compatible ordering C ∈ Ch(F), let C(i) = { j ∈ N : j ≤ i in C}. Let (N , F) be an augmenting system and let v : F → R a cooperative game. We define the Shapley value for the player i ∈ N as Shi (N , v, F) :=

 1 [v (C(i)) − v (C(i) \ i)] , c (N ) C∈Ch(F )

where c (N ) := |Ch(F)| is the total number of maximal chains in F. Since C(i) \ i = S ∈ F we have that i ∈ au(S). Thus, ⎞ 1 ⎠ [v(S ∪ i) − v(S)] ⎝ Shi (N , v, F) = c (N ) {S∈F : i∈au(S)} {C∈Ch(F ) : C(i)\i=S} 

=







c(S) c (S ∪ i, N ) [v(S ∪ i) − v(S)] , c(N ) {S∈F : i∈au(S)}

where c(S) is the number of maximal chains from ∅ to S, and c (S ∪ i, N ) is the number of maximal chains from S ∪ i to N . As a consequence, we obtain the following formula for the Shapley value of games on augmenting systems. Definition 33 Let v : F → R be a game on an augmenting system (N , F) such that N ∈ F. The Shapley value for the player i ∈ N is given by Shi (N , v, F) =



c(S) c (S ∪ i, N ) [v(S ∪ i) − v(S)] . c(N ) {S∈F : i∈au(S)}

Note that the sum of the coefficients of the Shapley value is

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⎞  1 1 ⎠= ⎝ = 1, c (N ) c (N ) {S∈F : i∈au(S)} {C∈Ch(F ) : C(i)\i=S} C∈Ch(F ) 





and this implies that the Shapley value satisfies the dummy axiom. Moreover, for every game v ∈ (F) we have 









⎝ 1 [v (C(i)) − v (C(i) \ i)]⎠ c (N ) i∈N C∈Ch(F )     1 = [v (C(i)) − v (C(i) \ i)] c (N ) C∈Ch(F ) i∈N

Shi (N , v, F) =

i∈N

=

 1 [v (N ) − v (∅)] c (N ) C∈Ch(F )

= v (N ) , which implies the efficiency axiom. Since the classical axiom of symmetry does not work, we consider a new axiom in which there is a relationship between the number of chains and the value of the identity game. Chain axiom Let (N , F) be an augmenting system such that N ∈ F and  : (F) → Rn a value.  For any S ∈ F such that S = N and any i, j ∈ au(S), we have c(S ∪ i, N )  j δS∪ j = c(S ∪ j, N ) i (δS∪i ). Combining this axiom with the efficiency axiom, we obtain the probability that a player joins coalition S ∈ F over the set Ch(F) of all the maximal chains in F. By using the previous results we prove the following characterization of the Shapley value for games on augmenting systems. Theorem 34 The Shapley value is the unique value  : (F) → Rn that satisfies linearity, dummy, efficiency and chain axioms. Proof Clearly the Shapley value satisfies all the four axioms. Conversely, let  be a value that satisfies linearity, dummy, efficiency and chain axioms. It follows from Theorems 31 and 32 that for every game v ∈ (F) and every player i ∈ N , 

i (v) =

i aS∪i [v(S ∪ i) − v(S)] ,

{S∈F : i∈au(S)}

where the coefficients satisfy  i∈ex(N )

aNi = 1, and

 i∈ex(S)

aSi =

 i∈au(S)

i aS∪i

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for every nonempty S ∈ F such that S = N . Thus, it suffices to show that c(S) c (S ∪ i, N ) c(N )

i = aS∪i

(1)

for every S ∈ F such that S = N and i ∈ au(S). Note that the chain axiom is j

i c(S ∪ i, N ) aS∪ j = c(S ∪ j, N ) aS∪i

for all i, j ∈ au(S). Let us consider a fix coalition S ∈ F such that S = N with i ∈ au(S), and we compute 

j



c(S ∪ j, N ) i a c(S ∪ i, N ) S∪i { j∈au(S) : j=i} ⎤ ⎡ i  aS∪i ⎣c(S ∪ i, N ) + = c(S ∪ j, N )⎦ c(S ∪ i, N ) { j∈au(S) : j=i}

i aS∪ j = aS∪i +

j∈au(S)

i = aS∪i

c(S, N ) . c(S ∪ i, N )

For S = ∅ the above equality is 

j

i a{ j} = a{i}

j∈au(∅)

c(N ) , c({i} , N )

where { j ∈ N : j ∈ au(∅)} = { j ∈ N : { j} ∈ F}. By using recursively the efficiency equations   i aSi = aS∪i , i∈ex(S)

i∈au(S)

and the equivalence i ∈ au(S) ⇔ i ∈ ex (S ∪ i), we calculate the sum 

j

a{ j} =

 j∈ex({ j})

j∈au(∅)

=



=



=

i∈ex(N )

aNi



= 1.



aSi =

{S∈F : |S|=n−1} i∈ex(S)



i aS∪i

{S∈F : |S|=1} i∈au(S)



{S∈F : |S|=2} i∈ex(S)

.. .





j

a{ j} =



i aS∪i

{S∈F : |S|=2} i∈au(S)

aSi =





{S∈F : |S|=n−1} i∈au(S)

i aS∪i

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Therefore, we have showed formula (1) for S = ∅, that is, for every i ∈ au(∅), i = a{i}

c({i} , N ) . c(N )

We assume the following induction hypothesis: For every T ∈ F such that |T | = k, where 0 ≤ k ≤ n − 2, we have j

aT ∪ j =

c(T ) c (T ∪ j, N ) c(N )

for all j ∈ au(T ). The case |T | = 0 ⇔ T = ∅ has just been proved. Let now S ∈ F such that |S| = k + 1 ≤ n − 1. Then ∅ = S = N , and hence the efficiency equations imply that 

j

aS∪ j =

j∈au(S)

 j∈ex(S)

j

aS =



j

a(S\ j)∪ j

j∈ex(S)

 c(S \ j) c (S, N ) = c(N ) j∈ex(S) =

c(S) c (S, N ) , c(N )

where we have used the induction hypothesis for T = S \ j where j ∈ ex (S). Moreover,  j c(S, N ) i , aS∪ j = aS∪i c(S ∪ i, N ) j∈au(S) which implies the formula (1) for S ∈ F such that |S| = k + 1. This proves that i (N , v, F) = Shi (N , v, F) for every v ∈ (F) and every i ∈ N .  Remark 35 Note that if F = 2N then {S ∈ F : i ∈ au(S)} = {S ⊂ N : i ∈ / S}. Thus, for every game v : 2N → R and every i ∈ N , we have Shi (N , v) =

 {S⊂N : i∈S} /

|S|! (|N | − |S| − 1)! [v(S ∪ i) − v(S)] . |N |!

7 Another Axiomatization of the Shapley Value Given an augmenting system (N , F) we consider the partially ordered set (or poset) (F, ⊆). Let us denote by Int(F) the set of intervals of (N , F), that is the collections [S, T ] = {R ∈ F : S ⊆ R ⊆ T }, where S, T ∈ F and S ⊆ T . We define the zeta function ζ : Int(F) → R by ζ (S, T ) = 1 for all S, T ∈ F such that S ⊆ T . The i-

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dentity function δ : Int(F) → R is defined by δ (S, T ) = 1 if S = T and δ (S, T ) = 0 otherwise. The convolution of the functions f, g : Int(F) → R is 

( f ∗ g) (S, T ) =

f (S, R) g (R, T ) ,

{R∈F : S⊆R⊆T }

and the identity function satisfies f ∗ δ = δ ∗ f = f . Moreover, the zeta function ζ is invertible, its inverse is called the Möbius function and is denoted μ (see Stanley [26, Sect. 3.7]). Lemma 36 Let (N , F) be an augmenting system. Then the Möbius function of the poset (F, ⊆) is  (−1)|T |−|S| if T ⊆ S + , μ (S, T ) = 0 otherwise. Proof It suffices to show that  (μ ∗ ζ) (S, T ) = δ (S, T ) =

1 if S = T , 0 if S ⊂ T ,

for all S, T ∈ F such that S ⊆ T . By Proposition 25 we obtain 

(μ ∗ ζ) (S, T ) =

μ (S, R) ζ (R, T )

{R∈F : S⊆R⊆T }



=

μ (S, R)

{R∈F : S⊆R⊆T }



=

(−1)|R|−|S| .

{R∈2N : S⊆R⊆S + , R⊆T } If S = T then R = S and hence (μ ∗ ζ) (S, T ) = (−1)|S|−|S| = 1. If S ⊂ T then we consider two cases: 1. Assume S + ⊆ T and let C = R \ S. Then 

(μ ∗ ζ) (S, T ) = {

R∈2N

=

(−1)|R|−|S|

: S⊆R⊆S +



} (−1)|C|

{C∈2N : C⊆S + \S } = (1 − 1)|S

+

\S |

=



1 if S = S + , 0 otherwise.

Since S = S + implies that S = N , we obtain a contradiction with S ⊂ T , and hence (μ ∗ ζ) (S, T ) = 0.

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2. Assume S +  T and let au(ST ) = {i ∈ T \ S : S ∪ i ∈ F}. Then     R ∈ 2N : S ⊆ R ⊆ S + , R ⊆ T = R ∈ 2N : S ⊆ R ⊆ S ∪ au(ST ) . We now obtain 

(μ ∗ ζ) (S, T ) =

(−1)|R|−|S|

{R∈2N : S⊆R⊆S∪au(ST )}  1 if Sau(T ) = ∅, |au(ST )| = (1 − 1) = 0 otherwise. / F for any i ∈ T \ S. This contradicts Note that au(ST ) = ∅ implies that S ∪ i ∈ Property P3 of the augmenting system and we conclude that (μ ∗ ζ) (S, T ) = 0.  For any T ∈ F such that T = ∅, the unanimity game ζT : F → R is defined by  ζT (S) :=

1 if T ⊆ S, 0 otherwise.

The collections of the identity games {δS : S ∈ F, S = ∅} and the unanimity games {ζT : T ∈ F, T = ∅} are two different bases of the vector space (F). Faigle and Kern [13] observed that  δS . ζT = {S∈F : S⊇T }

Theorem 37 Let v : F → R be a game on an augmenting system. Then  there exists an unique set of coefficients {dv (T ) : T ∈ F, T = ∅} such that v = {T ∈F : T =∅} dv (T ) ζT . Moreover,  dv (S) = (−1)|S|−|T | v(T ). {T ∈F : T ⊆S⊆T + }

for every nonempty S ∈ F. Proof The collection of the unanimity games {ζT : T ∈ F, T = ∅} is a basis of the vector space (F). Then, for every game v ∈ (F), v=



dv (T ) ζT ,

{T ∈F : T =∅}

and hence for every nonempty S ∈ F we have that v (S) =

 {T ∈F : T =∅}

dv (T ) ζT (S) =

 {T ∈F : T ⊆S}

dv (T ) .

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Applying the Möbius inversion formula [26, Chap. 3] of the poset (F, ⊆) and Lemma 36, we obtain   μ (T , S) v (T ) = (−1)|S|−|T | v(T ). dv (S) = {T ∈F : T ⊆S}

{T ∈F : T ⊆S⊆T + }



Let (N , F) be an augmenting system such that N ∈ F. Following the work of Faigle and Kern [13], we define the hierarchical strength hS (i) of a player i ∈ S in a feasible coalition S ∈ F as follows: hS (i) :=

|{C ∈ Ch(F) : C(i) ⊇ S}| . c (N )

Note that hS (i) is the average number of maximal chains of (F, ⊆) in which player i ∈ S is the last member of S in the chain. By using these numbers we will obtain a new formula for the Shapley value. Proposition 38 Let v : F → R be a game on an augmenting system (N , F) such that N ∈ F. The Shapley value for the player i ∈ N is given by 

Shi (N , v, F) =

dv (S) hS (i) ,

{S∈F : i∈S}

where dv (S) are the coefficients associated to the unanimity basis.  Proof Since v = {S∈F : S=∅} dv (S) ζS the linearity of the Shapley value implies that Shi (N , v, F) =



dv (S) Shi (N , ζS , F) .

{S∈F : S=∅}

For every nonempty S ∈ F and i ∈ N we compute Shi (N , ζS , F) =

 1 [ζS (C(i)) − ζS (C(i) \ i)] . c (N ) C∈Ch(F )

If i ∈ / S then S ⊆ C(i) implies S ⊆ C(i) \ i, and hence  ζS (C(i)) − ζS (C(i) \ i) =

1 if S ⊆ C(i) and i ∈ S, 0 otherwise,

for every chain C ∈ Ch(F). Thus we obtain  Shi (N , ζS , F) = which completes the proof.

hS (i) if i ∈ S, 0 if i ∈ / S, 

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Now we are ready to introduce a new axiom which gives rise to another axiomatization of the Shapley value. Hierarchical strength axiom Let (N , F) be an augmenting system such that N ∈ F and  : (F) → Rn a value. For any nonempty S ∈ F and any i, j ∈ S, we have hS (i)  j (ζS ) = hS ( j) i (ζS ). This axiom implies that players in unanimity games be rewarded according to their relative hierarchical strengths. Moreover, it reflect that the Shapley value is the expected marginal contribution of an individual player to the game. Note also that the above proposition implies that the Shapley value satisfies the hierarchical strength axiom. Theorem 39 The Shapley value is the unique value  : (F) → Rn that satisfies linearity, dummy, efficiency and hierarchical strength axioms. Proof We know  that the Shapley value satisfies the four axioms. Since  is a linear map and v = {T ∈F : T =∅} dv (T ) ζT , it suffices to prove that  coincides with the Shapley value on any game ζT , where T ∈ F and T = ∅. Fix a nonempty T ∈ F and i ∈ N . We show that any i ∈ / T is a dummy player in the game ζT . For this, let S ∈ F such that i ∈ au(S). Since i ∈ / T we have that T ⊆ S ∪ i implies T ⊆ S, and hence ζT (S ∪ i) − ζT (S) = 0. Moreover, if {i} ∈ F then ζT ({i}) = 0. By using the / T. dummy axiom we obtain that i (ζT ) = 0 for every i ∈ By applying the efficiency axiom we have  i∈T

i (ζT ) =



i (ζT ) = ζT (N ) = 1.

i∈N

Now we fix i ∈ T and the hierarchical strength axiom gives  j (ζT ) =

hT ( j) i (ζT ) , hT (i)

for every j ∈ T such that j = i. Thus 1=



i (ζT ) = i (ζT ) +

i∈T

=

i (ζT )  hT ( j). hT (i) j∈T



hT ( j) i (ζT ) h (i) { j∈T : j=i} T

Since in every chain C ∈ Ch(F) there exists an unique j ∈ T such that C( j) ⊇ T , we have  hT ( j) = 1, j∈T

and hence we conclude that i (ζT ) = hT (i) for every i ∈ T . Therefore, i (ζT ) = Shi (ζT ) for every nonempty T ∈ F and i ∈ N . 

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8 The Banzhaf Value for Augmenting The classical Banzhaf value has been defined by Owen [22]. For a player i ∈ N the value has the following expression. βi (N , v) =

1 {S∈2N :i∈S} / [v(S ∪ i) − v(S)] 2|N |−1

First we begin with the Banzhaf value for augmenting systems. Let v ∈ (F) a game in the augmenting system (N , F). We will assume throughout that the augmenting system (N , F) is normal. Definition 40 In the above conditions we define the Banzhaf value for a player i ∈ N as 1 {S∈2N :i∈au(S)} [v(S ∪ i) − v(S)] βi (N , F, v) = A(i) where A(i) = |{S ∈ F : S ∪ i ∈ F}|. Note that if F = 2N then A(i) = 2|N1|−1 and our value coincides with the traditional Banzhaf value. In order to prove the formula we consider the identity game δT where T ∈ F and we introduce several axioms. The first is the linearity axiom. Proposition 41 Let i a value in (F) that satisfies the linearity axiom then it exists a unique set of coefficients {aSi : S ∈ F, S = ∅} such that i (v) = {S∈F ,S=∅} aSi v(S) for every v ∈ (F). Proof As the collection {δS : S ∈ F, S = ∅} is a basis of (F) we can write that v = {S∈F ,S=∅} v(S)δS . Let aSi = i (δS ) for each i ∈ N , and every nonempty S ∈ F. Applying the linearity axiom to v we obtain the result.  The following axiom, dummy player, is well-known. First notice that if a player is dummy then our value is  βi (N , F, v) =

1  v(i) A(i) {S∈F :i∈au(S)}

if i ∈ F

0 otherwise

i.e., our value verifies the dummy axiom. The following axiom is new and it gives an anonymity property with respect to its enlarged coalitions of the payoff received for a player. Axiom 42 (Augmenting axiom) In the above context let i ∈ N and S, T ∈ F such that i, j ∈ au(S), it holds i (δS∪i ) = i (δT ∪i )

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We show that our value verifies the augmenting axiom. Proposition 43 In the above conditions the Banzhaf value for augmenting systems verifies the augmenting axiom. Proof Let (N , F) a normal augmenting system and v ∈ (F). Let S ∈ F and i ∈ au(S), we compute βi (N , δS∪i , F) =

1 {R∈F :i∈au(R)} [δS∪i (R ∪ i) − δS∪i (R)] A(i)

Since i ∈ au(R) we have that i ∈ / R and hence S ∪ i = R, which implies that δS∪i (R) = 0. Also notice that δS∪i (R ∪ i) = 1 if and only if S = R, and hence i ∈ au(S). Then we can write  1 if i ∈ au(S) βi (N , δS∪i , F) = A(i) 0 otherwise 

which implies the augmenting axiom. We can now to prove the following theorem.

Theorem 44 Let (N , F) be a normal augmenting system. Then the Banzhaf value is the unique value in (F) that verifies linearity, dummy and augmenting axioms. Proof We know that our value verifies the three axioms. Consider now another value,  in (F), also verifying the same axioms. As this value verifies dummy and linearity by proposition 5.3. we can write for all i ∈ N i (v) = {S∈F ,i∈au(S)} aS∪i [v(S ∪ i) − v(S)] and if {i} ∈ F then

{S∈F ,S∪i∈F } aS∪i = 1

Let S, T ∈ F such that i ∈ au(S) ∩ au(T ). As  verifies the augmenting axiom we i = aTi ∪i . As consequence have (δS∪i ) = (δT ∪i ) and hence aS∪i i |{T ∈ F : i ∈ au(T )}| = 1 {T ∈F :i∈au(T )} aTi ∪i = aS∪i

i.e., i = aS∪i

and hence (v) = β(v) for all v ∈ (F)

1 A(i) 

In the following sections we will introduce the concept of decreasing systems. We will study its properties, its relation to the augmenting systems and we will give an axiomatization of both the value of Shapley and that of Banzhaf.

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9 Decreasing Systems We introduce the new structure. Definition 45 A decreasing system is a set system (N , D) with the following properties: 1. N ∈ D 2. for all S, T ∈ D, with S ∪ T = N then S ∩ T ∈ D 3. for S, T ∈ D with S ⊂ T there exists i ∈ T \ S such that T \ i ∈ D. A decreasing system will called normal if N ∈ D The relationships between the combinatorial structures above mentioned is given in the next propositions. Proposition 46 A normal decreasing system D is a convex geometry if and only if D is closed under the intersection operation. Proof The second axiom for convex geometries is satisfied because we suppose that D is closed under the intersection operation. Now, we consider A ∈ D, such that A = N . By Property 3 of decreasing systems we can obtain a sequence i1 , i2 , · · · , i|N −A| such that A ⊂ A ∪ i|N −A| ⊂ · · · ⊂ A ∪ {i1 , i2 , · · · , i|N −A| } and all the elements of the chain are in D. Then the axiom 3 for convex geometries hold.  Proposition 47 Let (N , D) a normal decreasing system, then for every {i} ∈ D, the set system (N \ i, Di ) is a convex geometry where Di = {S ∈ D : i ∈ / S}. Proof First notice that the sets N \ i and ∅ are in Di . To check the second axiom for convex geometries let S, T ∈ Di . Because S ∪ T = N we have that S ∩ T ∈ D and as consequence S ∩ T ∈ Di . The third axiom for convex geometries is a direct  consequence of the third for decreasing systems applied to Di . As examples of decreasing systems we have D = 2N , D = {N − i, ∅}, and D = {N } ∪ {S ⊂ N : |S| = n − 1}. Another example related with graphs is the following. Let G = (N , E) a graph. The system (N , D) given by D = {S ⊆ N : (N \ S, E(N \ S)) is a connected subgraph of G} is a decreasing system. We conclude this section with a proposition that relates the augmenting and decreasing system structures. First, we define a normal augmenting system as an augmenting system F where the sets ∅ and N are in F. Proposition 48 A family F of subsets of a set N is a normal augmenting system if and only if the family D of complement sets of sets belonging to F is a normal decreasing system (Fig. 2). Proof Suppose first that (N , F) is a normal augmenting system, then ∅ and N are in D. Let now S, T ∈ D such that S ∪ T = N , then (N \ S) ∩ (N \ T ) = ∅ and from axiom two for augmenting systems we have that (N \ S) ∪ (N \ T ) ∈ F. This implies

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Fig. 2 Decreasing system. Dual structure of the augmenting in Fig. 1

1234

123

234

12

13

24

34

1

2

3

4

that S ∩ T ∈ D. To check the last axiom for decreasing systems let S, T ∈ D, such that S ⊂ T . Then (N \ T ) ⊂ (N \ S). As these sets are in F and applying the third axiom for augmenting system, it exists i ∈ (N \ S) \ (N \ T ) such that (N \ T ) ∪ i ∈ F. As consequence T \ i ∈ D and the axiom follows. 

10 Dual Games In order to prove the following results, we will to form a isomorphism  between the vector spaces (F) and (D) where F is a normal augmenting system and D is its dual normal decreasing system. Then for v ∈ (F), (v) = v ∗ where v ∗ (S) = v(N ) − v(N \ S), namely, v ∗ is the dual game of v. Note that v ∗∗ = v. Proposition 49 Let F be a normal augmenting system, and D its dual decreasing system. Then for every S ∈ F such that S = N , ∅, it holds: 1. The dual of the unanimity game ζN \S ∈ (D) is the game μS ∈ (F) defined, for each T ∈ F as  0 if T ⊆S μS (T ) = 1 otherwise 2. The dual of the identity game δN \S ∈ (D) is the game ρS ∈ (F) defined, for each T ∈ F as  −1 i f T = S ρS (T ) = 0 otherwise 3. The dual of the unanimity game δN ∈ (D) is the game ρ∅ ∈ (F) defined, for each T ∈ F as ρ∅ (T ) = 1, for all T ∈ F and ρ∅ (∅) = 0.

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Proof Let T ∈ F. The dual game of ζN \S is determined by ζN∗ \S (T )

 = ζN \S (N ) − ζN \S (N \ T ) = 1 −

1, i f N \ S ⊆ N \ T 0, otherwise

Then we have that ζN∗ \S (T ) = μS (T ). For the identity game δN \S , where S = ∅, we obtain: δN∗ \S (T )

 = δN \S (N ) − δN \S (N \ T ) = 0 −

i.e.,

1, i f N \ S = N \ T 0, otherwise

δN∗ \S (T ) = ρS (T )

If S = ∅ we have δN∗ (T ) i.e.

 = δN (N ) − δN (N \ T ) = 1 −

1, i f N \ T = N 0, otherwise

δN∗ (T ) = ρ∅ (T )

 We introduce some results about monotony and convexity for games in augmenting systems that are translated, in the dual sense, to framework of decreasing systems. Definition 50 Let (N , F) a normal augmenting system. A game v : F → R is monotone if for all S, T ∈ F with S ⊆ T , it holds that v(S) ≤ v(T ). In same way we can define the concept of monotony for decreasing systems. Now, we introduce a new definition of convexity for augmenting and decreasing systems. Definition 51 Let (N , F) a normal augmenting system. A game v : F → R is said convex on F if for all S, T ∈ F, such that S ∪ T ∈ F then, v(S ∪ T ) + C∈CF (S∩T ) v(C)  v(S) + v(T ) where CF (S ∩ T ) = {C, maximal feasible sets, such that C ⊆ S ∩ T }. In other hand, v is said concave if the inequality exposed before change of sense. Definition 52 Let (N , D) a normal decreasing system. A game v : D → R is said convex on D if for all S, T ∈ D, such that S ∩ T ∈ D then, C∈CD (S∪T ) [V (N ) − v(C)] − v(S ∩ T )  v(S) + v(T )

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where CD (S ∪ T ) = {C ∈ D : S ∪ T ⊂ Cand C minimal}. In other hand, v is said concave if the reverse inequality holds. Proposition 53 Let (N , F) a normal augmenting system and (N , D) its dual decreasing system. Then it holds. 1. v ∈ (D) is monotone in (N , D) if and only if v ∗ is monotone in (N , F). 2. v ∈ (D) is convex in (N , D) if and only if v ∗ is concave in (N , F). Proof Let v ∈ (D) and suppose that v is monotone. If S, T ∈ D, with S ⊆ T , then N \ T ⊆ N \ S and then v(N \ T ) ≤ v(N \ S). Then, v ∗ (S) = v(N ) − v(N \ S) ≤ v(N ) − v(N \ T ) = v ∗ (T ). We suppose now that v is convex. Thus, for any S, T ∈ F, such that S ∪ T ∈ F, we have v ∗ (S ∪ T ) + C∈CF (S∩T ) v ∗ (C) = v(N ) − v(N \ (S ∪ T )) + C∈C(F ) (S∩T ) [v(N ) − v(N \ C)] ≥ = 2v(N ) − v(N \ S) − v(N \ T ) = v ∗ (S) + v ∗ (T )



11 The Shapley Value for Decreasing Systems In this section we will axiomatize the Shapley value for decreasing systems. In what follows we fix the set of players N , (N , F) as a normal augmenting system and v a game in (N , F). A value in F is a function  : F → RN . Each coordinate of , i : F → R, is understood as the payment of the corresponding player. We will introduce several axioms to axiomatize the Shapley value in decreasing systems, but before we give some necessary definitions for its correct definition. Definition 54 Let i ∈ N and S ∈ F such that i ∈ au(S). The marginal contribution of player i respect to S is defined as v(S ∪ i) − v(S). Analogously let (N , D) be a normal decreasing system and w ∈ (D). For j ∈ N and S ∈ D such that S \ i ∈ ex(S), the marginal contribution of i in S is defined as w(S) − w(S \ i). Remark 55 Let (N , F) a normal augmenting system and let (N , D) its dual decreasing system. Let v ∈ (F). Then for each S ∈ F and i ∈ N such that i ∈ au(S) it holds v(S ∪ i) − v(S) = v ∗ (N \ S) − v ∗ (N \ (S ∪ i)). Proof v ∗ (N \ S) − v ∗ (N \ (S ∪ i)) = v(N ) − v(S) − v(N ) + v(S ∪ i) = v(S ∪ i) − v(S). 

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We recall here the axioms and the result about augmenting systems, [7]. We will use them in a comparative way for to define our value on decreasing systems. A player i is called dummy for game v ∈ (F) if, for each S ∈ F such that i ∈ au(S), then  v(i), i f {i} ∈ F v(S ∪ i) − v(S) = 0, otherwise

The dummy axiom say that if i ∈ N is a dummy player in the augmenting system (N , F) for a game v in (F) and  is a value in (F) then  v(i), i f {i} ∈ F i = 0, otherwise We introduce some concepts that we need to formulate the next axiom. Let T , S ∈ F, with T ⊆ S. The interval [T , S] is defined as: [T , S] = {C ∈ F : T ⊆ C ⊆ S} Under the same above mentioned conditions a maximal chain between in [T , S] is a subsequence of feasible sets T ⊂ S1 ⊂ S2 ⊂ · · · S|S|−1 ⊂ S such that Sk ∈ [T , S] and Sk = Sk−1 ∪ ik for all k : 0, . . . , |S|. Notice that this is always possible applying the third axiom for augmenting system. The number of maximal chains for the interval [T , S] are denoted by c[T , S]. In the particular case T = ∅, will be denote by c(S) to the total number of maximal chains in the augmenting from the ∅ to S. With the previous definitions the chain axiom says that for each S ∈ F and i, j ∈ N such that i, j ∈ au(S), it holds c(S ∪ i) j (δT ) = c(S ∪ j)i (δT ) where  is a value in (F). The other two axioms are linearity for games in (F) and efficiency, i. e., for any value in (F) and any v ∈ (F) i∈N i (N , F, v) = v(N ) From these axioms [7] prove that, fixed an augmenting system (N , F), the Shapley value, Sh(N , F, v) is the unique value that satisfies them and that for all i ∈ N and for all game v ∈ (F), Shi (N , F, v) = {S∈F :i∈au(S)}

c(S)c(S ∪ i, N ) [v(S ∪ i) − v(S)] c(N )

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By Proposition 2.6 there exists a bijective correspondence between a normal augmenting system F and its dual decreasing system D. We denote this correspondence by . (S) = N \ S for all S ∈ F. This fact permit us to define a value in a normal decreasing systems derived of a value on augmenting systems. Definition 56 Let (N , F) be a normal augmenting system and F a value in (F). We define its derived value in (D) as D (N , D, v) = F (N , F, v ∗ ), where D is the dual decreasing system of F and v ∈ (D). Notice that the set of normal decreasing systems over N are the set of dual decreasing systems of the normal augmenting systems over N . Then the values in (D) are the derived values of the values over (F). Definition 57 Let v be a game in the decreasing system (N , D). A player i ∈ N is a dummy player for v if for all S ∈ D with i ∈ ex(S) it holds that v(S) − v(S \ i) = v(i) if {i} ∈ D and 0 otherwise. Notice that the definitions of dummy player for augmenting and decreasing are consistent with the structure of each one. The axiom of dummy player say the same in both cases. Notice also that if i ∈ au(S) then i ∈ ex(N \ S). The following proposition show that we can translate the axioms for augmenting system to decreasing systems in a versioned form. Proposition 58 Let F a value in the augmenting system (N , F) and v ∈ (F) and D the derived value for its dual decreasing system. Then, the following statements hold. • F satisfies linearity axiom if and only if D also verifies the same axiom. • F satisfies efficiency axiom if and only if D also verifies the same axiom. • F satisfies dummy axiom if and only if D also verifies the same axiom. Proof The linearity respect to the games and efficiency are easy to check. Then, let i ∈ N a dummy player for v ∈ (F). Let now N \ S ∈ D. Then v ∗ (S) − v ∗ (S \ i) = v(N \ S) ∪ i) − v(N \ S)). Then both conditions are equivalent and the results follows.  It remains to analyze as the chain axiom can be translated to decreasing systems. The chain axiom, fixed a value D in a decreasing systems is introduced in the following definition:

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Definition 59 (Chain axiom) Let D a decreasing system. Suppose S ∈ D and i, j ∈ N such that i, j ∈ ex(S) then D D cD ([S \ i, N ])D j (δS ) = c ([S \ j, N ])i (δS )

where cD indicates the number of chains in D. From now, in order not to create confusion, we will denote the number of chains of D as cD and those of F as cF Proposition 60 A derived value for decreasing systems verifies the modified chain axiom in D if and only if the correspond value in augmenting system also verifies the chain axiom in F Proof To prove that the derived value satisfies the above chain axiom suppose that S ∈ D, S = ∅, and i, j ∈ ex(S). Notice that ρ∗N \S = δS where N \ S ∈ F and N \ S = N . As, by construction cD ([S \ i, N ]) = cF ([(N \ S) ∪ i]) we obtain F F ∗ cD ([S \ i, N ])D j (δS ) = c ([(N \ S) ∪ i]) j (ρN \S ) ∗ = cF ([(N \ S) ∪ j])F i (ρN \S )

= cD ([S \ j, N ])D i (δS ) 

and the result follow.

This axiom can be interpreted as follows: the player that obtain positive marginal contributions in the game δS are those that are meet to S immediately, and their payments are in proportion to the number of feasible orders. We give the theorem which characterize to the Shapley value for a decreasing system. Theorem 61 Let (N , D) a normal decreasing system. There exists a unique value ShD in (D) that satisfies linearity, dummy, efficiency and modified chain axioms. Moreover, for all i ∈ N , and each v ∈ (D), ShD i = {S∈D:i∈ex(S)}

cD [S \ i]cD [S, N ] [v(S) − v(S \ i)] cD(N )

Proof The existence and uniqueness of value is a consequence of those of the augmenting system and the relations above exposed between both structures. It remains F ∗ only to prove the formula. By definition ShD i (N , D, v) = Shi (N , F, v ) and applying the formula [7] we obtain

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F ∗ ShD i (N , D, v) = Shi (N , F, v )

cF [T ]cF [T ∪ i, N ] ∗ [v (T ∪ i) − v ∗ (T )] cF (N ) cF [T ]cF [T ∪ i, N ] = {T ∈F :i∈au(T )} [v(N \ T ) ∪ i) − v(N \ (T ∪ i)] cF (N ) cD [S \ i]cD [S, N ] v(S) − v(S \ i) = {S∈D:i∈ex(S)} cD (N ) = {T ∈F :i∈au(T )}

Notice that there exists a biyecction between {T ∈ F : S ∪ i ∈ F} and {S ∈ D : S \ i ∈ D} and between the maximal chains of [∅, T ] and the maximal chains [N \ T , N ].  This value will be called the Shapley value for normal decreasing systems.

12 Banzhaf Value for Decreasing Systems As we did in the case of the augmenting systems we will use the duality between both structures to define the value of Banzhaf in decreasing systems. The first thing we will do is state an axiom for this new value which we will call decreasing axiom. Axiom 62 (Decreasing axiom) Let (N , D) be a decreasing system and  a value in (D). Then the value  verifies the decreasing axiom if for all i ∈ N . and for all S, T ∈ D such that i ∈ ext(S) ∩ ext(T ) it holds that (δS\i ) = (δT \i ) where δS is the identity game in S. Proposition 63 Let F a value in the augmenting system (N , F) and v ∈ (F) and F the derived value for its dual decreasing system. Then, the following statements hold. • F satisfies linearity axiom if and only if D also verifies the same axiom. • F satisfies dummy axiom if and only if D also verifies the same axiom. • F satisfies augmenting axiom if and only if D also verifies the decreasing axiom. Proof We would only have to try the last item since it is remainder is analogous to Proposition 58. By Proposition 49 we know that ∗ D (δS\i) = F (δS\i ) = F (ρ(N \S)∪i ) = F (ρ(N \T )∪i ) = D (δT \i )

This proves the proposition.



We are now able to prove the theorem that characterize the Banzhaf value in decreasing systems.

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Theorem 64 Let (N , D) a normal decreasing system. There exists a unique value β D in (D) that satisfies linearity, dummy and decreasing axioms. Moreover, for each i ∈ N , and each v ∈ (D), βiD (N , v, D) =

1 A (i)

{S∈D:i∈ex(S)} [v(S) − v(S \ i)]

where A (i) = |{S ∈ D : i ∈ ex(S)}|. Proof Suppose that it exists a value  ∈ (D) satisfying the axioms above. We compute

F ∗ D i (v) = i (v ) =

1 {T ∈F :i∈au(T )} v ∗ (T ∪ i) − v ∗ (T ) A(i)

1 {T ∈F :i∈au(T )} [v(N \ T ) ∪ i) − v(N \ (T ∪ i)] A(i) 1 = {S∈D:i∈ex(S)} [v(S) − v(S \ i)] A (i) =

Notice that the numbers A(i) and A (i) are equals and that it exists a biyecction between the sets {S ∈ D : i ∈ ex(S)} and {T ∈ F : i ∈ au(T )}. This finish the proof because our value satisfies the three axioms. 

13 Applications Let (N , D) a decreasing system. Then it is an accessible set system because every nonempty feasible set X contains an element x such that X \ x is feasible. A greedoid (N , F) is an accessible set system that satisfies the exchange property: for all X , Y ∈ F with Y ⊂ X , there is some x ∈ X \ Y such that Y ∪ x ∈ F. This property is not verified by a decreasing system but it holds for the dual set of theirs feasible sets. Then a decreasing system is a special type of representation of a greedoid closed by the intersection and verifying that the union of two any feasible sets is not the ground set. The greedoids are intensively used in graph theory. A greedy algorithm is just an iterative process in which a locally best choice, usually an input of minimum weight, is chosen each round until all available choices have been exhausted. It is well-known that greedy algorithm is optimal for every R-compatible linear objective function over a greedoid. The intuition behind this proposition is that, during the iterative process, each optimal exchange of minimum weight is made possible by the exchange property, and optimal results are obtainable from the feasible sets in the underlying greedoid. This result guarantees the optimality of many well-known algorithms.

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To respect to augmenting systems there are a special case of union stable systems. Let F ⊂ 2N . F is an union stable system is for all S, R ∈ F, such that S ∩ R = 0, S ∪ R ∈ F. An accessible union stable systems is a stable system whit the property of accessibility, that is, for all S ∈ F, S = 0 there exists i ∈ S such that S \ i ∈ F. This is an important question since a motivation to study accessible union stable systems is that they have a characterizing property from communication networks (union stability) as well as hierarchies (accessibility) systems. The augmenting systems are particular cases of stable union systems and their properties allows to study a large number of situations characterized by communications networks and hierarchies. In particular all the algorithms related whit union system structures are more easily applied to augmenting systems.

References 1. Algaba, E., Bilbao, J.M., van den Brink, R., Jiménez-Losada, A.: Cooperative games on antimatroids. Discrete Math. 282, 1–15 (2004) 2. Algaba, E., Bilbao, J.M., Slikker, M.: A value for games restricted by augmenting systems. Siam J. Discrete Math. 24–3, 992–1010 (2008) 3. Aumann, R.J., Dreze, J.H.: Cooperative games with coalition structures. Int. J. Game Theory 3, 217–237 (1974) 4. Bilbao, J.M.: Axioms for the Shapley value on convex geometries. Eur. J. Oper. Res. 110, 368–376 (1998) 5. Bilbao, J.M., Edelman, P.H.: The Shapley value on convex geometries. Discrete Appl. Math. 103, 33–40 (2000) 6. Bilbao, J.M.: Cooperative games under augmenting systems. SIAM J. Discrete Math. 17, 122– 133 (2003) 7. Bilbao, J.M., Ordóñez, M.: Axiomatizations of the Shapley value for games on augmenting systems. Eur. J. Oper. Res. 196, 1008–1014 (2009) 8. Bilbao, J.M., Ordóñez, M.: The core and the Weber set of games on augmenting systems. Discrete Appl. Math. 158, 180–188 (2010) 9. van den Brink, R.: An axiomatization of the disjunctive permission value for games with a permission structure. Int. J. Game Theory 26, 27–43 (1997) 10. Dilworth, R.P.: Lattices with unique irreducible decompositions. Ann. Math. 41, 771–777 (1940) 11. Edelman, P.H., Jamison, R.E.: The theory of convex geometries. Geom. Dedicata 19, 247–270 (1985) 12. Faigle, U.: Cores of games with restricted cooperation. ZOR-Meth. Mod. Oper. Res. 33, 405– 422 (1989) 13. Faigle, U., Kern, W.: The Shapley value for cooperative games under precedence constraints. Int. J. Game Theory 21, 249–266 (1992) 14. Faigle, U., Peis, B.: Cooperative games with lattice structure. Technical Report 516. Universität zu Köln, Zentrum für Angewandte Informatik (2006) 15. Gallardo, J.M., Jiménez, N., Jiménez-Losada, A., Lebrón, E.: Games with fuzzy authorization structures. Fuzzy Set Syst. 272, 115–125 (2015) 16. Gillies, D.B.: Solutions to general non-zero-sum games. In: Tucker, A.W., Luce, R.D. (eds.) Contributions to the Theory of Games IV. Annals of Mathematics Studies, vol. 40, pp. 47–85. Princeton University Press, Princeton (1959)

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17. Gilles, R.P., Owen, G., van den Brink, R.: Games with permission structures: the conjunctive approach. Int. J. Game Theory 20, 277–293 (1992) 18. Ichiishi, T.: Supermodularity: applications to convex games and to the greedy algorithm for LP. J. Econ. Theory 25, 283–286 (1981) 19. Korte, B., Lovász, L., Schrader, R.: Greedoids. Springer, Berlin (1991) 20. Myerson, R.B.: Graphs and cooperation in games. Math. Oper. Res. 2, 225–229 (1977) 21. Ordóñz, M., Jiménez-Losada, A.: Duality on combinatorial structures. An application to cooperative games. Int. J. Gen. Syst. 46, 839–852 (2017) 22. Owen, G.: Values of graph-restricted games, SIAM. J. Algebraic Discrete. Methods 7, 210–220 (1986) 23. Owen, G.: Multilinear extensions and the Banzhaf value. Naval Res. Logistic 22, 41–50 (1075) 24. Shapley, L.S.: A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games II, Annals of Mathematics Studies, vol. 28, pp. 307–317. Princeton University Press, Princeton, NJ (1953) 25. Shapley, L.S.: Cores of convex games. Int. J. Game Theory 1, 11–26 (1971) 26. Stanley, R.P.: Enumerative Combinatorics, vol. I. Wadsworth, Monterey, CA (1986) 27. Weber, R.J.: Probabilistic values for games. In: Roth, A.E. (ed.) The Shapley value: Essays in honor of Lloyd S, pp. 101–119. Cambridge University Press, Cambridge, Shapley (1988)

Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases Khalid Hattaf and Noura Yousfi

Abstract In this chapter, we propose and analyze a class of three spatiotemporal models describing infectious diseases caused by viruses such as the human immunodeficiency virus (HIV) and the hepatitis B virus (HBV). The first model with cellular immunity, the second with humoral immunity and the third with cellular and humoral immune responses. In the three proposed models, the disease transmission process is modeled by a general incidence function which includes several forms existing in the literature. In addition, the global analysis of the proposed models is rigorously investigated. Furthermore, biological findings of our analytical results are presented. Moreover, mathematical virus models and results presented in many previous studies are extended and generalized. Keywords Virus dynamics · Immunity · Diffusion · Lyapunov functional · Global stability

1 Introduction Recently, many mathematical models used partial differential equations (PDEs) have been developed to better describe the dynamics of infectious diseases caused by viruses such as HIV and HBV. In 2007, Wang and Wang [1] proposed a PDE model to describe the HBV infection. In 2008, Wang et al. [2] modeled the intracellular time delay between infection of a cell and production of new virus particles by incorporating a discrete time delay into [1]. In 2011, Brauner et al. [3] adapted the model [1] to HIV infection. The bilinear incidence rate in three above models K. Hattaf (B) Centre Régional des Métiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco e-mail: [email protected] K. Hattaf · N. Yousfi Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, P.O Box 7955, Sidi Othman, Casablanca, Morocco © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_16

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was replaced by saturated infection rate in [4], by standard incidence function in [5], by Beddington-DeAnglis functional response in [6] and by a specific functional response in [7]. In 2015, we generalized all the above models by proposing a PDE model with general incidence function [8]. This general incidence function includes the above incidence rates and many other types existing in the literature such as the Crowley-Martin functional response [9], the incidence function was used by Zhuo [10], and the Hattaf-Yousfi functional response introduced in [11] and used in [12, 13]. On the other hand, the adaptive immunity plays a crucial role in the control of viral infection. In fact, cytotoxic T lymphocytes (CTL) cells attack the infected cells, while the B cells produce antibodies to neutralize the viral particles. The first immune response exerted by CTL cells is called the cellular immunity. However, the second immune response mediated by antibodies is called the humoral immunity. The aim of this work is to model the role of both arms of immunity in viral infections for to improve and generalize our model presented in [8] and the other PDE models existing in many previous studies. To this end, the next section deals with the PDE model with cellular immunity. In Sect. 3, we present the model with humoral immunity. The dynamical behavior of the model with cellular and humoral immune responses is investigated in Sect. 4. The chapter ends with a conclusion in Sect. 5.

2 The PDE Model with Cellular Immunity In this section, we investigate the dynamical behavior of the following model ⎧   ∂u ⎪ ⎪ = λ − du(x, t) − f u(x, t), w(x, t), v(x, t) v(x, t), ⎪ ⎪ ∂t ⎪ ⎪ ∞   ⎪ ∂w ⎪ −α1 τ ⎪ = g (τ )e f u(x, t − τ ), w(x, t − τ ), v(x, t − τ ) v(x, t − τ )dτ ⎪ 1 0 ⎨ ∂t −aw(x, t) − pw(x, t)z(x, t), ⎪ ∞ ⎪ ∂v ⎪ −α2 τ ⎪ w(x, t − τ )dτ − μv(x, t), ⎪ ⎪ ∂t = dv v + k 0 g2 (τ )e ⎪ ⎪ ⎪ ∂z ⎪ ⎩ = cw(x, t)z(x, t) − bz(x, t), ∂t (1) where u(x, t), w(x, t), v(x, t) and z(x, t) denote the densities of uninfected cells, infected cells, free virus and CTL cells at position x and time t, respectively. Uninfected cells are produced at rate λ, die at rate du and become infected by free virus at rate f (u, w, v)v. The parameters a, μ and b are, respectively, the death rates of infected cells, free virus and CTL cells. The parameter p represents the rate at which infected cells are killed by CTL cells, k is the production rate of free virus by an infected cell, and c is the proliferation rate of CTL cells. In addition, we assume that the virus contacts an uninfected target cell at time t − τ and the cell becomes infected at time t, where τ is a random variable taken from a probability distribution

Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases

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g1 (τ ). The term e−α1 τ represents the probability of surviving from time t − τ to time t, where α1 is the death rate for infected but not yet virus-producing cells. In the same, we assume that the time necessary for the newly produced virions to become mature and infectious is a random variable with a probability distribution g2 (τ ). The term e−α2 τ denotes the probability of surviving the immature virions during the delay period, where α12 is the average life time of an immature virus. Therefore, the inte∞ gral 0 g2 (τ )e−α2 τ w(x, t − τ )dτ describes the mature viral particles produced at position x and time t, respectively. The probability distribution functions g1 (τ ) and ∞ g2 (τ ) are assumed to satisfy gi (τ ) ≥ 0 and 0 gi (τ )dτ = 1 for i = 1, 2. Finally, dV is the diffusion coefficient and  is the Laplacian operator. As in [8], we assume that the incidence function f (u, w, v) is continuously differentiable in the interior of IR3+ and satisfies the following hypotheses: (H1 ) f (0, w, v) = 0, for all w ≥ 0 and v ≥ 0, (H2 ) f (u, w, v) is a strictly monotone increasing function with respect to u, for any fixed w ≥ 0 and v ≥ 0, (H3 ) f (u, w, v) is a monotone decreasing function with respect to w and v, i.e, ∂f (u, w, v) ≤ 0 and ∂∂vf (u, w, v) ≤ 0 for all u ≥ 0, w ≥ 0 and v ≥ 0. ∂w Biologically, the three hypotheses are reasonable and consistent with the reality. For more details on the biological significance of these three hypotheses, we refer the reader to the works [14, 15]. It is very important to note that our model described by system (1) includes many special cases. For example, we get the diffused HBV infection model proposed in [16] when g1 (τ ) = g2 (τ ) = δ(τ ) and f (u, w, v) = βu q , where q > 0, β > 0 is a constant rate describing the infection process and δ(.) is the Dirac delta function. When βu , where 1 , 2 ≥ g1 (τ ) = δ(τ − τ1 ), g2 (τ ) = δ(τ ) and f (u, w, v) = 1 + 1 u + 2 v 0 are constants, we obtain the diffusive and delayed virus dynamics model with Beddington-DeAngelis incidence function [17]. Further, when g1 (τ ) = δ(τ − τ1 ), βu , we get the diffusive and g2 (τ ) = δ(τ − τ2 ) and f (u, w, v) = (1 + 1 u)(1 + 2 v) delayed virus dynamics model with Crowley-Martin incidence function [18]. We consider system (1) with initial conditions u(x, θ) = φ1 (x, θ) ≥ 0, w(x, θ) = φ2 (x, θ) ≥ 0, ¯ × (−∞, 0], v(x, θ) = φ3 (x, θ) ≥ 0, z(x, θ) = φ4 (x, θ) ≥ 0, (x, θ) ∈ 

(2)

and Neumann boundary condition ∂v = 0, on ∂ × (0, +∞), ∂ν

(3)

∂ denotes the ∂ν outward normal derivative on ∂. From the biological point of view, the Neumann

where  is a bounded domain in IRn with smooth boundary ∂, and

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boundary condition means that the free virus particles do not move across the boundary ∂. Obviously, system (1) has always an infection-free equilibrium E 0 (u 0 , 0, 0, 0), λ where u 0 = , which represents the healthy state. The basic reproduction number d of (1) is given by λ kG 1 G 2 f ( , 0, 0) d , (4) R0 = aμ

where



Gi =

gi (τ )e−αi τ dτ , i = 1, 2.

0

Similarly to [19], if R0 > 1, system (1) has another infection equilibrium (λ − du 1 )G 1 , v1 = without cellular immunity E 1 (u 1 , w1 , v1 , 0), where w1 = a k(λ − du 1 )G 1 G 2 λ and u 1 ∈ (0, ) is the unique root of the following equation aμ d

aμ (λ − du)G 1 k(λ − du)G 1 G 2 , − = 0. G 1 G 2 f u, a aμ k If the cellular immune response has not been established, we have cw1 − b ≤ 0. So, we define the reproduction number for cellular immunity as follows R1z =

cw1 , b

(5)

1 denotes the average life expectancy of CTL cells and w1 is the number of b infected cells at E 1 . Hence, R1z represents the average number of the CTL immune cells activated by infected cells. If R1z > 1, then system (1) has an infection equilibrium with cellular immunity b kbG 2 cG 1 (λ − d x3 ) − ab , z2 = and u 2 ∈ E 2 (u 2 , w2 , v2 , z 2 ), where w2 = , v2 = c μc pb ab λ (0, − ) is the unique root of the following equation d dcG 1

μc b kbG 2 − f u, , (λ − du) = 0. c μc kbG 2

which

Next, we establish the global stability of each equilibrium. First, we have the following result. Theorem 2.1 The infection-free equilibrium E 0 of system (1) is globally asymptotically stable if R0 ≤ 1.

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Proof Construct a Lyapunov functional as follows L0 =



u(x, t) − u 0 − u

u(x,t)

1 f (u 0 , 0, 0) a ds + w(x, t) + v(x, t) f (s, 0, 0) G1 kG 1 G 2

0∞ p 1 + z(x, t) + g1 (τ )e−α1 τ cG 1 G1 0 t   × f u(x, s), w(x, s), v(x, s) v(x, s)dsdτ t−τ

∞ t a −α2 τ + g2 (τ )e w(x, s)dsdτ d x. G1G2 0 t−τ

For convenience, we let ϕ = ϕ(x, t) and ϕτ = ϕ(x, t − τ ) for any ϕ ∈ {u, w, v, z}. The time derivative of L 0 along the solution of system (1) satisfies

f (u 0 , 0, 0) ∂u 1 ∂w 1− + f (u, 0, 0) ∂t G 1 ∂t    p ∂z a ∂v + + f u, w, v v + kG 1 G 2 ∂t cG 1 ∂t ∞   1 a − g1 (τ )e−α1 τ f u τ , wτ , vτ vτ dτ + w G1 0 G1

∞ a g2 (τ )e−α2 τ wτ dτ d x − G1G2 0



 aμ f (u, w, v) u f (u 0 , 0, 0) du 0 1 − + R0 − 1 1− v = u0 f (u, 0, 0) kG 1 G 2 f (u, 0, 0) 

pb z dx − cG 1

aμ u f (u 0 , 0, 0) pb du 0 (1 − ) 1 − + (R0 − 1)v − z d x. ≤ u0 f (u, 0, 0) kG 1 G 2 cG 1 

d L0 = dt

Since the function f (u, w, v) is strictly monotonically increasing with respect to u, we have



u f (u 0 , 0, 0) 1− 1− ≤ 0. u0 f (u, 0, 0) d L0 If follows from R0 ≤ 1 that ≤ 0. It is easy to show that the largest invariant dt d L0 set in {(u, w, v, z)| = 0} is {E 0 }. By LaSalle’s invariance principle [20], the dt infection-free equilibrium E 0 is globally asymptotically stable when R0 ≤ 1.  Finally, we establish the global stability of the two infection steady states E i of system (1). To do this, we assume that R0 > 1 and the incidence function f satisfies for each infection equilibrium E i the following further hypothesis

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f (u, w, v) f (u, wi , vi ) v 1− ≤ 0, for all u, w, v > 0 − f (u, wi , vi ) f (u, w, v) vi

(H4 )

Thus, we have the following result. Theorem 2.2 Assume R0 > 1 and (H4 ) holds for each E i . (i) The infection equilibrium without cellular immunity E 1 of system (1) is globally asymptotically stable if R1z ≤ 1. (ii) The infection equilibrium with cellular immunity E 2 of system (1) is globally asymptotically stable if R1z > 1. Proof For (i), consider the following Lyapunov functional L1 =

u w 1 f (u 1 , w1 , v1 ) ds + w1 H u − u1 − f (s, w , v ) G w 1 1 1 1 u 

1 p a v + + v1 H z kG 1 G 2 v1 cG 1 ∞ 1 + f (u 1 , w1 , v1 )v1 g1 (τ )e−α1 τ G1 0 

 t f u(x, s), w(x, s), v(x, s) v(x, s) dsdτ × H f (u 1 , w1 , v1 )v1 t−τ

∞ t a w(x, s) −α2 τ + dsdτ d x, w1 g2 (τ )e H G1G2 w1 0 t−τ

where H (ξ) = ξ − 1 − ln ξ, ξ > 0. Obviously, H : (0, +∞) → [0, +∞) attains its strict global minimum at ξ = 1 and H (1) = 0. Hence, H (ξ) ≥ 0. Calculating the time derivative of L 1 along the solution of system (1), we obtain d L1 = dt

By λ = du 1 +

1 f (u 1 , w1 , v1 ) ∂u w1 ∂w ) + ) (1 − (1 − f (u, w , v ) ∂t G w ∂t 1 1 1  p ∂z a v1 ∂v + + (1 − ) kG 1 G 2 v ∂t cG ∂t ∞ 1 1 f (u 1 , w1 , v1 )v1 g1 (τ )e−α1 τ + G1 0    

 f u τ , wτ , vτ vτ   f u, w, v v  −H dτ × H f (u 1 , w1 , v1 )v1 f (u 1 , w1 , v1 )v1

∞ w  wτ  a −H dτ d x. w1 g2 (τ )e−α2 τ H + G1G2 w1 w1 0

a w, G1 1

f (u 1 , w1 , v1 )v1 =

a w G1 1

and

μ w1 = , we get kG 2 v1

Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases

d L1 = dt

535



u f (u 1 , w1 , v1 ) du 1 1 − 1− u1 f (u, w1 , v1 ) 

f (u 1 , w1 , v1 ) v f (u, w, v) a + + w1 1 − G1 f (u, w1 , v1 ) v1 f (u, w1 , v1 )

∞ 1 w1 vτ f (u τ , wτ , vτ ) a dτ + w1 1 − g1 (τ )e−α1 τ G1 G1 0 w v1 f (u 1 , w1 , v1 )

∞ v wτ v1 a 1 dτ + w1 1 − − g2 (τ )e−α2 τ G1 v1 G2 0 w1 v    ∞  f u τ , wτ , vτ vτ  1 a −α1 τ + w1 dτ g1 (τ )e ln G1 G1 0 f (u 1 , w1 , v1 )v1    ∞  f u, w, v vw   wτ  1 −α2 τ dτ − ln g2 (τ )e ln + G2 0 w1 f (u 1 , w1 , v1 )v1 w1

2 d |∇v| av pb 1 v (R1z − 1)z d x − d x. + 2 cG 1 kG 1 G 2  v

Therefore, d L1 = dt



u f (u 1 , w1 , v1 ) du 1 1 − 1− u1 f (u, w1 , v1 ) 

v v f (u, w, v) a f (u, w1 , v1 ) + + w1 − 1 − + G1 v1 f (u, w, v) v1 f (u, w1 , v1 )



f (u, w1 , v1 ) f (u 1 , w1 , v1 ) a +H − w1 H G1 f (u, w1 , v1 ) f (u, w, v)

∞ 1 w1 vτ f (u τ , wτ , vτ ) −α1 τ dτ + g1 (τ )e H G1 0 w v1 f (u 1 , w1 , v1 ) 

∞ 1 wτ v1 dτ + g2 (τ )e−α2 τ H G2 0 w1 v

|∇v|2 pb av1 dv z + (R1 − 1)z d x − d x. cG 1 kG 1 G 2  v 2

Since f (u, w, v) is strictly monotonically increasing with respect to u, we have



u f (u i , wi , vi ) 1− ≤ 0. du i 1 − ui f (u, wi , vi ) Based on the hypothesis (H4 ), we have

(6)

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v f (u, w, v) v f (u, wi , vi ) + −1− + v f (u, w, v) v f (u, wi , vi ) i

i f (u, wi , vi ) v f (u, w, v) ≤ 0. − = 1− f (u, wi , vi ) f (u, w, v) vi

(7)

d L1 Since H (ξ) ≥ 0 and R1z ≤ 1, we have ≤ 0 with equality if and only if u = u 1 , dt w = w1 , v = v1 and z = 0. It follows from LaSalle’s invariance principle that E 1 is globally asymptotically stable. For (ii), consider the following Lyapunov functional L2 =

u 1 f (u 2 , w2 , v2 ) w u − u2 − ds + w2 H f (s, w2 , v2 ) G1 w2 u 

2

p a + pz 2 v z + + v2 H z2 H kG 1 G 2 v2 cG 1 z2 ∞ 1 + f (u 2 , w2 , v2 )v2 g1 (τ )e−α1 τ G1 0 

 t f u(x, s), w(x, s), v(x, s) v(x, s) dsdτ × H f (u 2 , w2 , v2 )v2 t−τ

∞ t a + pz 2 w(x, s) + dsdτ d x. w2 g2 (τ )e−α2 τ H G1G2 w2 0 t−τ

The time derivative of L 2 along the solution of system (1) satisfies d L2 = dt



f (u 2 , w2 , v2 ) ∂u 1 w2 ∂w + ) (1 − f (u, w2 , v2 ) ∂t G1 w ∂t  p 1 a + pz 2 v2 ∂v z 2 ∂z + + + (1 − ) (1 − ) f (u 2 , w2 , v2 )v2 kG 1 G 2 v ∂t cG 1 z ∂t G1    

∞  f u τ , wτ , vτ vτ   f u, w, v v  −α1 τ −H × g1 (τ )e H dτ f (u 2 , w2 , v2 )v2 f (u 2 , w2 , v2 )v2 0

∞ w  wτ  a + pz 2 −H dτ d x. w2 g2 (τ )e−α2 τ H + G1G2 w2 w2 0 1−

By λ = du 2 + Ga1 w2 + μ w2 = , we find kG 2 v2

p w z , G1 2 2

f (u 2 , w2 , v2 )v2 =

1 (aw2 G1

+ pw2 z 2 ), w2 =

b and c

Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases

d L2 = dt

537



u f (u 2 , w2 , v2 ) du 2 1 − 1− u2 f (u, w2 , v2 ) 

v f (u, w, v) 1 v f (u, w2 , v2 ) + + (aw2 + pw2 z 2 ) − 1 − + G1 v2 f (u, w, v) v2 f (u, w2 , v2 )



f (u, w2 , v2 ) 1 f (u 2 , w2 , v2 ) +H − (aw2 + pw2 z 2 ) H G1 f (u, w2 , v2 ) f (u, w, v)

∞ v f (u , w , v ) 1 w 2 τ τ τ τ dτ + g1 (τ )e−α1 τ H G1 0 w v2 f (u 2 , w2 , v2 ) 

∞ (a + pz 2 )v2 dv |∇v|2 1 wτ v2 dτ dx − + g2 (τ )e−α2 τ H d x. 2 G2 0 w2 v kG 1 G 2  v

d L2 ≤ 0 with equality if and only if u = u 2 , w = w2 , v = v2 and z = Consequently, dt z 2 . By LaSalle’s invariance principle, we deduce that E 2 is globally asymptotically stable. 

3 The PDE Model with Humoral Immunity In some infections, the humoral immunity is more effective than cellular immunity [21]. For this reason, many mathematical models for virus dynamics with humoral immunity have been developed (see, for example [22–27]). To improve these mathematical models described by ordinary differential equations (ODEs), we propose the following system ⎧   ∂u ⎪ ⎪ = λ − du(x, t) − f u(x, t), w(x, t), v(x, t) v(x, t), ⎪ ⎪ ∂t ⎪ ⎪ ∞   ⎪ ∂w ⎪ −α1 τ ⎪ f u(x, t − τ ), w(x, t − τ ), v(x, t − τ ) v(x, t − τ )dτ ⎪ ⎨ ∂t = 0 g1 (τ )e −aw(x, t), ⎪ ∞ ⎪ ∂v ⎪ ⎪ = dv v + k 0 g2 (τ )e−α2 τ w(x, t − τ )dτ − μv(x, t) − qv(x, t)y(x, t), ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎩ ∂ y = gv(x, t)y(x, t) − hy(x, t), ∂t (8) where y(x, t) denotes the density of antibodies at position x and time t, respectively. The parameter h is the death rate of antibodies, q is the neutralization rate of the virus by the antibodies, and g is the proliferation rate of antibodies. The other variables and parameters have the same biological meaning as in (1). In this section, we consider system (8) with initial conditions u(x, θ) = φ1 (x, θ) ≥ 0, w(x, θ) = φ2 (x, θ) ≥ 0, ¯ × (−∞, 0], v(x, θ) = φ3 (x, θ) ≥ 0, y(x, θ) = φ4 (x, θ) ≥ 0, (x, θ) ∈ 

(9)

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and Neumann boundary condition ∂v = 0, on ∂ × (0, +∞). ∂ν

(10)

Clearly, system (8) has always an infection-free equilibrium P0 (u 0 , 0, 0, 0), λ where u 0 = . Similarly to above, if R0 > 1, system (8) has infection equid (λ − du 1 )G 1 , librium without humoral immunity P1 (u 1 , w1 , v1 , 0), where w1 = a k(λ − du 1 )G 1 G 2 λ v1 = and u 1 ∈ (0, ) is the unique root of the following equation aμ d

aμ (λ − du)G 1 k(λ − du)G 1 G 2 , − = 0. G 1 G 2 f u, a aμ k If the humoral immune response has not been established, we have gv1 − h ≤ 0. Then we define the reproduction number for humoral immunity as follows y

R1 =

gv1 , h

(11)

1 denotes the average life expectancy of antibodies and v1 is the number of h y free viruses at P1 . For the biological significance, R1 represents the average number of the antibodies activated by virus. y If R1 > 1, then system (8) has an infection equilibrium with humoral (λ − du 2 )G 1 h immunity P2 (u 2 , w2 , v2 , y2 ), where w2 = , v2 = , y2 = a g kgG 1 G 2 (λ − du 2 ) − aμh aμh λ and u 2 ∈ (0, − ) is the unique root of the aqh d dkgG 1 G 2 following equation

which



g (λ − du)G 1 h , − (λ − du) = 0. f u, a g h

Theorem 3.1 The infection-free equilibrium P0 of system (8) is globally asymptotically stable if R0 ≤ 1. Proof Define a Lyapunov functional as follows

Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases

u(x, t) − u 0 − V0 = 

u(x,t) u

539

1 f (u 0 , 0, 0) a ds + w(x, t) + v(x, t) f (s, 0, 0) G1 kG 1 G 2

0 ∞ aq 1 + y(x, t) + g1 (τ )e−α1 τ gkG 1 G 2 G1 0 t   × f u(x, s), w(x, s), v(x, s) v(x, s)dsdτ t−τ

∞ t a −α2 τ + g2 (τ )e w(x, s)dsdτ d x. G1G2 0 t−τ

The time derivative of V0 along the solution of system (8) satisfies

f (u 0 , 0, 0) ∂u 1 ∂w a ∂v 1− + + f (u, 0, 0) ∂t G ∂t kG 1 1 G 2 ∂t    aq ∂ y + f u, w, v v + gkG 1 G 2 ∂t ∞   1 a − g1 (τ )e−α1 τ f u τ , wτ , vτ vτ dτ + w G1 0 G1

∞ a g2 (τ )e−α2 τ wτ dτ d x − G1G2 0

 u f (u 0 , 0, 0) du 0 1 − 1− = u0 f (u, 0, 0) 



aμ f (u, w, v) aqh R0 − 1 − + v y dx kG 1 G 2 f (u, 0, 0) gkG 1 G 2



u f (u 0 , 0, 0) du 0 1 − 1− ≤ u f (u, 0, 0) 0 

aμ aqh + (R0 − 1)v − y d x. kG 1 G 2 gkG 1 G 2

d V0 = dt

Since 1 −

u u0



1−

f (u 0 ,0,0) f (u,0,0)

≤ 0 and R0 ≤ 1, we have

d V0 ≤ 0. Further, the dt

d V0 largest invariant set in {(u, w, v, y)| = 0} is {P0 }. It follows from LaSalle’s dt invariance principle that P0 is globally asymptotically stable when R0 ≤ 1.  Theorem 3.2 Assume R0 > 1 and (H4 ) holds for each Pi . (i) The infection equilibrium without humoral immunity P1 of system (8) is globally y asymptotically stable if R1 ≤ 1. (ii) The infection equilibrium with humoral immunity P2 of system (8) is globally y asymptotically stable if R1 > 1.

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Proof For (i), consider the following Lyapunov functional

u 1 f (u 1 , w1 , v1 ) w u − u1 − ds + V1 = w1 H f (s, w , v ) G w 1 1 1 1  u

1 aq a v + + v1 H y kG 1 G 2 v1 gkG 1 G 2 ∞ 1 + f (u 1 , w1 , v1 )v1 g1 (τ )e−α1 τ G1 0 

 t f u(x, s), w(x, s), v(x, s) v(x, s) dsdτ × H f (u 1 , w1 , v1 )v1 t−τ

∞ t a w(x, s) −α2 τ + dsdτ d x, w1 g2 (τ )e H G1G2 w1 0 t−τ The time derivative of V1 along the solution of system (8) satisfies d V1 = dt



f (u 1 , w1 , v1 ) ∂u w1 ∂w 1 1− 1− + f (u, w1 , v1 ) ∂t G1 w ∂t 

v1 ∂v a aq ∂ y 1− + + kG 1 G 2 v ∂t gkG 1 G 2 ∂t ∞ 1 f (u 1 , w1 , v1 )v1 g1 (τ )e−α1 τ + G1     0

 f u τ , wτ , vτ vτ   f u, w, v v  −H dτ × H f (u 1 , w1 , v1 )v1 f (u 1 , w1 , v1 )v1

∞ w  wτ  a w1 g2 (τ )e−α2 τ H −H dτ d x. + G1G2 w w1 1 0

μ w1 = , we have kG 2 v1



u f (u 1 , w1 , v1 ) d V1 du 1 1 − 1− = dt u1 f (u, w1 , v1 ) 

f (u 1 , w1 , v1 ) v f (u, w, v) a + + w1 1 − G1 f (u, w1 , v1 ) v1 f (u, w1 , v1 )

∞ 1 a w1 vτ f (u τ , wτ , vτ ) dτ + w1 1 − g1 (τ )e−α1 τ G1 G1 0 w v1 f (u 1 , w1 , v1 )

∞ v a 1 −α2 τ wτ v1 dτ + w1 1 − − g2 (τ )e G1 v1 G2 0 w1 v    ∞  f u τ , wτ , vτ vτ  1 a −α1 τ + w1 dτ g1 (τ )e ln G1 G1 0 f (u 1 , w1 , v1 )v1

By λ = du 1 +

a w, G1 1

f (u 1 , w1 , v1 )v1 =

a w G1 1

and

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541

   ∞  f u, w, v vw   wτ  1 −α2 τ dτ − ln + g2 (τ )e ln G2 0 w1 f (u 1 , w1 , v1 )v1 w1

|∇v|2 av1 dv aqh y (R1 − 1)y d x − d x. + gkG 1 G 2 kG 1 G 2  v 2 Thus, d V1 = dt

Since H (ξ) ≥ 0,





u f (u 1 , w1 , v1 ) du 1 1 − 1− u1 f (u, w1 , v1 ) 

v v f (u, w, v) a f (u, w1 , v1 ) + + w1 − 1 − + G1 v1 f (u, w, v) v1 f (u, w1 , v1 ) 



f (u 1 , w1 , v1 ) f (u, w1 , v1 ) a +H − w1 H G1 f (u, w1 , v1 ) f (u, w, v)

∞ 1 w1 vτ f (u τ , wτ , vτ ) dτ + g1 (τ )e−α1 τ H G1 0 w v1 f (u 1 , w1 , v1 ) 

∞ 1 wτ v1 dτ + g2 (τ )e−α2 τ H G2 0 w1 v

|∇v|2 aqh av1 dv y + (R1 − 1)y d x − d x. gkG 1 G 2 kG 1 G 2  v 2 y R1



u f (u i , wi , vi ) 1− ≤ 0 and form (7), we ≤ 1, du i 1 − ui f (u, wi , vi )

d V1 deduce that ≤ 0 with equality if and only if u = u 1 , w = w1 , v = v1 and z = 0. dt It follows from LaSalle’s invariance principle that P1 is globally asymptotically stable. For (ii), consider the following Lyapunov functional

u 1 f (u 2 , w2 , v2 ) w ds + u − u2 − w2 H V2 = f (s, w , v ) G w 2 2 1 2  u

2

aq a v y + + v2 H y2 H kG 1 G 2 v2 gkG 1 G 2 y2 ∞ 1 + f (u 2 , w2 , v2 )v2 g1 (τ )e−α1 τ G1 0 

 t f u(x, s), w(x, s), v(x, s) v(x, s) dsdτ × H f (u 2 , w2 , v2 )v2 t−τ

∞ t a w(x, s) dsdτ d x. + w2 g2 (τ )e−α2 τ H G1G2 w2 0 t−τ The time derivative of V2 along the solution of system (8) satisfies

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d V2 = dt



f (u 2 , u 2 , v2 ) ∂u 1 w2 ∂w 1− + ) (1 − f (u, w , v ) ∂t G w ∂t 2 2 1  a v2 ∂v y2 ∂ y aq + (1 − ) (1 − ) + kG 1 G 2 v ∂t gkG 1 G 2 y ∂t ∞ 1 f (u 2 , w2 , v2 )v2 g1 (τ )e−α1 τ + G1 0    

 f xτ , yτ , vτ vτ   f u, w, v v  −H dτ × H f (u 2 , w2 , v2 )v2 f (u 2 , w2 , v2 )v2

∞ w  wτ  a −H dτ d x. w2 g2 (τ )e−α2 τ H + G1G2 w2 w2 0

By f (u 2 , w2 , v2 )v2 = we get d V2 = dt

a w G1 2

=

a h (μ + qy2 )v2 , v2 = and λ = dw2 + kG 1 G 2 g



u f (u 2 , w2 , v2 ) du 2 1 − 1− u2 f (u, w2 , v2 ) 

f (u 2 , w2 , v2 ) a v f (u, w, v) + w2 1 − + G1 f (u, w2 , v2 ) v2 f (u, w2 , v2 )

∞ 1 a −α1 τ w2 vτ f (u τ , wτ , vτ ) dτ + w2 1 − g1 (τ )e G1 G1 0 w v2 f (u 2 , w2 , v2 )

∞ v wτ v2 a 1 dτ + w2 1 − − g2 (τ )e−α2 τ G1 v2 G2 0 w2 v  

 ∞ f u τ , wτ , vτ vτ 1 a −α1 τ dτ + w2 g1 (τ )e ln G1 G1 0 f (u 2 , w2 , v2 )v2  



∞ f u, w, v vw 1 wτ −α2 τ dτ − ln dx + g2 (τ )e ln G2 0 w2 f (u 2 , w2 , v2 )v2 w2 av2 dv |∇v|2 − d x. kG 1 G 2  v 2

Hence, d V2 = dt

a w , G1 2



u f (u 2 , w2 , v2 ) du 2 1 − 1− u2 f (u, w2 , v2 ) 

v f (u, w, v) v a f (u, w2 , v2 ) + + w2 − 1 − + G1 v2 f (u, w, v) v2 f (u, w2 , v2 ) 



f (u 2 , w2 , v2 ) f (u, w2 , v2 ) a +H − w2 H G1 f (u, w2 , v2 ) f (u, w, v)

Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases

543



∞ 1 w2 vτ f (u τ , wτ , vτ ) dτ g1 (τ )e−α1 τ H G1 0 w v2 f (u 2 , w2 , v2 ) 

∞ av2 dv |∇v|2 wτ v2 1 dτ dx − g2 (τ )e−α2 τ H d x. + G2 0 w2 v kG 1 G 2  v 2

+

d V2 Consequently, ≤ 0 with equality if and only if u = u 2 , w = w2 , v = v2 and y = dt y2 . By LaSalle’s invariance principle, we deduce that P2 is globally asymptotically stable. 

4 The PDE Model with Cellular and Humoral Immune Responses In this section, we consider the following model ⎧ ∂u ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ∂w ⎪ ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎨ ∂v ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ∂z ⎪ ⎩ ∂t

  = λ − du(x, t) − f u(x, t), w(x, t), v(x, t) v(x, t), ∞   = 0 g1 (τ )e−α1 τ f u(x, t − τ ), w(x, t − τ ), v(x, t − τ ) v(x, t − τ )dτ −aw(x, t) − pw(x, t)z(x, t), ∞ = dv v + k 0 g2 (τ )e−α2 τ w(x, t − τ )dτ − μv(x, t) − qv(x, t)y(x, t), = gv(x, t)y(x, t) − hy(x, t), = cw(x, t)z(x, t) − bz(x, t), (12)

with initial conditions u(x, θ) = φ1 (x, θ) ≥ 0, w(x, θ) = φ2 (x, θ) ≥ 0, v(x, θ) = φ3 (x, θ) ≥ 0, ¯ × (−∞, 0], y(x, θ) = φ4 (x, θ) ≥ 0, z(x, θ) = φ5 (x, θ) ≥ 0, (x, θ) ∈  (13) and Neumann boundary condition ∂v = 0, on ∂ × (0, +∞). ∂ν

(14)

System (12) has always an infection-free equilibrium of the form Q 0 (u 0 , 0, 0, 0, 0), λ where u 0 = . d Similarly to [11], if R0 > 1, system (12) has another infection equilibrium without (λ − du 1 )G 1 k(λ − du 1 )G 1 G 2 , v1 = immunity Q 1 (u 1 , w1 , v1 , 0, 0), where w1 = a aμ λ and u 1 ∈ (0, ) is the unique root of the following equation d

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aμ (λ − du)G 1 k(λ − du)G 1 G 2 , − = 0. G 1 G 2 f u, a aμ k If both humoral and cellular immune responses have not been established, we have gv1 − h ≤ 0 and cw1 − b ≤ 0. Hence, these last two conditions are equivalent to y R1 ≤ 1 and R1z ≤ 1, respectively. y When R1 > 1, system (12) has an infection equilibrium with only humoral immu(λ − du 2 )G 1 h nity Q 2 (u 2 , w2 , v2 , w2 , 0), where w2 = , v2 = , y2 = a g kgG 1 G 2 (λ − du 2 ) − aμh aμh λ and u 2 ∈ (0, − ) is the unique root of the aqh d dkgG 1 G 2 following equation

g (λ − du)G 1 h , − (λ − du) = 0. f u, a g h If cellular immunity has not been established, we have cw2 − b ≤ 0. For this, we define the reproduction number for cellular immunity in competition as R2z =

cw2 , b

(15)

which implies that cw2 − b ≤ 0 is equivalent to R2z ≤ 1. When R1z > 1, system (12) has an infection equilibrium with only cellular immub kbG 2 cG 1 (λ − d x3 ) − ab , z2 = nity Q 3 (u 3 , w3 , v3 , 0, z 3 ), where w2 = , v2 = c μc pb ab λ and u 2 ∈ (0, − ) is the unique root of the following equation d dcG 1

μc b kbG 2 − (λ − du) = 0. f u, , c μc kbG 2 If humoral immunity has not been established, we have gv3 − h ≤ 0. In this case, we define the reproduction number for humoral immunity in competition as y

R3 =

gv3 . h

(16) y

which implies that gv3 − h ≤ 0 is equivalent to R3 ≤ 1. When R2z > 1 and R3w > 1, system (12) has an infection equilibrium with both celb h lular and humoral immune responses Q 4 (u 4 , w4 , v4 , y4 , z 4 ), where w4 = , v4 = , c g ab μ y cG 1 (λ − du 4 ) − ab λ and u 4 ∈ (0, − y4 = (R3 − 1), z 4 = ) is the unique q pb d dcG 1 root of the following equation

Spatiotemporal Dynamics of a Class of Models Describing Infectious Diseases

545

g b h f (u, , ) − (λ − du) = 0. c g h It is important to note that y

y

R3 =

R1 gkbG 2 μ 1 y y , w2 = (R2z R3 − 1) and R3 > z . z = R1 hμc q R2

(17)

Theorem 4.1 The infection-free equilibrium Q 0 of system (12) is globally asymptotically stable if R0 ≤ 1. Proof Based on our method proposed in [28], we construct the Lyapunov functional for system (12) at Q 0 as follows W0 =

u(x, t) − u 0 − 

u(x,t)

u0

1 f (u 0 , 0, 0) a ds + w(x, t) + v(x, t) f (s, 0, 0) G1 kG 1 G 2

aq p + y(x, t) + z(x, t) gkG 1 G 2 cG 1 ∞ t   1 + g1 (τ )e−α1 τ f u(x, s), w(x, s), v(x, s) v(x, s)dsdτ G1 0 t−τ

∞ t a −α2 τ g2 (τ )e w(x, s)dsdτ d x. + G1G2 0 t−τ The time derivative of W0 along the solution of system (12) satisfies

f (u 0 , 0, 0) ∂u 1 ∂w a ∂v 1− + + f (u, 0, 0) ∂t G ∂t kG 1 1 G 2 ∂t    p ∂z aq ∂ y + + f u, w, v v + gkG 1 G 2 ∂t cG 1 ∂t ∞   1 − g1 (τ )e−α1 τ f u τ , wτ , vτ vτ dτ G1 0

∞ a a −α2 τ g2 (τ )e wτ dτ d x + w− G1 G1G2 0



 aμ u f (u 0 , 0, 0) f (u, w, v) du 0 1 − + R0 − 1 1− v = u0 f (u, 0, 0) kG 1 G 2 f (u, 0, 0) 

pb aqh y− z dx − gkG 1 G 2 cG 1

aμ u f (u 0 , 0, 0) du 0 (1 − ) 1 − + (R0 − 1)v ≤ u0 f (u, 0, 0) kG 1 G 2 

pb aqh y− z d x. − gkG 1 G 2 cG 1

dW0 = dt

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Since 1 −

u u0



1−

f (u 0 ,0,0) f (u,0,0)

≤ 0 and R0 ≤ 1, we have

dW0 ≤ 0. In addition, the dt

dW0 = 0} is {Q 0 }. It follows from LaSalle’s dt invariance principle that Q 0 is globally asymptotically stable when R0 ≤ 1.  largest invariant set in {(u, w, v, y, z)|

When R0 > 1, system (12) has four infection steady states Q i . The following theorem characterizes the global stability of these steady states. Theorem 4.2 Assume R0 > 1 and (H4 ) holds for each Q i . (i) The infection equilibrium without immunity Q 1 of system (12) is globally asympy totically stable if R1 ≤ 1 and R1z ≤ 1. (ii) The infection equilibrium with only humoral immunity Q 2 of system (12) is y globally asymptotically stable if R1 > 1 and R2z ≤ 1. (iii) The infection equilibrium with only cellular immunity Q 3 of system (12) is y globally asymptotically stable if R1z > 1 and R3 ≤ 1. (iv) The infection equilibrium with both cellular and humoral immune responses y Q 4 of system (12) is globally asymptotically stable if R2z > 1 and R3 > 1.   Proof Let ψ(x, t) = u(x, t), w(x, t), v(x, t), y(x, t), z(x, t) be a solution of (12), and we put Wi =



Li (ψ(x, t))d x,

where Li is the Lyapunov functional for the ODE model presented in [11] at Q i , 1 ≤ i ≤ 4. It is easy to show that Li satisfies the condition (7) given in [28] for all i = 1, 2, 3, 4. According to Proposition 2.1 of [28], we deduce that Wi is a Lyapunov  functional for (12) at Q i . This completes the proof. The conditions of the global stability of Q 2 and those of Q 3 given in (ii) and (iii) of the Theorem 4.2 do not hold simultaneously. In fact, supposing the contrary, then 1 y y y R1 > 1 ≥ R2z and R1z > 1 ≥ R3 . Since R3 ≤ 1 and R2z > y , we have R2z > 1. This R3 is a contradiction with R2z ≤ 1. According to (17) and Theorem 4.2, we have the following important result. Remark 4.3 Assume R0 > 1. y

1 If max(R1 , R1z ) ≤ 1, then system (12) converges to Q 1 without immunity. y 2 If max(R1 , R1z ) > 1, two cases arise: y

y

(i) When max(R1 , R1z ) = R1 , the humoral immunity is dominant and model (12) converges to Q 2 if R2z ≤ 1, and to Q 4 if R2z > 1. y (ii) When max(R1 , R1z ) = R1z , the cellular immunity is dominant and model (12) converges to Q 3 without humoral immunity.

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From this important remark, we can define the over domination of humoral immuy nity when R2z > 1 and R3 > 1, and the over domination of cellular immunity when z w R2 > 1 and R3 < 1. Therefore, we can conclude that the over domination of cellular immunity leads to the absence of the humoral immunity, and the over domination of the humoral immunity leads to the persistence of viral infection. This biological finding can be an explanation of the dysfunction of the adaptive immunity in viral infections such as HBV, which is still largely incomplete [29].

5 Conclusion In this work, we have proposed and analyzed a class of three spatiotemporal models with infinite distributed delays and general incidence function that includes the classical bilinear incidence rate, the saturated incidence rate, the standard incidence rate, the Beddington-DeAngelis functional response and the Crowley-Martin functional response. By a rigorous mathematical analysis, we have proved that the global dynamics of the first and second models is characterized by two threshold parameters. However, the global dynamics of the third model is characterized by five threshold parameters that are the basic reproduction number R0 , the reproduction numbers for y humoral immunity R1 , for cellular immunity R1z , for cellular immunity in competition y z R2 , and for humoral immunity in competition R3 . Since these threshold parameters do not depend on the diffusion coefficient, dv , we conclude that the diffusion of virus has no effect on the global stability of the steady states of the three PDE models. In addition, our PDE models presented in this chapter improve and generalize the ODE and PDE models proposed in more recent studies with only cellular immunity [16–19], with only humoral immunity [22–25], and with both arms of immunity [11, 23, 30–34].

References 1. Wang, K., Wang, W.: Propagation of HBV with spatial dependence. Math. Biosci. 210, 78–95 (2007) 2. Wang, K., Wang, W., Song, S.: Dynamics of an HBV model with diffusion and delay. J. Theor. Biol. 253, 36–44 (2008) 3. Brauner, C.-M., Jolly, D., Lorenzi, L., Thiebaut, R.: Heterogeneous viral environment in a HIV spatial model. Discrete Continuous Dyn. Syst. Ser. S 15, 545–572 (2011) 4. Xu, R., Ma, Z.E.: An HBV model with diffusion and time delay. J. Theor. Biol. 257, 499–509 (2009) 5. Chan Chí, N., Ávila Vales, E., García Almeida, G.: Analysis of a HBV model with diffusion and time delay. J. Appl. Math. 2012, 1–25 (2012) 6. Zhang, Y., Xu, Z.: Dynamics of a diffusive HBV model with delayed Beddington-DeAngelis response. Nonlinear Anal.: Real World Appl. 15, 118–139 (2014) 7. Hattaf, K., Yousfi, N.: Global dynamics of a delay reaction-diffusion model for viral infection with specific functional response. J. Comput. Appl. Math. 34(3), 807–818 (2015)

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Approximation of Short-Run Equilibrium of the N-Region Core-Periphery Model in an Urban Setting Minoru Tabata and Nobuoki Eshima

Abstract The purpose of this chapter is to give an approximation of short-run equilibrium of the N-region core-periphery model in an urban setting. The approximation is sufficiently accurate and expressed explicitly in terms of the distribution of workers that is contained as known function in the model. Making use of this approximation, we can analyze the behavior of each short-run equilibrium. Keywords Discrete nonlinear equation · Krugman model · Spatial economics Mathematical Subject Classification 39B72 · 91B72

1 Introduction In natural and social sciences, a large number of discrete nonlinear equations (DNEs) are considered, and it is important to study them mathematically (see, e.g., [3, 4, 6, 10, 14]). However, if we attempt to conduct such studies, then we often encounter serious obstacles due to the lack of a general mathematical theory of DNEs (see, e.g., [11, pp. 13–15]). Hence, it is advisable and useful to study concrete and specific DNEs constructed in various sciences. Among such DNEs, we are concerned with ones in economics. In particular, noting that various new mathematical sciences (game theory, financial engineering, and so on) were born from Nobel Prize research in economics, we find it important to study DNEs constructed in a branch of economics for which the prize has been recently awarded. Hence, this chapter deals with DNEs in spatial economics.

M. Tabata (B) Department of Mathematical Sciences, Osaka Prefecture University, Osaka 599-8531, Japan e-mail: [email protected] N. Eshima Center for Educational Outreach and Admissions, Kyoto University, Kyoto 606-8501, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_17

551

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Spatial economics is an interdisciplinary field between economics and geography of which the purpose is to study the location, distribution, and spatial organization of economic activities. In about 1990, Krugman began to conduct seminal research by placing particular emphasis on the clustering of economic activities and the formation of economic agglomeration in this interdisciplinary area. Since then, his research has grown into one of the major branches of spatial economics, which is now known as the New Economic Geography (NEG). In 2008, Krugman was awarded the Nobel Memorial Prize in Economic Sciences for his great contribution to spatial economics (see, e.g., [8, 15, 17, 21]). A large number of interesting and impressive DNEs are constructed in the NEG, but many of them have not been studied fully in mathematics (see, e.g., [9, (7.1)– (7.8), (7.14)–(7.19), (15A.1)–(15A.10), (16.1)–(16.8)]). The DNEs in the NEG are promising objects for applied mathematics. Among such DNEs, the N-region coreperiphery model (NRCP model) is one of the most important models in the NEG, since many spatial-economic models are constructed as its extensions (see, e.g., [7, 9.2] and [9, Chap. 5]). In this model, economic activities (agriculture and manufacturing) are conducted in a set consisting of N points, where N is an arbitrary natural number and each point represents a region. The population consists of farmers and workers [9, pp. 61–77]. If N = 1, then the NRCP model is trivial in spatial economics. Hence, in what follows throughout the chapter, we assume that N is an arbitrary integer such that N ≥ 2.

(1)

We should note that difficulties encountered when analyzing the NRCP model increase rapidly as the number N of regions increases. In fact, there are ample analytical results when N = 2, but there are a fewer ones when N ≥ 3 (see, e.g., [1, 5, 19, 20, 22]). We impose no restriction on N in addition to (1). The NRCP model is described by the wage equation, which is the system of four DNEs of which the unknown function denotes the distribution of nominal wages [9, (5.3)–(5.6)]. Each short-run equilibrium is defined as the distribution of real wages that are calculated from a solution to the wage equation when the distributions of workers and farmers are known functions. There are several studies on the existence of short-run equilibrium of the NRCP model. Moreover, there are several related studies (see, e.g., [2, 7, 12, 13, 16], [24, Theorem 3.1], and [25, Theorem 3]). However, in contrast to the studies on the existence of short-run equilibrium, there are still many open issues on the behavior of short-run equilibrium. In particular, there are few studies on how each short-run equilibrium depends on the distributions of workers and farmers and the transport costs. This chapter deals with this important open problem. In this study, as a first step, we treat the NRCP model in an urban setting, that is, under the condition that the economy has no agriculture. This condition is employed frequently in the NEG (see, e.g., [9, p. 331]), and the NRCP model in an urban setting actually corresponds to the Krugman’s model of international trade with imperfect

Approximation of Short-Run Equilibrium . . .

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competition [18]. In what follows in this chapter, we simply refer to the NRCP model in an urban setting as the NRCP model. No confusion should arise. It was proved that the NRCP model has a short-run equilibrium [23, Theorem 3.1(i)]. However, it is difficult to analyze the behavior of each short-run equilibrium. If transport costs are low, then we can obtain numerical short-run equilibria [23, Sect. 4]. Observing the numerical results, we can understand how each short-run equilibrium depends on the distribution of workers (see [23, Figs. 4.1–4.6]). Hence, it is useful to take a numerical approach to the NRCP model. However, if transport costs are high, then it is very difficult to obtain a numerical short-run equilibrium [23, Remark 9.1]. Hence, it is difficult to understand how each short-run equilibrium depends on the distribution of workers by taking a numerical approach to the NRCP model when transport costs are high. In order to overcome the difficulty, we give an approximation of short-run equilibrium that is sufficiently accurate when transport costs are high (Theorem 1). This chapter consists of four sections in addition to this introduction. In Sect. 2, we introduce the NRCP model. In Sect. 3, we state the main result (Theorem 1), which gives an approximation of short-run equilibrium that is expressed explicitly in terms of the distribution of workers that is contained as known function in the NRCP model. We should note that even if transport costs are so high that no shortrun equilibrium can be obtained numerically, then the approximation is sufficiently accurate. In Sect. 4, we obtain upper and lower bounds for each short-run equilibrium. In Sect. 5, we prove Theorem 1. In this chapter, we do not make use of the methods developed in [23–25]. Moreover, we use neither advanced knowledge of DNEs nor of spatial economics. Hence, this chapter can be easily understood even without reading [23–25] carefully and even without having an advanced knowledge of DNEs and spatial economics.

2 Equation By D we denote a set consisting of N points [see (1)]. By L we denote the set of all real-valued functions of x ∈ D. We can regard this set of functions as a Euclidean space of which the dimension is equal to N . Hence, each v = v(x) ∈ L is regarded as a point of the Euclidean space. However, to avoid the confusion of elements of L with points of D, we refer to v = v(x) ∈ L as function of x ∈ D. By L 0+ we denote the set of all nonnegative-valued functions of L. By L + , we denote the set of all positive-valued functions of L. We define the following norms of v = v(x) ∈ L: (2) ||v|| := max |v(y)|, y∈D

|||v||| :=

 y∈D

|v(y)|.

(3)

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In order to describe the transport cost incurred in dispatching manufactured goods from point y ∈ D to point x ∈ D, we define the transport-cost function c = c(x, y). We assume the following conditions [9, p. 49]: c(x, y) > 0 for all x, y ∈ D such that x = y,

(4)

c(x, y) = c(y, x) for all x, y ∈ D,

(5)

c(x, x) = 0 for all x ∈ D.

(6)

We assume that the elasticity of substitution σ is a given constant, and that the distribution of workers λ = λ(x) is a known function of x ∈ D. We assume the following conditions: σ is a constant such that σ > 1, (7) λ = λ(x) ∈ L ,

(8)

|||λ||| = 1,

(9)

λ > 0,

(10)

λ := minλ(x).

(11)

where x∈D

The conditions (4)–(10) are the most general assumptions accepted in spatial economics [9, Chaps. 4 and 5]. We impose no restriction on the NRCP model in addition to these conditions. The NRCP model is described by the wage equation, which is the system of four DNEs [9, (5.3)–(5.6)]. It is proved in [23, Proposition 5.1, Remark 5.2] that the wage equation in an urban setting reduces to the following singular DNE referred to as the real wage equation: ω(x)1/α =



ρ(x, y)λ(y)(

y∈D

1 )1/β , ω(y)1/α

(12)

where ω = ω(x) is an unknown function that denotes the distribution of real wages, and ρ(x, y) := exp(−(σ − 1)c(x, y)), (13) α :=

2σ − 1 , σ(σ − 1)

(14)

σ . σ−1

(15)

β :=

Approximation of Short-Run Equilibrium . . .

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If a function ω = ω(x) ∈ L + satisfies (12) for each x ∈ D, then we say that the function is a solution. Each short-run equilibrium is defined as a solution to (12).

3 Result and Discussion In [23, Theorem 3.1(i)] we proved that (12) has a solution ω = ω(x) ∈ L + , that is, that the NRCP model has a short-run equilibrium. We discuss how each short-run equilibrium depends on the distribution of workers λ = λ(x) and the transport-cost function c = c(x, y). It follows from (4) that the minimum of transport costs, c :=

min

x,y∈D,x= y

c(x, y),

(16)

is positive. In Sect. 5 we prove the theorem below, which shows that if c is large, then each short-run equilibrium is sufficiently close to Λ = Λ(x) := Λ0 (x)α + ( where

Λ0 (x)α−1 )(ρ0 Λ0 )(x), σ−1

Λ0 = Λ0 (x) := λ(x)σ/(2σ−1) , 

ρ0 f = (ρ0 f )(x) :=

ρ(x, y) f (y).

(17)

(18) (19)

y∈D,y=x

Theorem 1 There exist positive constants k and K such that if (16) is so large that c ≥ K,

(20)

then each solution ω = ω(x) to (12) satisfies the following inequality: ||ω − Λ|| ≤ kε(c)2 ,

(21)

ε(c) := exp(−(σ − 1)c).

(22)

where Let us discuss this theorem. No condition is imposed on c = c(x, y) in addition to (4)–(6). In (85), the positive constants k and K are explicitly defined as functions of σ. N , λ , and λ := maxλ(x). (23) x∈D

Hence, k and K are independent of the transport-cost function c = c(x, y).

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It follows from (21) that (17) can be regarded as an ε(c)2 -order approximate short-run equilibrium. Applying (7) to (22), we deduce that 0 < ε(c) < 1,

(24)

ε(c) → 0 + 0 as c → +∞.

(25)

Hence, the approximation error becomes smaller and smaller as the minimum of transport costs c becomes larger and larger. As mentioned in Sect. 1, it is difficult to obtain numerical solutions to (12) when transport costs are high. Even in such a difficult case, (21) holds. Moreover, it is difficult to prove the uniqueness of solution to (12) without assuming that transport costs are sufficiently small [23, Theorem 3.1(i)(ii)]. Even if (12) has two different solutions, then both solutions satisfy (21). From (13), (18), and (19), we see that the approximate short-run equilibrium (17) is expressed explicitly in terms of λ = λ(x) and c = c(x, y). Observing (17), we can approximately understand how each short-run equilibrium ω = ω(x) depends on the distribution of workers and the transport-cost function. For example, we can consider that the first term Λ0 (x)α expresses the influence that the worker population λ(x) exercises on the distribution of real wages ω(x) at each point x, and that the second term Λ0 (x)α−1 (ρ0 Λ0 )(x)/(σ − 1) reflects the total influence that the worker population λ(y) (y = x) exert on the distribution of real wages ω(x). For these reasons we see that the inequality (21) is useful. In order to discuss (17) fully, we need estimates for the operator (19) that is contained in Λ. From (24) and (25), we see that the following lemma shows that ρ0 is a small operator when c is large: Lemma 1 If f = f (x) ∈ L 0+ and ri = ri (x) ∈ L 0+ , i = 1, 2, satisfy that r1 (x) ≤ f (x) ≤ r2 (x) for all x ∈ D,

(26)

0 ≤ ε(C)(|||r1 ||| − ||r1 ||) ≤ (ρ0 f )(x) ≤ ε(c)|||r2 ||| for all x ∈ D,

(27)

C := max c(x, y).

(28)

then

where x,y∈D

Proof Applying (7), (16), and (28) to (13), we deduce that ε(C) ≤ ρ(x, y) ≤ ε(c) for all x, y ∈ D such that x = y.

(29)

Applying this inequality and (26) to (19), we easily deduce that ε(C)

 y∈D,y=x

r1 (y) ≤ (ρ0 f )(x) ≤ ε(c)

 y∈D,y=x

r2 (y) for all x ∈ D.

(30)

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557

Comparing (2) with (3), we see easily that ||v|| ≤ |||v|||, for all v ∈ L .

(31)

From this inequality, (2), and (3) we see that 

r1 (y) ≥ |||r1 ||| − ||r1 || ≥ 0,

y∈D,y=x



r2 (y) ≤ |||r2 |||.

y∈D,y=x

Applying these inequalities to (30), we obtain (27). The following lemma shows the relation between (17) and (18): Lemma 2

1−λ

Λ0 (x)α {1 + (

)ε(C)} ≤ Λ(x),

(32)

1 )ε(c)}. (σ − 1)λ

(33)

(σ − 1)λ

Λ(x) ≤ Λ0 (x)α {1 + ( Proof Rewrite (18) as follows: Λ0 (y) = ( where γ :=

1 )λ(y), λ(y)γ

(34)

σ−1 . 2σ − 1

(35)

Applying (7) to (35), we obtain 0 1) accounts for the increased likelihood of TB infection in previously treated TB patients in comparison to wholly susceptible individuals [42, 45]. It is further assumed that β is the transmission rate of TB by persons with active TB cases, while β1 is the transmission rate due to exogenous reinfection. Based on the above assumptions, the optimal control model is given by

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   dS1 (I + ηJ ) =  − 1 − 1 − u1 (t) α1 S1 − βS1 + θS2 − μS1 , dt N    dS2 (I + ηJ ) = 1 − 1 − u1 (t) α1 S1 − σβS2 − (θ + μ)S2 , dt N  dE (I + ηJ )  (I + ηJ ) (I + ηJ ) = (1 − p) βS1 + σβS2 + εβT dt N N N (I + ηJ ) − (k + μ)E, − β1 E N  dI (I + ηJ )  (I + ηJ ) (I + ηJ ) = p βS1 + σβS2 + εβT + kE dt N N N  (I + ηJ )  + β1 E − 1 + u2 (t) να2 I − (d + μ)I , N   dJ = 1 + u2 (t) να2 I − (r + μ)J , dt dT (I + ηJ ) = rJ − εβT − μT . dt N

(2.4)

The objective functional to be minimized is given by  G(u1 , u2 ) = 0

tf

  B1 B2 S1 (t) + I (t) − J (t) + u1 2 (t) + u2 2 (t) dt. 2 2

(2.5)

In (2.5), we minimize the number of susceptible individuals in the high risk (i.e., low level of TB awareness) group through improved TB awareness campaign programmes, and also minimize the number of persons with infectious TB through case finding techniques. In the same vein, we seek to maximize the number of persons identified with chronic TB (and isolated for effective treatment under the DOTS programme). Hence, our interest is to minimize an objective functional that shows a trade-off needed in minimizing the number of susceptible high-risk individuals and infectious individuals while, at the same time, maximizing the number of identified persons with active TB cases (to be isolated), with minimal associated cost for achieving these. The associated cost of carrying out (and sustaining) the improved TB awareness campaign (and case finding techniques which uses active cough as a marker for identifying potential infectious TB cases) in a population affected with insurgency, are nonlinear and thus take a quadratic form [2, 8]. The constants, B1 and B2 , represents the weights on the benefit and cost of implementing the optimal TB control strategy. In this case, B1 and B2 balance the cost factors due to the size and relevance of the terms in the objective functional.

Optimal Control Measures for Tuberculosis in a Population …

609

Hence, we seek an optimal pair, u1 ∗ and u2 ∗ , such that G(u1 ∗ , u2 ∗ ) = min{G(u1 , u2 ) : u1 , u2 ∈ },

(2.6)

where  is the control set defined by  = {(u1 , u2 ) ∈ L1 (0, t f ) × L1 (0, t f ) | ai ≤ ui ≤ bi },

(2.7)

with ai , bi (i = 1, 2) being non-negative constants.

2.3 A Brief Note on Analysis of Model Without Controls In this section, we consider the TB model in (2.4) but without control functions incorporated in it. Hence, we obtain the uncontrolled model given in (2.8) (I + ηJ ) dS1 =  − α1 S1 − βS1 + θS2 − μS1 , dt N dS2 (I + ηJ ) = α1 S1 − σβS2 − (θ + μ)S2 , dt N  dE (I + ηJ )  (I + ηJ ) (I + ηJ ) = (1 − p) βS1 + σβS2 + εβT dt N N N (I + ηJ ) − (k + μ)E, − β1 E N  dI (I + ηJ )  (I + ηJ ) (I + ηJ ) = p βS1 + σβS2 + εβT + kE dt N N N (I + ηJ ) + β1 E − (να2 + d + μ)I , N dJ = να2 I − (r + μ)J , dt dT (I + ηJ ) = rJ − εβT − μT . dt N

(2.8)

where all the state variables and parameters take the same previous definitions.

2.3.1

Basic Properties of the Model

Since the model (2.8) monitors human population, it is necessary to show that all its state variables and as well as the associated parameters are non-negative for all time.

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Theorem 2.1 Let the initial conditions for the TB model (2.8) be S1 (0) > 0, S2 (0) > 0, E(0) > 0, I (0) > 0, J (0) > 0, and T (0) > 0 . Then the solutions (S1 (t), S2 (t), E(t), I (t), J (t), T (t)) of the model (2.8), with positive initial data, will remain positive for all time t > 0. Proof Let t1 = sup{t > 0 : S1 (0) > 0, S2 (0) > 0, E(0) > 0, I (0) > 0, J (0) > 0, T (0) > 0}. From the first equation of model (2.8), it follows that dS1 (t) ≥  − λS1 + ϕS1 dt

(2.9)

) where λ = β (I +ηJ and ϕ = α1 + μ. Hence, (2.9) can be rewritten as N

 d S1 (t) exp ϕt + dt



t

λ(τ )d τ







t

≥  exp ϕt +

0

λ(τ )d τ .

(2.10)

0

Thus,   S1 (t1 ) exp ϕt1 +

t1

 λ(τ )d τ − S1 (0) ≥

0

t1

    exp ϕy +

y

λ(τ )d τ



dy,

0

0

(2.11)

So that, 



t1

S1 (t1 ) ≥ S1 (0) exp − ϕt1 −







×

t1

    exp ϕy +

y

t1

λ(τ )d τ + exp − ϕt1 −

0





λ(τ )d τ



0



λ(τ )d τ dy > 0.

(2.12) (2.13)

0

0

Similarly, it can be shown that S2 (t) > 0, E(t) > 0, I (t) > 0, J (t) > 0 and T (t) > 0 for all time t > 0. Thus, all solutions of the system (2.8) remain positive for all non-negative initial conditions.

Lemma 2.2 The closed set   D = (S1 , S2 , E, I , J , T ) ∈ R6+ : N ≤ μ

(2.14)

is positively-invariant and attracts all positive trajectories of the TB model (2.8). Proof Adding up all equations of model (2.8) yields dN =  − μN − dI . dt

(2.15)

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611

This implies that dN ≤ λ − μN . Hence, it follows that dN ≤ 0 if N (t) ≥ μ . Using a dt dt standard comparison theorem [19], it can be shown that N (t) ≤ N (0)e−μt + μ (1 − e−μt ). Specifically, if N (0) ≤ μ , then N (t) ≤ μ for all t > 0. Thus, the domain D is positively invariant. Moreover, if N (0) > μ , then either the solution enters the domain D in finite time or the total population, N (t), asymptotically approaches μ as t → ∞. Thus, the domain D attracts all solutions in R6+ .

Since the domain D is positively-invariant, it is sufficient to study the dynamics of the orbit generated by the model (2.8) in D. Hence, it has been shown that the model (2.8) is mathematically and epidemiologically well posed.

2.3.2

Local Stability of the Disease-Free Equilibrium

The model (2.8) has a disease-free equilibrium (DFE), obtained by setting the righthand sides of the equations in the model to zero, which is given by E0 = (S10 , S20 , E 0 , I 0 , J 0 , T 0 ) =



  θ + μ   α1 , , 0, 0, 0, 0 . μ α1 + θ + μ μ α1 + θ + μ (2.16)

The local stability of E0 can be established using the next generation operator method on the model (2.8). Using similar notations in van den Driessche and Watmough [43], the matrices F and V of new infection terms and the remaining transfer terms are respectively given by ⎛

S10 +σS20 N S10 +σS20 pβ N

0 (1 − p)β

⎜ F =⎜ ⎝p0 0

0



S10 +σS20 N ⎟ ⎟ S10 +σS20 ⎠ pβη N

(1 − p)βη

(2.17)

0

and  V =

(k + μ) 0 0 −k (να2 + d + μ) 0 0 −να2 (r + μ)

 ,

(2.18)

Note that in the calculation of the F and V matrices, we made use of the state variables with infection, i.e., E, I and J as explained in van den Driessche and Watmough [43]. Hence, it follows that the reproduction number denoted by RT = ρ(F V −1 ), with ρ being the spectral radius of F V −1 , is given by RT =

 β k + pμ ηνα2 + r + μ  θ + μ σα1 . · · + να2 + d + μ k + μ r+μ α1 + θ + μ α1 + θ + μ (2.19)

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Following Theorem 2 in van den Driessche and Watmough [43], the following result follows. Lemma 2.3 The DFE of the model (2.8), given by E0 , is locally asymptomatically stable if RT < 1 and unstable if RT > 1. The threshold quantity, RT , is the reproduction number and it measures the average number of new TB infections generated by a single chronic TB case in a completely susceptible population in the presence of control measures such as isolation and treatment. Epidemiologically speaking, Lemma 2.3 is that TB can be eradicated from the population when the reproduction numnber is less than unity (RT < 1) if the initial sizes of the subpopulations of the model (2.8) are in the basin of attraction of the DFE. In [29], the authors provided some analytical results of a variant of the model (2.8). In particular, they investigated the analysis of the reproduction number, calculated the endemic equilibrium (if it exists), showed that the model undergoes the phenomenon of backward bifurcation in the presence of exogenous reinfection when σ = 1 and ε = 1.

2.4 Analysis of optimal control model The Pontryagin’s Maximum Principle [38] provides the necessary conditions that an optimal control pair must satisfy. This principle converts (2.4), (2.5) and (2.6) into a problem of minimizing a Hamiltonian, H , pointwisely with respect to the controls, u1 and u2 :  B1 B2 λi f i , H = S1 (t) + I (t) − J (t) + u1 2 (t) + u2 2 (t) + 2 2 i=1 6

(2.20)

where f i (i = 1, . . . , 6) is the right hand side of the system of differential equations of the i-th state variable. When the Pontryagin’s Maximum Principle is applied and the existence result for optimal control from Fleming and Rishel [14], we claim the following result: Theorem 2.4 There exists an optimal control pair u1 ∗ , u2 ∗ and the corresponding solution S1 ∗ , S2 ∗ , E ∗ , I ∗ , J ∗ and T ∗ that minimizes G(u1 , u2 ) over . Furthermore, there exists adjoint functions: λ1 , λ2 , λ3 , λ4 , λ5 and λ6 such that

Optimal Control Measures for Tuberculosis in a Population …

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 β(I ∗ + ηJ ∗ )  d λ1 ∗ = −1 + λ1 + α u (t) + μ − λ2 α1 u1 ∗ (t) 1 1 dt N∗  β(1 − p)(I ∗ + ηJ ∗ )   βp(I ∗ + ηJ ∗ )  − λ4 , − λ3 ∗ N N∗  βσ(I ∗ + ηJ ∗ )  β(1 − p)(I ∗ + ηJ ∗ )σ   d λ2 + θ + μ − λ = −λ1 θ + λ2 3 dt N∗ N∗  βp(I ∗ + ηJ ∗ )σ  , − λ4 N∗   β (I ∗ + ηJ ∗ )   ∗ ∗ β1 (I + ηJ ) d λ3 1 = λ3 + k + μ − λ + k , 4 dt N∗ N∗  βS ∗   βσS ∗   βεT ∗    d λ4 1 2 = −1 + λ1 + λ − λ 1 + u (t) να + λ 2 5 2 2 6 dt N∗ N∗ N∗   βεT ∗ βS1 ∗ βσS2 ∗  β1 E ∗  − λ3 (1 − p) − + ∗ + N∗ N N∗ N∗ ∗ ∗   βεT ∗ β1 E ∗ βS1 βσS2 − λ4 p + + ∗ + − (d + μ) N∗ N N∗ N∗    − ν 1 + u2 (t) α2 ,  βηS ∗   βησS ∗   βηεT ∗  d λ5 1 2 = 1 + λ1 + λ + λ r − μ − (r + μ) − λ 2 5 6 dt N∗ N∗ N∗ ∗ ∗  ∗  βηεT ∗ β1 ηE βηS1 βησS2 − λ3 (1 − p) − + + N∗ N∗ N∗ N∗ ∗ ∗   βηεT ∗ ∗ βηS1 βησS2 β1 ηE − λ4 p , + + + N∗ N∗ N∗ N∗   ∗ ∗  ∗ d λ6 βε(1 − p)(I + ηJ ) βεp(I + ηJ ∗ )  = −λ3 − λ 4 dt N∗ N∗  βε(I ∗ + ηJ ∗ )  , + λ6 N∗ (2.21) with transversality conditions λi (t f ) = 0, i = 1, . . . , 6

(2.22)

and N ∗ = S1 ∗ + S2 ∗ + E ∗ + I ∗ + J ∗ + T ∗ . Moreover, the following characterization holds     1 u1 ∗ (t) = min max a1 , α1 S1 ∗ (λ1 − λ2 ) , b1 , B1     1 u2 ∗ (t) = min max a2 , α2 νI ∗ (λ4 − λ5 ) , b2 . B2

(2.23)

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Proof By Corollary 4.1 in Fleming and Rishel [14], the convexity of the integrand (of the objective functional) in (2.5) with respect to (u1 , u2 ) guarantees the existence of an optimal pair, a priori boundedness of the state variables, and the Lipschitz property of the state system with respect to the state variables. The adjoint equations and transversality conditions can be obtained via the Pontryagin’s Maximum Principle such that: d λ1 dt d λ2 dt d λ3 dt d λ4 dt d λ5 dt d λ6 dt

∂H , ∂S1 ∂H =− , ∂S2 ∂H =− , ∂E ∂H =− , ∂I ∂H =− , ∂J ∂H =− , ∂T =−

λ1 (t f ) = 0, λ2 (t f ) = 0, λ3 (t f ) = 0,

(2.24)

λ4 (t f ) = 0, λ5 (t f ) = 0, λ6 (t f ) = 0.

Considering the optimality conditions given by ∂H ∂H = 0 and = 0, ∂u1 ∂u2

(2.25)

the optimal control pair (u1 ∗ , u2 ∗ ) can be solved for, subject to the state variables. Taking into account the bounds on the control, the characterization in (2.23) can be obtained. This yields, for the optimal control, u1 ∗ (t), ∂H = B1 u1 + α1 S1 (λ2 − λ1 ) = 0 ∂u1

(2.26)

which implies that u1 ∗ (t) =

1 α1 S1 ∗ (λ1 − λ2 ) B1

(2.27)

on the set {t : a1 < u1 ∗ (t) < b1 }. Similarly, for the optimal control, u2 ∗ (t), we have ∂H = B2 u2 − α2 νI (λ5 − λ4 ) = 0 ∂u2

(2.28)

Optimal Control Measures for Tuberculosis in a Population …

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which implies that u2 ∗ (t) =

1 α2 νI ∗ (λ4 − λ5 ) B2

(2.29)

on the set {t : a2 < u2 ∗ (t) < b2 }. It is worth noting that the optimality conditions (i.e., taking derivatives of the Hamiltonian with respect to the controls) only hold in the interior of the control set.

Observe that some restriction should be imposed on the length of time intervals, so as to ensure the uniqueness of the optimality system. This restriction is as a result of the opposite time orientation in (2.4), (2.21) and (2.22) [13, 17].

3 Simulations In this section, numerical simulation results of the model without control functions (2.8) as well as the optimal control model 2.4 are both presented. The results presented in this sections are all based on the parameter values in Table 3.

Table 3 Baseline values and ranges of the parameters of the optimal control model (2.4) Parameter Baseline values Ranges References μ  β β1 p k r d ν η α1 α2 θ σ

0.02041 year−1 μ × 105 year−1 8.557 year−1 1.5 year−1 0.1 year−1 0.0005 year−1 1.5 ind−1 year−1 0.365 year−1 0.5 year−1 0.4 year−1 5 year−1 5 year−1 1.2 year−1 1 year−1 0.5 year−1

(0.0143, 0.04) – (4.4769, 15.1347) (1.5, 3.5) (0.05, 0.3) (0, 0.005) (1.5, 2.5) (0.22, 0.39) (0, 1) (0, 1) – – (1, 2) – (0, 1)

UNAIDS (2004) [36] [27] [27] [39] [6] [29] [26] [29] [29] [29] [29] [45] [29] [29]

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3.1 Numerical Simulations of TB Model Without Controls Here, the TB model without controls (2.8) is numerically simulated with a view to investigating the impact of varying the values of certain key parameters describing TB awareness rate (α1 ), cough identification rate (α2 ), cost factor associated with medical test and treatment (ν), relative infectiousness of identified TB patients (η), and in the treatment rate for isolated TB patients (r). The simulations in this section are based on demographic parameters relevant to Nigeria. In January 2018, the total population of Nigeria was estimated to be 194,615,054 people [7]. It thus follows that, at the disease-free state, /μ = 194, 615, 054. Furthermore, the average mortality rate in Nigeria is μ = 0.02041 per year [41] so that the mean recruitment rate is  = 3,972,090 per year. In addition, TB incidence in Nigeria was estimated to be 407,000 in 2017 [37, 51]. Figure 1 shows the cumulative number of new TB cases as we vary the awareness rate of TB (α1 ) between 5 and 30. The simulation shows that the cumulative number of new TB cases decreases as more persons are made aware or ‘educated’ about TB in the population (i.e., as α1 → 30). This implies that an increase in TB awareness or enlightenment rate in a population could have a positive impact in reducing the number of new TB cases over time. Considering Fig. 2, we have the cumulative number of new TB cases as the cough identification rate (α2 ) is varied between 0 and 30. The result indicates that the cumulative number of new TB cases significantly drops as the cough identification rate is increased (i.e., as α2 → 30). This suggests that an increase in the rate at which active TB patients are identified (and placed under the DOTS treatment plan) could have a positive effect in bringing down the number of new TB cases in the 5

Cumulative number of new TB cases

3.5

x 10

α1 = 5

3 2.5 2 1.5

α = 30 1

1 0.5 0 0

10

20

30

40

50

60

70

80

90

Time (years)

Fig. 1 Simulation results showing the cumulative number of new TB cases with varied α1

100

Optimal Control Measures for Tuberculosis in a Population …

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5

Cumulative number of new TB cases

3.5

x 10

3 2.5

α =5 2

2 1.5 1

α = 30

0.5

2

0 0

10

20

30

40

50

60

70

80

90

100

Time (years) Fig. 2 Simulation results showing the cumulative number of new TB cases with varied α2

Cumulative number of new TB cases

5

3

x 10

2.5 2

ν=0 1.5 1

ν=1

0.5 0 0

10

20

30

40

50

60

70

80

90

100

Time (years)

Fig. 3 Simulation results showing the cumulative number of new TB cases with varied ν

population over time. Evidently, increasing the value of α2 (see Fig. 2) could lead to more significant reduction in the incidence of TB in a population as compared to an increase in the value of α1 (see Fig. 1). Figure 3 depicts the cumulative number of new TB cases as we vary the cost factor associated with medical test and treatment (ν) between 0 and 1. The plot in Fig. 3 suggests that a significant reduction in the cost of medical test and treatment or completely making both free (i.e., as ν → 1) could lead to a huge reduction in the incidence of TB in a population. In particular, the plot shows that the associated cost

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Cumulative number of new TB cases

5

2

x 10

η=1

1.5

1

0.5

η=0 0 0

10

20

30

40

50

60

70

80

90

100

Time (years)

Fig. 4 Simulation results showing the cumulative number of new TB cases with varied η

would have to be brought down really low before it could lead to a positive impact in reducing the incidence of TB in the population over time. Figure 4 shows the cumulative number of new TB cases as the relative infectiousness of identified TB cases is varied between 0 and 1. The simulation result suggests that a reduction in the relative infectiousness of identified TB cases (i.e., as η → 0) could lead to a significant reduction in the incidence of new TB cases in a population. This implies that a reduction in the infectiousness of identified TB cases (perhaps through isolation, prompt and proper treatment of such persons) could result in a positive population level effect in reducing the number of new TB cases in a population. Finally, the plot in Fig. 5 depicts the cumulative number of new TB cases as the treatment rate for chronic TB patients (r) is varied between 1.5 and 2.5. The simulation result suggests, as expected, that an increase in the treatment rate of persons with active TB (i.e., as r → 2.5) could result in a decrease in the number of new TB cases in a population. This simulation result this suggest that increasing the treatment rate of persons with chronic TB cases could lead to a decrease in the incidence of TB in a population.

3.2 Numerical Simulations of Optimal Control Model Here, the optimal strategy for effective TB control, consisting of improved TB awareness campaign programmes and case finding techniques (for chronic TB cases) in a population affected with insurgency are obtained by solving the optimality systems which is made up of the system of controlled ordinary differential Eq. (2.4) for

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Cumulative number of new TB cases

5

2

x 10

1.5

r = 1.5

r = 2.5

1

0.5

0 0

10

20

30

40

50

60

70

80

90

100

Time (years)

Fig. 5 Simulation results showing the cumulative number of new TB cases with varied r

the state system and their corresponding adjoint Eq. (2.21). The optimality system is solved numerically using the Runge-Kutta method, implemented in a forwardbackward sweep fashion. Hence, the state system, together with an initial condition, is solved forward in time using a guess for the controls over the simulated time, while the adjoint system, with values at the final time t f , is solved backward in time using the current iterative solution of the state system. The controls are then updated using a convex combination of the controls coupled with the value from the characterizations (2.23). The process as well as the iteration is stopped if the values of unknowns at the previous iteration are very close to the ones at the present iteration [15]. We note that the analysis as well the simulations carried out in this work are independent of the specific parameter values used for the simulation. The parameter values used for the numerical solution of the optimal control model are given in Table 3, and for the initial conditions, we made use of the following values: S1 (0) = (70/120)N , S2 (0) = (35/120)N , E(0) = (14/120)N , I (0) = (35/120)N , J (0) = (20/120)N , T (0) = (0/120)N , where N takes the value 100 000. The value for N is so chosen because most epidemiological data on TB (as well as other infectious diseases in human population) are reported in 100,000s [48– 50]. And for the bounds on the control functions, we have made use of 0 ≤ u1 ≤ 0.95 and 0 ≤ u2 ≤ α12 ν . Figure 6 depicts the controls plotted as a function of time when the awareness rate (α1 ) is varied. For both values of α1 , we observe (in the first frame) that the control u1 remained close to the lower bound for almost the entire 5-year period of simulation, with a momentary increase (and decrease) that lasted for less than a year; the control still remained very close to the lower bound, resulting in a very small fraction of susceptible individuals benefiting from the awareness programme. This indicates that insurgency has a strong negative impact on TB awareness campaign.

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Control u1

0.2 0.15

α1 = 5

0.1

α = 20 1

0.05 0 0

2.5

2

1.5

1

0.5

3

3.5

4

4.5

5

3

3.5

4

4.5

5

Time (years)

Control u2

0.4 0.3

α1 = 5

0.2

α = 20 1

0.1 0 0

0.5

1

1.5

2

2.5

Time (years)

Fig. 6 The controls u1 and u2 for the case μ = 0.02041,  = μ × 105 , β = 8.557, β1 = 1.5, p = 0.1, k = 0.005, r = 1.5, d = 0.365, ν = 0.5, η = 0.4, α1 = 5, α2 = 5, = 1.2, θ = 1, σ = 0.5, B1 = 50, B2 = 100, when α1 is varied

The second frame shows that for both values of α1 , the control u2 , which represent case finding techniques for active TB, was at the upper bound for almost the entire period of simulation, before a steep decline to the lower bound. Figure 7 represents the effect on some of the epidemiological classes after the optimal control strategy (presented in Fig. 6) was implemented, with a variation in

x 10

Infectious (I)

Exposed (E)

4

4 3

α =5 1

2

α = 20

3000

α =5 1

α = 20

2000

1

1000

1

1 0

1

2

3

4

0 0

5

1

α1 = 20

2000 1000 0 0

1

2

3

Time (years)

3

4

5

10000

α1 = 5

Treated (T)

Isolated Infectious (I)

3000

2

Time (years)

Time (years)

4

5

5000

α =5 1

α1 = 20 0 0

1

2

3

4

5

Time (years)

Fig. 7 Optimal control strategies for the case μ = 0.02041,  = μ × 105 , β = 8.557, β1 = 1.5, p = 0.1, k = 0.005, r = 1.5, d = 0.365, ν = 0.5, η = 0.4, α1 = 5, α2 = 5, = 1.2, θ = 1, σ = 0.5, B1 = 50, B2 = 100, when α1 is varied

Optimal Control Measures for Tuberculosis in a Population …

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the awareness rate (α1 ). The plots show that increasing the value of α1 did not lead to a significant reduction in the number of latent and active TB cases. In particular, implementing this optimal control strategy resulted in a marginal reduction of infectious TB cases in the affected population. This is in agreement with the fact that u1 , the fraction of susceptible individuals who benefitted from the educational and enlightenment campaign programmes, remained very low for almost the 5-year period and did not even get any close to the upper bound (see Fig. 6). However, a 5-year simulation period of this optimal control measure results in the aversion of about 4639/100,000 latent TB infections, and 121/100,000 active TB cases. In Fig. 8, the controls are also plotted as functions of time when the effect of the educational/awareness programme is varied. Recall that the TB enlightenment programme is assumed to reduce the likelihood of TB infection of susceptible individuals (with high level of TB awareness) by a factor, σ (0 ≤ σ ≤ 1). The case σ = 0 implies that the enlightenment programme is completely effective, whereas σ = 1 indicates that the programme is completely ineffective in reducing TB incidence. The first frame in Fig. 8 shows that when σ = 0.9, the control (u1 ) remained close to the lower bound for the entire 5-year period of simulation; indicating that insurgency had a negative impact on the effectiveness of the enlightenment programme. Similarly, when σ = 0.1, the second frame shows that the control also remained close to the lower bound for a little over 4 years with a momentarily increase (and decrease that also lasted less than a year). The behavior of the second control is similar to that in Fig. 6. In Fig. 9, we have the number of persons in the various epidemiological compartments after the optimal control measure (presented in Fig. 8) is implemented.

Control u1

0.4 0.3 σ = 0.9 σ = 0.1

0.2 0.1 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3

3.5

4

4.5

5

Time (years)

Control u2

0.4 0.3

σ = 0.9 σ = 0.1

0.2 0.1 0 0

0.5

1

1.5

2

2.5

Time (years)

Fig. 8 The controls u1 and u2 for the case μ = 0.02041,  = μ × 105 , β = 8.557, β1 = 1.5, p = 0.1, k = 0.005, r = 1.5, d = 0.365, ν = 0.5, η = 0.4, α1 = 5, α2 = 5, = 1.2, θ = 1, σ = 0.5, B1 = 50, B2 = 100, when σ is varied

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x 10

3000

Infectious (I)

Exposed (E)

4 3 2

1 0

σ = 0.9 σ = 0.1 1

2

3

4

1000 0 0

5

σ = 0.9 σ = 0.1

2000

1

3

4

5

8000

3000 σ = 0.9 σ = 0.1

2000 1000 0 0

2

Time (years) Treated (T)

Isolated Infectious (I)

Time (years)

1

2

3

Time (years)

4

5

6000 4000 σ = 0.9 σ = 0.1

2000 0 0

1

2

3

4

5

Time (years)

Fig. 9 Optimal control strategies for the case μ = 0.02041,  = μ × 105 , β = 8.557, β1 = 1.5, p = 0.1, k = 0.005, r = 1.5, d = 0.365, ν = 0.5, η = 0.4, α1 = 5, α2 = 5, = 1.2, θ = 1, σ = 0.5, B1 = 50, B2 = 100, when σ is varied

When compared to the plots in Fig. 6, the plots in Fig. 9 show that the optimal control measure results in the aversion of more latent and active TB cases. In particular, this control measure results in the prevention of 7537/100,000 latent TB infections and 190/100,000 active cases, after a 5-year simulation period. In Fig. 10, the controls are also plotted as functions of time when the cough identification rate (α2 ) is varied. For both values of α2 , the control u1 , behaved very similar to the previous two cases (see Figs. 6 and 8), i.e., it remained close to the lower bound for almost the entire 5-year period of simulation. However, the control u2 , which represent case finding techniques for detecting active TB cases to be isolated, behaved very differently from the previous two cases earlier considered. With a low cough identification rate (e.g., α2 = 5), we observe that u2 remained at the upper bound for almost the entire 5-year period of simulation, in order to achieve optimal TB control strategy. On the other hand, with a higher cough identification rate (e.g., α2 = 20), observe that u2 was initially at the upper bound, before it experienced a gradual decrease with time, implying that an increase in the cough identification rate will require less case finding efforts to achieve control of TB in the population. Finally in Fig. 11, we present the results of implementing the optimal control measure (presented in Fig. 10) on some of the classes when the cough identification rate (α2 ) is varied. The plots show that increasing the value of α2 will results in the reduction of both latent and active TB cases, just like in the previous two cases considered in Figs. 7 and 9. In particular, we observe that for higher values of cough identification rate (e.g., α2 = 20) there is a sharp decrease in the number of infectious

Optimal Control Measures for Tuberculosis in a Population …

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Control u

1

0.2 α2 = 5

0.15

α = 20 2

0.1 0.05 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3

3.5

4

4.5

5

Time (years)

Control u2

0.4 0.3 0.2

α =5

0.1

α = 20

2 2

0 0

0.5

1

1.5

2

2.5

Time (years)

Fig. 10 The controls u1 and u2 for the case when μ = 0.02041,  = μ × 105 , β = 8.557, β1 = 1.5, p = 0.1, k = 0.005, r = 1.5, d = 0.365, ν = 0.5, η = 0.4, α1 = 5, α2 = 5, = 1.2, θ = 1, σ = 0.5, B1 = 50, B2 = 100, when α2 is varied

4

x 10

3000

Infectious (I)

Exposed (E)

4 3

α =5

2

2

α2 = 5 α = 20

2000

2

1000

α = 20 1 0

2

1

2

3

4

0 0

5

1

2

3

4

5

Time (years) 8000

4000 α =5

Treated (T)

Isolated Infectious (I)

Time (years)

2

3000

α = 20 2

2000 1000

6000 4000

α =5 2

2000

α = 20 2

0 0

1

2

3

Time (years)

4

5

0 0

1

2

3

4

5

Time (years)

Fig. 11 Optimal control strategies for the case μ = 0.02041,  = μ × 105 , β = 8.557, β1 = 1.5, p = 0.1, k = 0.005, r = 1.5, d = 0.365, ν = 0.5, η = 0.4, α1 = 5, α2 = 5, = 1.2, θ = 1, σ = 0.5, B1 = 50, B2 = 100, when α2 is varied

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persons in the population. Implementing this optimal control measure for a 5-year period will result in the aversion of about 7729/100,000 latent TB infections, as well as some 232/100,000 active TB cases.

4 Discussion and Conclusion In this work, we have modified a mathematical model describing the transmission dynamics TB [29] by incorporating time-dependent control functions which represents the fraction of susceptible persons who received TB enlightenment massage in the midst of insurgency, as well as case finding techniques for detecting chronic TB patients to be placed on isolation for effective treatment. Hence, the resulting model is formulated as an optimal control problem. Consequently, optimal control theory was applied to analyze the formulated problem. Using demographic data relevant to Nigeria, numerical simulations of the TB model 2.8 without control, suggest that the cumulative number of new TB cases could be reduced in a population with an increase in the TB awareness rate (α1 ), cough identification rate (α2 ) as well as the treatment rate of chronic TB cases in a population. Moreover, the cumulative incidence of TB could further be reduced with a significant reduction in the associated cost for medical test and treatment (ν), and a reduction in the relative infectiousness of identified TB patients (η). The results from numerical simulations of the optimal control model 2.4 suggest that in order to immediately tackle the surge in latent and active TB cases in a population affected by insurgency, then attention must shift to increasing the cough identification rate (for identifying active TB cases to be isolated for effective treatment), coupled with improving the effectiveness of the TB awareness campaign programmes. As the intensity of insurgency increases, less information about TB (such as its cause, transmission paths, signs and symptoms, diagnostic procedures, and treatment regimen) becomes available in the population. In Sub-Saharan African countries, such ignorance on the part of individuals infected with TB often lead to the postponement in seeking appropriate medical attention, and in some cases, such infected persons might even adopt other treatment methods such as visiting traditional healers or resulting to self-medications before contacting an appropriate health-care center, such as a DOT facility [23, 44]. For example, during periods of intense ‘Boko Haram’ insurgency in the northeastern part of Nigeria both human and vehicular movements are restricted due to the imposition of curfew by the military, suicide bombing, and fear of being attacked by the armed groups [2, 8]. The decimal performance of TB control programmes during periods of intense insurgency may be ascribed to the limited access to health care facilities as well as internal displacement of persons in the affected communities to perceived areas of safety, which is often associated with malnutrition, overcrowding, and limited health facilities [16]. The limited resources coupled with the humanitarian crises in the perceived areas of safety has further fueled the growing incidence of

Optimal Control Measures for Tuberculosis in a Population …

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TB such communities [5, 11]. This often leads to delay in prompt diagnosis of TB and incomplete treatment among TB patients (which is a catalyst for the emergence of multi-drug resistant TB strains) in such affected communities. During severe insurgency, there is an important need to work towards greater partnership between national TB programmes and the non-governmental organizations (NGOs) or United Nations High Commissioner for Refugees (UNHCR) managed programmes in the affected populations. One apparent step towards achieving this would be the systematic inclusion of refugees and internally displaced persons (IDPs) in the Global Fund to Fight AIDS, Tuberculosis, and Malaria by their host countries [40]. The creation and expansion of user-friendly DOTS in areas affected with insurgency should be encouraged. Moreover, there should be training and empowering of indigenes in the affected communities to serve as local community public health workers especially with respect to early detection of suspected TB cases, and for distribution and observation of treatment. This can be realized by the promotion of peace, creating and expansion of user-friendly DOTS in conflict regions, as well as training and empowering of local communities to qualify as local community public health workers especially with respect to early detection suspected TB case, and for distribution and observation of treatment.

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Insurance Model to Estimate the Financial Risk Due to Direct Medical Cost on Dengue Outbreaks S. S. N. Perera

Abstract Dengue fever is among the most critical infectious disease in tropical and subtropical countries of the world, and represents a significant financial and disease burden in endemic regions. In recent decades, it is observed that significant increase in the recorded infected cases and hence an increase in the public health burden. Mathematical models with modern financial features have become invaluable management tools for epidemiologists, both to understand the underlying observed dynamics as well as making quantitative predictions of the disease spared risk and how such a risk depends on the effectiveness of different control measures. This chapter is an attempt to build a bridge between epidemiological and insurance modeling and set up an actuarial based tool that provides financial arrangements to cover the future medical expenses resulting from the medical treatments of dengue disease. Dynamics of the Dengue transmission can be expressed using classical compartment Susceptible, Infected and Recovered (SIR) model. Introducing the fractions of the compartment of the host population as probability densities, we convert classical SIR model into probability model. Taking fractions as probability densities, we then develop the insurance plan to cover the future financial burden due to direct medical expenses. The premium, the present financial burden due to future expenses is defined via the equivalence principle and sensitivity of it with respect to model parameters and external variables is discussed. By introducing several control measures, the variability of the present financial burden with respect to such measurers are discussed. Further, the efficiency of the controls are analyzed. By defining the reserve function, necessary and sufficient criterion for the existence of insurance plans is also discussed. Key Words and Phrases: Dengue · Dynamical system · Expected value · Equivalence principle · Actuarial evaluation AMS Subject Classification: 34D20 · 92B05 · 92D30

S. S. N. Perera (B) Research and Development Centre for Mathematical Modeling, Faculty of Science, University of Colombo, Colombo 03, Sri Lanka e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_20

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1 Introduction Dengue Fever (DF), its critical form, Dengue Hemorrhagic Fever (DHF) and its high critical form, Dengue Shock Syndrome (DSS) is one of the most common widespread vector borne disease in the tropical and subtropical countries [1]. It is a leading cause of morbidity and mortality in developing countries and one of the key public health issues in such countries. According to the World Health Organization (WHO), dengue disease is ranked as one of the most critical infectious disease with severe impact on public health and well-being in the society and it estimates that about 40% of the world’s population is at risk of being infected with dengue [2]. The disease has spread to almost all the tropical and sub-tropical parts of the world and it has been estimated that nearly 3.9 billion people in more than 128 countries in Africa, the Americas, the Eastern Mediterranean, South-East Asia and the Western Pacific are at risk [3]. Of these, worldwide, annually, 50–100 million are infected and half a million are hospitalized with over 20,000 deaths are reported [3–6]. More than 80% of the annual cases belong to the milder DF form whereas the rest of the cases fall under DHF or DSS which reflect a high level of disease severity requiring hospitalization and intensive medical care. In South-East Asia region, 1.3 billion people in nearly 10 countries are vulnerable to dengue [2, 21]. Before the year 2003, only eight countries in South-East Asia had reported dengue cases however by the year 2009, the number increased after Nepal and Bhutan also reported their dengue outbreaks. In past decades, nearly all the countries in South-East Asia reported dengue cases every year. The timing of the epidemics in these dengue endemic countries tends to vary with seasonal cycles. Dengue is mainly an urban disease; however, studies have suggested that dengue has expanded beyond its own boundaries from urban to rural as well as tropics to non-tropics region too. According to the WHO reports, SouthEast Asian countries such as Bangladesh, India, Indonesia, Maldives, Myanmar, Sri Lanka, Thailand and Timor-Leste are highly endemic for dengue disease due to high number of hospitalizations and deaths [8]. The number of DF/DHF cases in Indonesia has been reported since 2004 and it follows an endemic pattern. Moreover, it is observed that in Bangladesh, Myanmar, and the Maldives, DF/DHF follows an endemic pattern. Further, it is observed that there has been an increase in the proportion of severe DF cases, particularly in Thailand, Indonesia, and Myanmar [7]. The first epidemics of DF were reported in latter part of 18th century and subsequently occurred every 10–30 years. Further, Dengue is the fastest-growing mosquito-borne viral infection, and its impact today is 30 times greater than 50 years ago. For example, in1970s, less than 10 countries had reported epidemics of severe dengue, however, at present, dengue appears more than 150 countries all over the world [2, 5, 8]. Figure 1 displays worldwide dengue distribution during the year 2013. At present, dengue is a major public health concern in Sri Lanka and DF/DHF both are endemic. In Sri Lanka, DF was first reported in 1965. During last five decades, several dengue outbreaks were observed and created severe damages to the public health sector of Sri Lankan society. The pattern of dengue changed in Sri Lanka after 1989, with an exponential increase in the incidence of DF as well as DHF. During

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Fig. 1 Countries or areas at risk. Source World Health Organization 2013

the period from 2012–2017, the trend of dengue cases peaked in the year 2017. In the year 2012, a total of 44,456 cases of dengue were reported from around the country. However in year 2013, number of cases dropped to 32,063 but increased to 47,512 cases in year 2014. In the year 2015, a total of 29,777 cases of dengue were reported and in year 2016, number of cases increased to 55,150 and in 2017, it further increased to 184,442 which is the highest number of dengue cases ever reported in the history of Sri Lanka [10]. Approximately, 40% cases were reported from the western province which has the highest population as well as the largest population density in Sri Lanka. Figure 2 displays the number of reported dengue cases from 1988 to 2017 from all over the island. Of them the highest number of cases were reported from the Colombo District which is the most urbanized and most densely populated district of the country [11]. Colombo is the capital and the largest city in Sri Lanka, its rapid urbanization and increased human movements have created Colombo to be a highly vulnerable geographic area for dengue disease. Generally, the nation experiences a higher number of dengue cases in the second half of each year, with peak periods occurring between May and October. The exact reasons for the re-emergence of dengue are not yet fully understood. However, the successful spread of dengue has been attributed to various factors including rapid increases in population growth, unplanned urbanization, lack of effective of vector control systems, global travel, climate change and extreme weather conditions, poor public health infrastructure, socio-economic status and expansion of the geographical distribution of the vector. Of them, the climate factors (mainly; temperature and rainfall) play a major role and although, mainly dengue spreads in tropical and sub-tropical regions, there is a risk for it to spread even in other regions

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Fig. 2 Dengue reported cases during 1988–2017 in Sri Lanka. Source: National Epidemiology Unit, Sri Lanka and [26]

due to the global climate change [12]. The vectors are sensitive to weather conditions, which can affect survival and reproduction rates, therefore, climate change influences the prevalence and distribution of vector-borne diseases like dengue. Especially, increased temperature would allow the vector to establish itself in newer locations and may increase the length of the transmission season in locations where the vector is already established [11, 12]. The transmission of dengue is influenced by temperature and rainfall due to the effects of temperature and rain on the rates of biological development, feeding, reproduction, population density, and survival of vectors. Previous studies have shown that dengue viruses may reduce incubation time in vectors from (approximately) two weeks to one week at temperatures of 32 ◦ C and above, however, below 36 ◦ C [11, 12]. In briefly, vector population increase rapidly during warmer periods due to short life cycle development at higher temperature. Further, high temperature reduces the duration of the incubation period of dengue viruses, prolongs the infective days of a vectors, hence, increases the dengue transmission rate. Though heavy rainfall could reduces the life time of outdoor vector, and potentially destroy immature stage, it also creates many temporary breeding sites for vectors, which in turn impacts dengue transmission. Climate factors may create favorable habitat for Aedes asgypti and Aedes albopictus vectors to survive and hence increase the risk of dengue spread. At higher temperature (above 32 ◦ C), vectors emerge from eggs to adults in a shorter period and also

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experience increasing mortality rates of adult mosquitoes with increasing temperature above 32 ◦ C [15]. Aedes (Stegomyia) aegypti (Linneaus) and Aedes albopictus are the major urban vectors that responsible to transmit dengue worldwide [5, 13]. From the literature [5, 13], it is noted that, Aedes aegypti is the principal vector in most of the dengue endemic regions. Further, it is noted that the majority of the severe dengue cases were reported in regions where Aedes aegypti was found. Aedes aegypti is closely associated with humans and human habitations, whereas Aedes albopictus is found mainly in outdoor premises. Both are day-time (mainly; morning and late afternoon) biting mosquito, which feeds on multiple hosts (Aedes aegypti prefers to bite humans, whereas Aedes albopictus also bites domestic animals ) during a single gonotrophic cycle (feed, convert a blood meal into eggs, lay the eggs and then seek out a new blood meal). Usually, female mosquitoes lay their eggs on or near the surface of water in natural, artificial or human-made containers [14]. The embryonated eggs are environmentally resistant and can withstand dry conditions for up to a year. During the last two decades, it is observed that significant increase in the global distribution of Aedes asgypti and dengue epidemics [5, 13, 14]. The average life time of an adult vectors ranges from 14 to 28 days and during this life span female vector usually, lay about 3 batches of eggs with around 100 eggs per batch [15]. It is common for Aedes aegypti to bite several humans to complete a blood meal and therefore to infect several people in the same household or approximately 100 m diameter area [5]. The horizontal as well as vertical dispersions of vectors in a geographical area influences dengue transmission. The extensive dispersion in distance and height allows vectors to lay eggs and transmit dengue rapidly, not only in the same household, but also in the neighboring communities. To spread the dengue transmission all over the island, greater dispersion of vectors could possibly be contributed to usual human transportation modes, such as, cars, busses, and trains, as it is common for local residents to travel from one end of the island to another for education, work, health and other matters. Dengue virus (DENV) is a wide spread arborvirus that belongs to Flaviviridae family, which groups over 60 arboviruses [16]. DENV appeases as four different serotypes, namely, DENV 1, DENV 2, DENV 3 and DENV 4 which are serologically and genetically related [5, 6, 16, 17]. These serotypes exhibit approximately 70% sequence similarity and are antigenically distinct [16]. An infection from one serotype gives life-long immunity against that serotype, but it confers only a Temporary Cross Immunity (TCI) against the three remaining serotypes. Because of the phenomenon known as Antibody-Dependent Enhancement (ADE), more severe forms of the dengue disease can be commonly observed in secondary heterologous infections while tertiary and quaternary infections are very rare. ADE can simply be defined as the increase in virulence in the presence of immunity conferred by a previous infection of a different serotype of the same virus group. In fact, this has become a serious issue with the possibility of multiple DENV to co-circulate in endemic settings. It is reported that a majority of the secondary infections occur at least six months apart the first infection. Further, the propensity of a tertiary or

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quaternary exposure to aggravate to the DHF form, relative to a primary or secondary infection, still remains unknown [5, 6, 16, 17]. Once the dengue virus enters into the human blood, it is subject to an incubation period approximately around 4–7 days and in completion of this period, most patients experience a sudden onset of fever and is often accompanied with symptoms such as myalgia, arthralgia, anorexia, sore throat, headaches, and a macular skin rash [16, 17]. The majority of patients show a self-limiting clinical course so that the fever does not transform into more severe levels such as dengue hemorrhagic fever (DHF), or dengue shock syndrome (DSS). The patients whose fever progressed into DHF or DSS could end up with serious damages such as organ failure pleural effusion ascites [16, 17]. Yet there are no vaccines or any specific treatments f for dengue and currently many research studies are focusing on developing a vaccine [2, 3, 5]. For the time being, the only mechanism for preventing and controlling DF/DHF is to ensure prompt diagnosis of onset of fever and appropriate clinical management during hospitalization, to reduce human-vector contact, and to control larval habitats [2, 3, 11, 12]. Dengue fever is among the most economic burden infectious diseases in tropical and subtropical regions and represents a significant economic and public health burden especially in developing countries [18]. For example, the annual average economic burden (with 95% certainty levels) due to dengue in Bhutan, Brunei, Cambodia, East-Timor, Indonesia, Laos, Malaysia, Myanmar, Philippines, Singapore, Thailand, and Vietnam (12 Southeast Asian Countries) was US$ 950 million (US$ 610 million to US$ 1384 million) or about US$ 1.65 (US$ 1.06 to US$ 2.41) per capita [19]. The total health system cost of dengue inward case management in the Colombo district hospitals in 2012 was around US$ 3.45 million (US$1.50 per capita). The cost of dengue prevention activities in Colombo exceeded a total of US$ 1.27 million in 2012 and this particular item has now become a higher burden to the national budget [18]. In Thailand, the average total cost of a patient was estimated at US$ 109.16 and in Puerto Rico, the cost was US$ 125 [20]. In southeast Asia, a study showed the severity of the disease is significantly correlated with the economic burden as the estimated cost for DHF was US$ 139 compared to US$ 4.29 for DF [20]. Economic cost in the Americas due to Dengue was estimated as US $2.1 billion per year on average (in 2010), with a range of US$ 14 billion in sensitivity analyses and substantial year to year variation [21]. The UNICEF-UNDP-World Bank-WHO Special Programme for Research and Training in Tropical Diseases (TDR) has summarized the costs of epidemic outbreaks of DF/DHF in several countries and such summary can be found from [22]. Table 1 presents the dengue direct cost according to the various regions and their health status. Considering all these significant numbers and facts, it has been observed and noted in recent years that, the dengue epidemic outbreaks make a critical impact on socio-economic indices around the world. The economic burden due to dengue can be classified as direct and indirect costs. Direct costs are those within health-care system

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Table 1 Dengue Direct Medical Costs in 2010 PPP-Adjusted $US - Source World Health Organization (WHO-CHOICE); A very low mortality for children and adults, B low mortality for children and adults, C low mortality for children, high mortality for adults, D high mortality for children and adults, E very high mortality for children and adults WHO subregion Inpatient cost Clinic cost African Region - D African Region - E Region of the Americas - A Region of the Americas - B Region of the Americas - D Eastern Mediterranean Region - B Eastern Mediterranean Region - D European Region - A European Region - B European Region - C South-East Asian Region - B South-East Asian Region - D Western Pacific Region - A Western Pacific Region - B

$71.24 $29.22 $665.45 $162.96 $50.68 $432.45 $28.84 $502.65 $95.17 $184.37 $46.64 $26.82 $660.59 $72.57

$12.52 $5.73 $65.50 $22.83 $9.66 $45.41 $6.70 $51.21 $14.92 $24.69 $9.27 $6.04 $65.35 $12.37

including cost of diagnosis, treatment, medical care, surveillance and prevention. Indirect costs are linked to economic value (human-hours) lost due to sickness and premature death including loss of productivity. There are three major direct cost categories; namely, medical care, surveillance and reporting, and prevention. The cost of medical care includes the cost for ambulatory and inpatient care. Surveillance and reporting costs take into account efforts by governments and non-government agencies to monitor and disseminate information about cases, outbreaks, and deaths. Prevention costs include activities to prevent dengue, such as vector control including inspections, management of disposable, awarness/education/media campaigns, and community mobilization. The economic impact of the dengue outbreaks is difficult to assess due to complex nature of it. However, quantifying the epidemiological and economic burden of dengue is key to formulating policy decisions on research priorities, prevention programmes and public health training. From a social point of view, the effectiveness of the health care system also depends on the affordable level of individual medical care cost. Hence, setting up conceptual process to estimate the future direct medical cost due to dengue is an essential task and which leads to establishing the efficient and effective disease control strategies. We propose the insurance based model to estimate the direct cost due to medical care expenditure. From the management and business point of views, estimating the direct cost may provide opportunities for the insurance industry to develop new business proposals (medical insurances).

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This chapter is organized as follows. Section 2 presents description of mathematical model related to dynamics of dengue. General introduction of dynamical model can be found at the beginning of the section. Dimensionless form of the dynamical system and mathematical analysis of the model can found in the middle of the section. Latter part of Sect. 2 focuses on the stability criterion of the prescribed model. Insurance based investment model for dengue dynamics is presented in Sect. 3. Introducing the probabilistic model definition and actuarial pricing model is discussed in the middle of the section. Premium computing algorithm is presented in the latter part of Sect. 3. Actuarial analysis of the proposed insurance based investment model is performed in Sect. 4. The benefit reserve curve and its properties and conditions related to cover up future financial risk are also presented in this section. Further, some practical issues and strategies to overcome such issues are discussed in Sect. 4. Implementing algorithms is also presented in Sect. 4. Section 5 focuses on possibilities to identify the efficiency of the control strategies which are supposed to minimize the spread of disease. Conclusions, remarks and future possible paths can be found in Sect. 6.

2 Mathematical Model 2.1 SIR Compartment Model Mathematical modelling process can be considered as a tool to understand, describe, simulate and predict the behavior of a population affected by an epidemic, the environment where it takes place or the pathogenic agents involved. The history of mathematical modeling in epidemiology goes to the 18th century; in 1760 Daniel Bernoulli developed the model aiming to evaluate the impact of variolation on human life expectancy [23]. However, there was no evidence found till the early 20th century, that the concept of dynamical systems approaches were applied to epidemiology. In 1927, Kermack and Mckendrick introduced the classical deterministic simple epidemic model and which gave a benchmark result for later sophisticated researches to develop more complex models [23]. Kermack and Mckendrick focused on acute infections, assuming life-long immunity, presented classical compartment model to describe the dynamics of epidemics. This classical model is known as SIR (Susceptible - Infected - Recovered) model and it categorises hosts within a population as Susceptible (if previously unexposed to the disease), Infected (if currently colonised by the pathogen) and Recovered (if they had successfully cleared the infection). We also start the development of dynamical system to describe the dynamics between host-vector population by introducing SIR compartments for host and SI compartments for vector [10–12, 23–25]. The host population is divided in to three classes; namely, Sh susceptible humans, Ih infectious humans and Rh recovered

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Fig. 3 The schematic diagram for SIR model of dengue disease transmission which describes the interactions between host and vector populations

humans. Since vectors’ infective period ends with their death, the vector population is divided into two classes; namely, Sv susceptible vectors, and Iv infectious vectors. The model assumes all new-born are susceptible in both populations (no vertical transmission) and a uniform birth rate. Let μ be the per capita birth rates for host population and D be the constant recruitment rate for vectors. A susceptible, infectious or recovered human can disappear from the respective compartment with a natural death, at a per capita rate of μh and an infectious human can recover from the infection and join the recovered population with a per capita rate of r. The natural death rate of vectors is denoted by μv . The effective rates between the host and vector populations can be defined as the average number of contacts per time leads to the infection of one party if the other party is infectious. The host-vector and vector-host rates depend on the host biting rate of the vectors, b, the transmission probabilities between vectors and hosts namely the transmission probability of the virus from vectors to host, βh , and the transmission probability of the virus from host to vectors, βv . In addition to mentioned factors, the effective contact rates between the two populations depend on the total number of individuals in each host and vector population, Nh and Nv respectively. The number of potentially infectious bites given to susceptible host per time can be defined as bSNhhIv and a fraction of these bites, namely, βh , successfully infect host. Thus, the infected host per unit time is defined h Iv v Ih . Similarly, the infected vectors per time is defined as βv bS . Figure 3 shows as βh bS Nh Nh the interaction between host and vector populations.

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Considering the inflow and outflow movements of each compartments, the system (1) of nonlinear differential equations is defined to describe the dynamics of dengue between two populations; dSh dt dIh dt dRh dt dSv dt dIv dt

= μNh − =

bβh Sh Iv − μh Sh Nh

bβh Sh Iv − (μh + r)Ih Nh

= rIh − μh Rh =D− =

bβv Sv Ih − μv Sv Nh

bβv Sv Ih − μv Iv Nh

(1a) (1b) (1c) (1d) (1e)

with the conditions Nh = Sh + Ih + Rh Nv = Sv + Iv and Sh (0) ≥ 0, Ih (0) ≥ 0, Rh (0) ≥ 0, Sv (0) ≥ 0, Iv (0) ≥ 0.

2.2 Dimensionless Model Assuming that rate of change of host and vector populations are zero, it can be easily obtained that μ = μh and D = μv Nv . Introducing the dimensionless quantities, S = NShh , I = NIvv , V = NIvv and imposing Nh = Sh + Ih + Rh and Nv = Sv + Iv , the system (1) can be reduced to system (2); dS = μh (1 − S) − γh nSV dt dI = γh nSV − (μh + r)I dt dV = γv I (1 − V ) − μv V dt

(2a) (2b) (2c)

where γv = bβv , γh = bβh and n is a measure of vector density per human and is denoted by NNvh with S(0) ≥ 0, I (0) ≥ 0 and V (0) ≥ 0. Figure 4 displays the numerical simulation of the model described in system (2).

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1 Susceptible Infected Recovered

0.9 0.8

Fractions

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

50

100

150

200

250

300

350

400

Time in days

Fig. 4 Numerical Solution to the model in (2) with S(0) = 0.9999, I (0) = 0.0001, V (0) = 0. Model parameters are μh = 0.00003424, μv = 0.04762, n = 10, γh = 0.02333, γv = 0.2333, r = 0.0714.

2.3 Mathematical Analysis of the Model Let  be the set of solutions to the system of nonlinear differential equations in (2) and is given as  = {(S, I , V ) ∈ R3+ : S + I ≤ 1, S ≥ 0, I ≥ 0, V ≥ 0}. By considering the stationary point of system (2), two equilibrium points of the model can be obtained and are given as; E 0 = (1, 0, 0) and E e = (S e , I e , V e ) where Se =

P+Q R20 − 1 Q(R20 − 1) e e , , I = , V = Q + PR20 Q + PR20 R20 (P + Q)

with P=

μh + r γv , Q= and R0 = μh μv



nγh γv . μv (μh + r)

The equilibrium point E 0 is known as disease free equilibrium (FE) point which always exists in the absence of infective population whereas the second point E e if exists is called endemic equilibrium point (EE) and R0 is known as the basic reproduction number. The basic reproduction number is a responsible measure or index to reflects the number of cases generated from one case on average over the course of its infectious period [11, 12, 27]. It can also be considered as a risk measurement index

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Infected Host Fraction

0.5 0.4 0.3 0.2 0.1 0

0

50

100

150

200

250

300

350

400

Time in days

Fig. 5 Variation of infected host fraction profile with respect to R0 (1.26 ≤ R0 ≤ 5.66)

of disease spread. Global and local stability of FE and EE depend on nature of the basic reproduction number, R0 [27]. Figure 5 displays the variation of infected host fraction profile with respect to R0 . One can see that when R0 increases the infected host profile quickly reaches the peak. Further, it can be noted that smaller the R0 lower the intensity of the peak. Theorem 1 FE point is locally stable if R0 < 1. Proof Considering the Jacobian matrix of the dynamical system, the characteristic equation corresponding to FE point is obtained and given as; λ3 + a2 λ2 + a1 λ + a0 = 0,

(3)

where a0 = μ2h μv + rμh μv − μh γv γh , a1 = μ2h + 2μh μv + (μh + μv )r − γh γv , a2 = r + μv + 2μh . By solving the Eq. (3) three eigenvalues can be obtained and given as λ1 = −μh , λ2 =

−(μh + μv + r) +



(μh + μv + r)2 − 4μv (μh + r)(1 − R20 ) 2

,

Insurance Model to Estimate the Financial Risk Due …

λ3 =

−(μh + μv + r) −



(μh + μv + r)2 − 4μv (μh + r)(1 − R20 ) 2

641

.

It can be clearly seen that three eigenvalues have negative real parts if R0 < 1.



Theorem 2 FE point is globally asymptotically stable if R0 < 1. Proof The global stability can be proved using the Lyapunov function given in (4): f (I , V ) =

γh n V + I. μv

(4)

Consider f˙ (V, I ):   f˙ (V, I ) = − γh n(1 − S)V + μv (μh + r)(1 − R20 )(1 + R20 V )I . It can be clearly seen that if R0 < 1, f˙ (V, I ) < 0. Thus FE point is globally  asymptotically stable if R0 < 1. Theorem 3 EE point is locally asymptotically stable if R0 > 1. Proof Considering the Jacobian matrix of the dynamical system, the characteristic equation corresponding to EE point is obtained and given as; λ3 + c2 λ2 + c1 λ + c0 = 0,

(5)

where c0 = μv μ2h P(R20 − 1)     Q + PR20 μv μh PQ , + μh μv R20 + (R20 − 1) c1 = μ2h P P+Q Q + PR20     Q + PR20 P+Q c2 = μh . + μh P + μv R20 P+Q Q + PR20 According to Routh-Hurwitz criterion if third order polynomial (i.e. if P(λ) = λ3 + c2 λ2 + c1 λ + c0 ) satisfies conditions given in (6) then the equilibrium point is locally stable; (6) c1 > 0, c2 > 0 and c1 c2 > c0 . It is clear that coefficients of (5) satisfy the conditions given in (6) as long as  R0 > 1. Thus the endemic equilibrium EE is locally stable whenever R0 > 1. Figures 6 and 7 display the solution trajectories of system (2) on (S, I ) plane with respective different R0 values. Figure 6 reflects the case that R0 < 1 whereas Fig. 7 reflects the case that R0 > 1.

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Fig. 6 The solution trajectory projected onto (S, I ) with R0 < 1 and S(0) = 0.75, I (0) = 0.25

0.25

Infected Host Fraction

0.2

0.15

0.1

0.05

0 0.5

0.55

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1

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Susceptible Host Fraction 0.6

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Infected Host Fraction

Fig. 7 The solution trajectory projected onto (S, I ) with R0 > 1 and S(0) = 0.9999, I (0) = 0.0001

0.4

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3 Insurance/Investment Epidemiological Model for Dengue Transmission Classically, insurance process can be defined as a tool of protection from financial losses. Further, it can be defined as a social and management tool giving financial benefits for the effects of misfortune, the payment being made from the accumulated contributions of all parties participating in the scheme. The idea behind developing an insurance model is to determine how much we need to allocate at present (today) to coverup future financial loses. Thus, the insurance model is the process to develop a form of contract between two parties, namely insure and insured whereby insurer undertakes in exchange for a fixed premium to pay insured, a fixed or variable benefits on basis of financial losses due to the happening of prescribed event/s. People who

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are facing the financial losses due to some particular event would like to form a market that could contribute premiums to an insurance fund, to obtain the future financial losses if they are affected by prescribe event. This insurance concept may be used to determine the present economical burden of future financial losses due to the prescribed risk event. In other words, insurance model can be defined as a financial tool to determine, at present, how much we need to invest or allocate in order to coverup future expenses. Utilizing the classical insurance concept, we develop insurance based investment model to predict the future direct medical expenses due to dengue epidemics. SIR compartment model described in Sect. 2, can be redefined to provide different roles under the insurance/investment perspective manner [18, 28, 29]. The individuals who are in susceptible compartment are facing the risk of being infected with dengue form a market that could contribute premiums or invest to an insurance fund, to obtain the future medical expenses (benefits) if they move to infected compartment. During a dengue outbreak, the infected policyholders/investors would benefit from the claim payments provided by the insurance fund. The individuals who are in infected class are removed from it due to either death or recovery. Such individual group is now counted as recovered compartment. The systematics diagram of insurance SIR dengue transmission is displayed in Fig. 8.

Fig. 8 The schematic diagram for SIR insurance model of dengue disease transmission

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3.1 Probabilistic SIR Dengue Model Consider the system of non-liner differential equations given in (2). In here, S(t) and I (t) represent the fractions of the host population in susceptible and infected compartments respectively. One can interpret the fraction of S(t) and I (t) as the probability of an individual being in susceptible, infected compartments, respectively, at the time t. Further, V (t) as the probability of an individual vector being in infected compartment. Now, taking S(t), I (t) and V (t) as the probabilities, rate of change of such probabilities can be described in (7); dS = μh (1 − S) − γh nSV dt dI = γh nSV − (μh + r)I dt dV = γv I (1 − V ) − μv V dt

(7a) (7b) (7c)

with S(0) ≥ 0, I (0) ≥ 0, V (0) ≥ 0 and S(0) + I (0) = 1. Considering the probability density functions, S(t) and I (t), and utilizing actuarial techniques, the components of insurance model / investment package is formulated to determine the future direct medical expenses due to dengue epidemic.

3.2 Actuarial Pricing Model We assume that in dengue insurance/investment plan, individuals who are known as susceptibles invest premiums in the form of continuous level payments. Simply speaking, the policyholders/investors are committed to pay/invest premiums as long as they remain as susceptible individuals. We also assume that the benefits/medical expenses are continuously claimed for each infected policyholder/investor during the entire period of treatment. Further we assume that, once the individual recovers or dies from the disease, the insurance/investment plan terminates immediately. At time, t = t, the present value of the future level payments may be computed as the discounted value of unit amount of money for level payments multiply by the corresponding probability of occurrence of such a payment [28–30]. Considering the expenditure side of the insurance plan or benefit side of the investment plan, expected total present value or actuarial present value of t period unit benefit (claim) payment is given as  E[B(t)] =

t

exp(−δt)Probability of an individual being infected at t dt 0

(8)

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where δ > 0 denotes the force of interest and B(t) denotes the present value of future benefit payments and E[B(t)] represents the expected or mean present value of the benefit payments. Similarly, one can define the income side of the insurance or investment plan as the expected total present value or actuarial present value of t period premium payment as  E[P(t)] =

t

exp(−δt)Probability of an individual being susceptible at t dt. (9) 0

where P(t) denotes the present value of the premium payments and E[P(t)] represents the expected or mean present value of the premium payments. Imposing I (t) and S(t) are as the probability of an individual being in infected and susceptible, respectively, at time t, the actuarial present value of benefit payment given in (8) and the actuarial present value of premium payment given in (9) can be expressed in (10)–(11); 

t

E[B(t)] = 

exp(−δt)I (t)dt

(10)

exp(−δt)S(t)dt.

(11)

0 t

E[P(t)] = 0

Considering the present value of benefit payments, B(t), and premium payments, P(t), the investor’s net random future loss, Ł, is defined as Ł = B(t) − πP(t), where π denotes the premium payment. According to the equivalence principle the requirement to exist such an investment or insurance plan is the expected value of loss random variable should equal to zero [30]. By applying the equivalence principle the premium payment, π, can be obtained as; E[Ł] = 0, E[B(t) − πP(t)] = 0, πE[P(t)] = E[B(t)]. Thus, the level premium payment, π, can be given as in (12) π=

E[B(t)] . E[P(t)]

(12)

The level premium, π, denotes the expected or average present value of premium payment which may cover the future medical expenses due to dengue outbreaks. In

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other words, π is the average present economical burden of future medical expenses due to the risk of dengue spread. Further, this parameter can also be considered as the present financial risk of the future expenses due to dengue epidemics.

3.3 Financial Risk (Premium) Computing Algorithm By coupling (7) together with two integral equations in (10) and (11), the financial risk computing model can described as in (13); π=

E[B(t)] , E[P(t)]

(13a)

where E[B(t)] and E[P(t)] given in  E[B(t)] =

t

exp(−δt)I (t)dt,

(13b)

exp(−δt)S(t)dt,

(13c)

0

 E[P(t)] =

t

0

where S(t) and I (t) given in dS = μh (1 − S) − γh nSV, dt dI = γh nSV − (μh + r)I , dt dV = γv I (1 − V ) − μv V, dt

(13d) (13e) (13f)

with S(0) ≥ 0, I (0) ≥ 0, V (0) ≥ 0 and S(0) + I (0) = 1. To solve the system (13), MATLAB ode45 solver is used. The function ode45 is a variable-step solver and is based on an explicit Runge- Kutta (4, 5) formula, the Dormand-Prince pair [31]. From the simulation results, we obtain a piecewise constant approximation for the continuous functions of I (t) and S(t) given as (14a) and (14b), Ij , j − 1 < t ≤ j (14a) I˜ (t) = 0, otherwise,

˜ = Sj , j − 1 < t ≤ j S(t) 0, otherwise,

(14b)

where Ij and Sj are the rate of infection and susceptible at the time interval (j − 1, j].

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Then the evaluation of both expected values of B(t) and P(t) are numerically approximated and given in (15a) and (15b), E[B(t)] ≈

1

(exp(−(j − 1)δ) − exp(−jδ)) Ij , δ j=1

(15a)

E[P(t)] ≈

1

(exp(−(j − 1)δ) − exp(−jδ)) Sj . δ j=1

(15b)

t

t

3.4 Numerical Simulation Results Figure 9 visualizes the variation of premium (present financial risk) with respect to basic reproduction number (R0 ). One can see that when R0 increases the value of premium also increases. Present financial risk depends on future medical expenses and which depends on risk of disease spread. When R0 increases, risk of disease spread also increases (see Fig. 5) thus the premium too. Figure 10 summarizes the variation of premium with respect to transmission probability of the virus from vector to host and rate of recovery. One can see that higher the transmission probability and lower the recovery rate directs to higher premium. This phenomena can be justified using Fig. 11. Corresponding variation of basic reproduction number with respect to prescribed two parameters is presented in Fig. 11. Since, R0 , is also high when high βh and low r values, it can be easily justified the variation of premium described in Fig. 10. Figure 12 presents the variation of present financial risk (premium) with respect to basic reproduction number and force of interest. Figure 13 displays the variation of premium with respect to force of interest. One can see that when the force of interest changes from 1–20% the premium reduces from 0.58 to 0.02. It can be easily understood that as the discounting factor (force of interest) increases present value of the future sum decreases.

4 Actuarial Analysis of the Model Insurance based model is proposed to predict the present financial risk in order to cover the future direct medical expenses due to dengue epidemics. Actuarial analysis is an essential procedure to be performed to identify the feasibility and or existence of insurance models in an investment risk environment. This analysis allows to predict the premium with a justifiable degree of accuracy which is sufficient to coverup future benefit payments [30].

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14

16

18

R0

Fig. 9 Variation of premium (present financial risk) with respect to basic reproduction number Fig. 10 Variation of premium (present financial risk) with respect to transmission probability of the virus from vector to host, βh , and rate of recovery, r

0.5 0.4 0.3 0.2 0.1 0 0.6 0.3

0.4 0.2

0.2 h

Fig. 11 Variation of basic reproduction number with respect to transmission probability of the virus from vector to host, βh , and rate of recovery, r

0.1 0

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r

15

R0

10

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0 0.6 0.3

0.4 0.2

0.2 h

0.1 0

0

r

Insurance Model to Estimate the Financial Risk Due … Fig. 12 Variation of premium (present financial risk) with respect to basic reproduction number and force of interest, δ

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0 15 0.2

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Fig. 13 Variation of premium with respect to force of interest, δ at R0 = 9.56

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4.1 Reserve Function Like most of insurance models, the premium which is determined by proposed dynamical system, may not be sufficient to coverup future medical expenses if the disease spread very rapidly. To verify this, the actuarial analysis is performed by introducing the reserve function. The reserve function is build by taking difference of the total discounted premium payments and total discounted benefit payments [30]. ˆ and B(t) ˆ as the accumulated sum of premium obtained Consider the quantities P(t) up to time, t, and the accumulated sum of benefits paid till time, t, respectively. The rate of change in the accumulated value of total premiums may be expressed as sum of the rate of change of premium income and the rate of change of interest obtained from the current total premium income and is given in (16):

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ˆ d P(t) ˆ = πS(t) + δ P(t), dt

(16)

ˆ with P(0) = πS(0) > 0. Similarly, the rate of change in the accumulated value of total benefits may be expressed as the sum of the rate of change of benefit payments and the rate of change of interest obtained from the current total benefit payments and is given in (17): ˆ d B(t) ˆ = I (t) + δ B(t), dt

(17)

ˆ with B(0) = I (0) > 0. The reserve function, R(t), which provides the accumulated benefit reserve at time, t, and it can be expressed as the difference of accumulated value of premium and benefit payment and is given in (18); ˆ − B(t). ˆ R(t) = P(t)

(18)

Thus the rate of change of the reserve function can be obtained by considering difference of (16) and (17); ˆ ˆ d P(t) d B(t) dR = − dt dt dt ˆ − B(t)). ˆ = πS(t) − I (t) + δ(P(t) Hence the rate of reserve function is given in (19): dR = πS(t) − I (t) + δR(t), dt

(19)

with R(0) = πS(0) − I (0) > 0.

4.2 Behaviour of the Benefit Reserve Function To obtain the reserve function, numerical simulation is performed to solve the differential equation given in (19) by coupling with (13). From (13), corresponding S(t) and I (t) profiles and the premium, π, can be obtained. One can see from Figs. 14 and 15, premium rates are significantly low due to quite low risk of disease spread. This is quite obvious since the basic reproduction number, R0 , is close to one. Further, it can be seen that undesirable negative benefit reserve functions during the policy period of these two cases, which means in these two cases, the premium which determined by the dynamical system is not sufficient

Insurance Model to Estimate the Financial Risk Due … Fig. 14 Benefit reserve function with R0 = 1.35, π = 0.008, δ = 0.05.

1.5

651

× 10 -3

1

Reserve Function

0.5 0 -0.5 -1 -1.5 -2

R0 = 1.35

-2.5

= 0.008

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20

30

40

50

60

Time in weeks

Fig. 15 Benefit reserve function with R0 = 1.17, π = 0.0067, δ = 0.05. Reserve Function

0

-0.005

R0 = 1.17 = 0.00667

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-0.015

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Time

to cover the future expenditures. By considering Figs. 14 and 15, one can conclude that the premium which determined by the dynamical system does not guarantee to coverup the future expenditures. In other words, proposed premium is not a sufficient condition to coverup future medical expenses. Figures 16 and 17 display the two profiles benefit reserve function with relatively high premiums. One can observe that in both cases the basic reproduction number is also high (R0 = 3.02 in Fig. 16 and R0 = 6.75 in Fig. 17). In other words, the risk of disease spread corresponding to Figs. 16 and 17 is significantly high. Though the benefit reserve function is positive throughout the policy period, the premium is relatively high. On the other hand, from Figs. 16 and 17, one can conclude that the proposed premium is not a minimal condition to coverup the future financial burden.

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Fig. 16 Benefit reserve function with R0 = 3.02, π = 0.026, δ = 0.05.

7

6

Reserve Function

5

4

3

R0 = 3.02 2

= 0.026

1

0

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40

60

80

100

120

Time

Fig. 17 Benefit reserve function with R0 = 6.75, π = 0.22, δ = 0.05.

10 9

Reserve Function

8 7 6 5 4 3

R0 =6.75

2

= 0.22

1 0

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Time

By observing Figs. 14, 15, 16 and 17, it can be concluded that the premium determined by equivalence principle together with dynamical system given by (13) is not in optimal under two cases: either • it may create undesirable negative reserves during the policy term - (There is a high possibility that an insurer may not be able to collect enough premiums. Hence the need to increase the premium to a level that guarantees positive benefit reserve); or • it may create quite high positive reserves during the policy term - (Since premiums are relatively high, there is a high possibility that healthy policyholders may not choose this options, ultimately which will increase the insurance costs. Hence the need to decrease the premium to a level that guarantees positive benefit reserve).

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Thus, it is essential to optimize the premium which guarantees minimal positive reserves during the policy duration.

4.3 Adjusted Premium Premiums should be determined in a way such that it guarantees that the investment reserves are sufficient to make future benefit payment. Mathematically, in this process, premium, π, need to be determined by imposing, the reserve function should be positive for entire policy term duration (i.e. R(t) ≥ 0 ∀t). Theorem 4 If π >

t I (0)+ 0 exp(−δt)I (t)dt t S(0)+ 0 exp(−δt)S(t)dt

then R(t) > 0 ∀t.

Proof Consider the derivative of R(t); dR(t) = πS(t) − I (t) + δR(t), dt d (exp(−δt)R(t)) = exp(−δt)(πS(t) − I (t)) dt  t  t d (exp(−δt)R(t))dt = exp(−δt)(πS(t) − I (t))dt 0 dt 0  t exp(−δt)(πS(t) − I (t))dt exp(−δt)R(t) = R(0) + 0    t  t exp(−δt)R(t) = π S(0) + exp(−δt)S(t)dt − I (0) − exp(−δt)I (t)dt 0

0

To have a R(t) > 0 ∀t    t  t exp(−δt)S(t)dt − I (0) − exp(−δt)I (t)dt > 0. π S(0) + 0

0

Which means π>

I (0) + S(0) +

t 0t 0

exp(−δt)I (t)dt exp(−δt)S(t)dt

.

(20a)

Therefore, adjusted premium, πA , can be set as the minimum of π as described in (20a) and given in (20b) or (20c); πA =

I (0) + S(0) +

t 0t 0

exp(−δt)I (t)dt exp(−δt)S(t)dt

,

(20b)

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Reserve Function

0.1 0.08 0.06 0.04 0.02 0 -0.02

0

10

20

30

40

50

60

70

80

Time

Fig. 18 Variation of benefit reserve function with respective premium (blue) and adjusted premium (red). Here π = 0.00667, πA = 0.0083638 and R0 = 1.1694.

or in discrete form as πA =

δI (0) + δS(0) +

t j=1

(exp(−(j − 1)δ) − exp(−jδ)) Ij

j=1

(exp(−(j − 1)δ) − exp(−jδ)) Sj

t

.

(20c) 

Figure 18 displays the variation of benefit reserve function with respect to the premium (π = 0.00667) and the corresponding adjusted premium (πA = 0.0083638). In both cases, the basic reproduction number is 1.1694. It can be easily observed that the benefit reserve curve is negative with basic premium. However, with corresponding adjusted premium, it is positive throughout the policy period. Figure 19 presents the variation of basic premium and adjusted premium with respect to the basic reproduction number. It can be easily understood that since the risk of disease spread increases, both premiums rise up. However, one can see that the basic premium which determined by the dynamical system via equivalence principle is not sufficient to cover up future medical expenditure at early stage of R0 . Further, when R0 exceed some threshold, one can see that adjusted premium is below the basic premium. Thus, the adjusted premium given in (20c) guarantees that the reserve function is positive throughout the policy period and which is the minimum condition to satisfies the positiveness of reserve function.

4.4 Implementation of the Model in Practice Implementing the discrete model derived in (15a) and (15b) is a difficult task in practical point of view. The infected table described in (14a), can be developed by

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0.045 0.04 0.035

A

Premium

0.03 0.025 0.02 0.015 0.01 0.005 0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

R0

Fig. 19 Variation of basic premium (blue) and adjusted premium (red) with respect to basic reproduction number

considering recorded public data from government or non government health sources. However, it is very difficult to establish the susceptible table. In practice, it is hard to keep record of susceptible individuals due to non existence of any scientific method to count them. This is partly because of the large numbers in a population and partly because of the difficulty in distinguishing a person susceptible to a disease from one with immunity as well as silent infectious cases. Hence it is essential to keep on the function I (t), instead of S(t), for all premium-rating computations. Consider the expected value of premium payment given in (11):  E[P(t)] =

tF

exp(−δt)S(t)dt  tF d −1 S(t) exp(−δt)dt = δ 0 dt    tF 1 = exp(−δt) (μh (1 − S(t)) − γh nS(t)V (t)) dt −S(t) exp(−δt)|t0F + δ 0      tF 1 1 tF E[P(t)] = −S(t) exp(−δt)|0 + μh 1+ exp(−δt)dt δ δ 0   tF   dI (t) 1 exp(−δt) (μh + r)I (t) + − dt δ dt 0 0

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   tF 1 μh exp(−δt)I (t)dt −S(t) exp(−δt)|t0F − exp(−δt)|t0F − (μh + r) δ δ 0  tF    1 dI (t) − dt exp(−δt) δ dt 0

=

  1 1 1+ E[P(t)] = ((S(0) + I (0)) − exp(−δtF )(S(tF ) + I (tF )) δ δ

μh + (1 − exp(−δtF )) δ  μh + r + δ E[B(t)] − δ Since S(0) + I (0) = 1;  1+

  1 1 μh 1 μh E[P(t)] = 1+ − exp(−δtF ) S(tF ) + I (tf ) + δ δ δ δ δ   μh + r + δ − E[B(t)]. δ

Thus, E[P(t)] =

  μh μh 1 1+ − exp(−δtF ) S(tF ) + I (tF )) + (1 + δ) δ δ − (μh + r + δ)E[B(t)]) . (21a)

and the basic premium π is given by π=

1 (1+δ)

 1+

μh  δ

E[B(t)]  − exp(−δtF ) S(tF ) + I (tF )) +

and

μh  δ

; − (μh + r + δ)E[B(t)] (21b)

F 1

(exp(−(j − 1)δ) − exp(−jδ)) Ij . δ j=1

t

E[B(t)] ≈

Here tF represents the final time point. Further simplification is possible by taking tF as an infinity. Thus π can be expressed as in (22): π=

1 (1+δ)

 1+

μh  δ

E[B(t)] . − (μh + r + δ)E[B(t)]

(22)

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Fig. 20 Comparison of premium calculation using (21b) (Top ) and (22) (Bottom) with respect to basic reproduction number

Thus, adjusted premium given in (20c) can now be expressed as in (23): πA =

S(0) +

1 (1+δ)

I (0) + E[B(t)]   . 1 + μδh − (μh + r + δ)E[B(t)]

(23)

Figure 20 presents the variation of premium with respect to basic reproduction number. Figure 20 - top summarizes the premium which is computed using formula given in (21b) whereas Fig. 20 - bottom visualizes the variation of premium which is computed using formula given in (22). One can easily see that there is no significant difference between both premium profiles. Thus, simplified premium computing formula given in (22) can be used for further simulation.

4.5 Implementing Algorithm In practice, one can follow the following steps to compute adjusted premium which provides the present financial risk due to future epidemic outbreak. Further, which is the optimal premium which is sufficient to cover the future expenses.

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0.5 0.4 0.3 0.2 0.1 0 -0.1

0

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Fig. 21 Variation of infected host fraction profile with 0 < u1 < 1 and 0 < u2 < 1

1. Develop the infected table given in (14a) using available recorded data. 2. Compute the average present benefit amount using (15a). 3. Compute the adjusted premium using (23).

5 Efficiency of Controllability Since there is no specific vaccine and specific drugs treatment for the dengue, imposing control strategies is the only mitigating technique to control the transmission of the disease. In Sect. 1, details information and annual per capita cost in various countries are discussed with providing corresponding references. Since these annual per capita cost is significantly high, it is worth enough to discuss the efficiency of the control stretergies. For such purpose, the dynamical system given in (2) is modified. Two control functions, u1 (t) (0 ≤ u1 (t) ≤ 1) and u2 (t) (0 ≤ u2 (t) ≤ 1) are imposed to the system (2) to minimize host-vector interaction. With two controls system (2) now read as dS = μh (1 − S) − γh n(1 − u1 (t))SV dt dI = γh n(1 − u1 (t))SV − (μh + r)I dt dV = γv (1 − u2 (t))I (1 − V ) − μv V dt

(24a) (24b) (24c)

with S(0) ≥ 0, I (0) ≥ 0, V (0) ≥ 0, 0 ≤ u1 (t) ≤ 1 and 0 ≤ u2 (t) ≤ 1. The control function, u1 , may represent the use of alternative preventive measures to minimize or eliminate host-vector contacts (such as the use of mosquito repellents

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or bed nets, conducting public awareness campaign programs and etc). Thus, the susceptible host population is now reduced by a factor of (1 − u1 ). The control function, u2 , may represent the level of larvacide and adulticide used for vector control administered at vector breeding sites to eliminate specific breeding areas. Consequently, the reproduction rate of the infected vector population is now reduced by a factor of (1 − u2 ). The function, (1 − u1 (t)), represents the effort that prevents transmission of dengue virus in order to reduce the contact between the vector and host populations, thus the number of infectious individuals may be decreased. If u1 = 1, the prevention of dengue virus is 100% effective, whereas if u1 = 0, transmission occurs without any control. Similarly, the function, (1 − u2 (t)), represents the effort that minimize the vector breeding sites so as to reduce the vector population. If u2 = 1, the efficient of minimizing the vector breeding sites is 100% effective, whereas if u2 = 0, there is no such effect. Figure 21 visualizes the variation of infected host population fraction with respect to two controls. Figure 22 summarizes the corresponding variation of basic reproduction number with respect to two controls u1 and u2 . One can see that the variation of basic reproduction number is not significant. Figure 22 provides evidence that the basic reproduction number varies in small range and which can be considered as an approximately constant and which is close to 2. However, the corresponding variation of infected host population is quite significant. This is due to impact of the two controls. The basic premium can be computed using formula (12) with coupling control system (24). Figure 23 displays the variation of basic premium with respect to 0 < u1 < 1 and 0 < u2 < 1 at the different basic reproduction number. Figure 23 top-left, top-right, middle-left, middle-right, bottom-left and bottom-right displays variation of the basic premium with R0 = 4.78, 6.68, 8.28, 9.56, 10.56, 11.71 respectively. Figure 24 displays corresponding basic premium with respect to

8

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Fig. 22 Variation of the basic reproduction number with respect to 0 < u1 < 1 and 0 < u2 < 1

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Fig. 23 Variation of the basic premium with 0 < u1 < 1 and 0 < u2 < 1 under different basic reproduction number. Top-left R0 = 4.78, Top-right R0 = 6.78, Middle-left R0 = 8.28, Middleright R0 = 9.56, Bottom-left R0 = 10.69, Bottom-right R0 = 11.71

particular basic reproduction number without controls. By comparing Figs. 23 and 24, one can see that basic premium can be reduced with controls. This reduction occurs due to the efficiency of control strategies. Thus, this reduction ratio may be a good index to measure the efficiency of the control strategies which is imposed to control the transmission of the disease. It is also possible to compute the adjusted premium with respect to two controls. It is also possible to develop a conceptual model to carry on comparison between future expenditures due to medical cost and present expenditures due to control strategy cost. However, those tasks are beyond the aim of this chapter. Hence, such analysis is not included in this chapter.

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0.35 0.3

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Fig. 24 Variation of basic premium with respect to basic reproduction number without controls

6 Conclusions According to the World Health Organization information, dengue fever is ranked as the most critical and the most rapidly spreading vector borne disease in the world. Further, dengue fever is currently endemic in more than hundred and twenty five countries all over the world. Reasons for such spread can be observed as in complex and multifactorial status. They may include climate change, virus evolution, and societal factors such as rapid urbanization, population growth and development, socioeconomic factors, as well as global travel and trade. The level of mentioned factors may increase during next decade and hence the risk of spread of dengue may also increase. Since there is no specific antiviral therapy or vaccination available against dengue at present time, the effective vector control approach is the only risk mitigating technique to minimize the spread of the disease. Thus, economical burden due to dengue can be categorized as direct and non-direct. Medical treatment cost can be considered as a direct dengue cost and all other expenditures can be considered as non-direct dengue cost. In this chapter, we focus only direct medical cost. Considering classical compartment concept, SIR model was proposed to identify the dynamical behaviour of the dengue transmission. Stability of the model was carried out to understand the behaviour of the model. The basic reproduction number was taken as a risk measurement index to reflects the risk of disease spread. Introducing the fraction of host and vector population, dimensionless form of the model was developed. By considering the fractions as probability of relevant fractions, probabilistic SIR model was introduced. Taking the susceptible and infected host fractions as probability distribution functions, insurance based SIR model was proposed. Susceptible host who are facing the financial losses due to medical expenses would like to form a market that could contribute premiums to an insurance fund, to obtain future

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financial losses if they become infected. Thus, we utilized this insurance concepts to determine the present economical burden of future direct financial losses due to the dengue outbreak. Considering the probability density functions, S(t) and I (t), and utilizing actuarial techniques, the components of dengue disease insurance model was formulated to determine future direct medical expenses due to dengue epidemic. Considering expected present value of continuous level benefit and continuous level premium payments and equivalence principle, present premium was computed to cover the future unit medical cost due to dengue outbreak. Actuarial analysis was performed to identify feasibility of the proposed model and sensitivity of the premium (present burden) with respect to the risk of disease spread was also discussed. Introducing the benefit reserve function, feasibility of the determined premium was carried out. Further, by defining the minimum condition for premium to guarantee positive reserve function throughout the policy term, the adjusted premium was determined. For the implementation of the model in practice, the infected table was constructed using reported dengue cases. Using the infected table, discreet actuarial pricing model was obtained. Thus, the premium and adjusted premium were also determined using discrete model. To see the efficiency of the control strategies, the proposed model was modified by introducing two control functions. Further, the efficiency of control strategies was analysed and sensitivity of such strategies were also discussed. For further implementation, the proposed model can be extended by considering the secondary infections and perform present financial risk due to secondary infections too. Actuarial pricing model may be extended to carry on comparison between fund allocation for control strategies and investment packages. Further, it is possible to develop portfolio to find out optimum fund allocation between the control strategies and investment plan.

References 1. Racloz, V., Ramsey, R., Tong, S., Hu, W.: Surveillance of dengue fever virus: a review of epidemiological models and early warning systems. Plos Negl.Ed Trop. Dis. 6(5), 1–9 (2012) 2. WHO: Comprehensive Guidelines for Prevention and Control of Dengue and Dengue Hemorrhagic Fever, Revised and expanded edition, Printed in India (2011) 3. Bhatia, R., Dash, A.P., Sunyoto, T.: Changing epidemiology of dengue in South-East Asia. WHO South-East Asia J. Public Health 23–27 (2013) 4. Murray, E.A., Quam, M., Smith, A.W.: Epidemiology of dengue: past, present and future prospects. Clin. Epidemiol. 5, 299–309 (2013) 5. Gubler, D.J.: Dengue and dengue hemorrhagic fever. Clin. Microbiol. Rev. 11(3), 480–496 (1998) 6. Guzman, G., Kouri, G.: Dengue and dengue hemorrhagic fever in the Americas: lessons and challenges. J. Clin. Virol. 27(1), 1–13 (2003) 7. Sirisena, P.D.N.N., Noordeen, F.: Evolution of dengue in Sri Lanka changes in the virus, vector, and climate. Int. J. Infect. Dis. 19, 6–12 (2014) 8. Eng-Eong, O., Duane, J.: Dengue in Southeast Asia: epidemiological characteristics and strategic challenges in disease prevention. Cad. Sade. Pblica, Rio J. 25(1), 115–124 (2008)

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9. National Plan of Action for Prevention and Control of Dengue Fever 2005–2009, Epidemiology Unit, Ministry of Health, Sri Lanka 10. Perera, S.S.N.: Host-vector model for dengue dynamics without immunity. Int. J. Pure Appl. Math. 118(6), 417–424 (2018) 11. Wickramaarachchi, W.P.T.M., Perera, S.S.N.: The nonlinear dynamics of the dengue mosquito reproduction with respect to climate in urban Colombo: a discrete time density dependent fuzzy model. Int. J. Math. Model. Numer. Optim. 8(2), 145–161 (2017) 12. Wickramaarachchi, W.P.T.M., Perera, S.S.N.: Developing a two dimensional climate risk model for dengue disease transmission in Urban Colombo. J. Basic Appl. Res. Int. 20(3), 168–177 (2017) 13. Gubler, D.J., Meltzer, M.: Impact of dengue/dengue hemorrhagic fever on the developing world. In: Maramorosch, K., Murphy, F.A., Shatkin, A.J. (eds.) Advances in Virus Research, vol. 53, pp. 35–70. Academic Press Inc, San Diego (1999) 14. Jansen, C.C., Beebe, N.W.: The dengue vector Aedes aegypti: what comes next. Microbes Infect. 12, 272–279 (2010) 15. Yang, H.M., Macoris, M.L.G., Galvani, K.C., Andrighetti, M.T.M., Wanderley, D.M.V.: Assessing the effects of temperature on the population of Aedes aegypti, the vector of dengue. Epidemiol Infect 137, 1188–1202 (2009) 16. Murrell, S., Wu, S.C., Butler, M.: Review of dengue virus and the development of a vaccine. Biotechnol. Adv. 29, 239–247 (2011) 17. Perera, S.D., Perera, S.S.N.: Impact of Humoral Immune Response and Absorption Effect on Dynamics of Dengue Virus. Eur. Sci. J. 13(12), 157–174 (2017) 18. Perera, S.S.N.: Pricing a Epidemiological Diseases Economic Burden. In: Proceedings, 6th International Conference on Computational Mathematics, Computational Geometry & Statistics (CMCGS 2017), pp. 5–9. Singapore (2017) 19. Shepard, D.S., Undurraga, E.A., Halasa, Y.A.: Economic and Disease Burden of Dengue in Southeast Asia. PLOS Negl.Ed Trop. Dis. 7(2), 1–12 (2013) 20. Shepard, D.S., Suaya, J.A., Halstead, S.B., Nathan, M.B., Gubler, D.J., et al.: Cost-effectiveness of a pediatric dengue vaccine. Vaccine 22, 1275–1280 (2004) 21. Shepard, D.S., Coudeville, L., Halasa, Y.A., Zambrano, B., Dayan, G.H.: Economic Impact of Dengue Illness in the Americas. Am. J. Trop. Med. Hyg. 84(2), 200–207 (2011) 22. WHO, Tdr, : Strategic direction for research: dengue. World Health Organization, Geneva (2002) 23. Derouich, M., Boutayeb, A.: Dengue fever: Mathematical Modeling and Computer Simulation. Appl. Math. Comput. 177(2), 528–544 (2006) 24. Pongsumpun, P.: Mathematical Model of Dengue Disease with the Incubation Period of Virus. World Acad. Sci. Eng. Technol. 44, 328–332 (2008) 25. Smith, H.L., Wang, L., Li, M.: Global dynamics of an SEIR epidemic model with verticle transmission. Soc. Ind. Appl. Math. 62(1), 58–69 (2001) 26. Wickramaarachchi, W.P.T.M.: Developing a Mathematical Model to Study the Dynamics of Dengue Epidemics and Controllability of Transmission of Dengue in Colombo, Ph.D. Thesis, University of Colombo (2015) 27. Derouich, M., Boutayeb, A., Twizell, E.H.: A Model of Dengue Fever. BioMedical Eng. OnLine 2(4), 1–10 (2003) 28. Perera, S.S.N.: Analysis of economic burden of seasonal influenza: an actuarial based conceptual model. J. Appl. Math., 2017. Article ID 4264737, 1–6 (2017) 29. Perera, S.S.N.: An Insurance Based Model to Estimate the Direct Cost of General Epidemic Outbreaks. Int. J. Pure Appl. Math. 117(14), 183–189 (2017) 30. Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbitt, C.J.: Actuarial Mathematics. Society of Actuaries, Schaumburg, IL (1997) 31. Senan, N.A.F.: A brief Introduction to using ODE 45 in MATLAB. University of California at Berkeley, USA (2007)

Dynamics of Zika Virus Epidemic in Random Environment Yusuke Asai, Xiaoying Han and Peter E. Kloeden

Abstract A mathematical model for Zika virus dynamics under randomly varying environmental conditions is developed, in which the birth and loss rates for mosquitoes, and environmental influence are modeled as random processes. The resulting system of random ordinary differential equations are studied by the theory of random dynamical systems and dynamical analysis. First the existence, uniqueness, positiveness and boundedness of solutions are discussed. Then the long term dynamics in terms of existence and geometric structures of random attractors and forward omega limit sets are investigated. Moreover, sufficient conditions under which the prevalence of Zika virus among human beings decreases monotonically to zero, as well as conditions under which an epidemic occurs are established.

1 Introduction Zika virus has been known as a mosquito-borne disease, that is transmitted to humans by the bites of infected mosquitoes [6]. Evidence indicates that Zika can also be transmitted by sexual contact, yet the role of sexual transmission is not wellunderstood. Recently Gao et al. introduced a deterministic mathematical model that considered Zika as a mosquito-borne and sexually transmitted disease, and investigated the spread and control of Zika by analyzing the proposed model [10]. The analysis done in [10] provided the crucial information that prevention and control efY. Asai Department of Hygiene, Graduate School of Medicine, Hokkaido University, Sapporo 060-8638, Japan e-mail: [email protected] X. Han (B) · P. E. Kloeden Department of Mathematics and Statistics, 221 Parker Hall Auburn University, Auburn, AL 36849, USA e-mail: [email protected] P. E. Kloeden e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_21

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forts against Zika should target not only mosquito-borne but also sexual transmission routes. Concerning the spread and transmission of Zika virus involve uncertainties, the goal of this work is to develop and study a mathematical model of Zika virus epidemic taking into account randomness due to fluctuations of environments. In the model developed in [10], an SEIR type of structure was used for human population and an SEI type of structure was used for the mosquito population. More precisely, the population of human is divided into six classes: susceptible, exposed, symptomatically infected, convalescent, asymptomatically infected and recovered; and the population of mosquitoes is divided into three classes: susceptible, exposed, and infectious. Symptomatically infected humans are contagious to both humans and mosquitoes during the incubation period, after which infected humans develop symptoms and become symptomatic. After the viremia period symptomatic humans enter the convalescent stage, which ends with lifelong immunity. Exposed, symptomatic and convalescent humans are all able to transmit the virus through sex, while only exposed and symptomatic humans can infect mosquitoes. Since the Zika virus can be transmitted between humans, from humans to mosquitoes, and from mosquitoes to humans during the incubation period, we consider an SIR type of structure for humans and an SI type of structure for mosquitoes by combining their respective exposed and infected groups. In addition, since sexual transmission of Zika from asymptomatically infected humans has not been documented and mosquitoes are not infected by biting them, we consider both the asymptomatically infected humans and the recovered humans as the removed. Relevant parameters are listed in the following table. α≥0 p ∈ [0, 1] q ∈ [0, 1] κ≥0 γ≥0 δ ∈ [0, 1] β≥0

Mosquito biting rate per human Transmission probability from an infectious mosquito to a susceptible human per bite Transmission probability from a symptomatically infected human to a susceptible mosquito per bite Average transmission rate from symptomatically infected humans to susceptible humans The recovery rate of infected humans The proportion of symptomatic infections The production and the death rate of mosquitoes

Let x1 (t), x2 (t) and x3 (t) be the population of susceptible, symptomatically infected and removed humans, respectively, and let y1 (t), y2 (t) be the population of susceptible and infectious mosquitoes, respectively. Then a simplified version of the model proposed in [10] reads

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dx1 dt dx2 dt dx3 dt dy1 dt dy2 dt

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= −α px1 y2 − κx1 x2 ,

(1)

= δ (α px1 y2 + κx1 x2 ) − γx2 ,

(2)

= (1 − δ) (α px1 y2 + κx1 x2 ) + γx2 ,

(3)

= β(y1 + y2 ) − αqy1 x2 − β y1 ,

(4)

= αqy1 x2 − β y2 .

(5)

In the above model the total numbers of humans and mosquitoes are both assumed to be constant over time. While such an assumption is acceptable for humans, it is less well-justified for mosquitoes as the population of mosquitoes is more sensitive to the season and the environment (e.g., weather changes, mosquito treatments) and thus clearly varies with respect to time. In this work we assume that both the reproduction rate and loss rate of mosquitoes vary randomly and can be different, modeled by two random processes β(θt ω) and ν(θt ω), respectively. Note that here ν(θt ω) stands for the collective loss rate of mosquitoes including natural death, mosquito treatments, migration out from the region, etc. In addition we include random external forces g1 (θt ω) and g2 (θt ω) to incorporate all non-reproduction type increase in the mosquito population, such as migration into the region. Equations (1)–(5) then become the following system of random ordinary differential equations (RODEs) [9]: dx1 (t, ω) dt dx2 (t, ω) dt dx3 (t, ω) dt dy1 (t, ω) dt dy2 (t, ω) dt

= −α px1 y2 − κx1 x2 ,

(6)

= δ (α px1 y2 + κx1 x2 ) − γx2 ,

(7)

= (1 − δ) (α px1 y2 + κx1 x2 ) + γx2 ,

(8)

= β(θt ω)(y1 + y2 ) − αqy1 x2 − ν(θt ω)y1 + g1 (θt ω),

(9)

= αqy1 x2 − ν(θt ω)y2 + g2 (θt ω).

(10)

Alternatives of the random system (6)–(10) can be obtained by allowing more parameters in system (1)–(5) to vary randomly with respect to time. For example, the transmission probabilities of the virus from infectious mosquitoes to susceptible humans and from infectious humans to susceptible humans can also be modeled as random parameters. While doing so can result in a more realistic model, it will create a major overload of analysis and make the chapter unacceptable long. Hence for simplification of exposition, we assume all other parameters are constant as in the deterministic model. In addition, we assume that β(θt ω), ν(θt ω) and g j (θt ω) ( j = 1, 2) are continuous, non-negative, and essentially bounded, i.e., there exist nonnegative constants m β , Mβ , m ν , Mν , m g and Mg such that

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m β ≤ β(θt ω) ≤ Mβ , m ν ≤ ν(θt ω) ≤ Mν , m g ≤ g j (θt ω) ≤ Mg ( j = 1, 2). (11) The goal of this work is to investigate properties of solutions to the RODE system (6)–(10). In particular, we will first study basic properties of solutions to (6)–(10). We then study the long term dynamics of the RODE system and investigate conditions under which the Zika virus is endemic or epidemic. The rest of this chapter is organized as follows. In Sect. 2 we discuss the existence, uniqueness, positiveness and boundedness of solutions. In Sect. 3 we provide preliminaries on concepts and theory of random dynamical systems (RDSs) needed for later analysis. In Sect. 4 we investigate long term dynamics of both mosquito population and human being populations, including existence of random attractors and forward omega limit sets for mosquito populations and conditions under which the prevalence of Zika virus among human beings is controlled and breaks out, respectively. Numerical simulations are presented in Sect. 5 to illustrate theoretical results.

2 Basic Properties of Solutions In this section, we prove the existence and uniqueness of positive solutions to the RODE system (6)–(10). Boundedness of solutions will also be discussed. Through the rest of this chapter denote   Rn+ := (x1 , . . . , xn ) ∈ Rn : xi ≥ 0, i = 1, · · · , n , and for simplicity write u(t, ω) = (x1 (t, ω), x2 (t, ω), x3 (t, ω), y1 (t, ω), y2 (t, ω))T . Theorem 2.1 For any ω ∈ , t0 ∈ R and initial data (x1 (t0 ), x2 (t0 ), x3 (t0 ), y1 (t0 ), y2 (t0 ))T = (x10 , x20 , x30 , y10 , y20 )T := u0 ∈ R5+ , the RODE system (6)–(10) admits a unique non-negative global solution u(·; t0 , ω, u0 ) ∈ C([t0 , ∞), R5+ ) with u(t0 ; t0 , ω, u0 ) = u0 . Proof First notice that

d (x1 (t) dt

+ x2 (t) + x3 (t)) = 0, hence

x1 (t) + x2 (t) + x3 (t) = x10 + x20 + x30 , ∀ t ≥ t0 , and it suffices to study the subsystem on v(t, ω) := (x1 (t, ω), x2 (t, ω), y1 (t, ω), y2 (t, ω))T with v 0 = (x10 , x20 , y10 , y20 )T . First rewrite Eqs. (6)–(10) as dv = Lv + f (θt ω, v), dt

(12)

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where ⎛

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0 0 β(θt ω) − ν(θt ω) β(θt ω) 0 0 0 −ν(θt ω)



⎟ ⎟, ⎠



−α px1 y2 − κx1 x2 ⎜δ (α px1 y2 + κx1 x2 )⎟ f = ⎝−αqy x + g (θ ω)⎠. 1 2 1 t αqy1 x2 + g2 (θt ω)

Since both β(θt ω) and ν(θt ω) are continuous and bounded, the operator L(θt ω) generates an evolution system on R4 . In addition since g1 (θt ω) and g2 (θt ω) are continuous with respect to time t, function f is continuous with respect to t and locally Lipschitz with respect to v. Hence by classical existence and uniqueness theorems for ODEs (see, e.g., [8, 9]), system (12) has a unique local solution v(·; t0 , ω, v 0 ) ∈ C([t0 , T ), R4 ). By continuity of solutions, each solution has to take value 0 before it reaches a negative value. Notice that dx1 |x =0,x2 ≥0,y1 ≥0,y2 ≥0 = 0, dt 1 dx2 |x ≥0,x2 =0,y1 ≥0,y2 ≥0 = δα px1 y2 ≥ 0, dt 1 dy1 |x ≥0,x2 ≥0,y1 =0,y2 ≥0 = β(θt ω)y2 + g1 (θt ω) ≥ 0, dt 1 dy2 |x ≥0,x2 ≥0,y1 ≥0,y2 =0 = αq x2 y1 + g2 (θt ω) ≥ 0. dt 1 This implies that v(t) ∈ R4+ for all t ∈ [t0 , T ). We next show that the solution to system (12) is bounded from above for any finite T , such that it is actually a global solution. First notice that x1 (t) + x2 (t) ≤ x10 + x20 + x30 , for all t ∈ [t0 , T ).

(13)

Let Y (t) = y1 (t) + y2 (t) with Y0 := y10 + y20 , and g(θt ω) := g1 (θt ω) + g2 (θt ω). Then by summing Eqs. (9) and (10) we obtain dY (t, ω) = (β(θt ω) − ν(θt ω)) · Y (t) + g(θt ω), dt which can be solved analytically to obtain Y (t, ω) = Y0 e

t

t0 (β(θs ω)−ν(θs ω))ds

+ t0

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t τ

(β(θs ω)−ν(θs ω))ds

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Hence by the Assumption (11) and direct integration:

2Mg Y (t, ω) ≤ Y0 + Mβ − m ν

e(Mβ −m ν )(t−t0 ) −

2Mg , ∀ t ≥ t0 , Mβ − m ν

(14)

i.e., Y (t, ω) is bounded at any finite time t ≤ T < ∞. Inequalities (13) and (14) together imply that system (12) has a unique nonnegative global solution v(·; t0 , ω, v 0 ) ∈ R4+ . It follows immediately that the RODE system (6)–(10) has a unique non-negative global solution u(·; t0 , ω, u0 ) ∈ R5+ . The proof is complete.  Remark 2.1 It is straightforward to check that when m ν > Mβ , the solution u(·; t0 , ω, u0 ) is globally bounded with 0 ≤ x1 (t, ω) + x2 (t, ω) + x3 (t, ω) ≤ x10 + x20 + x30 , 2Mg 0 ≤ y1 (t, ω) + y2 (t, ω) ≤ y10 + y20 + . m ν − Mβ Boundedness of solutions when m ν ≤ Mβ is more complicated and will be further investigated in Sect. 4.

3 Preliminaries on Random Dynamical Systems In this section we present some basic concepts and theory [1, 3, 4, 7] related to RDSs and random attractors that we require in the sequel. Our situation is in fact simpler, but to facilitate the reader’s access to the literature we give more general definitions here (for less abstract surveys see, e.g., [2] and [5]). Let (X,  ·  X ) be a separable Banach space and let (, F, P) be a probability space where F is the σ−algebra of measurable subsets of  (called “events”) and P is the probability measure. To connect the state ω in the probability space  at time 0 with its state after a time of t elapses, we define a flow  = {θt }t∈R on  with each θt being a mapping θt :  →  that satisfies (1) (2) (3) (4)

θ0 = Id , θs ◦ θt = θs+t for all s, t ∈ R, the mapping (t, ω) → θt ω is measurable and the probability measure P is preserved by θt , i.e. θt P = P.

This set-up establishes a time-dependent family  = (θt )t∈R that tracks the noise, and (, F, P, θ) is called a metric dynamical system [1, 2, 5]. Definition 3.1 A stochastic process {ϕ(t, ω)}t≥0,ω∈ is said to be a continuous RDS over (, F, P, (θt )t∈R ) with state space X if ϕ : [0, +∞) ×  × X → X is (B[0, +∞) × F × B(X ), B(X ))- measurable, and for each ω ∈ ,

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(1) the mapping ϕ(t, ω) : X → X , x → ϕ(t, ω, x) is continuous for every t ≥ 0; (2) ϕ(0, ω) is the identity operator on X ; (3) (cocycle property) ϕ(t + s, ω, ·) = ϕ(t, θs ω, ϕ(s, ω, ·)) for all s, t ≥ 0. Definition 3.2 A set-valued mapping B : ω → 2 X \∅ is said to be a random set if the mapping ω → dist X (x, B(ω)) is measurable for any x ∈ X . A random set B(ω) is said to be bounded (compact, closed, resp.) if B(ω) is bounded (compact, closed, resp.) for a.e. ω ∈ . Definition 3.3 A bounded random set B(ω) ⊂ X is said to be tempered with respect to  if for a.e. ω ∈ , lim e−βt

t→∞

sup

x∈B(θ−t ω)

x X = 0, for all β > 0;

a random variable ω → r (ω) ∈ R is said to be tempered with respect to  if for a.e. ω ∈ , lim e−βt sup |r (θ−t ω)| = 0, for all β > 0. t→∞

t∈R

In what follows we use D(X ) to denote the set of all tempered random sets of X . Definition 3.4 A random set K (ω) ⊂ X is called a random absorbing set in D(X ) if for any B ∈ D(X ) and a.e. ω ∈ , there exists TB (ω) > 0 such that ϕ(t, θ−t ω, B(θ−t ω)) ⊂ K (ω), ∀t ≥ TB (ω). Definition 3.5 Let {ϕ(t, ω)}t≥0,ω∈ be an RDS over (, F, P, (θt )t∈R ) with state space X and let A(ω)(⊂ X ) be a random set. Then A(ω) is called a global random D attractor (or pullback D attractor) for {ϕ(t, ω)}t≥0,ω∈ if ω → A(ω) satisfies (1) (random compactness) A(ω) is a compact set of X for a.e. ω ∈ ; (2) (invariance) for a.e. ω ∈  and all t ≥ 0, it holds ϕ(t, ω, A(ω)) = A(θt ω); (3) (attracting property) for any B ∈ D(X ) and a.e. ω ∈ , lim dist X (ϕ(t, θ−t ω, B(θ−t ω)), A(ω)) = 0,

t→∞

where dist X (G, H ) = sup inf g − h X g∈G h∈H

is the Hausdorff semi-metric for G, H ⊆ X.

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Remark 3.1 The convergence described above is the pullback type convergence, which starts earlier and earlier in the past and comes forward to the current time. On the contrary, the forward type convergence starts at current time and goes to further and further into the future (see e.g., [2, 5] for detailed comparison between the two types of convergence). Proposition 3.1 Let B ∈ D(X ) be an absorbing set for the continuous RDS {ϕ(t, ω)}t≥0,ω∈ which is closed and satisfies the asymptotic compactness condition for a.e. ω ∈ , i.e., each sequence xn ∈ ϕ(tn , θ−tn , B(θ−tn ω)) has a convergent subsequence in X when tn → ∞. Then the cocycle S has a unique global random attractor with component subsets A(ω) =

 

ϕ(t, θ−t ω, B(θ−t ω)).

τ ≥t B (ω) t≥τ

If the pullback absorbing set is positively invariant, i.e., ϕ(t, ω, B(ω)) ⊂ B(θt ω) for all t ≥ 0, then  ϕ(t, θ−t ω, B(θ−t ω)). A(ω) = t≥0

For a state space X = Rn as in this paper, the asymptotic compactness follows trivially. Note that the random attractor is pathwise attracting in the pullback sense, but need not be pathwise attracting in the forward sense, although it is forward attracting in probability, due to some possible large deviations, see e.g., Arnold [1] and Crauel and Kloeden [5].

4 Long Term Dynamics of Zika Virus It is straightforward to check that the unique solution u(t; t0 , ω, u0 ) proved in Sect. 2 satisfies u(t + t0 ; t0 , ω, u0 ) = u(t; 0, θt0 ω, u0 ) for all t0 ∈ R, t ≥ t0 , ω ∈  and u0 ∈ R5+ . This allows us to define a mapping ϕ(t, ω, ·), which is a RDS, as ϕ(t, ω, u0 ) = u(t; 0, ω, u0 ), ∀t ≥ 0, u0 ∈ R5+ , ω ∈ . From now on, we will simply write u(t; ω, u0 ) instead of u(t; 0, ω, u0 ). The goal of this section is to study the long term dynamics of the solution u(t; ω, u0 ). In particular, we first prove that the RDS ϕ(t, ω) defined above possesses a random attractor. Then we will investigate in greater details the dynamics of mosquito populations, which is used to study the prevalence of Zika virus among human beings. Throughout this section, denote by H be the total population (sus-

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ceptible, symptomatically infected, removed) of human beings in the region of Zika virus prevalence under consideration. To simplify notations, let μ := αq H + m ν ,

y1 (t) + y2 (t) = Y (t),

y10 + y20 = Y0 .

Theorem 4.1 The RDS {ϕ(t, ω)}t≥0,ω∈ possesses a random attractor A H = {A H (ω) : ω ∈ } provided Mβ < m ν . Proof Define the simplex  H by    H := (x1 , x2 , x3 ) ∈ R3+ : x1 + x2 + x3 ≡ H . Then  H is compact and positive invariant for the x-component solution mapping (x1 , x2 , x3 )(t; ω, u0 ). Following similar calculations to reach (14) to get

t

0 (β(θs ω)−ν(θs ω))ds

Y (t) ≤ Y0 e ≤ Y0 +

2Mg Mβ − m ν



t

+

g(θτ ω)e

t τ

(β(θs ω)−ν(θs ω))ds



0

e(Mβ −m ν )t +

2Mg . m ν − Mβ

(15)

Given any > 0 define  K := (y1 , y2 ) : y1 ≥ 0, y2 ≥ 0, y1 + y2 ≤

 2Mg + . m ν − Mβ

Since Mβ < m ν , then for any (y10 , y20 ) ∈ R2+ there exists a time T such that (y1 (t), y2 (t)) ∈ K for t ≥ T . Recall that (x1 (t), x2 (t), x3 (t)) ∈  H for all t ≥ 0. Thus for every > 0,  H × K is a compact and positive invariant absorbing set for the RDS {ϕ(t, ω)}t≥0,ω∈ , which implies that {ϕ(t, ω)}t≥0,ω∈ has a random attractor A H = {A H (ω) : ω ∈ } with component sets in  H × K . The proof is complete. The above theorem proves the existence of a random attractor for the full RODE system (6)–(10) under the assumption m ν > Mβ . In the subsections below we will discuss detailed dynamics of mosquito and human being populations, respectively.

4.1 Dynamics of Mosquito Population In this subsection we will discuss dynamics of mosquito population, in particular, population of infectious mosquitoes y2 (t, ω), in two scenarios: (1) m ν > Mβ ; and (2) m β > Mν .

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Theorem 4.2 Assume that m ν > Mβ . Then the population of infectious mosquitoes satisfies mg Mg 2αq H Mg + . ≤ lim y2 (t, ω) ≤ t→∞ Mν μ(m ν − Mβ ) μ Proof First notice that since y1 (t), x2 (t) ≥ 0 for all t ≥ 0, Eq. (10) implies that dy2 (t, ω) ≥ −ν(θt ω)y2 (t) + g2 (θt ω), dt which can be integrated explicitly to obtain t

t + g2 (θτ ω)e− τ ν(θs ω)ds dτ 0 t e−Mν (t−τ ) dτ ≥ y20 e−Mν t + m g 0

mg mg 0 e−Mν t + . ≥ y2 − Mν Mν

y2 (t) ≥ y20 e−

t 0

ν(θs ω)dt

(16)

Second the inequality (15) implies that

2Mg y1 (t) ≤ Y0 − m ν − Mβ

e−(m ν −Mβ )t +

2Mg − y2 (t), ∀ t ≥ 0. m ν − Mβ

And together with Eq. (10) and the fact that x2 (t) ≤ H for all t we have dy2 (t) ≤ −(ν(θt ω) + αq H )y2 (t) + ζ(θt ω), ∀ t ≥ 0 dt

(17)

where

2αq H Mg 2Mg e−(m ν −Mβ )t + g2 (θt ω) ≥ 0. ζ(θt ω) = + αq H Y0 − m ν − Mβ m ν − Mβ Integrating the differential inequality (17) we get y2 (t) ≤

t y20 e−αq H t− 0 ν(θs ω)ds

≤ y20 e−μt +



t 0



t

+

τ

ζ(θτ ω)eαq H (τ −t)+

0

ζ(θτ ω)eμ(τ −t) dτ .

t

ν(θs ω)ds



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The terms in the integral

t 0

675

ζ(θτ ω)eμ(τ −t) dτ satisfy, respectively,

 2αq H Mg μ(τ −t) 2αq H Mg  1 − e−μt ; e dτ = μ(m ν − Mβ ) 0 m ν − Mβ

t 2Mg e−(m ν −Mβ )τ eμ(τ −t) dτ αq H Y0 + Mβ − m ν 0

 (Mβ −m ν )t  αq H 2Mg = e Y0 + − e−μt ; Mβ + αq H Mβ − m ν t  Mg  1 − e−μt . g(θτ ω)eμ(τ −t) dτ ≤ μ 0 t

Therefore y2 (t) satisfies y2 (t) ≤ C1 e−μt + C2 e−(m ν −Mβ )t + C3 ,

(18)

where αq H 2αq H Mg − μ(m ν − Mβ ) Mβ + αq H

αq H 2Mg C2 = , Y0 + Mβ + αq H Mβ − m ν Mg 2αq H Mg C3 = + . μ(m ν − Mβ ) μ C1 = y20 −

Y0 +

2Mg Mβ − m ν



Mg , μ

The lower bound (16) and the upper bound (18) of y2 (t) together imply that for any > 0, there exists T > 0 such that Mg mg 2αq H Mg + + , ∀ t ≥ T . − ≤ y2 (t) ≤ Mν μ(m ν − Mβ ) μ

(19)

The proof is complete. Corollary 4.1 Assume that m ν > Mβ . Then the attractor A H for the RDS generated by solutions to (6)–(10) consists of nontrivial component sets provided mg Mg 2αq H Mg + . < Mν μ(m ν − Mβ ) μ Proof It follows directly from (19) and (15) that Mg (m ν + Mβ ) 2Mg mg − ≤ y1 (t) ≤ − + . μ(m ν − Mβ ) m ν − Mβ Mν

(20)

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Define Kˆ to be  Kˆ = (y1 , y2 ) ∈

R2+

:

Mg (m ν +Mβ ) 2Mg m − ≤ y1 ≤ m ν −M − Mgν μ(m ν −Mβ ) β mg 2αq H M M − ≤ y2 ≤ μ(m ν −Mβg ) + μg + Mν

+

 .

Then by the Assumption (20),  H × Kˆ is a nonempty and nontrivial component set for the attractor A H . The proof is complete. The above results require the minimum collective loss rate m ν to be larger than the maximum reproduction rate Mβ of mosquitoes, under which a compact absorbing set can be constructed. The absorbing set depends on the total population of human beings H , but does not depend on the initial condition. When m ν ≤ Mβ , the system may not have such an absorbing set, and hence has no attractor. Nevertheless, since the set  H ∩ R5+ is compact, a forward omega limit set does exist for each initial condition. These sets are nonempty and compact subsets of  H ∩ R5+ but not invariant, as they depend on the noise sample path under consideration. Their properties are similar to the nonautonomous limit sets (see [11, 12] for more details). In the theorem below we construct such sets for infectious mosquitoes. Theorem 4.3 Assume that for every ω ∈  there exist σ > 0 and Ig > 0 such that e

t

0 (β(θs ω)−ν(θs ω))ds



t

≤ σ,

g(θs ω)ds ≤ Ig

∀ t ≥ 0.

(21)

0

Then the population of infectious mosquitoes satisfies  Mg mg αq H σ  Y0 + Ig + + , ∀ t ≥ T . − ≤ y2 (t) ≤ Mν μ μ Proof The lower bound (16) for y2 (t) still holds. The inequality (15) now becomes

t

y1 (t) + y2 (t) ≤ Y0 σ +

g(θτ ω)e

t τ

(β(θs ω)−ν(θs ω))ds

dτ .

0

Notice that τ

t



t−τ

(β(θs ω) − ν(θs ω))ds =

(β(θr (θτ ω)) − ν(θr (θτ ω))dr.

0



Thus

t

y1 (t) + y2 (t) ≤ Y0 σ + σ

g(θτ ω)dτ ≤ σ(Y0 + Ig ).

0

Using the above inequality in (10), and again using x2 (t) ≤ H , we have   dy2 (t) ≤ −(ν(θt ω) + αq H )y2 (t) + g2 (θt ω) + αq H σ Y0 + Ig , dt

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which can be integrated to obtain

   g2 (θτ ω) + αq H σ Y0 + Ig eμ(τ −t) dτ 0

Mg + αq H σ(Y0 + Ig ) −μt Mg + αq H σ(Y0 + Ig ) 0 e . + ≤ y2 − μ μ

y2 (t) ≤

y20 e−μt

+

t

This implies that for any > 0, there exists T > 0 such that y2 (t) ≤ αq H σ(Y0 + Ig )/μ + for all t > T . The proof is complete. In summary we have the following corollary on the population of infectious mosquitoes. Corollary 4.2 For any t ≥ 0 and ω ∈ , the following bounds for y2 (t, ω) hold:

mg mg e−Mν t + y2 (t, ω) ≥ − , Mν Mν 2Mg , if Mβ < m ν y2 (t, ω) ≤ Y0 + m ν − Mβ  t αq H σ(Y0 + Ig ) 0 0 t g(θs ω)ds ≤ Ig and . , if y2 (t, ω) ≤ y2 + μ e 0 (β(θs ω)−ν(θs ω))ds ≤ σ

y20

(22) (23) (24)

4.2 Dynamics of Human Being Populations In this subsection we investigate the long term dynamics for human beings. For simplicity, let   xˆ10 = x10 e

 α p 0 mg   Mν (y2 − Mν )

.

(25)

Lemma 4.1 The population of susceptible human beings satisfies mg

x1 (t, ω) ≤ xˆ10 · e−α p Mν t , ∀ ω ∈ .

(26)

Proof First by using the lower bound of y2 (t) in Corollary 4.2, we have dx1 (t) ≤ −α px1 dt

y20 −

mg Mν

e−Mν t +

mg Mν

which can be integrated directly to obtain αp

mg

mg

x1 (t) ≤ x10 e−α p Mν t e Mν (y2 − Mν )(e ≤ xˆ10 · e−α p The proof is complete.

mg Mν

t

.

0

−Mν t

−1)

,

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The above Lemma states that the populations of susceptible human beings will exponentially decay to zero. This will be used to investigate the sufficient conditions under which population of infected human beings decreases monotonically to zero. To simplify notations, let  2Mg αq H σ(Y0 + Ig ) 0 . ,y + yˆ2 = max Y0 + m ν − Mβ 2 μ 

(27)

Throughout the rest of this subsection it is assumed that either (I) m ν > Mβ , or (II) there exist σ > 0 and Ig > 0 such that the Assumption (21) holds. Remark 4.1 Scenario (II) is not based on direct comparison of the magnitudes of ν(θt ω) and β(θt ω). Instead, it puts a restriction on the aggregated effects of ν(θt ω) ω). Scenario (I) can be regarded as a and β(θt ω), as well as the immigration g(θ

t t special scenario of Scenario (II), when e 0 (β(θs ω)−ν(θs ω))ds is not only bounded but also decays exponentially, g(θt ω) only needs to be bounded. Theorem 4.4 The population of infected human beings x2 (t, ω) always decays to zero as t → ∞. In particular x2 (t, ω) → 0 monotonically provided

α p yˆ2 mg ≤ γ < αp δ xˆ10 κ + . Mν x20

(28)

Proof First due to Corollary 4.2, y2 (t) ≤ yˆ2 for all t ≥ 0. Then by Lemma 4.1 and Eq. (7), mg dx2 ≤ δ xˆ10 · e−α p Mν t (α p yˆ2 + κx2 ) − γx2 , dt where xˆ10 is as defined in (25). Given any ∈ (0, γ/κ), there exists T1 ( ) such that dx2 ≤ α p yˆ2 + (κ − γ)x2 , dt

∀t ≥ T1 ( ),

which implies that x2 (t) ≤

x20

α p yˆ2

α p yˆ2 e−(γ−κ )t + . − γ − κ γ − κ

Since κ < γ, there exists T2 ( ) such that

α p yˆ2 , ∀ t ≥ max{T1 ( ), T2 ( )}, x2 (t) < 1 + γ − κ i.e., x2 (t) → 0 as t → ∞.

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We next show that x2 (t) → 0 monotonically under Assumption (28). By Lemma 4.1, Corollary 4.2 and Eq. (7), mg mg dx2 ≤ δα p xˆ10 yˆ2 e−α p Mν t + (δκxˆ10 e−α p Mν t − γ)x2 dt mg ≤ δα p xˆ10 yˆ2 e−α p Mν t + (δκxˆ10 − γ)x2 .

On the other hand, the Eq. (7) implies directly that x2 (t) ≥ x20 e−γt , ∀ t ≥ 0.   By Assumption (28), γ ≥ xˆ10 κ + α p yˆ2 /x20 > δ xˆ10 . Therefore mg dx2 ≤ δα p xˆ10 yˆ2 e−α p Mν t + x20 (δκxˆ10 − γ)e−γt . dt mg

By Assumption (28) again, e−α p Mν t ≤ e−γt , and hence   dx2 ≤ e−γt δα p xˆ10 yˆ2 + x20 (δκxˆ10 − γ) < 0. dt The proof is complete. The theorem above states that when m ν > Mβ , or the Assumption (21) holds for some σ > 0 and Ig > 0, the Zika virus will always die out given a long enough time. Moreover if Assumption (28) holds, the prevalence of virus decreases monotonically to zero, i.e., the Zika virus is controlled. In the theorem below we construct conditions under which there is an outbreak or epidemic of Zika virus, in the sense that infected human beings increase for a certain period of time before eventually decaying to zero. Theorem 4.5 The population of infected human beings x2 (t, ω) keeps increasing for at least up to some TM > 0, provided  α p 0 y . γ < δx10 κ + H 2 Proof First by Corollary 4.2 and Eq. (6),   dx1 ≥ −α px1 yˆ2 − κx1 x2 ≥ − α p yˆ2 + κH x1 , dt where yˆ2 is as defined in (27). Thus x1 (t, ω) ≥ x10 e−(α p yˆ2 +κH )t , ∀ t ≥ 0, ω ∈ . Using the above inequality in (7) and by Corollary 4.2 again we obtain

(29)

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  dx2 x2 ≥ δx10 e−(α p yˆ2 +κH )t α py2 + κx2 − γx2 dt H

αp m g −Mν t mg 0 −(α p yˆ2 +κH )t 0 ≥ δx1 e x2 )e + + κ − γx2 (y2 − H Mν Mν 

α pm g ≥ x2 −γ + δx10 + κ e−(α p yˆ2 +κH )t H Mν

 αp mg e−(α p yˆ2 +κH +Mν )t . y20 − + δx10 H Mν Since e−(α p yˆ2 +κH )t > e−(α p yˆ2 +κH +Mν )t , then by the Assumption (29) it follows immediately from the above differential inequality that   dx2 α p 0  −(α p yˆ2 +κH +Mν )t  > 0, ∀ t ≤ TM > x2 −γ + δx10 κ + y e dt H 2 where TM = −

1 γ ln 0  α p yˆ2 + κH + Mν δx1 κ +

α p 0 y H 2

> 0.

The proof is complete.

5 Numerical Experiments In this section, we will simulate the system (6)–(10) numerically and verify that the long term dynamics of infected mosquitoes satisfy Corollary 4.2 and the population of susceptible human being satisfies Lemma 4.1. In addition, we confirm that the population of infected humans x2 (t, ω) decreases monotonically under the condition (28) and increases for a while before decreases under the condition (29). To this end we transform the system (6)–(10) with four independent Ornstein– Uhlenbeck (OU) processes Z j (t) into a system of RODE–SODE pair: ⎛











x1 (t) −α px1 y2 − κx1 x2 0 δ(αx1 y2 + κx1 x2 ) − γx2 ⎜ 0 ⎟ ⎜ x2 (t) ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ x3 (t) ⎟ ⎜ ⎟ (1 − δ)δ(αx1 y2 + κx1 x2 ) + γx2 ⎜ ⎟ ⎟ ⎜ ⎟ d⎜ ⎜ y1 (t) ⎟ = ⎜β(Z 2 )(y1 + y2 ) − αqy1 x2 − ν(Z 1 )y1 + g1 (Z 3 )⎟dt + ⎜ 0 ⎟dWt , ⎝ 0 ⎠ ⎝ y (t) ⎠ ⎝ ⎠ αqy1 x2 − ν(Z 1 )y2 + g2 (Z 4 ) 2 θ j,3 Z j (t) θ j,1 − θ j,2 Z j

where j = 1, 2, 3, 4. The loss and the production rates of mosquitoes ν(Z 1 ) and β(Z 2 ) are assumed to distribute in a finite interval and they are randomized by

2r1 2r2 arctan Z 1 , β(Z 2 ) = ν(Z 1 ) + a¯ 1 − arctan Z 2 e−t , ν(Z 1 ) = ν¯ 1 − π π

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in Scenario (I) and

2r1 2r2 ν(Z 1 ) = ν¯ 1 − arctan Z 1 , β(Z 2 ) = β¯ 1 − arctan Z 2 , π π in Scenario (II). The external forces modelling random environment g1 (Z 3 ) and g2 (Z 4 ) are assumed to have a switching effect (see [9]) and they are randomized by the following equations: g1 (Z 3 ) = g¯1 1 − 2r3

Z3 1 + Z 32

, g2 (Z 4 ) = g¯2 1 − 2r4

Z4 1 + Z 42

,

¯ a, where ν, ¯ β, ¯ g¯1 and g¯2 are positive and 0 < r1 , r2 , r3 , r4 < 1. Thus the randomized parameters are continuous and essentially bounded and they are defined as in intervals ν ∈ ν¯ · [1 − r1 , 1 + r1 ], g1 ∈ g¯1 · [1 − r3 , 1 + r3 ], g2 ∈ g¯2 · [1 − r4 , 1 + r4 ], 

and β∈

[ν(1 ¯ − r1 ), ν(1 ¯ + r1 ) + a(1 ¯ + r2 )] in Scenario (I) . ¯ in Scenario (II) β · [1 − r2 , 1 + r2 ]

The OU processeses Z j (t) can be generated independently and we solve the RODE part, i.e., x1 , x2 , x3 , y1 and y2 compartments, of the above system. The explicit 1.5order explicit RODE-Taylor scheme was applied in the following simulation [9]. The coefficients for OU processes are fixed to θ11 = 1, θ12 = 3, θ13 = 0.8, θ21 = 0, θ22 = 1, θ23 = 0.5, θ31 = 3, θ32 = 2, θ33 = 2.2, θ41 = 4, θ42 = 2 and θ43 = 2.4 for all examples. Scenario (I) In this scenario, initial conditions for x1 , x2 , x3 , y1 and y2 compartments are set as x10 = 60, x20 = 40, x30 = 0, y10 = 90 and y20 = 10, respectively. We set the parameters to be α = 0.1, p = 0.2, q = 0.3, κ = 0.05, δ = 0.01 and γ = 0.16, and the randomized parameters ν¯ = 0.3, r1 = 0.1, a¯ = 0.1, r2 = 0.1, g¯1 = 3.5, r3 = 0.2, g¯2 = 3.5 and r4 = 0.2. The assumption Mβ < m ν does not hold for this set of parameters, however, one can find σ and Ig to satisfy (24) for some time T > 0. These parameter values satisfy Assumption (28). The left and the middle panels in Fig. 1 shows that the population of infected human beings decreases monotonically. In addition, the middle panel shows that the population of susceptible human beings is bounded from above with its upper bound (26). Moreover, the number of infectious mosquitoes on the right panel stays between the lower and upper bounds, i.e., (22) and (23).

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Fig. 1 With parameters α = 0.1, p = 0.2, q = 0.3, κ = 0.05, δ = 0.01, γ = 0.16, ν¯ = 0.3, r1 = 0.1, a¯ = 0.1, r2 = 0.1, g¯ 1 = 3.5, r3 = 0.2, g¯ 2 = 3.5 and r4 = 0.2 satisfying Assumption (28), the population of infected human beings decreases monotonically

Fig. 2 With parameters α = 0.3, p = 0.2, q = 0.3, κ = 0.00005, δ = 0.9 and γ = 0.08, ν¯ = 0.9, r1 = 0.1, β¯ = 0.1, r2 = 0.1, g¯ 1 = 3, r3 = 0.9, g¯ 2 = 5 and r4 = 0.9 satisfying Assumptions Mβ < m ν and (29), the population of infected human beings first increases before it decreases

Scenario (II) In this scenario, initial conditions for x1 , x2 , x3 , y1 and y2 compartments are set as x10 = 80, x20 = 20, x30 = 0, y10 = 10 and y20 = 2, respectively. We set the parameters to be α = 0.3, p = 0.2, q = 0.3, κ = 0.00005, δ = 0.9 and γ = 0.08, and the randomized parameters ν¯ = 0.9, r1 = 0.1, β¯ = 0.1, r2 = 0.1, g¯1 = 3, r3 = 0.9, g¯2 = 5 and r4 = 0.9. The assumption Mβ < m ν is satisfied by this set of parameters. In addition, the above parameter values satisfy Assumption (29). The left and the middle panels in Fig. 2 shows that the population of infected human beings first increases before it decreases. The middle panel shows that the population of susceptible human beings is bounded from above with its upper bound (26). Furthermore the number of infectious mosquitoes on the right panel stays between the lower and upper bounds, i.e., (22) and (23). Now we compare trajectories under different recovery rate γ and the initial conditions for human population x10 , x20 and x30 . In the following simulation, the parameter values are fixed to α = 0.3, p = 0.2, q = 0.3, κ = 0.00005 and δ = 0.9 and the ran-

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Fig. 3 The comparison under different recovery rate γ (left and right figures) and initial conditions (right figure). The parameter values are fixed at α = 0.3, p = 0.2, q = 0.3, κ = 0.00005 and δ = 0.9 with the initial conditions for mosquitoes y10 = 10 and y20 = 2. 1) Left and middle panels: The recovery rate γ is set to 0.02, 0.04, 0.06, 0.08, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9 with the initial conditions for human population (x10 , x20 , x30 ) = (80, 20, 0). 2) Right panel: The initial conditions for human population are set to (x10 , x20 , x30 ) = (86, 0, 14), (78, 10, 12), (70, 20, 10), (62, 30, 8), (54, 40, 6), (46, 50, 4), (38, 60, 2) and (30, 70, 0), respectively, with fixed γ = 0.05

domized parameters ν¯ = 0.9, r1 = 0.1, β¯ = 0.1, r2 = 0.1, g¯1 = 3, r3 = 0.9, g¯2 = 5 and r4 = 0.9. In addition, the initial population of mosquitoes are set to y10 = 10 and y20 = 2 for both comparison. 1. Comparison under different γ γ is set to 0.02, 0.04, 0.06, 0.08, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9, respectively, and the trajectories are compared with the initial conditions x10 = 80, x20 = 20 and x30 = 0 on the left and the middle panels in Fig. 3. The right hand side of (29), i.e., δx10 (κ + α py20 /H ), is 0.09 under the given parameter value and the left panel shows that the number of infected population x2 increases when γ < 0.09. On the other hand, x2 decreases first for γ > 0.5. The Assumption (29) is strong and we can confirm that x2 increases before it starts decreasing even for γ > 0.09. 2. Comparison under different initial conditions for human population (x10 , x20 , x30 ) The right panel in Fig. 3 shows the trajectories under different initial conditions, (x10 , x20 , x30 ) = (86, 0, 14), (78, 10, 12), (70, 20, 10), (62, 30, 8), (54, 40, 6), (46, 50, 4), (38, 60, 2) and (30, 70, 0) with γ = 0.05. In this simulation, (29) is satisfied only when (x10 , x20 , x30 ) = (86, 0, 14) and the other trajectories show small decrease before they increase.

References 1. Arnold, L.: Random Dynamical Systems. Springer-Verlag, Berlin (1998) 2. Caraballo, T., Han, X.: Applied Nonautonomous and Random Dynamical Systems. SpringerVerlag, Cham, BCAM SpringerBrief (2016)

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3. Caraballo, T., Kloeden, P.E., Schmalfuss, B.: Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Appl. Math. Optim. 50, 183–207 (2004) 4. Caraballo, T., Łukaszewicz, G., Real, J.: Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear Analysis TMA 6, 484–498 (2006) 5. Crauel, H., Kloeden, P.E.: Nonautonomous and random attractors. Jahresber. Dtsch.En Math.Ver.Igung 117, 173–206 (2015) 6. Fauci, A.S., Morens, D.M.: Zika Virus in the Americas - yet another arbovirus threat. N. Engl. J. Med. 374, 601–604 (2016) 7. Flandoli, F., Schmalfuss, B.: Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise. Stochast. Stochast. Rep. 59, 21–45 (1996) 8. Hale, J.K.: Ordinary Differential Equations. Robert E. Krieger Publishing Co., N.Y. (1980) 9. Han, X., Kloeden, P.E.: Random Ordinary Differential Equations and their Numerical Solution. Springer Nature, Singapore (2017) 10. Gao, D., Lou, Y., He, D., Porco, T.C., Kuang, Y., Chowell, G., Ruan, S.: Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: a mathematical modeling analysis, Scientific Reports 6. Article number 28070, (2016) 11. Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. American Mathematical Society, Providence, RI (2011) 12. Kloeden, P.E., Lorenz, T.: Construction of nonautonomous forward attractors. Proc. Amer. Mat. Soc. 144, 259–268 (2016)

Incidence Graph Models for the Analysis of Active Illegal Immigration Routes and Human Loss Sunil Mathew and John N. Mordeson

Abstract Connectivity in fuzzy incidence graphs (FIG) is studied in this article. Different connectivity aspects of fuzzy incidence graphs such as bonding pair, doubly bonding pair and incidence cut of pairs are discussed. Incidence connectivity and incidence connectivity of pairs are introduced and results similar to Whitney’s Theorem are presented. The concept of t-connected fuzzy incidence graphs are also studied and some characterizations are obtained. An application related with illegal migration is presented. The most vulnerable routes in the Mexican-US border are focussed on and corresponding risks are evaluated using t-conorms. Keywords Fuzzy incidence graph · Incidence connectivity · Cutpair · Bonding pair · Complete fuzzy incidence. AMS Classification: 05C22 · 05C40

1 Introduction and Preliminaries Fuzzy graph theory is grown as as one of the major areas of research today. It was introduced by Rosenfeld [25] and Yeh and Bang [31] simultaneously in 1975. Several authors including Mordeson [22, 23], Bhutani [4, 5], Sunitha [26, 27], Akram [1, 2] and Mathew [14, 15] have contributed much to the growth of both theory and applications of the field. Large number of variants of fuzzy graphs like intuitionistic fuzzy graphs [1] and bipolar fuzzy graphs [2] were also developed during this period. Lately, Dinesh S. Mathew (B) Department of Mathematics, National Institute of Technology Calicut, Calicut 673601, India e-mail: [email protected] J. N. Mordeson Department of Mathematics, Center for Mathematics of Uncertainty, Creighton University, Omaha, NE 68178, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_22

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[6, 7] introduced the concept of fuzzy incidence graphs (FIG) in 2012. Mathew and Mordeson analyzed FIGs in detail and found their applications in problems related with illicit flows and human trafficking [17, 19]. FIG is a reliable model to certain natural systems with extra supportive flows. The basic definitions and results of fuzzy graphs used in this paper are from [13] and [21]. The basic ideas of fuzzy incidence graphs can be seen in [11, 16, 18]. Applications of FIGs related to human trafficking are available in [8, 17, 19, 20, 28]. The symbol ∧ denote the minimum and ∨ denote the maximum. Let G = (V, E) be a graph with σ , a fuzzy subset of V and μ, a fuzzy subset of V × V. Consider a fuzzy subset  of V × E. If (v, e) ≤ σ (v) ∧ μ(e) for all v ∈ V and e ∈ E, then  = (σ, μ, )  is called a fuzzy incidence of G. If  is a fuzzy incidence, then G is called a fuzzy incidence graph (FIG) of G [6]. An example of a fuzzy incidence graph is given in Fig. 1. Let xy ∈ Supp(μ) (That is, μ(xy) > 0). Then xy is an edge of  = (σ, μ, ) if (x, xy), (y, xy) ∈ Supp(). Then (x, xy) the fuzzy incidence graph G and (y, xy) are called pairs. Two nodes of a fuzzy incidence graph are said to be connected if they are joined by an incidence path. The incidence strength of a fuzzy  = (σ, μ, ) is defined to be ∧{(v, e)|(v, e) ∈ Supp()} [7]. incidence graph G  = (σ, μ, ) is called a cycle if (Supp(σ ), Supp(μ), The fuzzy incidence graph G Supp()) is a cycle and a fuzzy cycle if (Supp(σ ), Supp(μ), Supp()) is a cycle and  there exists no unique xy ∈ Supp(μ) such that μ(xy) = ∧{μ(uv)|uv ∈ Supp(μ)}. G is called a fuzzy incidence cycle if it is a fuzzy cycle and there exists no unique (x, yz) ∈ Supp() such that (x, yz) = ∧{(u, vw)|(u, vw) ∈ Supp()} [16].  = (σ, μ, ) is a tree if (Supp(σ ), Supp(μ), The fuzzy incidence graph G Supp()) is a tree and is a forest if (Supp(σ ), Supp(μ), Supp()) is a forest.  if τ ⊆ σ, ν ⊆ μ, and  ⊆ . A  = (τ, ν, ) is a fuzzy incidence subgraph of G H   if  fuzzy incidence subgraph H of G is a fuzzy incidence spanning subgraph of G τ = σ [7]. Define  ∞ (s, t), also denoted as ICONNG (s, t) to be the incidence strength of a path from s to t of greatest incidence strength, where s, t ∈ σ ∗ ∪ μ∗ [11]. Consider u0 , (u0 , e1 ), e1 , (u1 , e1 ), u1 , . . . , (un , en+1 ), un . Since (ui−1 , ei ) ≤ σ (ui−1 ) ∧ μ(ei ), the strength of the incidence path is (u0 , e1 ) ∧ · · · ∧ (un , en+1 ).

Fig. 1 A fuzzy incidence graph

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An edge xy ∈ μ∗ is called a bridge if there exist u, v ∈ σ ∗ such that ICONNG−xy  (u, v) < ICONNG (u, v). If w ∈ σ ∗ , then w is called a cutnode if ICONNG−w (u, v) <  ICONNG (u, v) for some u, v ∈ σ ∗ such that u = w = v. w is called a fuzzy incidence cutnode, or simply an incidence cutnode if  ∞ (u, uv) <  ∞ (u, uv) for some (u, uv) ∈ V × E such that u = w = v, where  =  restricted to V × E . A pair (x, xy) is called a fuzzy incidence cutpair or simply an incidence cut where  =  repair if  ∞ (u, uv) <  ∞ (u, uv) for some pair (u, uv) in G,  = (σ, μ, ) be a fuzzy incidence graph stricted to (V × E)\{(x, xy)} [16]. Let G (FIG). An incidence pair (x, xy) is said to be a strong incidence pair if ψ(x, xy) ≥ (x, xy) where ICONNG−(x,xy) (x, xy) represents the greatest incidence ICONNG−(x,xy)    − (x, xy). In particular, (x, xy) is called an strength of x − xy incidence paths in G (x, xy) and β-strong incidence α-strong incidence pair if ψ(x, xy) > ICONNG−(x,xy)  (x, xy). An incidence pair (x, xy) is said to be a pair if ψ(x, xy) = ICONNG−(x,xy)  δ-incidence pair if ψ(x, xy) < ICONNG−(x,xy) (x, xy). (x, xy) is said to be δ ∗ , if it is  ∗  a delta pair, but ψ(x, xy) > ∧{ψ(u, uv) : (u, uv) ∈ ψ }. An incidence path ρ in G is said to be a strong incidence path if all pairs of ρ are strong. A strong incidence cycle is a closed strong incidence path [11].  is a fuzzy incidence forest if G  has a fuzzy incidence spanning subgraph  G F= (σ, ν, ) which is also a forest such that ∀(u, vw) ∈ Supp()\Supp(), (u, vw) < ∞ (u, vw). A connected fuzzy incidence forest is called a fuzzy incidence tree. In  = (σ, μ, ψ) is said to be a fuzzy incidence other words, A fuzzy incidence graph G  which is a tree, such tree if there exists a spanning subgraph  F = (τ, ν, ) of G, ∗ ∗ that for every pair (u, vw) ∈ ψ −  , ψ(u, vw) < ICONN F (u, vw) [16]. In other words, there exists an incidence path in  F from u to vw so that each of its pairs has greater ψ value than ψ(u, vw). See the example in Fig. 2. Note that the existence of an incidence path between any two elements of σ ∗ ∪ μ∗ is automatically guaranteed in this definition. It is established in [6] that  F in the definition of a fuzzy incidence forest is unique.  G is said to be fuzzy incidence complete if for all (u, vw) ∈ V × E, (u, vw) =

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 is fuzzy incidence complete, then (u, uv) = σ (v) ∧ σ (u) ∧ μ(vw). Note that if G μ(uv) = μ(uv) = σ (v) ∧ μ(uv) = (v, uv).  = (σ, μ, ) be a fuzzy incidence graph. The incidence degree of a node Let G  u ∈ σ ∗ is defined as di (u) = uv∈μ∗ ψ(u,  uv). Also the strong incidence degree of a node u ∈ σ ∗ is defined as dsi (u) = uv∈μ∗ ψ(u, uv), where (u, uv) is a strong ˜  is defined pair of G[17]. The incidence degree of an edge uv ∈ μ∗ of an FIG G as di (uv) = ψ(u, uv) + ψ(v, uv). The strong incidence degree of an edge uv is defined as the sum of ψ values of strong pairs incident on it. If there are no strong pairs incident on the edge uv, then define dsi (uv) = 0. Minimum and maximum of  i (G),  δsi (G),  si (G)  respectively [17]. node degrees are denoted as δi (G), Even though the incidence pairs are generally of the form (u, vw), we consider only pairs of the form (u, uv) in this article.

2 Incidence Cuts of Fuzzy Incidence Graphs Fuzzy incidence cuts and fuzzy incidence cuts of pairs in fuzzy incidence graphs were introduced in [17]. In this section, we shall have a closer look at these incidence cuts. Incidence analogues of bonds and cutbonds in FIGs are also discussed. Results similar to Whitney’s theorem in graph theory are obtained towards the end. We recollect some of the definitions from [17]. Note that elements of σ ∗ and μ∗ can be considered as the nodes of a fuzzy incidence graph. So incidence cutsets and connectivity can be defined similarly as in fuzzy graphs.  = (σ, μ, ψ) is a set D ⊆ σ ∗ ∪ μ∗ A disconnection of a fuzzy incidence graph G   whose removal disconnects G. The weight of D is defined to be min{μ(vu), u ∈ σ ∗} +



v∈D

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 denoted by Definition 1 [17] The node connectivity of a fuzzy incidence graph G,  is defined to be the minimum weight of a disconnection in G.  (G),  = (σ, μ, ψ) be a connected FIG. A set X = {s1 , s2 , . . . , sm } Definition 2 [17] Let G  if, ICONNG−X (s, t) < ⊂ σ ∗ ∪ μ∗ is said to be a fuzzy incidence cut (FIC) of G ICONNG (s, t) for some pair of elements s, t ∈ (σ ∗ ∪ μ∗ ) \ X . When, X consists of a single node s ∈ σ ∗ , it is a fuzzy incidence cutnode and when  s ∈ μ∗ , is an edge, it is an incidence bridge. If X is a fuzzy incidence  cut in G, then the strong weight of X , denoted by s(X ) is defined as s(X ) = {ψ(s, su) + s,xy∈X ,s∈σ ∗ ,xy∈μ∗

∧{ψ(x, xy), ψ(y, xy)}, where (s, su) is a strong pair of minimum weight at s, and ∧ is taken over all strong pairs incident at xy.

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Definition 3 [17] The incidence connectivity of a connected fuzzy incidence graph  is defined as the minimum strong weight of fuzzy incidence cuts of G.  It is denoted G  by κi (G).  = (σ, μ, ψ) be a connected FIG. A set of of strong pairs Definition 4 [17] Let G  S = {t1 , t2 , . . . , tm } ⊂ ψ∗ is said to be a fuzzy incidence cut of pairs (FICP) of G ∗ ∗ (s, t) < ICONN (s, t) for some pair of elements s, t ∈ (σ ∪ μ ) \ X . if, ICONNG−S   G  Note that a FICP of one element is a cutpair of G.  If S represents a fuzzy incidence cut of pairs in G, then the weight of S, denoted ψ(u, uv). by w(X ) is defined as w(X ) = (u,uv)∈S

Definition 5 [17] The incidence connectivity of pairs of a connected fuzzy inci is defined as the minimum weight of all fuzzy incidence cuts of pairs dence graph G   of G. It is denoted by κi (G).  = (σ, μ, ) be a fuzzy incidence graph. An edge uv ∈ Definition 6 [17] Let G ∗ μ is said to be a fuzzy incidence bond if ICONNG−uv (s, t) < ICONNG (s, t) for  some s, t ∈ σ ∗ ∪ μ∗ such that at least one of s, t is different from u and v. uv is (s, t) < ICONNG (s, t) for some called a fuzzy incidence cutbond if ICONNG−uv  s, t ∈ σ ∗ ∪ μ∗ , s, t = u, v. It is obvious that any fuzzy incidence cutbond is a fuzzy incidence bond, which is also a fuzzy incidence bridge. Analogous to the above definition, we have the following.  = (σ, μ, ) be fuzzy incidence graph. A pair (u, uv) ∈ ψ ∗ is Definition 7 Let G (s, t) < ICONNG (s, t) for some said to be a bonding pair (b-pair) if ICONNG−(u,uv)  s, t ∈ σ ∗ ∪ μ∗ such that at least one of s, t is different from u and v. (u, uv) is called (s, t) < ICONNG (s, t) for some a doubly bonding pair(db-pair) if ICONNG−(u,uv)  s, t ∈ σ ∗ ∪ μ∗ , s, t = u, v. Clearly, any db-pair is a b-pair and is a cutpair. Proposition 1 [17] At least one of the end nodes of a fuzzy incidence bond is a fuzzy incidence cutnode. Next we have an obvious proposition analogous to Proposition 1. Proposition 2 If (u, uv) is a b-pair, then either u or uv is a fuzzy incidence cut of  If (u, uv) is a db-pair, then both u and uv are fuzzy incidence cuts of G.  G.  = (σ, μ, ) Theorem 1 (Characterization of b-pairs of a fuzzy incidence tree) Let G be a fuzzy incidence tree (FIT) with |σ ∗ | ≥ 3. (x, xy) ∈ ψ ∗ is a b-pair if and only if (x, xy) is a pair of the unique fuzzy incidence spanning tree  F = (σ, ν, ) in the  definition of G.  = (σ, μ, ) be a fuzzy incidence tree with |σ ∗ | ≥ 3 and unique fuzzy Proof Let G incidence spanning tree  F = (σ, ν, ). First suppose that (u, uv) ∈ ψ ∗ is a b-pair of G. Then,

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ICONNG−(u,uv) (s, t) < ICONNG (s, t)  for some s, t ∈ σ ∗ ∪ μ∗ such that at least one of s, t is different from u and v. Then  passes through (u, uv). Since G  is all s − t incidence paths of greatest strengths in G a FIT, there exists a unique incidence path of greatest strength between every pair of F. Thus every elements s, t of σ ∗ ∪ μ∗ . Precisely it is the unique incidence path in   is in  s − t incidence path of greatest strength in G F. Since all such paths contain (u, uv), it follows that (u, uv) is a pair in  F. F is an incidence tree, Conversely suppose that (u, uv) ∈ ∗ . Since |σ ∗ | ≥ 3, and  both u and uv cannot be a fuzzy incidence end node and a fuzzy incidence end arc of  F together. So one of them say u is a fuzzy incidence cut. Being an FIT, there exists a unique incidence path of greatest strength between every pair of elements F. Let w be a strong neighbor from σ ∗ ∪ μ∗ of u other of σ ∗ ∪ μ∗ and it belongs to  than uv. Then there exists a unique strongest incidence path between w and uv in  Clearly this incidence path passes through the pair (u, uv) and  F and hence in G. hence it follows that (w, uv) < ICONNG (w, uv) ICONNG−(u,uv)   Thus (u, uv) is a b-pair in G.

 

 is a b-pair as all strong pairs In fact every strong pair of a fuzzy incidence tree G  belongs to its unique incidence spanning tree  of G F. Next we characterize fuzzy incidence trees using b-pairs.  be a fuzzy incidence graph. Then G  is a FIT if and only if every Theorem 2 Let G   strong pair of G is a b-pair of G.  = (σ, μ, ) be a fuzzy incidence graph. First suppose that G  is a FIT Proof Let G with  F = (σ, ν, ), its unique spanning fuzzy incidence subgraph in the definition.  By Theorem 3 of [17], (u, uv) is a pair of ∗ . By Let (u, uv) be a strong pair of G. Theorem above, every (u, uv) ∈ ∗ is a b-pair.  is a b-pair of G.  Clearly every Conversely suppose that every strong pair of G b-pair is a cutpair and hence by Theorem 2 of [17] every b-pair is an α - strong pair.  has no β - strong pairs. Hence by Proposition 7 of [17], it follows that G  That is G is an FIT.    = (σ, μ, ) be an FIT. Theorem 3 (Characterization of db-pairs in an FIT) Let G  is a db-pair if and only if u and uv are fuzzy incidence cuts (FIC) A pair (u, uv) of G  of G.  = (σ, μ, ) be an FIT and (u, uv), a db-pair of G.  Then, Proof Let G (s, t) < ICONNG (s, t) ICONNG−(u,uv)  for some s, t ∈ σ ∗ ∪ μ∗ such that s = u and uv = t or vice versa. Thus every s − t  passes through (u, uv). Since the deletion incidence path of greatest strength in G

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of either u or uv annihilates all such paths, it follows that both u and uv are fuzzy  incidence cuts of G.  such that u and uv are fuzzy inConversely suppose that (u, uv) is a pair of G  We have to show that (u, uv) is a db-pair. Clearly (u, uv) is an cidence cuts of G.  Also P : u(u, uv)uv is the internal pair of  F, the unique incidence spanning tree of G.  with strength ψ(u, uv). unique strongest u − uv incidence path in  F and hence in G  such that Now let s1 , s2 , s3 , s4 ∈ σ ∗ ∪ μ∗ be vertices of G ICONNG−u  (s1 , s2 ) < ICONNG  (s1 , s2 ) and (s3 , s4 ) < ICONNG (s3 , s4 ). ICONNG−uv 

Let P1 be the unique strongest incidence s1 − s2 path passing through u and P2 be the unique strongest s3 − s4 path passing through uv. Let s ∈ σ ∗ ∪ μ∗ be in P1 adjacent to u and t ∈ σ ∗ ∪ μ∗ be in P2 adjacent to uv. Then since P is unique, (s, t) < ICONNG (s, t) ICONNG−(u,uv)   where u = s and uv = t. Thus it follows that (u, uv) is a db-pair of G.

 

Corollary 1 A complete fuzzy incidence graph has no b-pairs. A fuzzy incidence graph without fuzzy incidence cuts of cardinality one is called an incidence block. Clearly, the following result holds.  is an incidence block, then no pair of G  is a b-pair of G.  Corollary 2 If G Definition 8 A node common to two or more db-pairs is called a critical node(c node) of a fuzzy incidence graph G. Clearly any critical node is an incidence cutnode.  be a fuzzy incidence graph. Let s ∈ σ ∗ ∪ μ∗ . The strong inciDefinition 9 Let G  − s are the maximal strong pair(in G)  induced incidence dence components of G  − x. fuzzy subgraphs of G  = (σ, μ, ) be an FIT and let w be a c-node G.  Then G −w Theorem 4 Let G will have at least two strong incidence components. Proof Since w is a c-node, it must be a common node of at least two db-pairs of  Since G  is a FIT, c must be the common node of at least two db-pairs of  G. F, the  Hence w is a c-node of  incidence spanning tree in the definition of G. F. Then since  F is an incidence tree  F − w will have at least two non trivial components. All pairs in  F are strong. Also by definition, pairs which are in ψ ∗ − ∗ are not strong. Hence the non trivial components of  F − w are precisely the strong incidence components  − w and the conclusion follows. of G  

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Now we turn to the incidence connectivity of different fuzzy incidence graphs. It is easy to calculate the incidence conductivity of a fuzzy incidence tree using the following result from [17].  = (σ, μ, ψ) be a fuzzy incidence tree. Then, κi (G)  = Theorem 5 [17] Let G  = ∧{ψ(u, uv) : (u, uv) is a strong pair in G}.  κi (G) Similar to Whitney’s inequality for connectivity in graphs, we have the following inequality true in any fuzzy incidence graph.  = (σ, μ, ψ), κi (G)  ≤ κi (G)  ≤ Theorem 6 In a connected fuzzy incidence graph G  δsi (G).  = (σ, μ, ψ) be a connected Proof First we prove the second inequality. Let G  Let P be the set fuzzy incidence graph. Let v ∈ σ ∗ be such that dsi (v) = δsi (G).  − P is of strong pairs incident at v. If these are the only pairs incident at v, then G disconnected. If not, let (v, vu) be a pair which is not strong at v. Then vu is an edge different from the edges of the pairs in P. By definition of a strong pair, ψ(v, vu) < ICONNG (v, vu)  which which implies that there exists a strongest v − vu incidence path say Q in G should definitely pass through one of the strong pairs at v. Thus the removal of P  will reduce the incidence strength between v and u. Thus in both cases, P from G  Hence it is a fuzzy incidence cut of pairs. The strong weight of this FICP is δsi (G).   follows that κi (G) ≤ δsi (G).  ≤ κi (G).  Let P be a FICP with weight κi (G).  We have the Next to prove κi (G) following cases. Case 1. Every pair in P has a common node v (say). In this case, let P = {pi = (v, vvi ), i = 1, 2, . . . , n}. Let S = {vv1 , vv2 , . . . , vvn }. Then clearly P is a fuzzy incidence cut. Now, minu∈σ∗ ψ(vi , uvi ) ≤ ψ(vi , vvi ). Therefore,  (minu∈σ∗ ψ(vi , uvi )) ≤ ψ(v, vv1 ) + ψ(v, vv2 ) + · · · + ψ(v, vvn ). i

 ≤ κi (G).  That is, κi (G) Case 2. Not all pairs in P have a node in common. Let P = {pi = (ui , ui vi ), i = 1, 2, . . . , n} for some n. Let X1 = {u1 , u2 , . . . , un } and X2 = {u1 v1 , u2 v2 , . . . , un vn }. By assumption, ICONNG−P  (s, t) < ICONNG  (s, t) for some s, t ∈ σ ∗ ∪ μ∗ . Sub Case 1. s and t are not members of X1 ∪ X2 . In this case, take X = X1 or X = X2 . Then clearly X is a fuzzy incidence cut since  reduces the incidence strength between s and t and, its deletion from G  ≤ weight of X ≤ weight of P = κi (G).  κi (G)

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Sub Case 2. Either s or t is in X1 ∪ X2 . Without loss of generality suppose that s is in X1 ∪ X2 . Suppose s ∈ X1 . Then take  will reduce X = X2 . Clearly X is a fuzzy incidence cut, for; the deletion of X from G the incidence strength between s and t. Thus,  ≤ weight of X ≤ weight of P = κi (G).  κi (G) Sub Case 3. Both s and t are in X1 ∪ X2 . Clearly s ∈ σ ∗ and t ∈ μ∗ . Hence t = rv where r, v ∈ σ ∗ . The removal of P reduces the incidence strength between s and v and the conclusion follows.  ≤ κi (G)  ≤ δsi (G).    Thus in all cases, κi (G) As in the case of incidence fuzzy trees, we can easily determine the incidence connectivity parameters of a complete fuzzy incidence graph, using the following result.  = (σ, μ, ψ), κi (G)  = κsi (G)  = Theorem 7 In a complete fuzzy incidence graph, G     κsi (G) = κi (G) = δsi (G) = δi (G).  = (σ, μ, ψ), be a CFIG such that |σ ∗ | = n. Since ψ is a complete fuzzy Proof Let G  will not reduce the incidence, the deletion of any set P of at most n − 2 pairs from G ∗ ∗ incidence strength between any pair of elements of σ ∪ μ . But a set of n − 1 pairs incident at a node u ∈ σ ∗ is a FICP with weight dsi (u) = di (u). Let v ∈ σ ∗ be such that di (v) = δi (G). Clearly the set of edges incident at v is a FICP with minimum weight.  = di (v) = δi (G).  Therefore, κi (G)   Now we prove that κi (G) = δi (G).  = δi (G).  By Theorem 6, κi (G)  ≤ κi (G)  ≤ δsi (G).  If possible suppose that κi (G)   Hence κi (G) < δsi (G) Note that any incidence cut of elements from σ ∗ will have cardinality n − 1. Among such incidence cuts, the one which does not contain v such that dsi (v) = δsi (G), say S1 will have the minimum weight since the set of pairs adjacent with elements in S1 with one end at v are the pairs with minimum ψ value among elements of S1 .  = s(S1 ) < δsi (G).  Thus, κi (G) Now let E1 be the set of pairs incident with v. Then E1 is a FICP such that,  which contradicts the fact that κi (G)  = δsi (G).  Hence, s (E1 ) = s(S1 ) < δsi (G),    κi (G) = κi (G) = δsi (G). The conclusion of the theorem follows from the fact that all pairs of a complete fuzzy incidence are strong.  

3 t-Connected and t-Pair Connected Fuzzy Incidence Graphs In this section, we introduce t-connected and t-pair connected fuzzy incidence graphs. Consider the following definitions of s − t incidence connectivity reducing sets.

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Definition 10 [17] A set S ⊂ (σ ∗ ∪ μ∗ ) \ {s, t} is said to be an s − t incidence  − S is the connectivity reducing set if ICONNG−S  (s, t) < ICONNG  (s, t), where G  obtained by removing all elements in S and the fuzzy incidence subgraph of G associated pairs. Definition 11 [17] A set P ⊂ ψ ∗ is said to be an s − t incidence connectivity  − P is the fuzzy reducing set of pairs if ICONNG−P  (s, t) < ICONNG  (s, t), where G  obtained by removing all pairs in P. incidence subgraph of G Definition 12 [17] An s − t incidence connectivity reducing set with n elements of σ ∗ ∪ μ∗ is said to be a minimum s − t incidence connectivity reducing set if there exist no s − t incidence connectivity reducing set with less than n elements. A minimum s − t incidence connectivity reducing set is denoted by IG (s, t). Minimum s − t incidence connectivity reducing set of pairs can be similarly defined and is denoted by PG (s, t). Characterizations of s − t incidence connectivity reducing sets are given in the following theorems.  = (σ, μ, ψ) be a connected fuzzy incidence graph and Theorem 10 [17] Let G s, t ∈ σ ∗ ∪ μ∗ . Then, a set S ⊂ σ ∗ ∪ μ∗ is an s − t incidence connectivity reducing set if and only if every strongest incidence path from s to t contains at least one element of S.  = (σ, μ, ψ) be a fuzzy incidence graph. For any two eleTheorem 11 [17] Let G ments s, t ∈ σ ∗ ∪ μ∗ such that (s, t) not a strong incidence pair, the maximum number  is equal to the number of of internally disjoint strongest s − t incidence paths in G elements in a minimal s − t incidence connectivity reducing set.  = (σ, μ, ψ) be a fuzzy incidence graph. For any two elTheorem 12 [17] Let G ements s, t ∈ σ ∗ ∪ μ∗ the maximum number of internally disjoint strongest s − t  is equal to the number of elements in a minimal s − t incidence incidence paths in G connectivity reducing set of pairs.  be a connected fuzzy incidence graph and t ∈ (0, ∞). G  is Definition 13 Let G    called t -connected if κi (G) ≥ t and G is called t - pair connected if κi (G) ≥ t.  is t - connected if there exist no fuzzy In other words a fuzzy incidence graph G incidence cut with strong weight less than t and is t - pair connected if there exist no fuzzy incidence cut of pairs with strong weight less than t  be a connected fuzzy incidence graph. G  is a fuzzy incidence Theorem 13 [12] Let G block if and only if sum of incidence strengths of all internally disjoint strongest incidence paths is at least 2CONNG (s, t) for every pair of elements s, t ∈ σ ∗ ∪ μ∗ such that (s, t) is not a cutpair. Now we give characterizations of t - connected incidence fuzzy graphs and t pair connected fuzzy incidence graphs below.

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 be a connected incidence fuzzy graph. Then G  is t - incidence Theorem 14 Let G connected if and only if mICONNG (s, t) ≥ t for every pair of elements s, t ∈ σ ∗ ∪ μ∗  of greatest where m is the number of internally disjoint s − t incidence paths in G incidence strength.  is t-incidence connected for some t ∈ (0, ∞). Then Proof First assume that G  ≥ t. To prove that for every pair of elements s, t ∈ σ ∗ ∪ μ∗ , the sum of inκi (G) cidence strengths of all internally disjoint fuzzy incidence paths of greatest incidence strength is at least t. If possible, assume that there exists a pair of nodes s1 , s2 ∈ σ ∗ ∪ μ∗ such that mCONNG (s1 , s2 ) < t where m is the number of internally disjoint s1 − s2 incidence paths of greatest incidence strength. Let S be a minimal s1 − s2 incidence connectivity reducing set of elements in in (σ ∗ ∪ μ∗ ) \ {s, t} with minimum strong weight. Let |S| = m. If P1 , P2 , . . . , Pm denote the internally disjoint s1 − s2 incidence paths of greatest strengths, each Pi must contain at least one of the elements from S and no element can appear in more than one Pi , i = 1, 2, . . . , m. Thus each incidence path Pi contains exactly one element ti (say) from S. Also if there exists another s1 − s2 incidence path Q of greatest incidence strength other than Pi , i = 1, 2, . . . , m, Q has to share an element of S with Pi for some i. Since S is an s1 − s2 incidence connectivity reducing set, it is a fuzzy incidence cut. Also since S is an s1 − s2 incidence connectivity reducing set of minimum strong weight, one of the elements incident at ti in Pi must have strength equal to ICONNG (s1 , s2 ). Therefore, strong weight of S = |S|ICONNG (s1 , s2 ) = mICONNG (s1 , s2 ) < t. Thus there exists a fuzzy incidence cut with strong weight less than t and hence  < t, a contradiction to our assumption. Thus the sum of incidence strengths κi (G) of all internally disjoint incidence paths of greatest strength is at least t. Conversely suppose that the sum of incidence strengths of all internally dis ≥ t. If joint incidence paths of greatest strengths is at least t. To show that κi (G)  < t. Then there exists a fuzzy incidence cut S such possible suppose that κi (G)  that strong weight of S is less than t. Also for some pair of elements t1 , t2 ∈ G, ICONNG−S  (t1 , t2 ) < ICONNG  (t1 , t2 ) and hence S is a t1 − t2 incidence connectivity reducing set. By Theorem 11, m = Number of elements in a minimal incidence connectivity reducing set ≤ |S| Therefore, mICONNG (t1 , t2 ) ≤ |S|ICONNG (t1 , t2 ) ≤ Strong weight of S < t. That is the sum of incidence strengths of all internally disjoint t1 − t2 incidence paths of greatest strength is less than t, which is a contradiction. Hence the theorem.   be a connected fuzzy incidence graph. Then G  is t - pair conTheorem 15 Let G nected if and only if mICONNG (s1 , s2 ) ≥ t for every pair of elements s1 and s2 in σ ∗ ∪ μ∗ where m is the number of pair disjoint s1 − s2 incidence paths of greatest  strength in G.  is t- pair connected. Then κi (G)  ≥ t. To prove that for Proof First assume that G every pair of elements s1 , s2 ∈ σ ∗ ∪ μ∗ , the sum of incidence strengths of all pair disjoint s1 − s2 incidence paths of greatest strength is at least t. On the contrary assume

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that there exists a pair of elements t1 , t2 ∈ σ ∗ ∪ μ∗ such that mICONNG (t1 , t2 ) < t where m is the number of pair disjoint t1 − t2 incidence paths of greatest strengths.  Let E be a minimum t1 − t2 incidence connectivity reducing set of pairs in G with minimum strong weight (This happens only when any pair (x, xy) in E has strength ICONNG (t1 , t2 )). Let |E| = m. If P1 , P2 , . . . , Pm denote the pair disjoint t1 − t2 incidence paths, each incidence path Pi must contain at least one of the pairs from E and no pair can appear in more than one incidence path. Thus each path Pi contains exactly one incidence pair from E. Also if there exists another t1 − t2 incidence path Q of greatest incidence strength other than Pi , i = 1, 2, . . . , m, it has to share a pair of E with Pi for some i. Since E is a t1 − t2 connectivity reducing set  and hence, of pairs,it is fuzzy incidence cut of pairs of G strong weight of E = mICONNG (t1 , t1 ) < t. Thus there exist a fuzzy incidence cut of pairs with strong weight less than t and  < t, a contradiction to our assumption. hence κi (G) Conversely suppose that the sum of strengths of all pair disjoint t1 − t2 incidence paths of greatest strength is at least t for every pair of elements t1 and t2 in σ ∗ ∪ μ∗ . To  ≥ t. If κi (G)  < t, then there exists a fuzzy incidence cut of pairs say show that κi (G) E with strong weight less than t. Also by definition of an FIC, ICONNG−E  (s1 , s2 ) < ICONNG (s1 , s2 ) for some elements s1 and s2 in σ ∗ ∪ μ∗ . Thus E is an s1 − s2 strength reducing set of pairs. By Theorem 12, m = Number of pairs in a minimal s1 − s2 incidence connectivity reducing set of pairs. Hence, m ≤ |E|. Therefore, mICONNG (s1 , s2 ) ≤ |E|ICONNG (s1 , s2 ) ≤ strong weight of E < t, which is a contradiction to our assumption. The proof is now complete.  

4 Modeling the Most Risky Immigration Routes The work in this section is based on [3, 28–30]. There are inherent challenges involved in tracking the deaths of irregular migrants and even the best counts will have gaps. The very nature of irregular travel - that the objective is to avoid detection - makes tracing deaths, and identifying bodies, extremely challenging. Even when the most “newsworthy”sinking occur, often the number of passengers on board is unknown, making accurate estimates of deaths near impossible. In addition to the clandestine nature of irregular migration, the topography through which migrants travel, again the choice of routes being motivated by the desire to remain undetected, presents challenges for documenting deaths. Migrant trails often pass through remote areas. The tough ecologies of land passages can mean that remains may be quickly destroyed by arid climates and wild animals, or may be lost in crevices or swept down rivers. According to the Colibri Center for Human Rights, which works to trace missing migrants, an estimated 2000 missing people are reported along the United States Mexican border. Countless others remain missing and are never reported. They are

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simply not heard of again, their remains lost in the sand, buried in unmarked graves or washed on remote shores. An additional complicating factor arises from the fact that irregular migration is frequently intertwined with the actions of smugglers, traffickers, other criminals and corrupt State officials. Countless numbers of deaths have occurred either through direct murder or indirect consequences of poor care, abuse, torture, or abandonment. Finally, at issue in documenting border-related deaths is determining which migrant deaths are deemed “border-related.” Migrant Deaths in United States and Mexico For an overview of the available data on migrant fatalities in the United States and in Mexico and an analysis of the reasons for incomplete counts see [3, 30]. We provide a short discussion. The lack of data on migrant fatalities reflects official disregard for the safety and security of irregular migrants in both countries. With hundreds of deaths each year in the single US state of Arizona, the death of migrants is a major public concern. Yet there continues to be no reliable complete count of the number of fatalities in either the United States or Mexico. According to a 2009 report by the CNDH, an estimated 9578 migrants were kidnapped in Mexico across 198 cases between September 2008 and February 2009 (CNDH, 2009). Findings from this report suggest that migrants are being kidnapped an masse, with 55 per cent of the victims being kidnapped in the southern part of Mexico - mostly in the states of Veracruz and Tabasco (ibid). Around 12 percent were kidnapped in the north and 1 per cent in the central region (ibid.). The region where the kidnapping occurred was unknown for 32 per cent of the victims (ibid.). Among 95,878 migrants kidnapped, the person’s country of origin was only determined for 552 individuals; 372 Hondurans, 101 Salvadorans, 74 Guatemalans, and 5 Nicaraguans (ibid.). Although the threats migrants experience in Mexico are primarily due to organized crime, other risks exist. Migrants are reported to drown each year in the Suchiate River which runs between Mexico and Guatemala (United States Conference of Catholic Bishops, 2003). Many are reported to accidently fall from moving trains, or misstep as they attempt to jump on or off [10, 24]. There are reports of migrants getting lost and dying of dehydration or heat stroke on the northern states of Mexico, as they traverse remote geographies approaching the United States-Mexico border [30]. Migrant deaths across the United States-Mexico border have not been evenly distributed across time and geography. The portion border where the most migrants have died is in the Tucson sector of Arizona. A significant number of deaths have also occurred in California and Texas, with the fewest recorded in New Mexico. We model the flow of immigrants from Mexico to the United States using a fuzzy graph or fuzzy incidence graph. For all cities, we assign the value 1. We assign the risk value in the tables below to the edge uv. Another approach is to assign the risk value to a pair (u, uv). The risk values were determined by the study of many reports concerning immigration from Mexico to the United States, for example [1–6].

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In the following table G is for Guadalupe, Ar is for Arrigo, Ix for Ixtepec, Gu for Guadalaja, Ma for Mazatlan, Ti for Tijuana, No for Nogalas, Sa for Saltillo, Re for Reynossa, Nu for Nuevo Laredo, Te for Texas, To for Torreon, Ten for Tenosique, Ve for Veracruz, Ne for New Mexico, Be for Belize, Ju for Juarez, Ca for California and Ari for Arizona. G G G G G G G Be

0.7 0.7 0.7 0.7 0.7 0.7 0.3 0.7

Ar Ar Ar Ar Ar Ar To Ten

0.5 Ix 0.5 Ix 0.5 Ix 0.5 Ix 0.5 Ix 0.5 Ix 0.5 Ju 0.5 Ve

0.5 0.5 0.5 0.5 0.5 0.9 0.5 0.5

Gu Gu Sa Sa Sa Te Ne Re

0.5 0.5 0.3 0.5 0.9

Ma Ma Re Nu Te

0.7 0.3 0.9 0.9

Ti 0.9 Ca No 0.9 Ari Te Te

0.9 Te

Let L denote the set of all cities in Mexico involved in this study. We provide an application of the notion of disconnection given in Sect. 2. Let Du = {uv|v ∈ L}. Now we provide an application of the notion of disconnection given in Sect. 2. The weight of Du denotes the highest possible risk from a city. Let u = Saltillo. Then the weight of Du = 0.3 + 0.5 + 0.9 = 1.7. This is the largest such weight of all u ∈ L. The μ-value for an edge uv gives a measure of the risk for a immigrant traveling from u to v. However there is an accumulative factor to the risk as an immigrant travels along a path. This can be modelled by “adding” the risk values in a route. This is best done by using a t -conorm. Let P : u0 = u, u0 u1 , u1 , . . . , un−1 , un−1 un , un = v be a path from u to v. Define the strength of the path s(P) to be μ(uu1 ) ◦ μ(u1 u2 ) ◦ · · · ◦ μ(un−1 un ), where ◦ is a t-conorm. Consider the path P : Guadalupe, Torreon, Juarez, New Mexico. Let ◦ denote algebraic sum. Then s(P) = 0.825. It is easy to see that this route has the smallest strength compared with all routes from Mexico to United States under consideration. That is, this is the safest route. The route with the most risk is the first one in the Guadalupe table. We also note that the removal of node Arrigo or edge ArrigoIxtepec or pair (Arrigo, ArrigoIxtepec) disconnects all routes with Guadalupe as the origin except for the route Guadalupe, Torreon, Juarez, New Mexico, the safest route. Consequently, Arrigo and edge ArrigoIxtepec and pair (Arrigo, ArrigoIxtepec) are fuzzy incidence cuts of their particular nature.

5 Conclusion Fuzzy incidence graphs are used to model relationships with influences. Several real world phenomena like human flows, information and traffic flow, etc. can be modeled using FIGs. We have analyzed different connectivity parameters of FIGs and derived relationships between them. Also, the concept of t-connected and t-pair connected fuzzy incidence graphs are introduced and some characterizations are obtained.

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