Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory: ICRAPAM 2018, New Delhi, India, October 23–25 (Springer Proceedings in Mathematics & Statistics, 307) 981151156X, 9789811511561

Table of contents :
Dedicated to Professor Niranjan Singh
Preface
Contents
About the Editors
Exact Solution for Mixed Integral Equations by Method of Bernoulli Polynomials
1 Introduction
2 Bernoulli Polynomial Method
3 Matrix Representation of Bernoulli Polynomial
4 Method of Solution for Mixed Volterra–Fredholm Integral Equation of Second Kind
5 Algorithm of Bernoulli Polynomial Method
6 Numerical Examples
7 Conclusion and Result
References
Turing Patterns in a Cross Diffusive System
1 Introduction
2 Model Formulation
3 Stability Analysis
3.1 Temporal Model
3.2 Model with Self Diffusion
3.3 Model with Self Diffusion and Cross Diffusion
4 Numerical Simulations
5 Conclusion
References
On Multi-objective Optimization Problems and Vector Variational-Like Inequalities
1 Introduction
2 Preliminaries
3 Main Results
4 Conclusions
References
Controllability of Semilinear Control Systems with Fixed Delay in State
1 Introduction
2 Preliminaries
3 Main Results
4 Application
References
Computational Performance of Server Using the Mx/M/1 Queue Model
1 Introduction
2 Notations and Mathematical Description
2.1 Notations
2.2 Mathematical Model
3 Numerical Approach
4 Conclusion
References
Quantum Codes from the Cyclic Codes Over mathbbFp[v,w]/langlev2-1,w2-1,vw-wvrangle
1 Introduction
2 Preliminary
3 Linear Codes Over R
4 Cyclic Codes Over R
5 Quantum Codes Over mathbbFp
6 Conclusion
References
Effect of Sterile Insect Technique on Dynamics of Stage-Structured Model Under Immigration
1 Introduction
2 Mathematical Model Formulation
3 Dynamical Behaviour
3.1 Boundedness of Model
3.2 Equilibrium Points
4 Conclusion
References
Strict Practical Stability of Impulsive Differential System in Terms of Two Measures
1 Introduction
2 Preliminaries
3 Main Section
4 Conclusion
References
Free Vibration Analysis of Rigidly Fixed Axisymmetric Viscothermoelastic Cylinder
1 Introduction
1.1 Formulation of Problem
2 Boundary Conditions
2.1 Solution of Mathematical Problem
3 Series Solution
4 Dispersion Relations
4.1 Deduction of Results
5 Numerical Results and Discussion
6 Conclusion
References
Study on a Free Boundary Problem Arising in Porous Media
1 Introduction
2 Flow Problem
3 Solution of the Problem
4 Concluding Remarks
References
Effect of Habitat on Dynamic of Native and Exotic Prey–Predator Population
1 Introduction
2 Biological Background
3 Mathematical Model
4 Boundedness
5 Basic Feasible Equilibrium Points
6 Local Stability Analysis
7 Global Stability
8 Numerical Example
9 Conclusion
References
On Cliques and Clique Chromatic Numbers in Line, Lict and Lictact Graphs
1 Introduction
2 Main Results
References
Friendship-Like Graphs and It's Classiffication
1 Introduction
2 Main Results
2.1 Friendship-Like Graphs
3 Conclusion
References
Chaotic Maps: Applications to Cryptography and Network Generation for the Graph Laplacian Quantum States
1 Introduction
2 Analysis of Chaotic Maps
3 Modelling of One-Dimensional Chaotic Map into Networks
4 Quantum States
5 Conclusion
References
Consumer Behaviour Analysis for Purchasing a Passenger Car in Indian Context
1 Introduction
2 Problem Statement and Data Collection
3 Methodology
3.1 Crosstab Analysis
3.2 AHP
3.3 TOPSIS
4 Illustration
4.1 Crosstab Analysis
4.2 AHP
4.3 TOPSIS
5 Conclusion
References
A New Hybrid Model Based on Triple Exponential Smoothing and Fuzzy Time Series for Forecasting Seasonal Time Series
1 Introduction
2 Triple Exponential Smoothing
3 Basic Notions of Fuzzy Time Series
4 The Proposed Hybrid Model
5 Numerical Results
6 Conclusion
References
New Fuzzy Divergence Measure and Its Applications in Multi-criteria Decision-Making Using New Tool
1 Introduction
2 Preliminaries
2.1 Information Measures
2.2 Divergence Measures
2.3 Fuzzy Set Theory
2.4 Fuzzy Divergence Measures
3 Proposed Generalized Fuzzy Divergence Measure
4 Series of Fuzzy Divergence Measure
4.1 Some New Other Fuzzy Divergence Measures
5 Application of New Fuzzy Divergence Measure in Multi-criteria Decision-Making
5.1 Numerical Illustration
5.2 Comparison with the TOPSIS Method
5.3 Numerical Illustration
6 Conclusion
References
An SIRS Age-Structured Model for Vector-Borne Diseases with Infective Immigrants
1 Introduction
2 The SIRS Model Formulation
3 Feasibility of Solution
4 Existence of Equilibrium Points
4.1 Disease-Free Equilibrium Point
4.2 Endemic Equilibrium Point
5 Basic Reproduction Number R0
6 Local Stability Analysis of the Equilibrium Points
7 Numerical Simulation
8 Conclusion
References
Numerical Study of Conformable Space and Time Fractional Fokker–Planck Equation via CFDT Method
1 Introduction
2 Preliminaries
2.1 Description of CFDTM
3 Analytical Study of Conformable Fractional Fokker–Planck Equation
4 Conclusion
References
Multispectral Bayer Color Image Encryption
1 Introduction
2 Review of Literature Survey
2.1 Multispectral Color Image and Bayer Image
2.2 Elliptic Curve Cryptography for Images
2.3 Generalized Arnold Transformation for Images
3 Proposed Model for Image Encryption and Decryption
3.1 Image Encryption Procedure
3.2 Image Decryption Procedure
4 Security Analysis and Experimental Results
4.1 Statistical Analysis
4.2 Numerical Analysis
5 Conclusions
References
Investigation of Prospective Elementary Teachers’ Opinions About Problem Concept
1 Introduction
1.1 What Is a Problem?
1.2 Problem-Solving and Process of Problem-Solving
1.3 Nonroutine Problems
1.4 Importance of the Study
2 Method
2.1 Sample
2.2 Instruments
2.3 Reliability and Validity
3 Results
3.1 What Is a Problem?
3.2 Types of Problem
3.3 Conditions of Being Problem
3.4 Problem-Solving Steps of Polya
3.5 Right/Wrong Items
3.6 Multiple-Choice Items
4 Discussion and Conclusion
References

Citation preview

Springer Proceedings in Mathematics & Statistics

Naokant Deo Vijay Gupta Ana Maria Acu P. N. Agrawal   Editors

Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory ICRAPAM 2018, New Delhi, India, October 23–25

Springer Proceedings in Mathematics & Statistics Volume 307

Springer Proceedings in Mathematics & Statistics This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

More information about this series at http://www.springer.com/series/10533

Naokant Deo Vijay Gupta Ana Maria Acu P. N. Agrawal •

•

•

Editors

Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory ICRAPAM 2018, New Delhi, India, October 23–25

123

Editors Naokant Deo Department of Applied Mathematics Delhi Technological University New Delhi, India

Vijay Gupta Department of Mathematics Netaji Subhas University of Technology New Delhi, India

Ana Maria Acu Department of Mathematics Lucian Blaga University of Sibiu Sibiu, Romania

P. N. Agrawal Department of Mathematics Indian Institute of Technology Roorkee Roorkee, Uttarakhand, India

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-981-15-1156-1 ISBN 978-981-15-1157-8 (eBook) https://doi.org/10.1007/978-981-15-1157-8 Mathematics Subject Classification (2010): 34-XX, 49-XX, 68-XX, 85-XX, 90-XX, 97-XX © Springer Nature Singapore Pte Ltd. 2020 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, expressed 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Dedicated to Professor Niranjan Singh

“Not everything that can be counted counts, and not everything that counts can be counted” —Albert Einstein

Mathematics is a game of logic and there was no one better at deciphering it than Prof. Niranjan Singh—a pioneer in his field with a heart of an innovator. Always eager to learn and teach, Prof. Singh taught at Kurukshetra University, Haryana, for 32 years, focusing on areas of analysis and related fields. He was instrumental in the development of innovative and creative culture in a large number of academic institutions in and around Kurukshetra, thereby bringing a significant change in the teaching methodologies of postgraduate courses. Apart from being an exemplary mathematician, Prof. Singh was a dedicated social reformist. He was a firm advocate for the use of Hindi language, as a result of which he taught and wrote in Hindi, inspiring its use. Professor Singh authored many books on mathematics of which Beej Ganit, written in 1979, was awarded the gold medal by Sahitya Academi—a national organization dedicated to the promotion of literature in the languages of India. Insisting on the fact that money shouldn’t define the caliber of any student, he firmly promoted the optimal utilization of resources and cost reduction in higher education. His ideology became his strength and helped many educationalists understand the relevance of an

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Dedication

unerring education system. His journey let him to be the head of Bhartiya Shikshan Mandal, through which he travelled around India and influenced society. Professor Singh is remembered by his peers for his academic excellence, research, aptitude, dedication to work, human values, and behavior. His will always be an inspiration to budding professors and mathematicians as he was the embodiment of everything mathematics. “Carve your name on hearts, not tombstones. A legacy is etched into the minds of others and the stories they share about you.” —Shannon Alder

Preface

We are delighted that the Department of Applied Mathematics, Delhi Technological University, Delhi, organised the international conference on ‘Recent Advances in Pure and Applied Mathematics 2018’ (ICRAPAM-2018 during 23–25 October 2018). This international conference was organised in the memory of our beloved late Dr. Niranjan Singh (1939–2014), who was a Professor at the Department of Mathematics, Kurukshetra University, India. The proceedings consist of two volumes, the second volume comprising of papers mainly on applied areas related to Optimisation, Differential Equations and Graph Theory. These proceedings consist of research literature on the latest developments in various branches of mathematics. It is an outcome of the research papers presented at the conference and some invited talks. In total, 180 research papers were presented by researchers and academicians in different areas. After peer review, 21 research/ survey papers were selected for inclusion in Volume II of the proceedings. The main areas covered in this volume are devoted to multi-objective optimisation problems, control theory, impulsive differential equations, mathematical modelling, fuzzy mathematics, graph theory, cryptography, coding theory especially on quantum codes, free vibration analysis, age-structured model and mathematical education. The authors of these papers have carefully described the problem and discussed appropriate methods to obtain the solution. Besides many plenary speakers as indicated in Volume I, some of the invited speakers who delivered talks on different topics in applied areas were: Prof. Ramesh Tikekar, IUCAA, Pune, India; Prof. Mani Mehra, IIT Delhi, India; Prof. A. K. B. Chand, IIT Madras, India; Prof. V. Sree Hari Rao, Foundation for Scientific Research and Technological Innovation, Hyderabad, India; Prof. B. Mishra, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad, India; Prof. D. R. Sahu, Banaras Hindu University, Varanasi, India and Prof. A. S. Ranadive, G. G. University, Bilaspur, Chhattisgarh, India. We are thankful to all the speakers who very kindly accepted our invitation to deliver talks at the conference. The thanks are due to all the funding agencies: Science and Engineering Research Board (SERB), Government of India, New Delhi; Third phase of vii

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Preface

Technical Education Quality Improvement Programme (TEQIP-III), Government of India and Government of NCT, New Delhi, for the partial financial support, without which it was difficult to host such a big event. Professor Yogesh Singh, Vice-Chancellor, DTU, Delhi, India and Pro-Vice Chancellors, Prof. S. K. Garg and Prof. Anu Singh Lather helped and supported a lot at each step to make the conference successful. We are thankful to Prof. H. C. Taneja, Prof. L. N. Das, Prof. Sangeeta Kansal, Prof. R. Srivastava, Prof. Anjana Gupta, Dr. C. P. Singh, Dr. Aditya Kaushik and Dr. Vivek Kumar Aggarwal. We give special thanks to Ms. Neha for helping in the preparation of the volume. We are grateful to the members of various committees, who put in a lot of hard work to make this event a huge success. We express our heartfelt gratitude to the reviewers for spending their invaluable time for careful review of the papers. Finally, we are grateful to Springer team for publishing the proceedings of the conference. New Delhi, India New Delhi, India Sibiu, Romania Roorkee, India

Naokant Deo Vijay Gupta Ana Maria Acu P. N. Agrawal

Contents

Exact Solution for Mixed Integral Equations by Method of Bernoulli Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mithilesh Singh, Nidhi Handa and Shivani Singhal Turing Patterns in a Cross-Diffusive System . . . . . . . . . . . . . . . . . . . . . Nishith Mohan and Nitu Kumari

1 11

On Multi-objective Optimization Problems and Vector Variational-Like Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vivek Laha and Harsh Narayan Singh

29

Controllability of Semilinear Control Systems with Fixed Delay in State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abdul Haq and N. Sukavanam

41

Computational Performance of Server Using the Mx/M/1 Queue Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jitendra Kumar and Vikas Shinde

51

Quantum Codes from the Cyclic Codes Over Fp ½v; w=hv2  1; w2  1; vw  wvi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Habibul Islam, Om Prakash and Ram Krishna Verma

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Effect of Sterile Insect Technique on Dynamics of Stage-Structured Model Under Immigration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sumit Kaur Bhatia, Sudipa Chauhan and Priyanka Arora

75

Strict Practical Stability of Impulsive Differential System in Terms of Two Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pallvi Mahajan, Sanjay Kumar Srivastava and Rakesh Dogra

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Free Vibration Analysis of Rigidly Fixed Axisymmetric Viscothermoelastic Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Himani Mittal and D. K. Sharma

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Study on a Free Boundary Problem Arising in Porous Media . . . . . . . . 113 Bhumika G. Choksi and Twinkle R. Singh Effect of Habitat on Dynamic of Native and Exotic Prey–Predator Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Namita Goel, Sudipa Chauhan and Sumit Kaur Bhatia On Cliques and Clique Chromatic Numbers in Line, Lict and Lictact Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Rashmi Jain and Anuj Kumar Jain Friendship-Like Graphs and It’s Classiffication . . . . . . . . . . . . . . . . . . . 145 K. Nageswara Rao, P. Shaini and K. A. Germina Chaotic Maps: Applications to Cryptography and Network Generation for the Graph Laplacian Quantum States . . . . . . . . . . . . . . 155 Anoopa Joshi and Atul Kumar Consumer Behaviour Analysis for Purchasing a Passenger Car in Indian Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Tanmay Agarwal, Nivedika Saroha and Girish Kumar A New Hybrid Model Based on Triple Exponential Smoothing and Fuzzy Time Series for Forecasting Seasonal Time Series . . . . . . . . 179 A. J. Saleena and C. Jessy John New Fuzzy Divergence Measure and Its Applications in Multi-criteria Decision-Making Using New Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Ram Naresh Saraswat and Adeeba Umar An SIRS Age-Structured Model for Vector-Borne Diseases with Infective Immigrants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Nisha Budhwar, Sunita Daniel and Vivek Kumar Numerical Study of Conformable Space and Time Fractional Fokker–Planck Equation via CFDT Method . . . . . . . . . . . . . . . . . . . . . 221 Brajesh Kumar Singh and Anil Kumar Multispectral Bayer Color Image Encryption . . . . . . . . . . . . . . . . . . . . . 235 Binay Kumar Singh and Jagat Singh Investigation of Prospective Elementary Teachers’ Opinions About Problem Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Danyal Soybaş and Sevim Sevgi

About the Editors

Naokant Deo is a professor at the Department of Applied Mathematics, Delhi Technological University, India. He completed his Ph.D. in Mathematics from Guru Ghasidas University, Bilaspur, India. His areas of research include approximation theory and real analysis. Professor Rao is a recipient of the CAS-TWAS Fellowship awarded by the Chinese Academy of Sciences, Beijing, China, and International Centre for Theoretical Physics, Trieste, Italy. He is an active member of academic bodies such as Indian Mathematical Society, India, Research Group in Mathematical Inequalities and Applications, Australia, and World Academy of Young Scientists, Hungary. His research papers have been published in national and international journals of repute. Vijay Gupta is a professor in the Department of Mathematics at the Netaji Subhas University of Technology, New Delhi, India. He holds a Ph.D. from the Indian Institute of Technology Roorkee (formerly, the University of Roorkee), and his area of research is approximation theory, with a focus on linear positive operators. The author of 5 books, 15 book chapters, and over 300 research papers, he is actively involved in editing over 25 international scientific research journals. Ana Maria Acu is a professor at the Department of Mathematics and Computer Science, Lucian Blaga University of Sibiu, Romania. She earned her Ph.D. in Mathematics from the Technical University of Cluj-Napoca, Romania. Professor Acu is an active member of various scientific organizations, editorial boards of scientific journals, and scientific committees, and her main research interest is approximation theory. P. N. Agrawal is a professor at the Department of Mathematics, Indian Institute of Technology Roorkee, India. He received his Ph.D. degree from the Indian Institute of Technology Kanpur, India, in 1980. Having published his research papers in various journals of repute, Prof. Agarwal has delivered invited lectures and presented papers at a number of international conferences in India and abroad. His research interests include approximation theory, numerical methods, and complex analysis.

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Exact Solution for Mixed Integral Equations by Method of Bernoulli Polynomials Mithilesh Singh, Nidhi Handa and Shivani Singhal

Abstract In this article, a new method has been developed for solving the mixed second kind Volterra–Fredholm integral equations numerically. A method is introduced in this paper is known as the Bernoulli matrix method. It is applied for solving mixed VFIE’s integral equations. The one property of this method is that it reduces the degree of the problem for solving a structure of algebraic equations. Our proposed method is introduced and it is applied to convert the integral equation into the algebraic equation using of Bernoulli matrix equation. Finally, there are some numerical results that have been given for illustrating the efficiency and exactness of this method. Keywords Bernoulli polynomial method · Linear Volterra–Fredholm Integral Equations of second kind

1 Introduction Integral equations have great significance in the area of Science. The concept of Integral equations is a well-known mathematical tool in both applied and pure Mathematics. There are several mathematicians and physicists, who have used integral equations in formulating boundary value problems of gravitation, electrostatics, fluid dynamics and scattering. They have various applications in mass and heat transfer, approximation theory, electrodynamics, queuing theory, electrical engineering, economics, etc. Many problems in the fields of ordinary and partial differential equations can be solved by integral equations [1–4]. The theory of linear Volterra–Fredholm integral equations come from parabolic boundary value problems, from the mathematical modelling of the spatio-temporal M. Singh Department of Applied Science, Rajkiya Engineering College, Sonbhadra, U.P., India N. Handa · S. Singhal (B) Department of Mathematics and Statistics, Gurukula Kangri Vishwavidyalaya, Haridwar, UK, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_1

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M. Singh et al.

development of an epidemic, different types of physical and biological models. Taylor series and the Adomian decomposition methods [5, 6], the modification of hat function methods [7] and other methods [8–10] have been used for solving linear Volterra–Fredholm integral equations. Many researchers such as Majeed and Omran [11] and Al-A’asam [12] used some numerical methods to solve LVFIE’s of the first and second kinds, namely the repeated Trapezoidal method, the repeated Simpson’s 1/3 method and Bernstein polynomials for deriving the modified Simpson’s 3/8 and the composite modified Simpson’s 3/8 rule to solve one-dimensional linear second kind Volterra integral equations. Al-Jarrah and Lin [13] used the scaling function interpolation method to solve linear Volterra–Fredholm integral equations (VFIE’s). Scaling functions and wavelet functions are the foundation of wavelet methods, which are a useful tool for solving the integral equation. Computational methods for Volterra–Fredholm integral equations are investigated in H˛acia [14]. Mohammadi [15] used a Chebyshev wavelet operational method for solving stochastic Volterra– Fredholm integral equations. Numerical solution of Fredholm–Volterra Linear Integral Equations using Lagrange Polynomials in Muna and Iman [16] and Bernstein polynomials [17] are employed. Bernoulli polynomials are useful in the scientific theory of number and in numerical and classical analysis. The job of Bernoulli polynomials is significant in different expansions and approximation formulae. These polynomials can be defined by lots of methods depending on the applications. Specifically, Bernoulli polynomials having six approaches to the theory are known; these are attached with Bernoulli [18], Euler [19], Appell [20], Lucas [21] and Lehmer [22]. There are some works that used Bernoulli polynomials as the basis for numerically solving integral, differential and integro-differential equations such as [23, 24]. Many authors applied the differential transform method to obtain the solution of a class of linear and non-linear two-dimensional Volterra integral equations. There are various specific properties and uses of Bernoulli polynomials in multiple areas of mathematics apart from the analytic theory of numbers to the classical and numerical analysis [22, 25, 26]. The numerical solutions of the linear and non-linear integral solutions are solved by Bernoulli polynomials [27]. Moreover, use of Bernoulli polynomials yields more accurate and exact solution of a given problem, which is one of the advantages. In this report, the numerical method has been used which is known as the Bernoulli polynomial method to solve mixed second kind Volterra–Fredholm integral equations. The first matrix representation of Bernoulli polynomials has been made which is the linear combination of its basis function. Then transform the integral equation into the structure of the algebraic equation, which gives the numerical result. Moreover, this method gives the exact result of the problems. Here the method of Bernoulli polynomial is used to solve Volterra–Fredholm mixed integral equations of the second kind x

b k1 (x, t)y(t)dt + λ2

y(x) = f (x) + λ1 a

k2 (x, t)y(t)dt, a

(1)

Exact Solution for Mixed Integral Equations by Method …

3

where a ≤ x ≤ b, λ1 and λ2 are real numbers f (x), k1 (x, t), and k2 (x, t) are continuous and analytic functions, respectively. Also, k1 (x, t) and k2 (x, t) are known as kernel of the integral equation, y(x) is unknown function which is to be determined.

2 Bernoulli Polynomial Method Polynomial is used in various problems of integral equations. They can be directly clarified, computed easily and symbolize huge class of functions. Bernoulli polynomials (see, for instance [22, 25, 26]) have special applications in number theory and classical analysis. They are seen in the integral form of the differentiable periodic functions since they are applied for approximating such functions in terms of polynomials. Some properties of Bernoulli polynomials are given as follows: 

Property 1 (Differentiation [25]), Bn (x) = n Bn−1 (x), n = 1, 2 . . . Property 2 (Integral means conditions [25]),

1

Bn (x)dx = 0, n = 1, 2 . . .

0

Property 3 (Differences [25]), Bn (x + 1) − Bn (x) = nx n−1 , n = 1, 2 . . .. The Bernoulli polynomial of degree n is defined by B0 (x) = 1, n    n Bn (x) = C Bk x n−k , n ≥ 1. k=0

(2)

k

3 Matrix Representation of Bernoulli Polynomial A matrix formulation of Bernoulli polynomial is to develop at a linear combination in terms of the dot product. A linear combination of Bernoulli basis function has been given below. B(t) = c0 B0 (t) + c1 B1 (t) + c2 B2 (t) + · · · + cn Bn (t).

(3)

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M. Singh et al.

The dot product of two vectors ⎡

⎤ c0 ⎢ c1 ⎥ ⎥ ⎢  ⎢ ⎥ B(t) = B0 (t) B1 (t) B2 (t) · · · Bn (t) ⎢ c2 ⎥, ⎢. ⎥ ⎣ .. ⎦

(4)

cn would be converted in the following pattern: ⎡

b00 ⎢0  ⎢ ⎢ B(t) = 1 t t 2 · · · t n ⎢ 0 ⎢. ⎣ .. 0

b01 b11 0 .. .

b02 b12 b22 .. .

0

0

··· ··· ··· .. .

⎤⎡ ⎤ c0 b0n ⎢ c1 ⎥ b1n ⎥ ⎥⎢ ⎥ ⎢ ⎥ b2n ⎥ ⎥⎢ c2 ⎥, ⎥ ⎥ .. ⎦⎢ ⎣ ... ⎦ .

· · · bnn

(5)

cn

where bnm are the coefficients of the power basis, which are applied to solve the respective Bernoulli polynomial. Here above matrix is upper triangular.

4 Method of Solution for Mixed Volterra–Fredholm Integral Equation of Second Kind Here, the Bernoulli polynomials method is applied to the scheme of mixed Volterra–Fredholm integral equations of the second kind for finding the solution of these equations. Take the mixed Volterra–Fredholm integral equations of the second kind (1) x

b k1 (x, t)y(t)dt + λ2

y(x) = f (x) + λ1 0

k2 (x, t)y(t)dt

(6)

a

Let us assume y(x) = Bi (t) then ⎡

⎤ c0 ⎢ c1 ⎥ ⎥  ⎢ ⎢ ⎥ y(x) = B0 (x) B1 (x) B2 (x) · · · Bn (x) ⎢ c2 ⎥ ⎢. ⎥ ⎣ .. ⎦ cn Substituting (7a) in (6) yields

(7a)

Exact Solution for Mixed Integral Equations by Method …

5

⎡

⎤ c0 ⎢ c1 ⎥ ⎥ ⎢  ⎢ ⎥ B0 (x) B1 (x) B2 (x) · · · Bn (x) ⎢ c2 ⎥ = f (x) ⎢. ⎥ ⎣ .. ⎦ cn

⎡

⎤ c0 ⎢ c1 ⎥ x ⎥ ⎢  ⎢ ⎥ + λ1 k1 (x, t) B0 (t) B1 (t) B2 (t) · · · Bn (t) ⎢ c2 ⎥dt ⎢. ⎥ ⎣ .. ⎦ a ⎡

cn

⎤ c0 ⎢ c1 ⎥ x ⎥ ⎢  ⎢ ⎥ + λ2 k2 (x, t) B0 (t) B1 (t) B2 (t) · · · Bn (t) ⎢ c2 ⎥dt. ⎢. ⎥ ⎣ .. ⎦ a

(7b)

cn Applying Eq. (5) into Eq. (7b), the matrix representation has ⎡

⎤ c0 ⎢ c1 ⎥ ⎥  ⎢ ⎢ ⎥ B0 (x) B1 (x) B2 (x) · · · Bn (x) ⎢ c2 ⎥ = f (x) ⎢. ⎥ ⎣ .. ⎦ cn ⎡ b00 ⎢ x 0  ⎢  ⎢ + λ1 k1 (x, t) 1 t t 2 · · · t n ⎢ 0 ⎢. ⎣ .. a ⎡

0

b00 ⎢0 b ⎢  ⎢ + λ2 k2 (x, t) 1 t t 2 · · · t n ⎢ 0 ⎢. ⎣ .. a 0

⎤⎡ ⎤ c0 b0n ⎢ c1 ⎥ b1n ⎥ ⎥⎢ ⎥ ⎢ ⎥ b2n ⎥ ⎥⎢ c2 ⎥dt ⎢. ⎥ .. ⎥ . ⎦⎣ .. ⎦

b01 b11 0 .. .

b02 b12 b22 .. .

··· ··· ··· .. .

0

0

· · · bnn

b01 b11 0 .. .

b02 b12 b22 .. .

··· ··· ··· .. .

0

0

⎤⎡

cn

⎤ c0 b0n ⎢ ⎥ b1n ⎥ ⎥⎢ c1 ⎥ ⎥ ⎥ b2n ⎥⎢ ⎢ c2 ⎥dt. ⎢ ⎥ .. ⎦⎣ .. ⎥ . . ⎦

· · · bnn

(7c)

cn

Now to calculate all integration of equation and the value of c0 , c1 , c2 , . . . cn are to be determined. For this choose xi , i = 1, 2 . . . n in [a, b]. Solve these n equations. The succeeding algorithm summarizes the stages for getting the output for mixed Volterra–Fredholm integral equation.

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5 Algorithm of Bernoulli Polynomial Method Input: ( f (x), k(x, t), y(x), a, b, x, λ1 , λ2 ). Stage 1: Choose n the degree of Bernoulli polynomials

Bi (t) =

n    n C Bk t n−k For i = 0, 1, 2 . . . n. k=0

(8)

k

Stage 2: Set the Bernoulli polynomials in second kind Volterra–Fredholm linear mixed integral equations ⎡

⎤ c0 ⎢ c1 ⎥ x ⎥ ⎢  ⎢ ⎥ y(x) = f (x) + λ1 k1 (x, t) B0 (t) B1 (t) B2 (t) · · · Bn (t) ⎢ c2 ⎥dt ⎢. ⎥ ⎣ .. ⎦ a ⎡

b + λ2 a

⎤

cn

c0 ⎢ c1 ⎥ ⎥ ⎢  ⎢ ⎥ k2 (x, t) B0 (t) B1 (t) B2 (t) · · · Bn (t) ⎢ c2 ⎥dt. ⎢. ⎥ ⎣ .. ⎦ cn

(9)

Stage 3: Compute Volterra integral ⎡

b00 ⎢ x 0  ⎢  ⎢ λ1 k1 (x, t) 1 t t 2 · · · t n ⎢ 0 ⎢. ⎣ .. a 0

··· ··· ··· .. .

⎤⎡ ⎤ c0 b0n ⎢ c1 ⎥ b1n ⎥ ⎥⎢ ⎥ ⎢ ⎥ b2n ⎥ ⎥⎢ c2 ⎥dt. ⎢. ⎥ .. ⎥ . ⎦⎣ .. ⎦

b01 b11 0 .. .

b02 b12 b22 .. .

0

0

· · · bnn

b01 b11 0 .. .

b02 b12 b22 .. .

··· ··· ··· .. .

0

0

(10)

cn

Compute Fredholm integral ⎡

b00 ⎢ 0 b ⎢  ⎢0 2 n λ2 k2 (x, t) 1 t t · · · t ⎢ ⎢. ⎣ .. a 0

⎤⎡ ⎤ c0 b0n ⎢ ⎥ b1n ⎥⎢ c1 ⎥ ⎥ ⎢ ⎥ b2n ⎥ ⎥⎢ c2 ⎥dt. ⎢. ⎥ .. ⎥ . ⎦⎣ .. ⎦

· · · bnn

cn

(11)

Exact Solution for Mixed Integral Equations by Method …

7

Stage 4: Compute c0 , c1 , c2 , . . . cn , xi , i = 1, 2 . . . n, xi ∈ [a, b]. Stage 5: Put the value of c0 , c1 , c2 , . . . cn in Eq. (9), we get the result of the integral equation.

6 Numerical Examples In this part, apply the process of Bernoulli polynomial to solve Volterra–Fredholm integral linear mixed equations of the second kind. Also exemplify the process of Bernoulli polynomial, both examples were solved by modified adomian decomposition method and series solution method, respectively [5]. Solving the below two examples in accordance with given algorithm, we get the exact solutions of the numerical result. Example 1 Assume the linear mixed Volterra–Fredholm integral equation of the second kind [5]. For n = 2. x y(x) = 3x + 4x − x − x − 2 + 2

3

1 t y(t)dt +

4

0

t y(t)dt

(12)

−1

And the exact solution is y(x) = 3x + 4x 2 For n = 2, first establish the Bernoulli polynomials in Eq. (12) according to Eq. (9). Find the Bernoulli coefficient matrix which has been obtained with the support of Eq. (5). Now, Compute Volterra integral x 0

⎡ ⎤⎡ ⎤ b00 b01 b02 c0  2 ⎣ ⎦ ⎣ xt 1 t t b10 b11 b12 c1 ⎦dt b20 b21 b22 c2 

and Fredholm integral x 0

⎡ ⎤⎡ ⎤ b00 b01 b02 c0  xt 1 t t 2 ⎣ b10 b11 b12 ⎦⎣ c1 ⎦dt b20 b21 b22 c2 

After calculating above integral, we have got the value of c0 , c1 , c2 . After substituting the value of c0 , c1 , c2 , we have attained the exact solution of Eq. (12).

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Example 2 Consider the linear Volterra–Fredholm integral equation [5] x y(x) = 2 − x − x − 6x + x + 2

3

1 t y(t)dt +

5

0

(x + t)y(t)dt

(13)

−1

and the exact solution of Eq. (13) is y(x) = 2 + 3x − 5x 3 . The proposed method is applied in Eq. (13) for n = 2 to get the exact solution. By processing the algorithm of Bernoulli polynomial method, exact solutions of above these examples have been obtained.

7 Conclusion and Result The desired outcome of our task is using the Bernoulli Polynomial method for solving linear Fredholm–Volterra integral equations. It is clear that solving of integral equation analytically is usually difficult. The Bernoulli polynomial method converts the integral equation into an algebraic equation by helping matrix equations of Bernoulli polynomials. In this paper, two examples have been solved by this method. It can be easily seen that we get exact solutions after applying this method. Furthermore, it gives an exact solution for less value of n in comparison with any other method. A comparison with other procedures reveals that the Bernoulli polynomial method is too efficient and finer than any other method.

References 1. A.K. Borzabadi, A.V. Kamyad, H.H. Mehne, A different approach for solving the nonlinear Fredholm integral equations of the second kind. Appl. Math. Comput. 173, 724–735 (2006) 2. E. Babolian, F. Fattahzadeh, E. Golpar Raboky, A Chebyshev approximation for solving nonlinear integral equations of Hammerstein type. Appl. Math. Comput. 189, 641–646 (2007) 3. R.P. Agarwal, Boundary value problems for higher order integro-differential equations. Nonlin. Anal. Theory Methods Appl. 9, 259–270 (1983) 4. S. Youse, M. Razzaghi, Legendre wavelet method for the nonlinear Volterra-Fredholm integral equations. Math. Comput. Simul. 70, 1–8 (2005) 5. A.M. Wazwaz, “Linear and Nonlinear Integral Equations”: Methods and Applications (Springer, Saint Xavier University Chicago, USA, 2011) 6. K. Maleknejad, M. Hadizadeh, A new computational method for Volterra-Fredholm integral equations. Comput. Math Appl. 37, 18 (1999) 7. F. Mirzaee , E. Hadadiyan, Numerical Solution of Volterra–Fredholm integral equations via modification of hat functions. Appl. Math. Comput. 280, 110–123 (2016) 8. J.P. Kauthen, Continuous time collocation methods for Volterra-Fredholm integral equations. Numer. Math. 56, 409–424 (1989) 9. E. Yusufoglu, E. Erbas, Numerical expansion methods for solving Fredholm-Volterra type linear integral equations by interpolation and quadrature rules. Kybernetes 37(6), 768–785 (2008)

Exact Solution for Mixed Integral Equations by Method …

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10. M.A. Abdou, F.A. Salama, Volterra-Fredholm integral equation of the first kind and spectral relationships. J. Appl. Math. Comput. 153, 141–153 (2004) 11. S.J. Majeed, H.H. Omran, Numerical methods for solving linear Volterra-Fredholm integral equations. J. Al-Nahrain Univ. 11(3), 131–134 (2008) 12. J.A. Al-A’asam, Deriving the composite Simpson rule by using Bernstein polynomials for solving Volterra integral equations. Baghdad Sci. J. 11(3) (2014) 13. Y. Al-Jarrah, E.B. Lin, Numerical solution of Fredholm-Volterra integral equations by using scaling function interpolation method. Appl. Math. 4, 204–209 (2013) 14. L. H˛acia, Computational methods for Volterra-Fredholm integral equations. Comput. Methods Sci. Technol. 8(2), 13–26 (2002) 15. F. Mohammadi, A Chebyshev wavelet operational method for solving stochastic VolterraFredholm integral equations. Int. J. Appl. Math. Res. 4(2), 217–227 (2015) 16. M.M. Mustafa, I.N. Ghanim, Numerical solution of linear Volterra-Fredholm integral equations using lagrange polynomials. Math. Theor. Model. 4(5) (2014) 17. M.K. Shahooth, Numerical solution for solving mixed Volterra-Fredholm integral equations of second kind by using. Bernstein Polynomials AIP Adv. 7, 125123 (2017) 18. J. Bernoulli, Ars conjectandi, Basel, (1713), posthumously published, p. 97 19. L. Euler, Methodus generalis summandi progressiones. Comment. Acad. Sci. Petrop. 6(1738) 20. P.E. Appell, Sur une classe de polynomes. Annales d’ecole normale superieur,s. 2, 9 (1882) 21. E. Lucas, Th´eorie des Nombres, Paris (1891) (Chapter 1) 22. D.H. Lehmer, A new approach to Bernoulli polynomials. Am. Math. Month. 95, 905–911 (1988) 23. E. Tohidi, A.H. Bhrawy, K. Erfani, A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. Appl. Math. Model. 37(6), 4283–4294 (2013) 24. F. Toutounian, E. Tohidi, A new Bernoulli matrix method for solving second order linear partial differential equations with the convergence analysis. Appl. Math. Comput. 223, 298–310 (2013) 25. F.A. Costabile, F. Dell’ Accio, Expansions over a rectangle of real functions in Bernoulli polynomials and applications. BIT Numer. Math. 41, 451–464 (2001) 26. P. Natalini, A. Bernaridini, A generalization of the Bernoulli polynomials. J. Appl. Math. 3, 155–163 (2003) 27. S. Bazam, Bernoulli polynomials for the numerical solution of same classes of linear and non-linear integral equations. J. Comput. Appl. Math. (2014)

Turing Patterns in a Cross Diffusive System Nishith Mohan and Nitu Kumari

Abstract In this paper we investigate the role of cross diffusion in pattern formation for a tritrophic food chain model. In the formulated model the prey interacts with the mid level predator in accordance with Holling Type II functional response and the mid and top level predator interact via Crowley Martin functional response. We have proved that the stationary uniform solution of the system is stable in the presence of diffusion and absence of cross diffusion but unstable in the presence of cross diffusion. Moreover we carry out numerical simulations to understand the Turing pattern formation for various self and cross diffusivity coefficients of the top level predator.

1 Introduction In spatio temporal food chain model [1–11] predators tend to develop migratory strategies to take advantage over prey. The migratory movement of a predator is influenced by movement of its prey on which it predates. Classical, diffusion based food chain models [12–14] are inadequate to discuss this phenomenon as they take into account movements motivated by self characteristics of a species i.e. self diffusion. In order to model such a scenario we need to incorporate the role of cross diffusion along with self diffusion as in [17–20, 23]. Self diffusion represents the movement of species from a region of its higher concentration to a region of its lower concentration where as cross diffusion is the expression of population fluxes of one species due to the influence of other. The self diffusivity which is indicative of self tendency of a species to move is always positive but cross diffusivity which represents movement under the influence of other may be positive or negative. Positive N. Mohan · N. Kumari (B) School of Basic Sciences, Indian Institute of Technology Mandi, Mandi 175001, Himachal Pradesh, India e-mail: [email protected] N. Mohan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_2

11

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N. Mohan and N. Kumari

cross diffusivity denotes movement of a particular species in the direction of lower concentration of another species. Negative cross diffusivity represents the movement of one species in the direction of higher concentration of another. In this work we develop a cross diffusive tritrophic model to study the influence of cross diffusion on the stability of food chain. The phenomenon of pattern formation is frequently observed in spatiotemporal prey predator interactive systems. Patterns may be defined as ordered outcome of random phenomenon. Many authors in literature have investigated the role of self diffusion on pattern formation as in [15, 16] (and the references there in), but the role of cross diffusion which constitute a system of strongly coupled parabolic partial differential equations remains less explored. In literature [23] investigated the role of cross diffusion but for a model system based on classical Leslie Grower functional response. In this work we study cross diffusion induced pattern formation on a food chain in which prey and mid level predator interact via Holling type functional response and mid and top level predator interact in accordance with Crowley Martin functional response. In [24] pattern formation for such a model was studied but with perturbations of trigonometric functions which had linear arguments. In the present work we take a more complex perturbations and extend the study. In Sect. 2 we first present the proposed model system. We later extend this model system and include nonlinear cross diffusion at the third or top trophic level. In Sect. 3 stability analysis of both temporal and spatiotemporal model is carried out. After we proved that the positive equilibrium point is local asymptotically stable we have shown that the system is stable under the effect of diffusion and absence of cross diffusion but instable in the presence of both diffusion and cross diffusion. In Sect. 4 we perform a numerical simulation of the cross diffusive system and take into account Turing pattern formation for various values of self and cross diffusion coefficients of the top level predator.

2 Model Formulation In the present work we take into account a tritrophic food chain model with Crowley Martin functional response. The prey U1 (T ) and mid level predator U2 (T ) interact via Holling type–II functional response and the mid level predator and top level predator U3 (T ) interact in accordance with Crowley Martin functional response. The temporal formulation, defined by a system of ordinary differential equations is given below:   U1 wU1 U2 dU1 = a1 U 1 1 − − dT K U1 + D dU2 w1 U1 U2 w2 U2 U3 2 = −a2 U2 − EU2 + − dT U 1 + D1 1 + dU2 + bU3 + bdU2 U3 dU3 w3 U2 U3 = −cU3 + dT 1 + dU2 + bU3 + bdU2 U3

(1)

Turing Patterns in a Cross Diffusive System

13

Here, the prey U1 has an intrinsic growth rate of a1 , and carrying capacity K in the absence of predator U2 , D measures the extent to which environment can protect U1 from U2 , w describes maximum value, which per capita reduction rate of U1 can attain. As stated earlier U2 predates on U1 in accordance with Holling Type II functional response. The intermediate predator U2 has a natural death rate of a2 , w1 describes maximum rate at which U2 can predate over U1 , D1 measures the extent to which the environment can provide protection to U2 and E represents the internal competition coefficient among the members of U2 . The predation of U2 over U1 is governed by Holling Type II functional response and the top level predator U3 predates on U2 in accordance with Crowley Martin functional response. The constants w2 , b and d are the parameters that describe the effects of capture rate, handling time and magnitude of interference among predators on the feeding rate in Crowley Martin functional response. The top level predator U3 dies at a natural death rate of c. w3 is the saturating Crowley Martin type functional response parameter similar to w2 . All the parameters described above assume only positive values and the model system is a three species food chain model involving a hybrid type of prey dependent and predator dependent functional responses. The model system described by (1) consists of 13 parameters, which makes mathematical analysis quite complex, therefore we reduce the number of parameters by rescaling the model system. The model is rescaled, using the following variables and parameters: t = a1 T, u 1 = w8 =

U1 wU2 ww2 U3 D a2 w1 D1 , u2 = , u3 = 2 , , w6 = , w7 = , w4 = , w5 = K a1 K K a1 a1 K a1 d K

a 2 bd K Ew a1 b w c w3 , w11 = , e= 2 , w9 = 1 , w10 = , w12 = w2 ww2 a1 d K a1 a1 d a1 K

(2)

The rescaled system is given as follows [24]:   du 1 u2 = u1 1 − u1 − dt u 1 + w4   du 2 w6 u 1 u3 = u 2 −w5 + − eu 2 − dt u 1 + w7 u 2 + (w8 + w9 u 2 ) r − u 3 + w10   du 3 w12 u 2 = u 3 −w11 + dt u 2 + (w8 + w9 u 2 ) r − u 3 + w10

(3)

Both (1) and the rescaled form in (3) assumes a homogeneous distribution of species in space, but in nature species distribution is always inhomogeneous. A realistic food chain scenario describing the above prey predator interaction can be modeled by spatially extending the system (3) using reaction diffusion mechanism.

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At any location (x, y) and time t, the interaction of three species populations namely u 1 (x, y, t), u 2 (x, y, t) and u 3 (x, y, t) can be modeled with the reaction diffusion equation given by (4),  where Δ denotes two dimensional Laplacian operator  ∂2 ∂2 given by Δ ≡ 2 + 2 , where x, y ∈ Ω and t > 0. Ω is a bounded domain ∂x ∂y in R2 with a smooth boundary ∂Ω. ∂u 1 ∂t ∂u 2 ∂t ∂u 3 ∂t ∂u 1 ∂n

  u2 − d1 Δu 1 = u 1 1 − u 1 − u 1 + w4   w6 u 1 u3 − d2 Δu 2 = u 2 −w5 + − eu 2 − u 1 + w7 u 2 + (w8 + w9 u 2 ) u 3 + w10 (4)   w12 u 2 − d3 Δu 3 = u 3 −w11 + u 2 + (w8 + w9 u 2 ) u 3 + w10 ∂u 2 ∂u 3 = = =0 ∂n ∂n

∂u 2 ∂u 3 ∂u 1 = = = 0 are the homogeneous Neumann The boundary conditions ∂n ∂n ∂n boundary conditions or zero flux boundary conditions (n is the outward normal to Ω). The model system discussed in (4), assumes that the motilities or movement of the species in concern is governed solely by their own characteristics. In system (4) the movements of the species can be physically affected only by the population pressures arising out of the mutual interference between the individuals of the same species. The diffusivities d1 , d2 and d3 involved in (4) are referred to as self diffusion rates of the species u 1 , u 2 and u 3 respectively. However, in case of a model system describing spatiotemporal prey predator interactions motility of one species effects the motility of the other. The system described in (4) do not incorporate this effect and is inadequate to describe such species interaction. The predators have a tendency of developing migratory strategies to take an advantage over the prey. We, now include the effect of cross diffusion to describe such a situation. In addition of the self tendency to move i.e. self diffusion, the species also migrate or move under the influence of other. Such behavior considers the concentration of both the predator species i.e., u 2 , the mid level predator and u 3 , top level predator—constituting a cross diffusive system. The cross diffusive coefficient may be positive or negative. The positive cross diffusion coefficient represents that one species tends to move in the direction of lower concentration of another species. The negative cross diffusion terms represents that the population flux of one species is in the direction of higher concentration of the other. Now, we propose the interaction of the above three species population model with cross diffusion in the following form:

Turing Patterns in a Cross Diffusive System

15

  u2 − d1 Δu 1 = u 1 1 − u 1 − u 1 + w4   w6 u 1 u3 − d2 Δu 2 = u 2 −w5 + − eu 2 − u 1 + w7 u 2 + (w8 + w9 u 2 ) u 3 + w10 (5)   w12 u 2 − d3 Δ (u 3 + d4 u 2 u 3 ) = u 3 −w11 + u 2 + (w8 + w9 u 2 ) u 3 + w10 ∂u 2 ∂u 3 = = = 0. ∂n ∂n   ∂2 ∂2 where, Δ denotes two dimensional Laplacian Δ ≡ 2 + 2 , (x, y) ∈ Ω. ∂x ∂y The nonlinear diffusion terms in the equation governing the dynamics of top level predator u 3 implies that the direction of dispersion of top predator contains a self diffusion term by which it moves from a region of its higher concentration to a region of its lower concentration but also a cross diffusive term. The top predator u 3 diffuses with the flux ∂u 1 ∂t ∂u 2 ∂t ∂u 3 ∂t ∂u 1 ∂n

J = −∇(d3 u 3 + d3 d4 u 2 u 3 ) = −d3 d4 u 3 ∇u 2 − (d3 + d3 d4 u 3 )∇u 3 , where, −d3 d4 ∇u 3 < 0, the −d3 d4 r ∇u 2 part of the flux J is directed towards the decreasing population density of the mid level predator u 2 . This behavior can be justified by the fact that in various prey predator interacting systems, the prey tend to form groups as a protective measure against predation. As, in the proposed model system the predation of u 3 is impossible, therefore to extensively study cross diffusion induced instability in the system, we introduce it at the third or top trophic level.

3 Stability Analysis 3.1 Temporal Model In this section we assume that model system (3) has the unique positive stationary uniform solution, we denote it by u∗ = (u ∗1 , u ∗2 , u ∗3 ) and derive the conditions under which it is locally asymptotically stable. Theorem 1 The stationary uniform solution u∗ = (u ∗1 , u ∗2 , u ∗2 ) of (3) is locally asymptotically stable if the parameters satisfy, u ∗2 u ∗3 (1 + w9 u ∗2 ) u ∗2 < e & < 1.  ∗    2 (u ∗1 + w4 )2 u 2 + w8 + w9 u ∗2 u ∗3 + w10 Proof Through out the paper we denote,

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N. Mohan and N. Kumari ⎡

u 1 g1 (u) ⎡ ⎤ ⎢ ⎢ G 1 (u) ⎢ ⎢ G(u) = ⎣G 2 (u)⎦ = ⎢ u 2 g2 (u) ⎢ ⎢ G 3 (u) ⎣ u 3 g3 (u)

  ⎤ u2 ∼ = u1 1 − u1 − ⎥ u 1 + w4  ⎥ ⎥ w6 u 1 u2 ⎥ ∼ − eu 2 − ⎥ = u 2 −w5 + u 1 + w7 u 2 + (w8 + w9 u 2 ) u 3 + w10 ⎥ ⎥   ⎦ w12 u 2 ∼ = u 3 −w11 + u 2 + (w8 + w9 u 2 ) u 3 + w10

Calculating Gu (u∗ ), and putting, Gu (u∗ ) = 0.

(6)

⎤ u ∗1 u ∗2 −u ∗1 ∗ 0 ∗ ⎥ ⎢ (u ∗ + w4 )2 − u 1 u 1 + w4 ⎥ ⎢ 1 ⎥ ⎢ w6 w7 u ∗2 u ∗3 u ∗2 (1 + w9 u ∗3 ) u ∗2 (u ∗2 + w10 ) ∗ ⎥ ⎢ ∗ −eu + −         2 Gu (u ) = ⎢ (u ∗ + w )2 2 2⎥ ∗ ∗ ∗ ∗ ∗ ∗ 7 u 2 + w8 + w9 u 2 u 3 + w10 u 2 + w8 + w9 u 2 u 3 + w10 ⎥ ⎢ 1 ⎥ ⎢ ∗ ∗ ∗ ∗ ∗ −w12 u 2 u 3 (w8 + w9 u 2 ) u 3 w12 (w8 u 3 + w10 ) ⎦ ⎣ 0  ∗   ∗ 2  ∗   ∗ 2 ∗ ∗ u 2 + w8 + w9 u 2 u 3 + w10 u 2 + w8 + w9 u 2 u 3 + w10 ⎡

The characteristic polynomial of Gu (u∗ ) is

(7)

ρ(λ) = λ3 + H1 λ2 + H2 λ + H3 where,     w12 u ∗2 u ∗3 w8 + w9 u ∗2 u ∗3 u ∗2 (1 + w9 u ∗3 ) ∗ H1 =   2 + u 2 e −  ∗   2  u ∗2 + w8 + w9 u ∗2 u ∗3 + w10 u 2 + w8 + w9 u ∗2 u ∗3 + w10   u∗ + u ∗1 1 − ∗ 2 2 (u 1 + w4 )  H2 = u ∗1 1 −

u ∗2 (u 1 + w4 )2



  w12 u ∗2 u ∗3 (w8 + w9 u ∗2 ) u ∗3 u ∗2 (1 + w9 u ∗3 ) ∗ e − + u  ∗   2  ∗   2 2 u 2 + w8 + w9 u ∗2 u ∗3 + w10 u 2 + w8 + w9 u ∗2 u ∗3 + w10

∗ w6 w7 u ∗1 u ∗2 u ∗ u ∗ w12 (u ∗ + w10 )(w8 u ∗3 + w10 ) w12 u ∗2 2 u 3 (w8 + w9 ) + 2 3  2  2 +  ∗   2 (u ∗1 + w4 )(u ∗1 + w7 )2 u ∗2 + w8 + w9 u ∗2 u ∗3 + w10 u 2 + w8 + w9 u ∗2 u ∗3 + w10   u ∗3 u ∗2 (1 + w9 u ∗3 ) × e−   2  ∗ u 2 + w8 + w9 u ∗2 u ∗3 + w10

+

H3 =

 ∗ ∗ ∗ ∗ w6 w7 u ∗1 u ∗2 u 2 u 3 w12 (u ∗2 + w10 )(w8 u ∗3 + w10 ) 2 u 3 (w8 + w9 u 3 ) +  ∗     ∗   2 2 ∗ ∗ 2 (u 1 + w4 )(u 1 + w7 ) u 2 + w8 + w9 u 2 u 3 + w10 u 2 + w8 + w9 u ∗2 u ∗3 + w10    ∗ w12 u ∗2 u ∗3 u ∗2 (1 + w9 u ∗3 ) 2 u 3 (w8 + w9 ) +  e −  2  ∗   2  u ∗2 + w8 + w9 u ∗2 u ∗3 + w10 u 2 + w8 + w9 u ∗2 u ∗3 + w10     u∗ × u ∗1 1 − ∗ 2 2 (u 1 + w4 )

Turing Patterns in a Cross Diffusive System

17

By, using the criteria stated in Theorem 1 it is easy to verify that H1 , H2 , H3 > 0 and it has been verified that H1 H2 − H3 > 0. Therefore, it follows from the Routh– Hurwitz criteria that, the three roots λ1 , λ2 , λ3 of ρ(λ) = 0 have negative real parts. Hence, the stationary uniform solution u∗ of (3) is locally asymptotically stable under the stated condition.

3.2 Model with Self Diffusion In order to discuss the local asymptotically stability of the system of parabolic Eqs. 4 and 5 we lay down the following notation as in [19]. Notation 1 Let 0 = μ1 < μ2 < · · · → ∞ be the eigenvalues of −Δ on Ω under noflux boundary conditions, and E (μi ) be the space of eigenfunctions corresponding to μi . We define the following space decomposition     (i) Xij := c . φi j : c ∈ R3 where φi j are orthonormal basis of E (μi ) for j = 1, . . . ,  dimE (μi ).    ∞ ¯ 3 : ∂n u 1 = ∂n u 2 = ∂n u 3 = 0 on ∂Ω , and so X = i=1 (ii) Xij := u ∈ C 1 (Ω) dimE(μi ) Xi , where Xi = Xij . j=1 Theorem 2 The stationary uniform solution u∗ of (4) is locally asymptotically stable if, u ∗3 (1 + w9 u ∗3 ) u ∗2 < e & 0 and through calculation it has been verified that A1 A2 − A3 > 0. Therefore, it follows from the Routh–Hurwitz criteria that, for each i ≥ 1, the three roots λi,1 , λi,2 , λi,3 of ψi (λ) = 0 all have negative real parts. Hence, the stationary uniform solution u∗ of (4) is locally asymptotically stable under the stated condition. From Theorem 2, it is clear that on adding the self diffusion terms to the temporal model system (3) results in stable positive stationary uniform solution under the stated condition, therefore diffusion driven instability has not yet occurred. Therefore, we will now consider the effect of cross diffusion on the model system.

Turing Patterns in a Cross Diffusive System

19

3.3 Model with Self Diffusion and Cross Diffusion We now consider system (5), in detail. Theorem 3 Assuming d4 > 0 and the following condition holds, 

u ∗3 (1 + w9 u ∗3 ) e−    2 u ∗2 + w8 + w9 u ∗2 u ∗3 + w10

 1−

d4 u ∗3 (u ∗3 + w10 ) <    2 (1 + d4 u ∗2 ) u ∗2 + w8 + w9 u ∗2 u ∗3 + w10



u ∗2



(u ∗1 + w4 )2

1−

w6 w7 (u ∗1 + w4 )(u ∗1 + w7 )2 

+

u ∗2 ∗ (u 1 + w4 )2

if μ2 < μ, ˜ where μ2 is as explained in Notation 1 and  μ is given by (9). Then there exists a positive constant d3∗ . such that when d3 ≥ d3∗ , the stationary uniform solution u∗ of the cross diffusive system (5) is unstable. Proof We denote Φ(u) = (d1 u 1 , d2 u 2 , d3 u 3 (1 + d4 u 2 ))T . Linearizing system (5) at u∗ , we have   ut = Φu Δ + Gu (u∗ ) u, where,

⎡

⎤ d1 0 0 ⎦ 0 Φu = ⎣ 0 d2 0 d3 d4 u ∗3 d3 + d3 d4 u ∗2

The characteristic polynomial of −μi Φu + Gu (u∗ ) is given by, ψλ = λ3 + B1 λ2 + B2 λ + B3 where, w12 u ∗2 u ∗3 (w8 + w9 u ∗2 ) B1 = μi (d1 + d2 + d3 ) + μi d3 d4 u ∗2 +   2  u ∗2 + w8 + w9 u ∗2 u ∗3 + w10     ∗ ∗ ∗ u ∗2 u (1 + w u ) u 9 3 2 3 ∗ ∗ + u2 e −   2 + u 1 1 − ∗  (u 1 + w4 )2 u ∗2 + w8 + w9 u ∗2 u ∗3 + w10

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 w12 u ∗2 u ∗3 (w8 + w9 u ∗2 ) B2 = μi (d2 + d3 ) + μi d3 d4 u ∗2 +   2  ∗ u 2 + w8 + w9 u ∗2 u ∗3 + w10      u ∗3 u ∗2 (1 + w9 u ∗3 ) u ∗2 ∗ ∗ + u2 e −  d + u 1 − × μ i 1    1 2 (u ∗1 + w4 )2 u ∗2 + w8 + w9 u ∗2 u ∗3 + w10 ∗ ∗ w12 eu ∗2 2 u 3 (w8 + w9 u 2 ) + μi2 d2 d3 (1 + d4 u ∗2 ) + μi d3 eu ∗2 (1 + d4 u ∗2 ) +   2  u ∗2 + w8 + w9 u ∗2 u ∗3 + w10

μi d2 w12 u ∗2 u ∗3 (w8 + w9 u ∗2 ) w12 u ∗ u ∗ (u ∗ + w10 )(w8 u ∗3 + w10 ) +  2 3 2    4  2 u ∗2 + w8 + w9 u ∗2 u ∗3 + w10 u ∗2 + w8 + w9 u ∗3 u ∗2 + w10   μi d3 d4 u ∗3 u ∗2 (u ∗2 + w10 ) μi d3 u ∗3 u ∗2 (1 + w9 u ∗3 ) ∗ 1 + d − − u 4     2   2 2 u ∗2 + w8 + w9 u ∗2 u ∗3 + w10 u ∗2 + w8 + w9 u ∗2 u ∗3 + w10 +

−

B3 =

∗2 ∗ ∗ w12 u ∗2 2 u 3 (1 + w9 u 3 )(w8 + w9 u 2 )  ∗   ∗ 4 ∗ u 2 + w8 + w9 u 3 u 2 + w10

∗ ∗ μi d3 w6 w7 u ∗1 u ∗2 w6 w7 u ∗1 u ∗2 2 u 3 (w8 + w9 u 2 ) (1 + d4 u ∗2 ) +   2  ∗ 2 + w4 )(u 1 + w7 ) (u ∗1 + w4 )(u ∗1 + w7 )2 u ∗2 + w8 + w9 u ∗2 u ∗3 + w10     u∗ μi d1 + u ∗1 1 − ∗ 2 2 × μi2 d2 d3 (1 + d4 u ∗2 ) + μi d3 eu ∗2 (1 + d4 u ∗2 ) (u 1 + w4 )

(u ∗1

+

u ∗2

∗ ∗ w12 eu ∗2 μi d2 w12 u ∗2 u ∗3 (w8 + w9 u ∗2 ) 2 u 3 (w8 + w9 u 2 )  ∗ 2 +  ∗   2  ∗ u 2 + w8 + w9 u ∗2 u ∗3 + w10 + w8 + w9 u 2 u 3 + w10

w12 u ∗ u ∗ (u ∗ + w10 )(w8 u ∗3 + w10 ) μi d3 d4 u ∗3 u ∗2 (u ∗2 + w10 ) +  2 3 2  ∗ 4 −  ∗   2 ∗ ∗ u 2 + w8 + w9 u 3 u 2 + w10 u 2 + w8 + w9 u ∗2 u ∗3 + w10 −

  w12 u ∗2 u ∗2 (1 + w9 u ∗3 )(w8 + w9 u ∗2 ) μi d3 u ∗3 u ∗2 (1 + w9 u ∗3 ) 1 + d4 u ∗2 −  2 3    4  2 u ∗2 + w8 + w9 u ∗2 u ∗3 + w10 u ∗2 + w8 + w9 u ∗3 u ∗2 + w10

Let λ1 (μi ), λ2 (μi ), λ3 (μi ) be the three roots of ψ(λ) = 0, then, λ1 (μi ).λ2 (μi ).λ3 (μi ) = −B3 . For at least one Re(λi (μi )) > 0, it is sufficient to show that B3 < 0. Also, B3 = det(μi φi − Gu (u∗ )) Hence, we have, B3 = Q 3 μi3 + Q 2 μi2 + Q 1 μi + Q 0 where, Q 3 = d1 d2 d3 (1 + d3 u ∗2 )



Turing Patterns in a Cross Diffusive System

21

  u ∗3 u ∗2 (1 + w9 u ∗3 ) d1 d2 w12 (w8 + w9 u ∗2 )u ∗2 u ∗3 Q 2 = d1 d3 (1 + d4 u ∗2 )u ∗2 e −   ∗ 2 +  ∗   2  ∗ ∗ u 2 + w8 + w9 u 2 u 3 + w10 u 2 + w8 + w9 u ∗2 u ∗3 + w10   d1 d3 d4 u ∗3 u ∗2 (u ∗2 + w10 ) u ∗1 u ∗2 ∗ ∗ −  ∗ 2 + d2 d3 (1 + d4 u 2 ) u 1 −  ∗ ∗ (u 1 + w4 )2 u 2 + w8 + w9 u 2 u 3 + w10 

u ∗3 u ∗2 (1 + w9 u ∗3 )   2 ∗ u 2 + w8 + w9 u ∗2 u ∗3 + w10



w − 12u ∗2 u ∗3 (w8 + w9 u ∗2 )   2 + w8 + w9 u ∗2 u ∗3 + w10   d2 w12 u ∗2 u ∗3 (w8 + w9 u ∗2 ) u ∗1 u ∗2 d1 w12 u ∗2 u ∗3 (u ∗2 + w10 )(w8 u ∗3 + w10 ) u ∗1 − +   ∗      4 2 2 (u 1 + w4 ) u 2 + w8 + w9 u ∗3 u ∗2 + w10 u ∗2 + w8 + w9 u ∗2 u ∗3 + w10    ∗ ∗ ∗ ∗ ∗ u 3 u 2 (1 + w9 u 3 ) u1 u2 d3 (1 + d4 u ∗2 ) e −  u ∗1 −   ∗ 2 ∗ ∗ (u + w4 )2 1 u 2 + w8 + w9 u 2 u 3 + w10   u ∗1 u ∗2 w6 w7 u ∗1 u ∗2 d3 d4 u ∗3 u ∗2 (u ∗2 + w10 ) ∗ + d3 (1 + d4 u ∗2 ) −   ∗ 2 u 1 − 2 ∗ ∗ (u + w ) (u + w4 )(u 1 + w7 )2 1 4 1 u 2 + w8 + w9 u 2 u 3 + w10

Q 1 = d1 u ∗2 e − 



u ∗2

  w12 u ∗2 u ∗3 (w8 + w9 u ∗2 ) u ∗3 u ∗2 (1 + w9 u ∗3 ) e−      ∗   2 2 ∗ ∗ ∗ u 2 + w8 + w9 u 2 u 3 + w10 u 2 + w8 + w9 u ∗2 u ∗3 + w10   ∗ ∗ u∗u∗ u 2 u 3 w12 (u ∗2 + w10 )(w8 u ∗3 + w10 ) + u ∗1 − ∗ 1 2 2  ∗   4 (u 1 + w4 ) u + w8 + w9 u ∗ u ∗ + w10

 Q 0 = u ∗1 −

u ∗1 u ∗2 (u ∗1 + w4 )2

2

3

2

∗ ∗ w6 w7 w12 u ∗1 u ∗2 2 u 3 (w8 + w9 u 2 ) +  ∗   2 ∗ 2 (u 1 + w4 )(u 1 + w7 ) u 2 + w8 + w9 u 2 u ∗3 + w10

From above we see that Q 0 = −det(Gu (u∗ )). ˜ Let, Q(μ) = Q 3 μ3 + Q 2 μ2 + Q 1 μ − det(Gu (u∗ ) and let μ˜ 1 , μ˜ 2 , μ˜ 3 be the three ˜ roots of Q(μ) = 0, with Re(μ˜1 ) ≤ Re(μ˜ 2 ) ≤ Re(μ˜ 3 ). Now, μ˜ 1 μ˜ 2 μ˜ 3 = det(Gu (u∗ )). As det(Gu (u∗ )) < 0 because of condition specified in Theorem 2. Therefore, we have, μ3 < 0 μ˜ 1 μ˜ 2  As, d1 > 0, d2 > 0, d3 > 0 & d4 > 0 therefore, we have Q 3 > 0. From the theory 2 , μ 3 is real and negative and the product of other two is of equation one of μ 1 , μ positive. Consider the following limits, Q3 = d1 d2 (1 + d4 u ∗2 ) ∼ lim = b3 d3 →∞ d3 lim

d3 →∞

  u ∗3 u ∗2 (1 + w9 u ∗3 ) d1 d4 u ∗3 u ∗2 (u ∗2 + w10 ) Q2 = d1 (1 + d4 u ∗2 )u ∗2 e −  −    2   2 ∗ ∗ ∗ ∗ d3 u 2 + w8 + w9 u 2 u 3 + w10 u 2 + w8 + w9 u ∗2 u ∗3 + w10   u ∗1 u ∗2 ∼ + d2 (1 + d4 u ∗2 ) u ∗1 − = b2 (u 1 + w4 )2

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   u ∗3 u ∗2 (1 + w9 u ∗3 ) u ∗1 u ∗2 Q1 ∗− u = (1 + d4 u ∗2 )u ∗2 e −     1 2 d3 →∞ d3 (u 1 + w4 )2 u ∗2 + w8 + w9 u ∗2 u ∗3 + w10     ∗ ∗ ∗ ∗ ∗ w6 w7 u ∗1 u ∗2 d4 u 3 u 2 (u 2 + w10 ) u1 u2 ∗− ∗) ∼ − u u + (1 + d = b1    4 1 2 2 (u 1 + w4 )2 (u 1 + w4 )(u 1 + w7 )2 u ∗2 + w8 + w9 u ∗2 u ∗3 + w10 lim

If, b1 < 0, then from above the following holds, 

u ∗3 (1 + w9 u ∗3 ) e−   2  u ∗2 + w8 + w9 u ∗2 u ∗3 + w10

 1−

d4 u ∗3 (u ∗3 + w10 ) <    2 (1 + d4 u ∗2 ) u ∗2 + w8 + w9 u ∗2 u ∗3 + w10





u ∗2

(u ∗1 + w4 )2

1−

u ∗2

+

w6 w7 ∗ (u 1 + w4 )(u ∗1 + w7 )2



(u ∗1 + w4 )2

We have, b1 < 0 < b3 . Also, lim

d3 →∞

˜ Q(μ) = b3 μ3 + b2 μ2 + b1 μ = μ(b3 μ2 + b2 μ + b1 ) d3

As, b1 < 0 < b3 , which means that the equation, lim

d3 →∞

˜ Q(μ) =0 d3

μ is real have one positive root and one negative root. From continuity if d3 → ∞,  3 > 0, μ 2 μ 3 are real and positive. and negative. Further μ 2 μ lim μ˜ 1 =

−b2 −

d2 →∞

lim μ˜ 3 =

 b22 − 4b1 b3 2b3

−b2 −

 b22 − 4b1 b3

d2 →∞

2b3 lim μ˜ 2 = 0

d2 →∞

0= μ

(9) (10)

Therefore there exists a positive number d3∗ such that when d3 ≥ d3∗ the following ˜ ˜ holds, Q(μ) < 0 when μ ∈ (−∞, μ˜ 1 ) ∪ (μ˜ 2 , μ˜ 3 ). Therefore, when 0 < μ2 < μ, ˜ 2 ) < 0. Therefore B3 < 0, and the proof is then μ2 ∈ (μ˜ 2 , μ˜ 3 ), it follows that Q(μ complete. Therefore, from the above theorems we conclude that cross diffusion destabilizes the stationary uniform solution, so the spatial patterns for Crowley Martin type functional response can generate.

Turing Patterns in a Cross Diffusive System

23

4 Numerical Simulations Our goal in this study is to understand the impact of cross diffusion with different initial conditions. Hence, we consider an initial condition with trigonometric perturbation with quadratic argument. To understand the spatiotemporal dynamics of top predator u 3 we carry out numerical experiments with cross diffusion. For this purpose in this section we will do a detailed investigation of the patterns in the top level predator for different self and cross diffusivity coefficients. The system in (5) of partial differential equations is numerically solved using semi implicit finite difference technique, forward difference scheme is used for the reaction terms and standard five point explicit finite difference scheme is used for diffusion term. Turing patterns were obtained from the effect of nonlinear diffusion term for the top level predator u 3 as in [22, 23]. Discretization of the cross diffusive term of (5) has been carried out using Taylors series expansion about the non trivial equilibrium point (u ∗1 , u ∗2 , u ∗3 ). We get a system of the following form: ∂u 1 ∂t ∂u 2 ∂t ∂u 3 ∂t ∂u 1 ∂n

  u2 − d1 Δu 1 = u 1 1 − u 2 − u 1 + w4   w6 u 1 u3 − d2 Δu 2 = u 2 −w5 + − eu 2 − u 1 + w7 u 2 + (w8 + w9 u 2 ) u 3 + w10 (11)   w12 u 2 − d3 d4 u ∗3 Δu 2 − d3 (1 + d4 u ∗2 )Δu 3 = u 3 −w11 + u 2 + (w8 + w9 u 2 ) u 3 + w10 ∂u 2 ∂u 3 = = =0 (x, y) ∈ ∂Ω. ∂n ∂n

To ensure convergence the step length Δx = Δy and time step Δt has been chosen appropriately. Standard five-point approximation has been used for the 2D Laplacian with the zero-flux boundary conditions. Initially, the entire system is at the stationary state (u ∗1 , u ∗2 , u ∗3 ), and the perturbation introduced in the initial condition is of the order 5 × 10−4 as given in (12):    π(x − x0 ) 2 π(y − y0 ) 2 u(x, y) = u + ε1 sin + ε2 sin 0.2 0.2 2    π(x − x0 ) π(y − y0 ) 2 v(x, y) = v ∗ + ε1 sin + ε2 sin 0.2 0.2 2    π(x − x0 ) π(y − y0 ) 2 r (x, y) = r ∗ + ε1 sin + ε2 sin 0.2 0.2 ∗



(12)

where, ε1 = ε2 = 5 × 10−4 , x0 = y0 = 0.1. The, set of parameter values at which the system will yield a locally asymptotically stable solution is:

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N. Mohan and N. Kumari

Table 1 Values of diffusivity coefficients d1 , d2 , d3 and d4 used in the simulations Figure d1 d2 d3 d4 1 2(a) 2(b) 3(a) 3(b) 4(a) 4(b) 5(a) 5(b)

85 85 85 85 85 85 85 85 85

75 75 75 75 75 75 75 75 75

10 1.7 1.1 0.8 0.5 0.3 0.1 0.03 0.02

0 2.3 3 6 10 8 15 85 200

w4 = 0.25, w5 = 0.25, w6 = 0.8, w7 = 0.25, w8 = 0.01, w9 = 0.1, w10 = 0.28, w11 = 0.25, w12 = 0.78, e = 2. In order to analyze the role of cross diffusion on u 3 we consider the following set of parameters: w4 = 17.25, w5 = 17.25, w6 = 17.25, w7 = 17.25, w8 = 0.25, w9 = 0.25, w10 = 0.28, w11 = 18.26, w12 = 3.05, e = 22. We perform the simulation on a 50 × 50 grid with spatial step size h = 0.5 and time step size Δt = 0.1. In order to investigate the role of cross diffusion and self diffusion in the pattern formation of the top predator u 3 , we perform simulations for a wide range of self diffusive coefficient d3 and cross diffusivity coefficient d4 . The different values of self and cross diffusive coefficients used in numerical experiments of top level predator u 3 are given below in Table 1. We have carried out all the simulation for time level t = 10000 for the model system given in (11). In the absence of cross diffusion i.e. d4 = 0 no patterns were obtained for top predator u 3 even for higher values of d3 this is shown in Fig. 1. This experiment is indicative of the fact that in absence of cross diffusion patterning is not possible, as there is no destabilization in the system. For our second numerical experiment, we take into account the role of cross diffusion. We slowly increase cross diffusivity d4 and decrease self diffusivity d3 . Our objective is to observe the role of cross diffusion on the dynamics, when there is very low self diffusion. Therefore in Fig. 2a d4 is increased from 0 to 2.3 and d3 is decreased from 10 to 1.7. The introduction of cross diffusion destabilizes the dynamics of u 3 , the top predator and we obtain a mixture of hot spots and labyrinth patterns as seen in Fig. 2a. On further increasing d4 to 3 we get hot spot Turing patterns, observed in Fig. 2b.

Turing Patterns in a Cross Diffusive System

25

Fig. 1 No Patterns for top predator of the model system (11) were obtained at time level t = 10000 in the absence of cross diffusion i.e. self diffusion and cross diffusion coefficients being d3 = 10, d4 = 0 respectively

(a)

(b)

u3

50

r

50 8.61 8.6

40

8.59

40

8.58

8.59 8.58

30

8.57

30

8.57 8.56

20

8.56

20

8.55

8.55 8.54

10

8.54

10 8.53

8.53

0

0

10

20

30

40

50

0

8.52

0

10

20

30

40

50

Fig. 2 a A mix of Hot Spot and Labyrinth Turing patterns obtained at d3 = 1.7, d4 = 2.3. b Hot Spot Turing patterns obtained at d3 = 1.1, d4 = 3

Further increment in d4 and decrement in d3 , leads to hexagonal Turing patterns as observed in Fig. 3a at d4 = 6 and d3 = 0.8 and hot spot Turing pattern at d4 = 10 and d3 = 0.5 as seen in Fig. 3b. Again, increasing the cross diffusivity, d4 to 8 and decreasing the self diffusivity d3 to 0.3 we get floral Turing patterns as given in Fig. 4a. At d4 = 15 and d3 = 0.1 a mixture of hot spots and labyrinth patterns is obtained in Fig. 4b. For very high values of cross diffusivity we see hot spot patterns in Fig. 5a at d4 = 85 and d3 = 0.05 and for d4 = 200 and d3 = 0.01 we obtain pentagonal shaped patterns in Fig. 5b.

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N. Mohan and N. Kumari

(a)

(b)

u3

50

8.61

u3

50

8.6

8.6

40

8.59

40

8.59 8.58

8.58

30

8.57 8.56

20

30

8.57 8.56

20

8.55

8.55 8.54

10

8.54

10

8.53

8.53

0

0

10

20

30

40

0

50

0

10

20

30

40

50

Fig. 3 a Hexagonal Turing patterns obtained at d3 = 0.8, d4 = 6. b Hot spot Turing patterns obtained at d3 = 0.5, d4 = 10

(a)

(b)

u3

50

8.61

40

8.6 8.59

30

8.58 8.57

20

8.56

10

0

0

10

20

u3

50

8.62

30

40

50

45

8.65

40 35 8.6

30 25

8.55

20

8.55

15

8.54

10

8.53

5

8.52

0

8.5

0

10

20

30

40

50

Fig. 4 a Floral Turing patterns obtained at d3 = 0.3, d4 = 8. b A mix of Hot Spot and Labyrinth Turing patterns obtained at d3 = 0.1, d4 = 15

Above experiments help us in concluding that the role of cross diffusion is more significant in destabilizing the system than self diffusion. Figure 1 shows that no pattern formation can take place even for high values of self diffusivity, if cross diffusion is absent. Table 1 describes the various self and cross diffusivity coefficients used for obtaining the spatial patterns of the system (11). Table 1 indicates a negative correlation of −0.636 between self diffusion coefficient and cross diffusion coefficients which result in pattern formation.

Turing Patterns in a Cross Diffusive System

(a)

(b)

u3

50

27

8.64 8.62

40

u3

50

8.62 8.61 8.6

40

8.59

8.6

30

8.58

20

8.56

8.58

30

8.57 8.56

20

8.55 8.54

10

8.54

10

8.53

8.52

0

0

10

20

30

40

50

0

8.52

0

10

20

30

40

50

Fig. 5 a Hot Spot Turing patterns obtained at d3 = 0.05, d4 = 85. b Pentagonal Turing patterns obtained at d3 = 0.02, d4 = 200

5 Conclusion A tritrophic food chain model based on Holling Type–II and Crowley Martin functional response is considered in our study. As species interaction takes place in both space and time we spatially extend the model system in accordance with reaction diffusion mechanism. In order to understand the tendency of predators to develop migratory strategies to take advantage over prey we introduce cross diffusion, which takes into account movement of one species influenced by movement of another species. We proved that the positive equilibrium state of the model system in presence of self diffusion and absence of cross diffusion but unstable in presence of both self and cross diffusion. The patterns induced by cross diffusion for top level predator has been shown by perturbing the initial state with a trigonometric function with quadratic arguments. Through our study we conclude that such a model system is stable under self diffusion but whenever movement is under influence of other species it tends towards instability. Acknowledgements This work has further been extended and published in [24].

References 1. Y. Kuang, E. Beretta, Global qualitative analysis of a ratio-dependent predatorprey system. J. Math. Biol. 36(4), 389–406 (1998) 2. L.A. Segel, Modeling Dynamic Phenomena in Molecular and Cellular Biology (Cambridge University Press, Cambridge, 1984) 3. P.W. Price et al., Interactions among three trophic levels: influence of plants on interactions between insect herbivores and natural enemies. Annu. Rev. Ecol. Syst. 11(1), 41–65 (1980) 4. A. Hastings, T. Powell, Chaos in a three-species food chain. Ecology 72(3), 896–903 (1991)

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5. K. McCann, P. Yodzis, Biological conditions for chaos in a three-species food chain. Ecology 75(2), 561–564 (1994) 6. S. Gakkhar, R.K. Naji, Chaos in three species ratio dependent food chain. Chaos Solitons Fractals 14(5), 771–778 (2002) 7. V. Rai, R.K. Upadhyay, Chaotic population dynamics and biology of the top-predator. Chaos Solitons Fractals 21(5), 1195–1204 (2004) 8. P.H. Crowley, E.K. Martin, Functional responses and interference within and between year classes of a dragonfly population. J. N. Am. Benthol. Soc. 8(3), 211–221 (1989) 9. R.K. Upadhyay, R.K. Naji, Dynamics of a three species food chain model with Crowley-Martin type functional response. Chaos Solitons Fractals 42(3), 1337–1346 (2009) 10. G.T. Skalski, J.F. Gilliam, Functional responses with predator interference: viable alternatives to the Holling type II model. Ecology 82(11), 3083–3092 (2001) 11. Y. Dong et al., Qualitative analysis of a predator-prey model with crowley-martin functional response. Int. J. Bifurc. Chaos 25(09), 1550110 (2015) 12. A.M. Turing, The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B: Biol. Sci. 237(641), 37–72 (1952) 13. S. Kondo, T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation. Science 329(5999), 1616–1620 (2010) 14. S. Kondo, The reaction-diffusion system: a mechanism for autonomous pattern formation in the animal skin. Genes Cells 7(6), 535–541 (2002) 15. B. Dubey, N. Kumari, R.K. Upadhyay, Spatiotemporal pattern formation in a diffusive predatorprey system: an analytical approach. J. Appl. Math. Comput. 31(1–2), 413–432 (2009) 16. N. Kumari, Pattern formation in spatially extended tritrophic food chain model systems: generalist versus specialist top predator. ISRN Biomath. 2013, 12 (2013) 17. K. Kuto, Stability of steady-state solutions to a preypredator system with cross-diffusion. J. Differ. Equ. 197(2), 293–314 (2004) 18. K. Kuto, Y. Yamada, Multiple coexistence states for a preypredator system with cross-diffusion. J. Differ. Equ. 197(2), 315–348 (2004) 19. P.Y.H. Pang, M. Wang, Strategy and stationary pattern in a three-species predator-prey model. J. Differ. Equ. 200(2), 245–273 (2004) 20. M. Wang, Stationary patterns caused by cross-diffusion for a three-species prey-predator model. Comput. Math. Appl. 52(5), 707–720 (2006) 21. A.B. Medvinsky et al., Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 44(3), 311–370 (2002) 22. C. Tian, Z. Ling, Z. Lin, Spatial patterns created by cross-diffusion for a three-species food chain model. Int. J. Biomath. 7(02), 1450013 (2014) 23. C. Tian, Turing patterns created by cross-diffusion for a Holling II and Leslie-Gower type three species food chain model. J. Math. Chem. 49(6), 1128–1150 (2011) 24. N. Kumari, N. Mohan, Cross diffusion induced turing patterns in a tritrophic food chain model with crowley-martin functional response. Mathematics 7(3), 229 (2019)

On Multi-objective Optimization Problems and Vector Variational-Like Inequalities Vivek Laha and Harsh Narayan Singh

Abstract This paper deals with nonsmooth multi-objective optimization problems involving locally Lipschitz V − r -invexity using Michel–Penot subdifferential. We consider vector variational-like inequalities of Stampacchia and Minty type and establish some results, which give necessary and sufficient conditions for a feasible point to be Pareto optimal solution of the MOP. We also establish various results related to weak Pareto optimal solution of the MOP and corresponding weak versions of the vector variational-like inequalities. Keywords Multi-objective optimization · Clarke subdifferential · Michel–Penot subdifferential · V − r -invexity · Efficient solution · Variational inequalities

1 Introduction In recent years, generalized convexity plays an important role in the study of multiobjective optimization problems. One of these generalizations of convexity is the class of invex functions introduced by Hanson [9]. Mohan and Neogy [19] gave a relation between invex functions and preinvex functions on an invex set. Multiobjective optimization problems involving invex functions is difficult to deal because it requires the same kernel function for all the objective functions. In order to overcome such restrictions, Jeyakumar and Mond [10] introduce the notion of smooth V -invex functions which reduces to the class of invex functions for scalar case. Later, Egudo and Hanson [6] extended V -invexity concept for nonsmooth case. Antczak [3] introduced a new generalization of the idea of smooth r -invex functions and smooth V -invex functions and defined the class of differentiable V − r -invex V. Laha · H. Narayan Singh (B) Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, India e-mail: [email protected] V. Laha e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_3

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functions. Later, Antczak [4] introduced V − r -invexity concept for locally Lipschitz multi-objective optimization problems using Clarke Subdifferential. Giannessi [7, 8] extended the classical variational inequalities of Stampacchia and Minty type to the vectorial case. Later, many authors (see, e.g. [1, 2, 13]) established relationships between vector variational inequalities of Stampacchia and Minty type and multi-objective optimization problems involving various generalizations of invexity. We refer to the results [11, 12, 15–18] and references therein for more details. The outline of this paper is as follows: In Sect. 2, we provide some preliminary definitions and results. In Sect. 3, we consider multi-objective optimization problem and obtain necessary and sufficient conditions for Pareto optimality and weak Pareto optimality using vector variational-like inequalities. In Sect. 4, we conclude the results of this paper.

2 Preliminaries In this section, we provide some preliminary definitions and results that we shall use in the sequel. Throughout this paper, X is a real Banach space with norm . and X ∗ is dual ∗ space of X with the norm .∗ . We denote by 2 X , ., ., [x, y] and (x, y) the family of all non-empty subsets of X ∗ , the dual pair between X and X ∗ , the line segment for x, y ∈ X and the interior of [x, y], respectively. We consider the following multi-objective optimization problem: (MOP) min f (x) := ( f 1 (x), . . . , f m (x)) subject to x ∈ K ⊆ X, where f : X → Rm is a vector-valued function on a non-empty subset K such that f i : X → R for all i ∈ M, are locally Lipschitz on K . The concepts of Pareto optimal solutions and weak Pareto optimal solutions are given an follows: Definition 1 A vector x¯ ∈ K ⊆ X is said to be a Pareto optimal solution of MOP, iff for all x ∈ K , one has ¯ . . . , f m (x) − f m (x)) ¯ ∈ / −Rm f (x) − f (x) ¯ := ( f 1 (x) − f 1 (x), + \ {0} . Definition 2 A vector x¯ ∈ K ⊆ X is said to be a weak Pareto optimal solution of MOP, iff for all x ∈ K , one has ¯ . . . , f m (x) − f m (x)) ¯ ∈ / −intRm f (x) − f (x) ¯ := ( f 1 (x) − f 1 (x), +. Remark 1 A Pareto optimal solution is also a weak Pareto optimal solution of MOP, but the converse is not true in general.

On Multi-objective Optimization Problems …

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Now we recall the definitions of Clarke and Michel–Penot subdifferentials, which will be used in the sequel. Definition 3 [5] Let K be an open subset of X. The function h : K → R is said to be locally Lipschitz (of rank C) at x ∈ K , iff there exist a positive constant C and a neighbourhood N of x such that, for any y, z ∈ N , one has |h(y) − h(z)| ≤ C y − z . If the inequality above is satisfied for any x ∈ K , then h is said to be locally Lipschitz (of rank C) on K . Definition 4 [5] Let φ = K ⊆ X and let g : K → R is locally Lipschitz at x¯ ∈ K , then the Clarke’s directional derivative of g at x¯ ∈ K in the direction v ∈ X, denoted ¯ v), is given by g ◦ (x, ¯ v) = lim sup g ◦ (x, t↓0,x→x¯

g(x + tv) − g(x) , t

and the Clarke’s generalized Subdifferential of g at x¯ ∈ K , denoted ∂ ◦ g(x), ¯ is defined as follows:     ¯ := x ∗ ∈ X ∗ : x ∗ , v ≤ g ◦ (x, ¯ v), ∀v ∈ X . ∂ ◦ g(x) Definition 5 [14] Let φ = K ⊆ X and let g : K → R is locally Lipschitz at x¯ ∈ K , then the M-P directional derivative of g at x¯ ∈ K in the direction v ∈ X, denoted ¯ v), is given by g  (x, ¯ v) = sup lim sup g  (x, w∈X

t↓0

g(x¯ + tw + tv) − g(x¯ + tw) , t

and the M-P Subdifferential of g at x¯ ∈ K , denoted ∂  g(x), ¯ is defined as follows:     ¯ := x ∗ ∈ X ∗ : x ∗ , v ≤ g  (x, ¯ v), ∀v ∈ X . ∂  g(x) ¯ v) ≤ g ◦ (x, ¯ v) and Remark 2 It is clear that g  (x,  ◦ ¯ ⊆ ∂ g(x). ¯ ∂ g(x) Now, we recall the vector variational-like inequalities of Stampacchia and Minty type in terms of the M-P subdifferentials introduced by Laha et al. [12]. ¯ i ∈ M := (MP-SVVLI) To find x¯ ∈ K such that, for all x ∈ K , and x¯i∗ ∈ ∂  f i (x), {1, . . . , m} , one has 

   x¯1∗ , η(x, x) ¯ , . . . , x¯m∗ , η(x, x) ¯ ∈ / −Rm + \ {0} .

(MP-MVVLI) To find x¯ ∈ K such that, for all x ∈ K , and for all xi∗ ∈ ∂  f i (x), i ∈ M := {1, . . . , m}, one has

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   x1∗ , η(x, x) ¯ , . . . , xm∗ , η(x, x) ¯ ∈ / −Rm + \ {0} .

Also, we recall the weak version of vector variational-like inequalities of Stampacchia and Minty type in terms of M-P Subdifferentials. ¯ i ∈ M := (MP-SWVVLI) To find x¯ ∈ K such that, for all x ∈ K , and x¯i∗ ∈ ∂  f i (x), {1, . . . , m} , one has 

   x¯1∗ , η(x, x) ¯ , . . . , x¯m∗ , η(x, x) ¯ ∈ / −intRm +.

Remark 3 It is clear that every solution of MP-SVVLI is also a solution of MPSWVVLI, but the converse is not true in general. (MP-MWVVLI) To find x¯ ∈ K such that, for all x ∈ K , and for all xi∗ ∈ ∂  f i (x), i ∈ M := {1, . . . , m}, one has 

   ¯ , . . . , xm∗ , η(x, x) ¯ ∈ / −intRm x1∗ , η(x, x) +.

Remark 4 It is obvious from above definition that every solution of MP-MVVLI is also a solution of MP-MWVVLI, but the converse is not true in general.

3 Main Results In this section, we prove some results which relate solutions of the MOP to the solutions of vector variational-like inequalities of Stampacchia and Minty type. We introduce the class of locally Lipschitz V − r -invex functions in terms of M-P subdifferentials as follows. Definition 6 (M-P V − r -invex Function) Let φ = K ⊆ X and let f : X → Rm be a vector-valued function such that f i : X → R is locally Lipschitz near x¯ ∈ K ⊆ X, ∀i ∈ M and let r be an arbitrary real number. If there exist functions η : X × X → X and αi : X × X → R+ \ {0} such that   1 {r fi (x)} 1 ¯ e ≥ (>) e{r fi (x)} [1 + r αi (x, x) ¯ x¯i∗ ; η(x, x) ¯ ], r r when r = 0   ¯ ≥ (>)αi (x, x) ¯ x¯i∗ ; η(x, x) ¯ , f i (x) − f i (x)

when r = 0

¯ i ∈ M and x ∈ K , then f is said to be M-P V − r - invex holds for any x¯i∗ ∈ ∂  f i (x), (strictly M-P V − r -invex) function with respect to η and αi , i ∈ M at x¯ ∈ K . If the inequality is satisfied at any point u ∈ K , then f is said to be M-P V − r invex (strictly M-P V − r -invex) w.r.t. η and αi , i ∈ M on K . The following result states sufficient condition for a point to be a Pareto optimal solution of the MOP in terms of MP-SVVLI.

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Theorem 1 Let φ = K ⊆ X and let f : X → Rm be locally Lipschitz and M-P V − r -invex at x¯ over K with respect to η and αi for i ∈ M. If x¯ is a solution of MP-SVVLI for f over K with respect to η, then x¯ is a Pareto optimal solution of the MOP. Proof Suppose to the contrary that x¯ ∈ K ⊆ X is not a Pareto optimal solution of the MOP. Then, ∃x˜ ∈ K such that ˜ − f i (x) ¯ ≤ 0, ∀ i ∈ M, f i (x)

(1)

and ∃ at least one j ∈ M such that ˜ − f j (x) ¯ < 0. f j (x) We have the following cases: Case I: When r = 0. From (1), one has ˜ − f i (x)) ¯ ≤ 0, with r > 0 ∀ i ∈ M, r ( f i (x) ˜ − f i (x)) ¯ ≥ 0, with r < 0 ∀ i ∈ M, r ( f i (x) and ∃ at least one j ∈ M such that ˜ − f j (x)) ¯ < 0, with r > 0. r ( f j (x) ˜ − f j (x)) ¯ > 0, with r < 0. r ( f j (x) Since the exponential function is monotonically increasing, it follows that ˜ f i (x))} ¯ ≤ 1, with r > 0 ∀ i ∈ M, e{r ( fi (x)− ˜ f i (x))} ¯ ≥ 1, with r < 0 ∀ i ∈ M, e{r ( fi (x)−

and ∃ at least one j ∈ M such that ˜ f j (x))} ¯ < 1, with r > 0. e{r ( f j (x)− ˜ f j (x))} ¯ > 1, with r < 0. e{r ( f j (x)−

Since f is locally Lipschitz and M-P V − r -invex at x¯ ∈ K w.r.t. η and αi for i ∈ M, it follows that ˜ x) ¯ x¯i ∗ ; η(x, ˜ x) ¯ ≤ 0, ∀x¯i ∗ ∈ ∂  f i (x), ¯ ∀ i ∈ M, r αi (x, with r > 0

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∀ i ∈ M, r αi (x, ˜ x) ¯ x¯i ∗ ; η(x, ˜ x) ¯ ≥ 0, ∀x¯i ∗ ∈ ∂  f i (x), ¯ with r < 0 and ∃ at least one j ∈ M such that ˜ x) ¯ x¯j ∗ ; η(x, ˜ x) ¯ < 0, ∀x¯j ∗ ∈ ∂  f j (x), ¯ with r > 0. r α j (x, ˜ x) ¯ x¯j ∗ ; η(x, ˜ x) ¯ > 0, ∀x¯j ∗ ∈ ∂  f j (x), ¯ with r < 0. r α j (x, that is,

˜ x) ¯ ≤ 0, ∀x¯i ∗ ∈ ∂  f i (x) ¯ ∀ i ∈ M, x¯i ∗ ; η(x,

and ∃ at least one j ∈ M such that ˜ x) ¯ < 0, ∀x¯j ∗ ∈ ∂  f j (x). ¯ x¯j ∗ ; η(x, Case II: When r = 0. Since f is locally Lipschitz and M-P V − r -invex at x¯ ∈ K w.r.t. η and αi for i ∈ M, it follows that ˜ x) ¯ ≤ 0, ∀x¯i ∗ ∈ ∂  f i (x), ¯ ∀ i ∈ M, x¯i ∗ ; η(x, with strict inequality for at least one i ∈ M. From above cases I and II, we arrive at a contradiction that x¯ is not a solution of the MP-SVVLI. Hence, the result.  The following result states the necessary condition for a point to be a Pareto optimal solution of the MOP in terms of MP-MVVLI. Theorem 2 Let φ = K ⊆ X and let f : X → Rm be locally Lipschitz and M-P V − r -invex over K with respect to η and αi for i ∈ M such that η is skew-symmetric over K . If x¯ is a Pareto optimal solution of the MOP, then x¯ also solves MP-MVVLI with respect to η. Proof We suppose to the contrary that x¯ is not a solution of MP-MVVLI with respect to η. Then, ∃x˜ ∈ K and x˜i ∗ ∈ ∂  f i (x), ˜ i ∈ M such that ˜ x) ¯ ≤ 0, ∀i ∈ M, x˜i ∗ ; η(x, ˜ such that and ∃ at least one j ∈ M and x˜j ∗ ∈ ∂  f j (x) ˜ x) ¯ < 0. x˜j ∗ ; η(x, Since η is skew-symmetric over K , it follows that ¯ x) ˜ ≥ 0, ∀i ∈ M, x˜i ∗ ; η(x,

(2)

On Multi-objective Optimization Problems …

35

and ∃ at least one j ∈ M such that ¯ x) ˜ > 0. x˜j ∗ ; η(x, We have the following two cases: Case I: When r = 0. From (2), one has ¯ x) ˜ x˜i ∗ ; η(x, ¯ x) ˜ ≥ 1, with r > 0 ∀i ∈ M, 1 + r αi (x, ¯ x) ˜ x˜i ∗ ; η(x, ¯ x) ˜ ≤ 1, with r < 0 ∀i ∈ M, 1 + r αi (x, and ∃ at least one j ∈ M such that ¯ x) ˜ x˜j ∗ ; η(x, ¯ x) ˜ > 1, with r > 0. 1 + r α j (x, ¯ x) ˜ x˜j ∗ ; η(x, ¯ x) ˜ < 1, with r < 0. 1 + r α j (x, Since f is locally Lipschitz and M-P V − r -invex over K w.r.t. η and αi for i ∈ M, it follows that ¯ f i (x))} ˜ ≥ 1, with r > 0 ∀i ∈ M, e{r ( fi (x)− ¯ f i (x))} ˜ ≤ 1, with r < 0 ∀i ∈ M, e{r ( fi (x)−

and ∃ at least one j ∈ M such that ¯ f j (x))} ˜ > 1, with r > 0. e{r ( f j (x)− ¯ f j (x))} ˜ < 1, with r < 0. e{r ( f j (x)−

Taking logarithm on both the sides, one has ¯ − f i (x)) ˜ ≥ 0, with r > 0 ∀i ∈ M, r ( f i (x) ¯ − f i (x)) ˜ ≤ 0, with r < 0 ∀i ∈ M, r ( f i (x) and ∃ at least one j ∈ M such that ¯ − f j (x)) ˜ > 0, with r > 0. r ( f j (x) ¯ − f j (x)) ˜ < 0, with r < 0. r ( f j (x) which implies that ˜ − f i (x) ¯ ≤ 0, ∀i ∈ M, f i (x)

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and ∃ at least one j ∈ M such that ˜ − f j (x) ¯ < 0. f j (x) Case II: When r = 0. From (2), it follows that ˜ − f i (x) ¯ ≤ 0, ∀i ∈ M, f i (x) and ∃ at least one j ∈ M, such that ˜ − f j (x) ¯ < 0. f j (x) By the above cases I and II, we arrive at a contradiction to the fact that x¯ is a Pareto optimal solution of the MOP. Hence, the result.  The following theorem derives the sufficient condition for a point to be a weak Pareto optimal solution of the MOP in terms of the MP-SWVVLI. Theorem 3 Let φ = K ⊆ X, and let f be locally Lipschitz and M-P V − r -invex at x¯ ∈ K w.r.t. η and αi for i ∈ M. If x¯ is a solution of the MP-SWVVLI w.r.t. η, then x¯ is also a weak Pareto optimal solution of the MOP. Proof We suppose to the contrary that x¯ ∈ K is not a weak Pareto optimal solution of the MOP. Then, ∃x˜ ∈ K such that ˜ − f i (x) ¯ < 0, ∀i ∈ M, f i (x) which implies that ˜ − f i (x)) ¯ < 0, when r > 0. ∀i ∈ M, r ( f i (x) ˜ − f i (x)) ¯ > 0, when r < 0. ∀i ∈ M, r ( f i (x) Since the exponential function is monotonically increasing, it follows that ˜ f i (x))} ¯ < 1, when r > 0. ∀i ∈ M, e{r ( fi (x)− ˜ f i (x))} ¯ > 1, when r < 0. ∀i ∈ M, e{r ( fi (x)−

By M-P V − r -invexity of f at x¯ ∈ K w.r.t. η and αi for i ∈ M, it follows that ˜ x) ¯ < 0, ∀x¯i ∗ ∈ ∂  f i (x). ¯ ∀i ∈ M, x¯i ∗ ; η(x, which contradicts that x¯ is a solution of MP-SWVVLI. This completes the proof.  The following theorem derives the relation between the solution of the MPMWVVLI and weak Pareto optimal solution of the MOP.

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Theorem 4 Let φ = K ⊆ X and let f be locally Lipschitz and M-P V − r -invex function on K w.r.t. η and αi , i ∈ M such that αi , i ∈ M is symmetric and η is skewsymmetric. Then, x¯ solves the MP-MWVVLI if it is weak Pareto optimal solution of the MOP. Proof We assume that x¯ is not a solution of the MP-MWVVLI w.r.t. η. Then, ∃x˜ ∈ K and x˜i ∗ ∈ ∂  f i (x) ˜ such that ˜ x) ¯ < 0, ∀i ∈ M, x˜i ∗ ; η(x, ˜ x) ¯ > 0, i ∈ M, one Since η is skew-symmetric, αi , i ∈ M is symmetric and αi (x, has ¯ x) ˜ x˜i ∗ ; η(x, ¯ x) ˜ > 0. (3) ∀i ∈ M, αi (x, We have the following cases: Case I: From (3), for r = 0, one has ¯ x) ˜ x˜i ∗ ; η(x, ¯ x) ˜ > 1, with r > 0. ∀i ∈ M, 1 + r αi (x, ¯ x) ˜ x˜i ∗ ; η(x, ¯ x) ˜ < 1, with r < 0. ∀i ∈ M, 1 + r αi (x, Since f is locally Lipschitz and M-P V − r -invex at x¯ over K w.r.t. η and αi for i ∈ M, it follows that ¯ f i (x))} ˜ > 1, with r > 0. ∀i ∈ M, e{r ( fi (x)− ¯ f i (x))} ˜ < 1, with r < 0. ∀i ∈ M, e{r ( fi (x)−

Taking logrithm on both the sides, one has ¯ − f i (x) ˜ > 0. ∀i ∈ M, f i (x) Case II: When r = 0. From (3), it follows that ¯ − f i (x) ˜ > 0. ∀i ∈ M, f i (x) By the cases I and II, we arrive at a contradiction. Hence, the result.



The following result is a direct consequence of the fact that every solution of MP-MVVLI is also a solution of MP-MWVVLI and Theorems 1 and 2. Corollary 1 Let φ = K ⊆ X and let f : X → Rm be locally Lipschitz and M-P V − r -invex at x¯ over K with respect to η and αi for i ∈ M such that η is skewsymmetric over K . If x¯ is a solution of MP-SVVLI for f over K with respect to η, then x¯ is also a solution of MP-MWVVLI w.r.t. η.

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4 Conclusions In this paper, we have considered the Stampacchia and Minty vector variationallike inequalities in terms of Michel–Penot subdifferentials in Banach spaces. Under the assumptions of locally Lipschitz M-P V − r -invexity, we have obtained necessary and sufficient conditions for a point to be a Pareto optimal solution and weak Pareto optimal solution of the multi-objective optimization problem in terms of vector variational-like inequalities. Also, we have obtained the relationship between solutions of Stampacchia vector variational-like inequalities and Minty weak vector variational-like inequalities. Acknowledgements The research of the first author is supported by UGC-BSR start up grant by University Grant Commission, New Delhi, India (Letter No. F. 30-370/2017(BSR)) (Project No. M-14-40).

References 1. S. Al-Homidan, Q.H. Ansari, Generalized Minty vector variational-like inequalities and vector optimization problems. J. Optim. Theory Appl. 144, 1–11 (2010) 2. Q.H. Ansari, M. Rezaie, J. Zafarani, Generalized vector variational-like inequalities and vector optimization. J. Glob. Optim. 53, 271–284 (2012) 3. T. Antczak, V − r -invexity in multiobjective programming. J. Appl. Anal. 11(1), 63–80 (2005) 4. T. Antczak, Optimality and duality for nonsmooth multiobjective programming problems with V − r -invexity. J. Glob. Optim. 45, 319–334 (2009) 5. F.H. Clarke, Nonsmooth optimization (Wiley-Interscience, New York, 1983) 6. R.R. Egudo, M.A. Hanson, On Sufficiency of Kuhn-Tucker Conditions in Nonsmooth Multiobjective Programming, FSU Technical, Report No. M-888 (1993) 7. F. Giannessi, Theorems of the alternative, quadratic programs and complementarity problems, in Variational Inequalities and Complementarity Problems, ed. by R.W. Cottle, F. Giannessi, J.-L. Lions (Wiley, New York, 1980), pp. 151–186 8. F. Giannessi, On Minty variational principle, in New Trends in Mathematical Programming, ed. by F. Giannessi, S. Komlsi, T. Tapcsck (Kluwer, Dordrecht, 1998), pp. 93–99 9. M.A. Hanson, On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981) 10. V. Jeyakumar, B. Mond, On generalized convex mathematical programming, J. Austral. Math. Soc. Ser., B 34, 43–53 (1992) 11. V. Laha, S.K. Mishra, On vector optimization problems and vector variational inequalities using convexificators. Optimization 66(11), 1837–1850 (2017) 12. V. Laha, B. Al-Shamary, S.K. Mishra, On nonsmooth V -invexity and vector variational-like inequalities in terms of Michel-Penot subdifferentials. Optim. Lett. 8, 1675–1690 (2014) 13. S.K. Mishra, V. Laha, R.U. Verma, Generalized vector variational-like inequalities and Nonsmooth vector optimization of radially (η, α)-continuous functions. Adv. Nonlinear Variat. Ineq. 14(2), 1–18 (2011) 14. P. Michel, J.-P. Penot, Calcul sous-diffrentiel pour de fonctions Lipschitziennes et nonlipschitziennes, C. R. Acad. Sci. Paris Sr. I Math. 12, 269–272 (1984) 15. S.K. Mishra, V. Laha, On V − r -invexity nd vector variational-like inequalities. Filomat 26(5), 1065–1073 (2012)

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16. S.K. Mishra, V. Laha, On generalized Minty and Stampacchia vector variational-like inequalities and V -invex vector optimization in Asplund spaces. Adv. Nonlinear Var. Inequal 16(2), 43–60 (2013) 17. S.K. Mishra, V. Laha, On minty variational principle for nonsmooth vector optimization problems with approximate convexity. Optim Lett. 10, 577–589 (2016) 18. S.K. Mishra, V. Laha, On approximately star-shaped functions and approximate vector variational inequalities. J. Optim. Theory Appl. 156(2), 278–293 (2013) 19. S.R. Mohan, S.K. Neogy, On invex sets and preinvex functions. J. Math. Anal. Appl. 189, 901–908 (1995)

Controllability of Semilinear Control Systems with Fixed Delay in State Abdul Haq and N. Sukavanam

Abstract This work studies the controllability of a class of delay differential equations. Instead of C0 -semigroup associated with the mild solution of the system, we use the concept of fundamental solution. Approximate controllability of the system is shown using sequence method. Finally, an illustrative example has been provided. Keywords Delay system · Fundamental solution · Mild solution · Approximate controllability

1 Introduction   Let H and H be Hilbert spaces and U = L 2 [0, β]; H , Z t = L 2 ([−t, β]; H) be the function spaces. Consider the semilinear system of the form z  (t) = A1 z(t) + A2 z(t − b) + Bυ(t) + F(t, z(t − b), υ(t)), t ∈ (0, β] z(t) = ψ(t), t ∈ [−b, 0]

 (1)

where the state z(·) ∈ Z 0 , the control υ(·) ∈ U ; A1 : D(A1 ) ⊆ H → H is a closed and densely defined linear operator generating a C0 -semigroup T0 (t), A2 is the bounded linear operator defined on H, B : U → Z 0 is a bounded linear operator and F : [0, β] × H × H → H is a nonlinear operator. Controllability is one of the fundamental properties of dynamical systems, which was introduced by Kalman in 1960. Roughly speaking, a dynamical system is said to be state controllable during some finite time interval, over some space X , if we can steer that system from every initial state to every desired final state in X during that time interval, by using a set of control functions. There are various notions A. Haq (B) · N. Sukavanam Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_4

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of controllability such as complete state controllability, total state controllability, approximate controllability, trajectory controllability, null controllability, interior controllability, and exact controllability etc. In finite- dimensional spaces, controllability of linear and nonlinear systems of ordinary differential equations has been extensively studied. In infinite-dimensional spaces, several results for controllability can be seen in [1]. In deterministic setting, a set of sufficient conditions for approximate controllability of semilinear control system dominated by linear part is obtained by Naito [2]. A survey on controllability of nonlinear systems in Banach spaces is presented by Balachandran and Dauer [3]. Minimum energy control and relative controllability of finite-dimensional linear system is presented by Klamka [4]. Utilizing Schauder’s fixed point theorem Klamka [5] obtained the controllability result for finite-dimensional nonlinear systems. Approximate Controllability of semilinear control systems in Hilbert Spaces is presented by Mahmudov and Semi [6]. In [7], Wang and Du proved approximate controllability of semilinear degenerate systems with convection term. There are many problems in which the present rate of change of some unknown function depends upon past values of the same function. Such problems are modeled by the time-delay systems. Wang [8] extended the results of [2] and obtained a set of novel conditions for the controllability of semilinear delay systems. Sukavanam [9] presented the controllability of a delayed system with growing nonlinear term. Klamka [10, 11] studied the controllability of linear systems with fixed delay in control. In [12], Devies and Jackreece deduced the results for null and exact controllability of linear delay systems. Shukla et al. [13] proved the approximate controllability of semilinear systems with state delay in both linear and nonlinear terms. In this article, the work has been extended for systems with state delay and involving control function in nonlinear part, with suitable modification. The plan of the paper is like this: Sect. 1 contains the introduction. In Sect. 2, we have given the preliminaries. In Sect. 3, approximate controllability of the system (1) is deduced. Finally, an example is provided to illustrate the theory.

2 Preliminaries Consider the linear system z  (t) = A1 z(t) + A2 z(t − b), t ∈ (0, β] z(t) = ψ(t), t ∈ [−b, 0]

 (2)

where ψ ∈ L 2 ([−b, 0]; H). Let z ψ (t) be the unique solution of (2). Define the operator ζ(t) on H by  z ψ (0), t ∈ [0, β] ζ(t)ψ(0) = 0, t ∈ [−b, 0)

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then, we say that ζ(t) is the fundamental solution of (2) verifying t ζ(t) = T0 (t) +

T0 (t − s)A2 ζ(t − b)ds, t ∈ [0, β] 0

ζ(t) = 0 for

− b ≤ t < 0.

Definition 2.1 A function z(·) ∈ Z b is said to be the mild solution of (1) if it satisfies z(t) = ζ(t)ψ(0) +

t

ζ(t − s){Bυ(s) + F (s, z(s − b), υ(s))}ds, t ∈ [0, β]

0

z(t) = ψ(t), t ∈ [−b, 0).

⎫ ⎬ ⎭

(3)

Definition 2.2 The set given by K β (F) = {z(β) ∈ H : z ∈ Z b }, where z is a mild solution of (1) corresponding to control υ ∈ U is called the reachable set of (1). Definition 2.3 The system (1) is said to be approximately controllable on [0, β] if K β (F) = H. Define the bounded linear operator L : Z 0 → H by β L( p) =

ζ(β − s) p(s)ds,

p(·) ∈ Z 0 .

0

To prove our results, we need the following assumptions: [H1 ] There is a positive constant M1 such that T0 (t) ≤ M1 . [H2 ] There are positive constants L and lF such that   F(t, z 1 (t), υ1 (t)) − F(t, z 2 (t), υ2 (t)) ≤ L z 1 (t) − z 2 (t) + υ1 (t) − υ2 (t) .

and F(t, 0, 0) ≤ lF . [H3 ] For each p ∈ Z 0 , there is a q ∈ R(B) such that L( p) = L(q). Hence it follows that for any given  > 0 and p(·) ∈ Z 0 there is a υ ∈ U such that

44

A. Haq and N. Sukavanam

L( p) − L(Bυ) ≤ . [H4 ] Bυ(·) Z 0 ≤ λ p(·) Z 0 where λ is a positive constant independent of p(·). [H5 ] There is a positive constant τ such that υU ≤ τ Bυ Z 0 . From [14] utilizing Gronwall’s inequality, one can easily deduce that ζ(t) ≤  M1 exp M1 (β − b)A2  = M2 .

3 Main Results Lemma 3.1 Under the assumption [H2 ] the solution (υ)(·) satisfies (υ)(t)H ≤ K exp (M2 Lβ)  √ where K = M2 ψ(0) + β(1 + τ )Bυ L 2 + lF β and the solution map  is given by t ζ(t − s)[Bυ(s) + F(s, z(s − b), υ(s))ds.

(υ)(t) = z(t) = ζ(t)ψ(0) + 0

Proof t ζ(t − s)[Bυ(s) + F(s, z(s − b), υ(s))ds

z(t) ≤ ζ(t)ψ(0) + 0

t ≤ M2 ψ(0) + M2

Bυ(s) + F(s, 0, 0)ds 0

t F(s, z(s − b), υ(s)) − F(s, 0, 0)ds

+M2 0

≤ M2 ψ(0) + M2 βBυ Z 0 + M2 lF β t   z(s − b)H + υ(s)H ds +M2 L 0

 ≤ M2 ψ(0) + β(1 + τ )Bυ Z 0 + lF β t +M2 L z(s − b)H ds 0

Controllability of Semilinear Control Systems with Fixed Delay in State

45

t−b ≤ K + M2 L z(s)H ds −b

⇒ (υ)(t)H = z(t)H ≤ K exp(M2 Lβ).  Lemma 3.2 Let υ1 and υ2 be in U . Then z 1 − z 2  Z 0 ≤ M2 β(1 + τ )Bυ1 − Bυ2  Z 0 exp (M2 Lβ), where z i (t) = (υi )(t), i = 1, 2. Proof Let us define y(·, ψ) : [−b, β] → H as  ζ(t)ψ(0), t ∈ [0, β] y(t, ψ) = ψ(t), t ∈ [−b, 0]. Let z(t) = y(t) + x(t), t ∈ [−b, β]. Then z(·) satisfies (1) iff x(0) = 0 and for all t ∈ [0, β], t x(t) =

ζ(t − s)[Bυ(s) + F(s, y(s − b) + x(s − b), υ(s))]ds. 0

Now take z 1 (·), z 2 (·) ∈ Z 0 and υ1 (·), υ2 (·) ∈ U , then t x1 (t) − x2 (t) ≤ M2

t Bυ1 (s) − Bυ2 (s)ds + M2

0

F (s, y(s − b) + x1 (s − b), υ1 (s)) 0

−F (s, y(s − b) + x2 (s − b), υ2 (s))ds

≤ M2 βBυ1 − Bυ2  Z 0 t   +M2 L x1 (s − b) − x2 (s − b)H + υ1 (s) − υ2 (s)H ds 0

  ≤ M2 β Bυ1 − Bυ2  Z 0 + υ1 − υ2 U t +M2 L

x1 (s − b) − x2 (s − b)H ds 0

t−b

≤ M2 β(1 + τ )Bυ1 − Bυ2  Z 0 + M2 L x1 (s) − x2 (s)H ds. −b

Applying Gronwall’s inequality, we obtain

46

A. Haq and N. Sukavanam

x1 (t) − x2 (t)H ≤ M2 β(1 + τ )Bυ1 − Bυ2  Z 0 exp(M2 Lβ). Hence

z 1 − z 2  Z 0

⎛ β ⎞1/2  = ⎝ z 1 (s) − z 2 (s)2H ds ⎠ 0

⎛ β ⎞1/2  = ⎝ x1 (s) − x2 (s)2H ds ⎠ 0

≤ M2 β(1 + τ )Bυ1 − Bυ2  Z 0 exp(M2 Lβ).  Theorem 3.3 Under the assumptions [H1 ]–[H5 ], the system (1) is approximately controllable if  λL M2 β(1 + τ ) exp(M2 Lβ) + τ < 1. Proof Since D(A1 ) is dense in H, it is enough to show that D(A1 ) ⊆ K β (F). For this, we will prove that for any given  > 0 and g ∈ D(A1 ), we can find a control υ ∈ U satisfying g − ζ(β)ψ(0) − L[F(s, z  (s − b), υ (s))] − L(Bυ ) ≤ , where z  (t) = (υ )(t). Since ζ(β)ψ(0) ∈ D(A1 ), therefore we can find a p ∈ Z 0 such that L( p) = g − ζ(β)ψ(0). Now we construct a sequence as follows: Let υ1 ∈ U be arbitrary. Then by assumption [H3 ], we can find a υ2 ∈ U satisfying g − ζ(β)ψ(0) − L[F(s, z 1 (s − b), υ1 (s))] − L(Bυ2 ) ≤

 32

(4)

and for υ2 ∈ U , we can find a ω2 ∈ U satisfying L{F(s, z 2 (s − b), υ2 (s)) − F(s, z 1 (s − b), υ1 (s))} − L(Bω2 ) ≤ where z i (t) = (υi )(t), i = 1, 2.

 , 33

(5)

Controllability of Semilinear Control Systems with Fixed Delay in State

47

By assumption [H4 ], we have Bω2  Z 0 ≤ λF (s, z 2 (s − b), υ2 (s)) − F (s, z 1 (s − b), υ1 (s)) L 2 ([0,β];H) ⎛ ⎞1/2 β ⎜ ⎟ = λ ⎝ F (s, z 2 (s − b), υ2 (s)) − F (s, z 1 (s − b), υ1 (s))2H ds ⎠ 0

⎛ ⎜ ≤ λL ⎝

β

⎞1/2  2 ⎟ z 1 (s − b) − z 2 (s − b)H + υ1 (s) − υ2 (s)H ds ⎠

0

⎡⎛ ⎞1/2 ⎛ ⎞1/2 ⎤ β β ⎢⎜ ⎟ ⎜ ⎟ ⎥ ≤ λL ⎣⎝ z 1 (s − b) − z 2 (s − b)2H ds ⎠ + ⎝ υ1 (s) − υ2 (s)2H ds ⎠ ⎦ 0

 = λL z 1 − z 1  Z 0 + u 1 − u 2 U .

0

Utilizing assumption [H5 ] and Lemma 3.2, we obtain  Bω2  Z 0 ≤ λL M2 β(1 + τ ) exp(M2 Lβ) + τ Bυ1 − Bυ2  Z 0 Setting υ3 = υ2 − ω2 , we have g − ζ(β)ψ(0) − L[F(s, z 2 (s − b), υ2 (s))] − L(Bυ3 ) ≤ g − ζ(β)ψ(0) − L[F(s, z 1 (s − b), υ1 (s))] − L(Bυ2 ) +L(Bω2 ) − L[F(s, z 2 (s − b), υ2 (s)) − F(s, z 1 (s − b), υ1 (s))]. Using (4) and (5), we get  g − ζ(β)ψ(0) − L[F(s, z 2 (s − b), υ2 (s))] − L(Bυ3 ) ≤

1 1 + 3 2 3 3

 .

Utilizing mathematical induction, one can obtain a sequence {υn } in U such that  g − ζ(β)ψ(0) − L[F (s, z n (s − b), υn (s))] − L(Bυn+1 ) ≤

1 1 1 + 3 + · · · + n+1 32 3 3

 ,

(6) where z n (t) = (υn )(t), n = 1, 2, . . . and  Bυn+1 − Bυn  Z 0 ≤ λL M2 β(1 + τ ) exp(M2 Lβ) + τ Bυn − Bυn−1  Z 0 From above inequality, it is evident that the sequence {Bυn } is Cauchy in Z 0 . Therefore, it is convergent and for any given  > 0, there is a positive integer n 0 such that

48

A. Haq and N. Sukavanam

L(Bυn 0 +1 ) − L(Bυn 0 ) ≤

 . 3

Now g − ζ(β)ψ(0) − L[F(s, z n 0 (s − b), υn 0 (s))] − L(Bυn 0 ) ≤ g − ζ(β)ψ(0) − L[F(s, z n 0 (s − b), υn 0 (s))] − L(Bυn 0 +1 ) +L(Bυn 0 +1 ) − L(Bυn 0 )    1 1 1 + 3 + · · · + n +1  + ≤ 2 0 3 3 3 3 < . Which shows that g ∈ K β (F). This completes the proof.



Theorem 3.4 Under the assumption [H2 ] and [H5 ], the system (1) is approximately controllable if R(B) = Z 0 . Proof Let  > 0 be given. Since R(B) = Z 0 , therefore for any given p ∈ Z 0 and η > 0, we can find a Bυ ∈ R(B) satisfying Bυ − p Z 0 < η p Z 0 .

(7)

Now β L( p) − L(Bυ) ≤ M2

 p(s) − Bυ(s)ds 0

≤ M2 β p − Bυ Z 0

≤ M2 η β p Z 0 ≤ . It means [H3 ] is satisfied. Also Bυ Z 0 ≤ Bυ − p Z 0 +  p Z 0 ≤ η p Z 0 +  p Z 0 ≤ (η + 1) p Z 0 . Hence [H4 ] is satisfied. If we select η > 0 in such a way that the additional assumption of Equation (4) is satisfied, then the controllability of the system (1) follows. 

Controllability of Semilinear Control Systems with Fixed Delay in State

49

4 Application Consider the delay system governed by the heat equation ⎫ 2 z(t,x) ⎬ = ∂ ∂x + z(t − b, x) + Bυ(t, x) + F (t, z(t − b, x), υ(t, x)), t ∈ (0, β] ⎪ 2 z(t, 0) = z(t, π) = 0, t ∈ [0, β] ⎪ ⎭ z(t, x) = ψ(t, x), t ∈ [−b, 0], 0 < x < π ∂z(t,x) ∂t

where ψ(t, x) is continuous. Take H = L 2 (0, π) and define A1 : D(A1 ) ⊂ H → H by A1 z =  D(A1 ) = z ∈ H : z,

dz dx

(8)

d2z , with domain dx2

 d2z ∈ H and z(0) = 0 = z(π) . 2 dx

are absolutely continuous,

Let ξ j (x) = ( π2 )1/2 sin j x, 0 ≤ x ≤ π, j = 1, 2, . . ., then λ j = − j 2 are eigenvalues of A1 with corresponding eigenfunctions ξ j and the family {ξ j } j∈N form a complete orthonormal set for H and exp (λ j t) are eigenvalues of the C0 −semigroup T0 (t) generated by A1 [1]. Define   ∞ ∞   2 H = υ|υ = α j ξ j with αj < ∞ 

j=2

j=2

with the norm υH

 ∞  =! α2 . j

j=2

Define the bounded linear operator B : H → H by Bυ = 2α2 ξ1 +

∞  j=2

αjξj,

∞ 

α j ξ j ∈ H .

j=2

The abstract form of (8) is z  (t) = A1 z(t) + A2 z(t − b) + Bυ(t) + F(t, z(t − b), υ(t)), t ∈ (0, β] z(t) = ψ(t), t ∈ [−b, 0]

 (9)

where A2 = I . If all the assumptions of Equation (4) are satisfied, then the system (8) is approximately controllable.

50

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References 1. R.F. Curtain, H. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Texts in Applied Mathematics, vol. 21 (Springer, New York, 1995) 2. K. Naito, Controllability of semilinear control systems dominated by the linear part. SIAM J. Control Optim. 25, 715–722 (1987) 3. K. Balachandran, J.P. Dauer, Controllability of nonlinear systems in Banach spaces: a survey. J Optim. Theory Appl. 115(1), 7–28 (2002) 4. J. Klamka, Relative controllability of minimum energy control of linear systems with distributed delays in control. IEEE T. Automat. Contr. 21, 594–595 (1976) 5. J. Klamka, Schauder’s fixed point theorem in nonlinear controllability problems. Control Cybern. 29, 153–165 (2000) 6. N.I. Mahmudov, N. Semi, Approximate controllability of semilinear control systems in Hilbert spaces. TWMS J. App. Eng. Math. 2, 67–74 (2012) 7. C. Wang, R. Du, Approximate controllability of a class of semilinear degenerate systems with convection term. J. Differ. Equ. 254(9), 3665–3689 (2013) 8. L. Wang, Approximate controllability of integrodifferential equations with multiple delays. J. Optim Theory Appl. 143, 185–206 (2009) 9. N. Sukavanam, Approximate controllability of semilinear control systems with growing nonlinearity, in Mathematical theory of control proceedings of international conference (Marcel Dekker, New York, 1993), pp. 353–357 10. J. Klamka, Stochastic controllability of systems with variable delay in control. Bull. Pol. Ac. Tech. 56, 279–284 (2008) 11. J. Klamka, Stochastic controllability and minimum energy control of systems with multiple delays in control. Appl. Math. Comput. 206, 704–715 (2008) 12. I. Davies, P. Jackreece, Controllability and null controllability of linear systems. J. Appl. Sci. Environ. Manag. 9, 31–36 (2005) 13. A. Shukla, N. Sukavanam, D.N. Pandey, Approximate controllability of semilinear system with state delay using sequence method. J. Frankl. Inst. 352, 5380–5392 (2015) 14. N. Sukavanam, S. Tafesse, Approximate controllability of a delayed semilinear control system with growing nonlinear term. Nonlinear Anal. 74, 6868–6875 (2011)

Computational Performance of Server Using the Mx /M/1 Queue Model Jitendra Kumar and Vikas Shinde

Abstract In this paper, we present the algorithms for evaluating the most effective and efficient transient solution to MX /M/1 queueing model. The analytical results are expressed in modified Bessel functions and also use generalized Q-function. Numerical illustration has been obtained and compared with other algorithms by their own programs and results. Keywords Bessel and Bessel modified · Q-functions · Transient state probability · Queueing model

1 Introduction Analytical queue theoretical approaches are becoming of great importance in many real-world situations such as service systems, manufacturing systems, computer and communication systems. Especially in traffic congestion problems in communication networks. The packets are transmitted into batches then immediately queue size becomes reduce which helps to enhance the efficiency of the systems. Several authors have been focused on these issues in different frameworks for making a more effective system. Conolly and Langaris [1] and Tarabia [2] recognized methods that shorten the computational time for state probabilities for emphasizing the calculation of CPU time. Parthasarthy and Lenin [3] discussed the application of transient probabilities to multi-server queues. Jain et al. [4] designated the computing method with different expressions for the transient solution of M/M/1 queues using a probabilistic approach. Abate and Whitt [5] developed some new ways to analyze the transient behavior of the M/M/1 queue via Laplace transform. Abate and Whitt [6, 7] offered some new perspectives on the time-dependent behavior of the M/M/1 queue with J. Kumar (B) · V. Shinde Department of Applied Mathematics, Madhav Institute of Technology & Science, Gwalior, M. P., India e-mail: [email protected] V. Shinde e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_5

51

52

J. Kumar and V. Shinde

performance measures and also described factorial moments of the queue length as functions of time when queue starting at the origin point. Cantrell [8] suggested a numerical procedure for calculating time-dependent performance measures for the M/M/1 queue, such as the mean, the variance and cdf and pdf of the queue length at time t and also described Q-function and generalized Q-function. Jones et al. [9] obtained some method for computing transient state occupancy probabilities of the M/M/1 queue. Jain et al. [10] established a finite queueing model having a single and batch service model for the telecommunication system and also obtained performance measures average queue length, expected idle time and expected busy period. Bertsimas and Nakazato [11] calculated the transient behavior of mixed generalized Erlang distribution queueing system and use the method of stages combined with the separation of variables and root-finding techniques, also obtained the busy period distribution. Ausin et al. [12] studied the transient behavior and duration of a busy period in a single queueing system with both simulated and real data. Grftihs et al. [13] achieved the transient phase probabilities and new generalization of the modified Bessel’s function and also evaluated the mean waiting time in the queues. Kijma [14] studied the transient behavior of Markovian queue such as M/M/S queue and bulkarrival M/M/1 queue and also expressed the transient probabilities of bulk-arrival M/M/1 queue and its busy period density. Chaudhary [15] provided an explicitly transient solution for the multi-server queueing system using generalized Eigenvector matrices. Hanbali and Boxma [16] considered of the transient behavior of a state depended on M/M/1/K queues during the busy period. Parthasarathy and Sudhesh [17] attained transient solutions for M/M/C queueing model with the arrival of N customer. Sharma [18] investigated the transient behavior of different queue models and obtained the state probabilities in the closed from. The present article is arranged as follows: Notations and mathematical description of model are mentioned in Sect. 2. We carried out numerical results have been carried out by using different parameter and plot the graphs in Sect. 3. Finally, concluding remakes and notable features of this investigation are highlighted in Sect. 4.

2 Notations and Mathematical Description 2.1 Notations N α i≥0 t k λ μ Q(a, b) Ik

Number of customers. Position of customers (batch arrival). Initial number of customers. Dependent times. Number of states. Arrival service rate. Service rate. Circular Coverage Function. Bessel Function.

Computational Performance of Server Using the Mx /M/1 Queue …

53

2.2 Mathematical Model Different algorithms have been used for the computation of the transient-state probabilities of the MX /M/1 queueing model. Abate and Whitt [6, 7] have obtained the results for M/M/1 queue model. The modification of the model to reduce the state space that needs to be considered, and thus increases the computational efficiency of the algorithm, as given in Table 1, defines the variables and parameters that are used in the following section.        Q m (t) = e−(αλ+μ)t ρ (n−i)/2 In−i 2 αλμt − In+i 2 αλμt n −(αλ+μ)t

+ρ e

 kp −k/2   Ik 2 αλμt μt k=n+i+1 ∞ 

(1)

where I k is the modified Bessel’s function of order n, as follows:; Ik =

∞  m=0

(z/2)2k+n , k!(n+k)!

n = 0, 1, 2, . . .

and using the Bessel function and its property as given below:

          2k Ik 2 αλμt = Ik−1 2 αλμt − Ik+1 2 αλμt √ 2 αλμt

or        √αλμt    Ik−1 2 αλμt − Ik+1 2 αλμt Ik 2 αλμt = k The last term of the expression (1) can be rewritten as ρ n e−(αλ+μ)t

 kp −k/2   Ik 2 αλμt μt k=n+i+1 ∞ 

Table 1 (i = 4, t = 2, and α = 2) λ = 2, μ = 7, i = 4 and t = 2

λ = 5, μ = 13, i = 4 and t = 2

λ = 6, μ = 12, i = 4 and t = 2

λ = 8, μ = 17, i = 4 and t = 2

Algorithm 1

0.608662

0.204611

0.286026

0.0647789

Algorithm 2

0.428571

0.230769

0.20000

0.0588235

Algorithm 3

0.260855

0.047218

0.0953417

0.00381052

Algorithm 4

1.23294

0.197132

0.394096

0.03769

54

J. Kumar and V. Shinde n −(αλ+μ)t

=ρ e

−k/2 √ ∞        kp αλμt  Ik−1 2 αλμt − Ik+1 2 αλμt μt k k=n+i+1        p −k/2 p 1/2 Ik−1 2 αλμt − Ik+1 2 αλμt

∞ 

= ρ n e−(αλ+μ)t

k=n+i+1

⎡

∞ 

= ρ n e−(αλ+μ)t ⎣

p −(k−1)/2 I

k=n+i+1

 n −(αλ+μ)t

=ρ e

∞ 

∞ 

   k−1 2 αλμt − ρ



k=n+i+1

p

−k/2

   Ik 2 αλμt − ρ

k=n+i

 

p −(k+1)/2 Ik+1 2 αλμt

∞ 

p

−k/2

⎤  ⎦

    Ik 2 αλμt

k=n+i+2

       = e−(αλ+μ)t p (n−i)/2 In+i 2 αλμt + p (n−i−1)/2 In+i+1 2 αλμt  ∞     n −k/2 p Ik 2 αλμt +ρ (1 − ρ) k=n+i+2

⎡

 √  √   ⎤ p (n−i)/2 In+i 2 αλμt + p (n−i−1)/2 In+i+1 2 αλμt + ∞  √  ⎦  Q m (t) = e−(αλ+μ)t ⎣ p −k/2 Ik 2 αλμt ρ n (1 − ρ) k=n+i+2

(2) Identity I: (1 − ρ) =

e−(αλ+μ)t μt

 ∞

   np −n/2 In 2 αλμt

(3)

n=−∞

Proof Consider the generating function of I n (y), we have

1 y x+ x In (y) = exp 2 2 n=−∞ ∞ 

n

Differentiating both sides, we have ∞ 

nx

(n−1)

n=−∞

Substitute x = ∞  n=−∞

n



μ αλ



1 y 1 y x+ × 1− 2 In (y) = exp 2 2 2 x

 √  & y = 2 αλμt in Eq. (4), we get

  μ (n−1)   In 2 αλμt = − exp(αλ + μ) × μt (1 − ρ) αλ

(4)

Computational Performance of Server Using the Mx /M/1 Queue …

(1 − ρ) =

e−(αλ+μ)t μt

 ∞

55

   np −n/2 In 2 αλμt

n=−∞

Identity II: ∞ m+n+i     (αλt)m  (μt)k kp −k/2 Ik 2 αλμt = (k − m) m! k! m=0 k=−∞ k=0 n+i 

(5)

Proof Consider left-hand side, we have −k/2  n+i ∞     αλ (αλμ)(k+2m)/2 t k+2m kp −k/2 Ik 2 αλμt = k μ m!(k + m)! m=0 k=−∞ k=−∞ −k/2  n+i ∞  αλ (αλμ)(k+2m)/2 t k+2m = k (Using Identity I) μ m!(k + m)! m=0 k=−∞ n+i 

=

∞ n+i ∞ m+n+i   (μt)l (αλt)m  (μt)k+m (αλt)m  = , where l = k + m, k (l − m) m! k=−m (k + m)! m! (l)! m=0 m=0 l=0

Hence identity II is proved. Now Eq. (1) can be rewritten as using Identity II        Q m (t) = (1 − ρ)ρ n + e−(αλ+μ)t ρ (n−i)/2 In−i 2 αλμt − In+i 2 αλμt −

n+i    ρ n e−(αλ+μ)t  kp −k/2 Ik 2 αλμt μt k=−∞

(6)

Above equation can also be rewritten using Identity II        Q m (t) = (1 − ρ)ρ n + e−(αλ+μ)t ρ (n−i)/2 In−i 2 αλμt − In+i 2 αλμt −(αλ+μ)t  k+n+i ∞ e (αλt) k  (μt)m (7) − ρn (m − k) μt k! (m)! m=0 k=0 After a simplification of the above expression given by Sharma (see Ref. [18]) the two-dimensional state model and its solution is provided as Q m (t) = (1 − ρ)ρ n + e−(αλ+μ)t ρ n

k+n+i ∞  (αλt) k  k=0

+ e−(αλ+μ)t

∞ 

(αλt)n+k−i (μt)k

k=0

k!

m=0

(k − m)

(μt)m−1 (m)!

1 1 − k!(n + k − i) (n + k)!(k − i)

(8)

56

J. Kumar and V. Shinde

MATLAB programming is used to evaluate the infinite sum of Modified Bessel functions as it is a major cause of a large increase in processing time. For getting better CPU time, we get a finite sum, by introducing a function Q(a, b) with parameters a and b which is defined as ∞

Q(a, b) = ∫ e

a 2 +x 2 2

b

I0 (ax)xdx

(9)

It can also be written as Q(a, b) = e

a 2 +x 2 2

∞    a k k=0

b

Ik (ab)

(10)

Now, considering the right-hand side of Eq. (9), we have ∞

= ∫e ∞

= ∫e b

= =

b

 a 2 +x 2 2

a 2 +x 2 2

I0 (ax)xdx

 ∞  (ax/2)2m xdx (since k + n = m) (m!)2 m=0



∞  (a/2)2m ∞ a2 +x 2 2m 2 ∫ e x xdx (m!)2 b m=0

∞  m (a/2)2m ∞ −y  a2 + x 2 ∫ e 2y − a 2 dy where y = 2 (m!) a2 +b2 2 m=0 2

After successive integration, we have ∞   (a/2)2m  a2 +b2  2m 2(m−1) 2 2(m−2) 2 e b + 2mb + 2 m(m − 1)b + · · · (m!)2 m=0  ∞ ∞ ∞  (ab/2)2m  a   (ab/2)2m−1  a 2  (ab/2)2m−2 a 2 +b2 + + ··· + =e 2 (m!)2 b m=0 (m − 1)! m! b m=0 (m − 2)! m! m=0     a 1  a 2 a 0 a 2 +b2 2 =e I0 (ab) + I−1 (ab) + I−2 (ab) · · · b b b     a 1  a 2 a 0 a 2 +b2 I0 (ab) + I1 (ab) + I2 (ab) . . . (since Ik = I−k ) =e 2 b b b

=

In general Eq. (4), maybe written as

Computational Performance of Server Using the Mx /M/1 Queue …

=e

a 2 +b2 2

∞    a m m=0

Now, substitute a =

√

Q

√

2αλt and b =

√

b

57

Im (ab)

(11)

2μt, we get

∞ −k       2αλt, 2μt = e−(αλ+μ)t ρ 2 Ik 2 αλμt k=0

(12)

Now, Eq. (12) is used in Eq. (2), Jones et al. (see Ref. [9]) established the transient solution of M/M/1 queueing model as √   Q m (t) = ρ n (1 − ρ)Q 2αλt, 2μt ⎡      ⎤ p (n−i)/2 In−i 2 αλμt + p (n−i−1)/2 In+i+1 2 αλμt ⎢ ⎥ ⎢ ⎥ n+i+1 + e−(αλ+μ)t ⎢    ⎥  ⎣ − ρ n (1 − ρ) ⎦ −k/2 p I 2 αλμt k

k=0

(13) Now, Eq. (13) can rewrite in a more compact way by using it give Eq. (9), Generalized Q-function, which we define as Q m (a, b) =

1

∞

a m−1

b

∫e

a 2 +x 2 2

Im−1 (ax)x m dx

(14)

If we choose m = 1 which is equal to 1 − Q m (a, b) = e

∞ k  b

a 2 +b2 2

k=m

a

Ik (ab)

(15)

Now taking the R. H. S. of Eq. (8), we have 1

∞

a m−1

b

∫e

a 2 +b2 2

= =

1

∞

a m−1

b

1 a m−1

∫ xme

1

∞

a 2 +x 2 2

∞   (ax/2)2k+m−1 dx k!(k + m − 1)! k=0 

∫x e a m−1 b ∞  (ax/2)2k+m−1 a 2 +x 2

Im−1 (ax)x d x = m

2

k=0

m

k!(k + m − 1)!

dx

∞  (ax/2)2k+m−1 ∞ 2k+2(m−1) a2 +x 2 ∫x e 2 dx k!(k + m − 1)! b k=0

After simplification, we get

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= 

a 2 +b2 ∞ e 2  (ax/2)2k+m−1 a m−1 k=0 k!(k + m − 1)!



b2k+2(m−1) + 2(k + m − 1)b2k+2(m−2)

+ 22 (k + m − 1)(k + m − 2)b2k+2(m−3) + · · · ⎡ m−1 

m−2 

⎤ ∞ ∞ b (ax/2)2k+m−1 (ax/2)2k+m−2 b + ⎢ a k!(k + m − 1)! a k!(k + m − 2)! ⎥ ⎥ a 2 +b2 ⎢ k=0 k=0 ⎢ ⎥ 2 =e m−3  ⎢ ⎥ ∞ 2k+m−3

(ax/2) b ⎣ ⎦ + ··· + a k!(k + m − 3)! k=0  

m−3

m−2 b m−1 b b b b b a 2 +b2 =e 2 + + + ··· Im−1 Im−2 Im−3 a a a a a a ⎡ 1−m 2−m ⎤ b b b b + I1−m I2−m ⎢ a a 2 +b2 ⎢ a a a ⎥ ⎥ =e 2 ⎢ ⎥ (∵ Ik = I−k )

3−m ⎦ ⎣ b b + ··· + I3−m a a ∞    a k a 2 +b2 =e 2 Ik (ab) (16) b k=1−m Now Eq. (15) can be easily verified by Eq. (16) as Q m (a, b) = e  =e

a 2 +b2 2

e

a 2 +b2 2

∞   m−1    a k a k a 2 +b2 Ik (ab) = e 2 Ik (ab) b b k=1−m k=−∞  ∞   ∞   a k z −1 − Ik (ab) ∵ e 2 (t+t ) = (t)k Ik (z) b k=m k=−∞

a 2 +b2 2

1 − Q m (a, b) = e

a 2 +b2 2

∞ k  b

k=m

Now, substitute a = 1 − Qm

√

2αλt and b =

√

a

Ik (ab)

2μt in Eq. (15), we get

∞ √       −k 2αλt, 2μt = e−(αλ+μ)t ρ 2 Ik 2 αλμt

(17)

k=0

Using equation (17) in equation (2), we get  Q m (t) = e

−(αλ+μ)t

 √  √   2  + p (n−i−1)/2 αλμt p (n−i)/2 In−i 2 αλμt √ In+i+1  √ (18) +ρ n (1 − ρ) 1 − Q n+i+2 2αλt, 2μt

Computational Performance of Server Using the Mx /M/1 Queue …

59

Table 2 (i = 4, t = 2, and α = 3) λ = 0.7, μ = 5, i = 4 and t = 2

λ = 0.9, μ = 5.5, i = 4 and t = 2

λ = 1.5, μ = 6, i = 4 and t = 2

λ = 0.5, μ = 2.2, i = 4 and t = 2

Algorithm 1

1.0937

0.331098

0.358925

0.829124

Algorithm 2

0.58

0.509091

0.25

0.318182

Algorithm 3

0.634344

0.168559

0.0897313

0.263818

Algorithm 4

4.78348

0.980446

0.368401

1.07996

Table 3 (i = 2, t = 4, and α = 2) λ = 2, μ = 7, i = 2 and t = 4

λ = 5, μ = 13, i = 2 and t = 4

λ = 6, μ = 12, i = 2 and t = 4

λ = 8, μ = 17, i = 2 and t = 4

Algorithm 1

0.205734

0.0540504

−0.0690807

Algorithm 2

0.428571

0.230769

3.84875e–024

0.0588235

Algorithm 3

0.0881719

0.0168303

4.8122e–046

0.00308253

Algorithm 4

0.238141

0.0540504

0.0345403

0.0286959

0.0437681

Note Above calculation is executed on Celeron 1 G.Hz. and the CPU times are given in seconds and here, we assume, Algorithm 1: Using Eq. 2, Algorithm 2: Using Eq. 8, Algorithm 3: Using Eq. 13 and Algorithm 4: Using Eq. 18. Tables 1, 2 and 3 are showing CPU timings for the above algorithms.

The finite sum of Bessel function and Q-function in Eq. (13) is being replaced by the Generalized Q-function. Qm (t) at time (t = 0) is evaluated using Eq.(8) Q m (t) = (1 − ρ)ρ n + e−(αλ+μ)t ρ n

k+n+i ∞  (αλt) k  k=0

∞  + e−(αλ+μ)t (αλt)n+k−i (μt)k k=0

k!

m=0

(k − m)

(μt)m−1 (m)!

1 1 − k!(n + k − i) (n + k)!(k − i)

(19)

3 Numerical Approach Numerical illustration is carried out using MATLAB programming. We calculate the probability of observing n customers at time t, given that there were i customers at time 0, the service rate is μ, the interarrival rate is λ and α is bulk (batch size) using queueing model with different expressions of equations for computing the probability distribution for the MX /M/1 queueing system. We compare the computational performance efficiency of these different expressions given in the previous section (Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12).

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We plotted 12 figures between transient probability distribution versus the number of systems with different values of λ and μ, which are described in Tables 1, 2, and 3. We evaluate the performance of four algorithms for fixed values of i = 4, t = 2 and α = 2, i = 4, t = 2, and α = 3 and i = 2, t = 4, and α = 2 as mentioned in Tables 1, 2, and 3, respectively. All algorithms are implemented for computing the transient probability distribution using MATLAB (Package 6.2) and executed on the HP machine (Core i5) and we also evaluated CPU timing. Consequently, it has been noted that Algorithm 3 provides better outcomes. 0.35

Probabilty of Algorithm

0.3 0.25 0.2 0.15 0.1 0.05 0

25

20

15

10

5

0

Number of Units /Iterations

Fig. 1 Probability versus No. of iterations 0.05

Probabilty of Algorithm

0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0

0

5

10

15

20

25

30

35

Number of Units /Iterations

Fig. 2 Probability versus No. of iterations

40

45

Computational Performance of Server Using the Mx /M/1 Queue … x 10

9

61

-3

8

Probabilty of Algorithm

7 6 5 4 3 2 1 0

0

10

20

30

50

40

60

Number of Units /Iterations

Fig. 3 Probability versus No. of iterations 6

x 10

-3

Probabilty of Algorithm

5

4

3

2

1

0

0

10

20

30

40

50

Number of Units /Iterations

Fig. 4 Probability versus No. of iterations

60

70

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J. Kumar and V. Shinde 0.7

Probabilty of Algorithm

0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10

15

Number of Units /Iterations

Fig. 5 Probability versus No. of iterations 0.45 0.4

Probabilty of Algorithm

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

2

4

6

8

10

12

14

Number of Units /Iterations

Fig. 6 Probability versus No. of iterations

16

18

Computational Performance of Server Using the Mx /M/1 Queue …

63

0.09

Probabilty of Algorithm

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

5

10

15

20

25

Number of Units /Iterations

Fig. 7 Probability versus No. of iterations 0.35

Probabilty of Algorithm

0.3 0.25 0.2 0.15 0.1 0.05 0

0

10

5

15

20

Number of Units /Iterations

Fig. 8 Probability versus No. of iterations

25

64

J. Kumar and V. Shinde 0.09

Probabilty of Algorithm

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

5

0

10

15

25

20

Number of Units /Iterations

Fig. 9 Probability versus No. of iterations

Probabilty of Algorithm

0.025

0.02

0.015

0.01

0.005

0

0

5

10

15

20

25

30

35

Number of Units /Iterations

Fig. 10 Probability versus No. of iterations

40

45

Computational Performance of Server Using the Mx /M/1 Queue …

65

0.5

Probabilty of Algorithm

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

20

40

60

80

100

120

50

60

Number of Units /Iterations

Fig. 11 Probability versus No. of iterations

x 10

3.5

-3

Probabilty of Algorithm

3 2.5 2 1.5 1 0.5 0

0

10

20

30

40

Number of Units /Iterations

Fig. 12 Probability versus No. of iterations

4 Conclusion In this investigation, we derived the different expressions for computing the transient probability solution of the Mx /M/1 queueing model. Various results have been obtained for transient probability distribution using MATLAB programming. Generalized Q-function is the most efficient expression for all algorithms. We examine the four algorithms by tuning different parameters. Validation of results has been shown by using numerical illustrations and graphs. It has been observed that Algorithm

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3 is more efficient rather than the other algorithms for the computational performance and also, we applied this algorithm in traffic, manufacturing and obtained the performance of any complicate system.

References 1. B.W. Conolly, C. Langairs, On a new formula for the transient state probabilities for M/M/1 queues and computational implications. J. Appl. Probab. 30, 237–246 (1993) 2. A.M.K. Tarabia, A new formula for the transient behavior of a non-empty M/M/1/∞ queue. Appl. Math. Comput. 132(1), 1–10 (2002) 3. P.R. Parthasarhy, R.B. Lenin, On the numerical solution of transient probabilities of a statedependent multi-server queue. Int. J. Comput. Math. 66(3), 241–255 (1998) 4. J.L. Jain, S.G. Mohanty, A.J. Meitei, Transient Solution of M/M/1 Queues: A Probabilistic Approach. APORS-2003, vol. 1 (2004), pp 74–81 5. J. Abate, W. Whitt, Transient behavior of M/M/1 queue via Laplace Transform. Adv. Appl. Probab. 25, 145–178 (1988) 6. J. Abate, W. Whitt, Calculating time dependent performance measure of the M/M/1 queue. IEEE Trans. Commun. 37, 1102–1104 (1989) 7. J. Abate, W. Whitt, The transient behavior of the M/M/1 queue: starting at the origin. Queueing Syst. 2, 41–65 (1987) 8. P.E. Cantrell, Computation of the transient M/M/1 queue cdf, pdf and mean with generalized Q-function. IEEE Trans. Commun. 34(8), 814–817 (1986) 9. S.K. Jones, R.K. Cavin III, D.A. Johnston, An efficient computational procedure for the evaluation of the M/M/1 transient state occupancy probabilities. IEEE Trans. Commun. 28, 2019–2020 (1986) 10. M. Jain, G.C. Sharma, V.K. Saraswat, Rakhee, Transient analysis of a telecommunication system using state dependent Markovian queue under Bi-level control policy. J. King Saud Univ. Eng. Sci. 20(1), 77–90 (2009) 11. D.J. Bertsimas, D. Nakazato, Transient and busy period analysis of the GI/G/1 queue: the method of stages. Queueing Syst. 10, 153–184 (1992) 12. M.C. Ausin, M.P. Wiper, R.E. Lillo, Bayesion prediction of the transient behavior and busy period in short and long-tallied GI/G/1 queeing system. Comput. Stat. Data Anal. 52, 1615– 1635 (2008) 13. J.D. Grftihs, G.M. Leoneko, J.E. Willians, The transient solution to M/Ek /1 queue. Oper. Res. Lett. 34(3), 349–354 (2006) 14. M. Kijma, The transient solution to a class of Markovian queues. Comput. Math. Appl. 24(1 & 2), 17–24 (1992) 15. M.L. Chaudhary, Y.Q. Zhao, Transient solution of same multi-server queueing systems with finite spaces. Int. Trans. Oper. Res. 6(2), 161–182 (1999) 16. A.A. Hanbali, O. Boxma, Busy period analysis of the state dependent M/M/1/K queue. Oper. Res. Lett. 38(1), 1–6 (2010) 17. P.R. Parthasarathy, R. Sudhesh, Transient solution of multi-server Poison queue with N-policy. Int. J. Comput. Math. Appl. 55, 550–562 (2008) 18. O.P. Sharma, Markovian Queues (Eillis Hoewood Chichester Ltd, 1990)

Quantum Codes from the Cyclic Codes Over F p [v, w]/v 2 − 1, w2 − 1, vw − wv Habibul Islam, Om Prakash and Ram Krishna Verma

Abstract In this article, for any odd prime p, we study the cyclic codes over the finite ring R = F p [v, w]/v 2 − 1, w 2 − 1, vw − wv to obtain the quantum codes over F p . We obtain the necessary and sufficient condition for cyclic codes which contain their duals and as an application, some new quantum codes are presented at the end of the article. Keywords Cyclic code · Quantum code · Self-orthogonal code · Gray map. 2010 MSC 94B15 · 94B05 · 94B60.

1 Introduction The quantum codes are used in quantum communication and quantum computing to protect the quantum information from the noise occurred during the transmission in the channel. The quantum error-correcting codes from the classical errorcorrecting codes are rapidly developing after the remarkable work presented by Shor [17], in 1995. It is well studied over finite fields in [4, 7, 10, 11, 15]. In 2015, Gao [6] presented some new quantum codes over Fq from the cyclic codes over Fq + vFq + v 2 Fq + v 3 Fq . Meanwhile, Dertli et al. [5] obtained some binary quantum codes derived from the cyclic codes over F2 + uF2 + vF2 + uvF2 and Ashraf and Mohammad [1] studied quantum codes which are obtained from cyclic codes over Fq + uFq + vFq + uvFq . In 2016, Ozen et al. [14] explored several ternary H. Islam (B) · O. Prakash · R. K. Verma Department of Mathematics, Indian Institute of Technology Patna, Patna 801 106, India e-mail: [email protected] O. Prakash e-mail: [email protected] R. K. Verma e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_6

67

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H. Islam et al.

quantum codes from the cyclic codes over F3 + uF3 + vF3 + uvF3 . Recently, many researchers have obtained some new quantum codes over F p from the classical cyclic and constacyclic codes, we refer [2, 9, 12, 13, 16]. In this article, for any odd prime p, we consider the cyclic codes over F p [v, w]/ v 2 − 1, w 2 − 1, vw − wv to obtain some new quantum codes over F p . Also, we generalize the work of [14] for any odd prime p and present some new quantum codes over F7 at the end of this article.

2 Preliminary For the odd prime p, let F p be the field of order p and R = F p [v, w]/v 2 − 1, w 2 − 1, vw − wv. Then R is a finite commutative ring with p 4 elements and characteristic p. A linear code C of length n is defined as an R-submodule of R n and elements of C are called codewords. The dual (Euclidean) code of the linear code C is denoted by C ⊥ and defined as C ⊥ = {a ∈ R n | a · b = 0, ∀ b ∈ C}, where the inner product of any two codewords a = (a0 , a1 , . . . , an−1 ), b = (b0 , b1 , . . . , bn−1 ) ∈ C n−1 ai bi . The code C is said to be self-orthogonal if C ⊆ C ⊥ and is a · b = i=0 ⊥ self-dual if C = C. Also, the ring R can be expressed as F p + vF p + wF p + vwF p where v 2 = w 2 = 1, vw = wv and any element r ∈ R is of the form r = a + vb + wc + vwd where a, b, c, d ∈ F p . Let ξ ∈ F p such that 4ξ ≡ 1 (mod p) and μ1 = ξ(1 + v + w + vw), μ2 = ξ(1 − v + w − vw), μ3 = ξ(1 + v − w − vw), μ4 = ξ(1 − v − w + vw). Then 1 = μ1 + μ2 + μ3 + μ4 , and  μi μ j =

μi , if i = j. 0, if i = j.

Hence, by Chinese Remainder Theorem, we have R = μ1 R ⊕ μ2 R ⊕ μ3 R ⊕ μ4 R ∼ = μ1 F p ⊕ μ2 F p ⊕ μ3 F p ⊕ μ4 F p . Thus, every element of R can be uniquely expressed as a + vb + wc + vwd = k1 μ1 + k2 μ2 + k3 μ3 + k4 μ4 , for ki ∈ F p , i = 1, 2, 3, 4. We define a map φ : R −→ F4p by φ(a + vb + wc + vwd) = (a + b + c + d, a − b + c − d, a + b − c − d, a − b − c + d),

(2.1) for a, b, c, d ∈ F p . The map φ can be extended to R n naturally. Here, the Hamming weight w H (c) is defined as the number of nonzero components of the codeword c = (c0 , c1 , . . . , cn−1 ) ∈ C and the distance between two codewords d H (c, c ) = w H (c − c ). The Hamming distance for the code C is d H (C) = min{d H (c, c ) |

Quantum Codes from the Cyclic Codes Over F p [v, w]/v 2 − 1, w 2 − 1, vw − wv

69

c = c , ∀ c, c ∈ C}. The Gray weight of an element r = a + vb + wc + vwd ∈ R is define as wG (r ) = w H (φ(r )) = w H (a + b + c + d, a − b + c − d, a + b − c − d, a − b − c + d) and Gray weight for r = (r0 , r1 , . . . , rn−1 ) ∈ R n is wG (r ) =  n−1

i=0 wG (ri ). The Gray distance between any two codewords c, c is dG (c, c ) =

wG (c − c ) and Gray distance for the code C is dG (C) = min{dG (c, c ) | c = c , c, c ∈ C}.

3 Linear Codes Over R In this section, we present some useful results on linear codes over R which are taken from [8]. Therefore, we omit the proofs of these results here. Theorem 3.1 The map φ defined in Eq. (2.1) is linear and isometric from (R n , Gray distance) to (F4n p , Hamming distance). Theorem 3.2 Let C be a (n, k, dG ) linear code over R. Then φ(C) is a (4n, k, d H ) linear code over F p where d H = dG . Theorem 3.3 Let C be a linear code of length n over R. Then φ(C ⊥ ) = (φ(C))⊥ and φ(C) is self-orthogonal if C is self-orthogonal. Further, C is self-dual if and only if φ(C) is self-dual. Let Ai be the linear code for i = 1, 2, 3, 4. We denote A1 ⊕ A2 ⊕ A3 ⊕ A4 = {a1 + a2 + a3 + a4 | ai ∈ Ai ∀ i} and A1 ⊗ A2 ⊗ A3 ⊗ A4 = {(a1 , a2 , a3 , a4 ) | ai ∈ Ai ∀ i}. Let C be a linear code of length n over R. Define C1 = {a + b + c + d ∈ Fnp | a + vb + wc + vwd ∈ C}, C2 = {a − b + c − d ∈ Fnp | a + vb + wc + vwd ∈ C}, C3 = {a + b − c − d ∈ Fnp | a + vb + wc + vwd ∈ C}, C4 = {a − b − c + d ∈ Fnp | a + vb + wc + vwd ∈ C}. Then Ci s are linear codes of length n over F p for i = 1, 2, 3, 4. Theorem 3.4 Let C be a linear code of length n over R. Then φ(C) = C1 ⊗ C2 ⊗ C3 ⊗ C4 and | C |=| C1 || C2 || C3 || C4 |. Corollary 3.1 Let Mi be the generator matrix of the ⎛ linear⎞code Ci for i = 1, 2, 3, 4. μ1 M1 ⎜μ2 M2 ⎟ ⎟ Then the generator matrix for C is given by M = ⎜ ⎝μ3 M3 ⎠. μ4 M4 Corollary 3.2 If φ(C) = C1 ⊗ C2 ⊗ C3 ⊗ C4 , then C = μ1 C1 ⊕ μ2 C2 ⊕ μ3 C3 ⊕ μ4 C4 . Corollary 3.3 Let C = μ1 C1 ⊕ μ2 C2 ⊕ μ3 C3 ⊕ μ4 C4 be a linear code of length n over R where Ci s are [n, ki , dG (Ci )] linear codes for i = 1, 2, 3, 4. Then φ(C) is a 4 [4n, i=1 ki , min{dG (Ci ) | i = 1, 2, 3, 4}] linear code over F p .

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Theorem 3.5 Let C = μ1 C1 ⊕ μ2 C2 ⊕ μ3 C3 ⊕ μ4 C4 be a linear code of length n over R. Then C ⊥ = μ1 C1⊥ ⊕ μ2 C2⊥ ⊕ μ3 C3⊥ ⊕ μ4 C4⊥ . Moreover, C is self-dual if and only if Ci is self-dual for i = 1, 2, 3, 4.

4 Cyclic Codes Over R Definition 4.1 A linear code C of length n over R is said to be cyclic if for any c = (c0 , c1 , . . . , cn−1 ) ∈ C, we have σ (c) := (cn−1 , c0 , . . . , cn−2 ) ∈ C. Here, the operator σ is known as cyclic shift. Let C be a cyclic code of length n over R. We identify each codeword c = (c0 , c1 , . . . , cn−1 ) ∈ C with a polynomial c(x) ∈ R[x]/x n − 1 under the correspondence c = (c0 , c1 , . . . , cn−1 ) −→ c(x) = (c0 + c1 x + · · · + cn−1 x n−1 ) mod x n − 1. In this polynomial representation of C, one can easily verify the next result. Theorem 4.1 Let C be a linear code of length n over R. Then C is a cyclic code if and only if it is an ideal of the ring R[x]/x n − 1. Theorem 4.2 Let C = μ1 C1 ⊕ μ2 C2 ⊕ μ3 C3 ⊕ μ4 C4 be a linear code of length n over R. Then C is a cyclic code if and only if Ci is a cyclic code of length n over F p for i = 1, 2, 3, 4. Proof Let C be a cyclic code of length n over R. Let a = (a0 , a1 , . . . , an−1 ) ∈ C1 , b = (b0 , b1 , . . . , bn−1 ) ∈ C2 , c = (c0 , c1 , . . . , cn−1 ) ∈ C3 , d = (d0 , d1 , . . . , dn−1 ) ∈ C4 and ri = μ1 (ai + bi + ci + di ) + μ2 (ai − bi + ci − di ) + μ3 (ai + bi − ci − di ) + μ4 (ai − bi − ci + di ) for i = 0, 1, . . . , n − 1. Then r = (r0 , r1 , . . . , rn−1 ) ∈ C. Therefore, σ (r ) = (rn−1 , r0 , . . . , rn−2 ) ∈ C where σ (r ) = μ1 σ (a) + μ2 σ (b) + μ3 σ (c) + μ4 σ (d) ∈ C = μ1 C1 ⊕ μ2 C2 ⊕ μ3 C3 ⊕ μ4 C4 . Hence, σ (a) ∈ C1 , σ (b) ∈ C2 , σ (c) ∈ C3 , σ (d) ∈ C4 . Consequently, Ci is a cyclic code of length n over F p , for i = 1, 2, 3, 4. Conversely, let Ci be a cyclic code of length n over F p for i = 1, 2, 3, 4. Let r = (r0 , r1 , . . . , rn−1 ) ∈ C where ri = μ1 (ai + bi + ci + di ) + μ2 (ai − bi + ci − di ) + μ3 (ai + bi − ci − di ) + μ4 (ai − bi − ci + di ) for i = 0, 1, . . . , n − 1. Then a = (a0 , a1 , . . . , an−1 ) ∈ C1 , b = (b0 , b1 , . . . , bn−1 ) ∈ C2 , c = (c0 , c1 , . . . , cn−1 ) ∈ C3 , d = (d0 , d1 , . . . , dn−1 ) ∈ C4 . Therefore, σ (a) ∈ C1 , σ (b) ∈ C2 , σ (c) ∈ C3 , σ (d) ∈ C4 . Now, σ (r ) = μ1 σ (a) + μ2 σ (b) + μ3 σ (c) + μ4 σ (d) ∈ μ1 C1 ⊕ μ2 C2 ⊕ μ3 C3 ⊕  μ4 C4 = C. This shows that C is a cyclic code of length n over R. Theorem 4.3 Let C = μ1 C1 ⊕ μ2 C2 ⊕ μ3 C3 ⊕ μ4 C4 be a cyclic code of length n over R. Then there exists a polynomial g(x) ∈ R[x] such that C = g(x) and g(x) | x n − 1. Proof Since C is a cyclic code, by Theorem 4.2, Ci is a cyclic code of length n over F p for i = 1, 2, 3, 4. Let Ci = gi (x) and gi (x) | x n − 1 for i = 1, 2, 3, 4. Then μ1 g1 (x), μ2 g2 (x), μ3 g3 (x), μ4 g4 (x) are the generators of C. Let g(x) = μ1 g1 (x) +

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μ2 g2 (x) + μ3 g3 (x) + μ4 g4 (x). Then g(x) ⊆ C. Also, μ1 g(x) = μ1 g1 (x) ∈ g(x), μ2 g(x) = μ2 g2 (x) ∈ g(x), μ3 g(x) = μ3 g3 (x) ∈ g(x), μ4 g(x) = μ4 g4 (x) ∈ g(x). Therefore, C ⊆ g(x) and hence C = g(x). Further, since gi (x) | x n − 1, we have x n − 1 = gi (x)h i (x) for h i (x) ∈ F p [x], i = 1, 2, 3, 4. Now, g(x)[μ1 h 1 (x) + μ2 h 2 (x) + μ3 h 3 (x) + μ4 h 4 (x)] = μ1 g1 (x)h 1 (x) + μ2 g2 (x)h 2 (x) + μ3 g3 (x)h 3 (x) + μ4 g4 (x)h 4 (x) = x n − 1. This shows that g(x) |  x n − 1. Corollary 4.1 Every ideal of R[x]/x n − 1 is principally generated. Corollary 4.2 Let C = μ1 C1 ⊕ μ2 C2 ⊕ μ3 C3 ⊕ μ4 C4 be a cyclic code of length n over R and gi (x) be the generator of the code Ci such that x n − 1 = gi (x)h i (x) for i = 1, 2, 3, 4. Then 1. C ⊥ = μ1 C1⊥ ⊕ μ2 C2⊥ ⊕ μ3 C3⊥ ⊕ μ4 C4⊥ is a cyclic code of length n over R. 2. C ⊥ = μ1 h ∗1 (x) + μ2 h ∗2 (x) + μ3 h ∗3 (x) + μ4 h ∗4 (x) where h i∗ (x) is the reciprocal polynomial of h i (x), i.e., h i∗ (x) = x deg(h i (x)) h i (1/x) for i = 1, 2, 3, 4. 4 3. | C ⊥ |= p i=1 deg(gi (x)) . 4. C is a self-dual cyclic code if and only if Ci is a self-dual cyclic code over F p for i = 1, 2, 3, 4.

5 Quantum Codes Over F p Recall that a p-ary quantum code of length n and size K is a K -dimensional subspace of p n -dimensional Hilbert space (C p )⊗n , where p is a prime. The standard notation for a quantum code is [[n, k, d]] p , where n is the length, d is the minimum distance, and K = p k . Toward the quantum codes, we include two important results of [7] and [4] as below. Lemma 5.1 ([7], Theorem 3) Let C1 = [n, k1 , d1 ]q and C2 = [n, k2 , d2 ]q be two linear codes over G F(q) with C2⊥ ⊆ C1 . Then there exists a quantum error-correcting code C = [[n, k1 + k2 − n, d]] where d = min{wt (v) : v ∈ (C1 \C2⊥ ) ∪ (C2 \C1⊥ )} ≥ min{d1 , d2 }. Further, if C1⊥ ⊆ C1 , then there exists a quantum error-correcting code C = [[n, 2k1 − n, d1 ]], where d1 = min{wt (v) : v ∈ C1 \C1⊥ }. Lemma 5.2 ([4], Theorem 13) The cyclic code C of length n over F p with the generator polynomial g(x) contains its dual code if and only if x n − 1 ≡ 0 (mod g(x)g ∗ (x)), where g ∗ (x) is the reciprocal polynomial of g(x). Theorem 5.1 Let C = μ1 C1⊕ μ2 C2 ⊕ μ3 C3 ⊕ μ4 C4 be a cyclic code of length n 4 μi gi (x), where gi (x) is the generator of Ci , for over R and C = g(x) =  i=1 ⊥ i = 1, 2, 3, 4. Then C ⊆ C if and only if

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x n − 1 ≡ 0 (mod gi (x)gi∗ (x)), where gi∗ (x) is the reciprocal polynomial of gi (x), for i = 1, 2, 3, 4. Proof Let x n − 1 ≡ 0 (mod gi (x)gi∗ (x)), for i = 1, 2, 3, 4. Then, by Lemma 5.2, we 4 have Ci⊥ ⊆ Ci and hence μi Ci⊥ ⊆ μi Ci , for i = 1, 2, 3, 4. Thus, C ⊥ = i=1 μi Ci⊥ ⊆ 4 i=1 μi Ci = C. 4 4 Conversely, let C ⊥ ⊆ C. Then i=1 μi Ci⊥ ⊆ i=1 μi Ci . Since Ci is a cyclic code over F p such that μi Ci ≡ C (mod μi ), for i = 1, 2, 3, 4, then Ci⊥ ⊆ Ci , for i = 1, 2, 3, 4. Again, by Lemma 5.2, we have x n − 1 ≡ 0 (mod gi (x)gi∗ (x)), where gi∗ (x) is the reciprocal polynomial of gi (x), for i = 1, 2, 3, 4.



Corollary 5.1 Let C = μ1 C1 ⊕ μ2 C2 ⊕ μ3 C3 ⊕ μ4 C4 be a cyclic code of length n over R. Then C ⊥ ⊆ C if and only if Ci⊥ ⊆ Ci , for i = 1, 2, 3, 4. Now, the Theorem 5.1 and Lemma 5.1 help us to obtain quantum codes as given below: Theorem 5.2 Let C = μ1 C1 ⊕ μ2 C2 ⊕ μ3 C3 ⊕ μ4 C4 be a cyclic code of length n over R and [4n, k, dG ] be the parameter of φ(C), where dG is the minimum Gray distance of C. If C ⊥ ⊆ C, then there exists a quantum code with parameter [[4n, 2k − 4n, dG ]] over F p . Now, by using Theorem 5.2, we construct several examples to validate our technique. All computations are carried out by using the Magma software [3]. Example 5.1 Let p = 3, n = 33, and R = F3 [v, w]/v 2 − 1, w 2 − 1, vw − wv. Now, in F3 [x] x 33 − 1 = (x + 2)3 (x 5 + 2x 3 + x 2 + 2x + 2)3 (x 5 + x 4 + 2x 3 + x 2 + 2)3 . 4 Let gi (x) = x 5 + 2x 3 + x 2 + 2x + 2 for i = 1, 2, 3, 4. Then C =  i=1 μi gi (x) is a cyclic code of length 33 over R and [132, 112, 2] is the parameter of φ(C). Also, gi (x)gi∗ (x) divides x 33 − 1 for i = 1, 2, 3, 4, and hence C ⊥ ⊆ C. Thus, by Theorem 5.2, we have the quantum code [[132, 92, 2]]3 . Example 5.2 Let p = 5, n = 31, and R = F5 [v, w]/v 2 − 1, w 2 − 1, vw − wv. Now, in F5 [x] x 31 − 1 =(x + 4)(x 3 + x + 4)(x 3 + 2x + 4)(x 3 + x 2 + x + 4)(x 3 + x 2 + 3x + 4) (x 3 + 2x 2 + x + 4)(x 3 + 2x 2 + 4x + 4)(x 3 + 3x 2 + 4)(x 3 + 4x 2 + 4) (x 3 + 4x 2 + 3x + 4)(x 3 + 4x 2 + 4x + 4). 4 Let gi (x) = x 3 + 2x 2 + x + 4 for i = 1, 2, 3, 4. Then C =  i=1 μi gi (x) is a cyclic code of length 31 over R and φ(C) is a [124, 112, 3] linear code over F5 . Since

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Table 1 Some New Quantum Codes Over F7 g3 = g4 x2 + 4

φ(C)

12

g1 = g2 x +3

[48, 42, 2]

[[48, 36, 2]]7

21

(x + 3)2 (x + 5)

(x + 3)(x + 5)2

[84, 72, 3]

[[84, 60, 3]]7

24

(x + 3)(x 2 + 6x + 4)

(x + 5)(x 2 + 5x + 2)

[96, 84, 3]

[[96, 72, 3]]7

27

(x + 5)(x 9 + 5)

[108, 68, 3]

[[108, 28, 3]]7

(x + 3)(x 2 + 3x + 6)

[192, 180, 3]

[[192, 168, 3]]7

49

(x + 3)(x 9 + 3) (x + 4)(x 2 + 2x + 3) (x + 6)8

(x + 6)8

[196, 164, 3]

[[196, 132, 3]]7

49

(x + 6)15

(x + 6)15

[196, 136, 4]

[[196, 76, 4]]7

56

(x + 6)(x 2 + 4x + 1)3

(x + 1)(x 2 + 3x + 1)3

[224, 196, 4]

[[224, 168, 4]]7

57

(x + 5)(x 3 + 6x 2 + 5x + 6)

(x + 3)(x 3 + 5x 2 + 5x + 6)

[228, 212, 3]

[[228, 196, 3]]7

63

(x + 3)2 (x 3 + 5)

(x + 5)2 (x 3 + 3)

[252, 232, 3]

[[252, 212, 3]]7

n

48

[[n, k, d]] p

for i = 1, 2, 3, 4, gi (x)gi∗ (x) divides x 31 − 1, by Theorem 5.1, we have C ⊥ ⊆ C. Again, by Theorem 5.2, there exists a quantum code [[124, 100, 3]]5 . In Table 1, we present some new quantum codes over F7 . First column of Table 1 represents the length of the cyclic codes, fourth and fifth column represents the parameters of Gray images and associated quantum codes, respectively. Also, in the second and third columns, gi s represent the generator polynomials for the code C.

6 Conclusion In this article, we have studied self-orthogonal cyclic codes over R and by using their properties, we obtained many quantum codes in which some of them are new in the literature. In Table 1, some new quantum codes over F7 are obtained and similarly one can also find more new quantum codes over F p Acknowledgements The authors are thankful to the University Grants Commission (UGC) and the Council of Scientific & Industrial Research (CSIR), Govt. of India for financial supports and the Indian Institute of Technology Patna for providing research facilities. Further, the authors would like to thank the anonymous referee(s) for their valuable comments to improve the presentation of the article.

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References 1. M. Ashraf, G. Mohammad, Quantum codes from cyclic codes over Fq + uFq + vFq + uvFq . Quantum Inf. Process. 15(10), 4089–4098 (2016) 2. M. Ashraf, G. Mohammad, Quantum codes over F p from cyclic codes over F p [u, v]/u 2 − 1, v 3 − v, uv − vu. Cryptogr. Commun. (2018). https://doi.org/10.1007/s12095-018-0299-0 3. W. Bosma, J. Cannon, Handbook of Magma Functions (University of Sydney, 1995) 4. A.R. Calderbank, E.M. Rains, P.M. Shor, N.J.A. Sloane, Quantum error correction via codes over G F(4). IEEE Trans. Inform. Theory 44, 1369–1387 (1998) 5. A. Dertli, Y. Cengellenmis, S. Eren, On quantum codes obtained from cyclic codes over A2 . Int. J. Quantum Inf. 13(3), 1550031 (2015) 6. J. Gao, Quantum codes from cyclic codes over Fq + vFq + v 2 Fq + v 3 Fq . Int. J. Quantum Inf. 13(8) (2015), 1550063(1-8) 7. M. Grassl, T. Beth, On optimal quantum codes. Int. J. Quantum Inf. 2, 55–64 (2004) 8. H. Islam, R.K. Verma, O. Prakash, A family of constacyclic codes over F pm [v, w]/v 2 − 1, w 2 − 1, vw − wv. Int. J. Inf. Coding Theory (2018) 9. H. Islam, O. Prakash, Quantum codes from the cyclic codes over F p [u, v, w]/u 2 − 1, v 2 − 1, w 2 − 1, uv − vu, vw − wv, uw − wu. J. Appl. Math. Comput. 60(1–2), 625–635 (2019) 10. X. Kai, S. Zhu, Quaternary construction of quantum codes from cyclic codes over F4 + uF4 . Int. J. Quantum Inf. 9, 689–700 (2011) 11. R. Li, Z. Xu, X. Li, Binary construction of quantum codes of minimum distance three and four. IEEE Trans. Inf. Theory 50, 1331–1335 (2004) 12. J. Li, J. Gao, Y. Wang, Quantum codes from (1 − 2v)- constacyclic codes over the ring Fq + uFq + vFq + uvFq . Discrete Math. Algorithms Appl. 10(4), 1850046 (2018) 13. F. Ma, J. Gao, F.W. Fu, Constacyclic codes over the ring F p + vF p + v 2 F p and their applications of constructing new non-binary quantum codes. Quantum Inf. Process. 17, 122 (2018). https://doi.org/10.1007/s11128-018-1898-6 14. M. Ozen, N.T. Ozzaim, H. Ince, Quantum codes from cyclic codes over F3 + uF3 + vF3 + uvF3 . Int. Conf. Quantum Sci. Appl. J. Phys. Conf. Ser. 766, 012020-1-012020-6 (2016) 15. J. Qian, W. Ma, W. Gou, Quantum codes from cyclic codes over finite ring. Int. J. Quantum Inf. 7, 1277–1283 (2009) 16. A.K. Singh, S. Pattanayek, P. Kumar, On quantum codes from cyclic codes over F2 + uF2 + u 2 F2 . Asian-Eur. J. Math. 11(1), 1850009 (2018) 17. P.W. Shor, Scheme for reducing decoherence in quantum memory. Phys. Rev. A 52, 2493–2496 (1995)

Effect of Sterile Insect Technique on Dynamics of Stage-Structured Model Under Immigration Sumit Kaur Bhatia, Sudipa Chauhan and Priyanka Arora

Abstract A Stage-Structured Model has been proposed to study the effect of discharge of irradiated male insects and the immigration of wild insects using the Sterile Insect Technique (SIT). The release of sterile male insects will replace natural insects from the environment. The Logistic Growth Rate has been considered for the Larvae population. We have studied the dynamics of the model in three cases. In the first case, the model has been considered without the discharge of sterile male and immigration has not been taken into account. In the second case, the release of sterile male insect has been considered but immigration has not been taken into consideration. Lastly, in the third model, both immigration and discharge of sterile male have been taken into account. In all three cases, equilibrium points have been evaluated and stability analysis has been done. Thresholds for the sterile male population have been obtained which can be helpful in understanding and implementing the SIT. Keywords Immigration · Sterile insect technique · Prey–predator model · Equilibrium points · Stability analysis

1 Introduction Insect pests contribute to a high prevalence of undernourishment in the world. As part of environment-control tactics, SIT has proven to be a very effective tool for pest management. Knipling [1] formulated a model in which natural reproductive performances of insects are damaged by mutagens and thus rendering insects sterile. In order to copulate with indigenous insects in the current setting, they are released S. K. Bhatia · S. Chauhan · P. Arora (B) Amity Institute of Applied Sciences, Amity University, Noida, India e-mail: [email protected] S. K. Bhatia e-mail: [email protected] S. Chauhan e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_7

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in very large numbers in the setting. Although indigenous female mates with sterile males and will also generate eggs but such eggs will not hatch and if sterile insects are further expanded in big numbers then mostly crosses will be sterile, as time increases, the amount of indigenous insects will decrease thus increasing the sterile to fertile ratio, leading to extinction of indigenous population. This technique is known as Sterile Insect Technique. SIT has a distinctive biological benefit that suits them well with the notion of eradication, i.e., their efficacy improves as the population of pests decreases. In SIT, the amount of sterile males produced and their price is relative to population amount, and as the cost-to-account ratio declines with a reduction in the amount of the pest population, it can be a cost-effective method when integrated with a traditional pesticide control technique. In 1958 in Florida, SIT was first used efficiently to manage Screwworm fly (Cochliomya omnivorax) [2]. In the 1980s, Mexico and Belize eliminated their screwworm problems with SIT. A digit of mathematical models was made to support the efficacy of the SIT [3–7]. In [4, 8], the authors gave the theory of effect on the genetic control of immigration using SIT. In this paper, we have studied the model under two distinct life divisions of insects (Viz. larvae and adult) and under external immigration of adolescents. In Sect. 2, we have formulated our mathematical model and studied its dynamics in Sect. 3. Lastly, we have concluded our results in Sect. 4.

2 Mathematical Model Formulation In this paper, we have proposed the flow of life control, the life cycle of an insect is split into two divisions: youthful (egg, hatchlings) and grown-up one. The size of population of the infantine phase of insect at time t is denoted by A. For adults, we inspect the specified divisions: women before merging, I; merging unfertilized women, F; merging fertilized women, U; and male insects, M. The size of futile insect population at time t is M S . The death count of infantine forms per capita, unmating women, merging fertilized women, merging unfertilized women, wild and futile male insects are marked, respectively, by δ A , δ I , δ F , δU , δ M , and δT . The net oviposition rate per women insect is relative to its density, and is also controlled through an impact of carrying ability depending on the activity of accessible handler locations. We suppose that the rate of oviposition per capita is expressed by ξ(1 − A/C), where C is carrying quantity recognized with nutrient and space measurement, and ξ is inherent rate of oviposition. Infantine population is advancing towards becoming grown-up insects at a per capita rate η; a proportion of r are women and (1-r) are men. Let i and m be the amount of immigrated males and females, respectively. As M S is the amount of sterile males released in each species generation so that the (I+i) number of women take their mates from (M + M S + m). Streams from I to F and U compartments depend on number of female encounters with native and grownM+m is the probability of up males and matching rates. Here, we suppose that (M+M S +m) female meeting with natural insects. At that point, the per capita rate of fertilization β(M+m) , where β is the natural insects match rate. Since sterile of female insects is (M+M S +m)

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organisms are artificially positioned, it is natural to believe a sterile male’s likelihood encountering a woman confides not only on the ratio of such men (M S /(M + M S + m)), but also on how distant from the breeding locations they are positioned. We are going to let that this net probability is expressed by pM S /(M + M S + m), where parameter p, with 0 ≤ p ≤ 1, is the proportion of sterile insects that are sprinkled in the sufficient spaces. Efficient fertilization during matching can also be reduced by sterilization, that causes us to believe that an efficient match rate of sterile insects is qβ, 0 ≤ q ≤ 1. Combining all of the above, we get that βT M S /(M + M S + m) which is the per capita rate at which women insects are futile sperm fertilized, where βT = pqβ. Parameter p is linked to efficiency of futile male preface in regard with spatial distribution of women insects, where q is assumed as a physiological modification induced by sterilization technique. Element p defines a significant role in insects whose reproduction is heavily dependent on non-homogeneous distribution of breeding locations, such as Aedes aegypti mosquitoes [9, 10]. Like the natural insect population, we believe that expansion rate of futile male insects is managed by the capability impact. This element requires into account the restricting limit (of labs, for example) for the delivery and release of disinfected male insects, which permits us to depict the recruitment frequency of futile male insects by τ M S (1 − M S /K ), where K is maximal capacity associated with sterile bug production. Note: τ (1 − M S /K ) is rate of recruitment per capita and τ is the inborn rate of recruitment. With all considerations and assumptions, the following mathematical system is proposed:   dA A F − (η + δ A )A =ξ 1− dT C

(1)

dI β(M + m)(I + i) βT M S (I + i) = (r η A) − − − δI I + i dT M + MS + m M + MS + m

(2)

β(M + m)(I + i) dF = − δF F dT M + MS + m

(3)

dM = (1 − r )η A − δ M M + m dT

(4)

d MS MS = τ M S (1 − ) − δT M S dT K

(5)

dU βT M S (I + i) = − δU U dT M + MS + m

(6)

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Table: Biological Meaning of Parameters Parameter δA δF δM η τ

Biological meaning Death rate of immature forms Death rate of fertilized females Death rate of wild insects Conversion rate from immature to mature insects Recruitment rate of Sterile male

Parameter δI δU δT β ξ

Biological meaning Death rate of Unmating females Death rate of Unfertilized females Death rate of Sterile insects Mating rate of natural insects Intrinsic oviposition rate

3 Dynamical Behaviour In this section, we will prove that the model is bounded. We will obtain its equilibrium points. Also, we will study the global stability of the model.

3.1 Boundedness of Model Here to corroborate that the system described by Model 1 is bounded. We will begin with following Lemma. Lemma 1 All solutions of the model will lie in the region P1 (A, I, F, M, M S , U ) ∈ 6 :, 0 < A ≤ Amax , 0 < I ≤ Imax , 0 < F ≤ Fmax , 0 < M ≤ Mmax , 0 < M S ≤ R+ M Smax , 0 < U ≤ Umax as t → ∞, for all initial values (A0 , I0 , F0 , M0 , M S0 , U0 ) ∈ 6 . R+ Proof As A ≤ C, where C is Carrying Quantity. From (4) we get, M(t) ≤ (1 − r )η A + m − δ M M. Thus, M ≤ K = Mmax From (5) we get, M S (t) ≤ τ M S − Kτ M S2 and thus, M S ≤ K = M Smax From (2) we get, I (t) ≤ r ηC − δ I I + i. Therefore, I (t) ≤ r ηC+i = Imax δI β(M+m)(I +i) From (3) we get, F(t) ≤ M+MS +m − δ F F As M, I, M S are bounded, therefore, F is bounded and F ≤ FMax T M S (I +i) From Eq. (6) we get, U (t) ≤ βM+M − δU U S +m As M S , M, I are bounded, hence U is bounded and U ≤ UMax

3.2 Equilibrium Points 3.2.1

When M S = 0 and no immigration(i = m = 0)

The immigration scheme fosNote: τ (1 − M S /K ) is the rate of recruitment per capita and τ is the inborn rate of recruitments. Note: τ (1 − M S /K ) is the rate of recruitment per capita and τ is the inborn rate of recruitment futile male insects has M S0 = 0

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and M S1 = ((τ − δT )K )/τ as two balance points. If and only if τ > δT , Eq. (4) is biologically feasible. As population of futile male insects under immigration is managed without impact of the natural population, Eq. (5)’s stability assessment demonstrates that the M S0 , the trivial equilibrium, is stable if τ > δT otherwise it will be volatile and outcomes in the stable non-trivial balance M S1 . This behaviour must be heavily confirmed by the dynamics of interaction between populations generated naturally and artificially. The value M S0 = 0 creates two balance points: one is trivial equilibrium, P0 = (0, 0, 0, 0) and U0 = 0, when sterile and natural insects are not present and P1 is an equilibrium where only natural insects live. We equate the following equations to 0.   MS  − δT M S = 0 (7) MS = τ MS 1 − K Taking M S = 0 (sterile insects are absent). From Eq. (3) we get,

β(M + m)(I + i) = M +m

δ F F and hence F = β(Iδ F+i) From Eq. (2), we get I = r η A−β∗i+i . On substituting values of I and F in Eq. (1) β+δ I we get I +1) = (η + δ A )A ξ(1 − A/C) δβF ∗ r η A+i(δ β+δ I η η ⇒ A2 ( Cδ Fξβr ) + A(( δ F ξβr + (β+δ I ) (β+δ I ) which further implies,

A=

ξβi(δ I +1) cδ F (β+δ I )

− (η + δ A ))2 ) +

ξβi(δ I +1) δ F (β+δ I )

=0

ξβi(δ I +1) ξβi(δ I +1) ξβr η ξβr η δ F (β+δ I ) − cδ F (β+δ I ) −(η+δ A )±( δ F (β+δ I ) + cδ F (β+δ I ) −(η+δ A )) ξβr η 2 cδ (β+δ ) F I

Therefore,

  (ξβr η − δ F (η + δ A )(β + δ I )) 1 ¯ A=c =c 1− ξβr η R

(8)

¯ ¯ (1−r )η A+m ¯ η A )C A ¯ r η A+i(1−β) where R = δ F (η+δξβr . Thus, P1 = ( A, , (η+δ , 0) ¯ , β+δ I δM ξ(C− A) A )(β+δ I ) From (8), we can clearly see if and only if R > 1 and for R < 1 P1 is biologically feasible and there will be only trivial equilibrium P0

3.2.2

Stability Analysis

In this section, we will analyse stability conditions of P0 and P1 . From the Jacobian for the trivial equilibrium P0 , the characteristic equation for trivial equilibrium P0 (0, 0, 0, 0, 0) is given by ξF − (η + δ A ) − λ)]] − βr ηξ(1 − (−δ M − λ)[(−δ F − λ)[(−β − δ I − λ)(− C A/C) = 0. Linear analysis reveals that one eigen value is −δ M and other eigen values are the zeroes of the polynomial.

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ξF ξF + η + δ A ] + λ[βδ F + δ F δ I + δ F + δ F (η + δ A ) + λ3 + λ2 [δ F + β + δ I + C C ξF ξF ξF β + η + δ A + δI + δ I (η + δ A )] + βδ F + βδ F (η + δ A ) + δ F δ I (η + δ A ) C C C A ξF − βr ηξ(1 − ) = 0 + δF δI C C 3 λ + (η + δ A + β + δ I + δ F )λ2 + [(η + δ A )(β + δ I ) + δ F (η + δ A ) + β + δ I ]λ + βr ηξ ∗ (η + δ A )(β + δ I )δ F =0 δ F (η + δ A )(β + δ I ) − (η + δ A )(β + δ I )δ F λ3 + (η + δ A + β + δ I + δ F )λ2 + [(η + δ A )(β + δ I ) + δ F (η + δ A ) + β + δ I ]λ + δ F (η + δ A (β + δ I )(1 − R)) = 0 If R < 1 then product of roots is positive. By Descartes Rule, we can see that conditions for stability of this polynomial are satisfied if and only if R < 1, we can clearly say that if R < 1, P0 is stable and if R > 1 then it becomes unstable. For analyzing the stability of model at P1 . On considering Jacobian for the nontrivial equilibrium P1 , the characteristic equation for non-trivial equilibrium P1 is as follows: ξF (−δ M − λ)[(−δ F − λ)[(β − δ I − λ)( − (η + δ A ) − λ)]] − βr ηξ(1 − A/C) − C ξF − (η + δ A ) − λ)] (1 − r )η(−δ F − λ)[(β − δ I − λ)( C It shows that one eigen value is −δ M and the other eigen values are zeroes of the polynomial. ξF ξF + δ I + β] + λ[δ F (η + δ A ) + δ F + δ I (η + δ A ) + λ3 + λ2 [(η + δ A ) + δ F + C C ξF ξF ξF + δ I δ F + β(η + δ A ) + β + βδ F ] + δ F (β + δ I )( + (η + δ A ) + βr ηξ δI C C C A (1 − )) = 0 C ξF ξF 3 λ + λ2 [(η + δ A ) + δ F + + δ I + β] + λ[δ F (η + δ A ) + δ F + δ I (η + δ A ) + C C ξF ξF ξF + δ I δ F + β(η + δ A ) + β + βδ F ] + δ F (β + δ I )( + (η + δ A ) − δI C C C βr ηξ )=0 R λ3 + [(η + δ A + β + δ I + δ F )]λ2 + [(η + δ A )(β + δ I ) + δ F (η + δ A + β + δ I ) βr ηξ(η + δ A )(β + δ I )δ F =0 λ + δ F (η + δ A (β + δ I )R − ξr ηβ λ3 + [(η + δ A + β + δ I + δ F )]λ2 + [(η + δ A )(β + δ I ) + δ F (η + δ A + β + δ I )λ + δ F (η + δ A (β + δ I )(R − 1) = 0 Here P1 is stable equilibrium iff R > 1. Parameter R is biologically defined. As 1 is the mean moment of existence of the immature stage of insects at time t, η1 (η+δ A ) → mean time of permanence, r is the proportion of infantile forms that becomes women then, r η(η + δ A )−1 is a chance that the egg will succeed in becoming a woman insect. In this case, β(β + δ I )−1 is the probability of fertilizing a woman and ξ is average amount of eggs produced by each fertilized woman. Therefore, product δF of these amounts denoted by R, is therefore the average amount of secondary woman

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insect generated by a single woman insect. We can see that there is no impact on sterile insects due to immigration and that the condition R > 1 is essential to keep natural insects in nature.

3.2.3

When (M S = 0 and i = m = 0)

Equilibrium points under the discharge of futile male(without Immigration) when M S = 0. In this section, we will find the equilibrium points of the model under the discharge of futile male and without immigration. 

M S = τ M S (1 −

MS ) − δT M S K

(9)

Taking condition (9) with all the four condition taken in first case: )η A , also from From (3), we get, β M I = δ F F(M + M S ) and from (4), M = (1−r δM (1−r )η A (τ −δT )K . Putting value of M + M S in (11), we get, (9), we get, M + M S = δ M + τ I =

)η A r η A( (1−r + δτ ) δ M

(δ I +β)M+(δ I +βT )

T (τ −δT )K τ

δ F (η+δ A )C A ξ(C−A) )η A (1−r )η A (τ −δT )K 1 Now, β(1−r r η A( + ) (τ −δ )K δM δM τ (δ I +β)M+(δ I +βT ) τT (τ −δT )K + ) τ βr η A2 η +δ I )(τ −δT )K δ M − (1 − δ F (η+δβrAξ)(β+δ )A − (βT(1−r δ F (η+δ A (δ I +β)) )ητβ+δ I ) I)

From Eq. 10, we get, F =

=

δ F (η+δ A )C A ξ(C−A)

a A2 + b A + c = 0

)η A ∗ ( (1−r δM

(10)

η +δT )(τ −δT )K δ M b = (1 − δ F (η+δβrAξ)(β+δ ), c = (βT(1−r , As R = )ητβ+δ I ) I) + δ )δ (τ − δ )K (β T I M T ξr ηβ = M S∗ , and M S = (δ A +η A )(β+δ I )δ F (β + δ I )(1 − r )ηCτ Therefore, (10) can be written as

where a =

βr η , δ F (η+δ A (δ I +β))

P(A) =

R ∗ A2 − (R − 1)A + C M S = 0 C

(11)

and the conditions for biological existence of the non-trivial equilibria becomes R > 1, (R − 1)2 − 4R ∗ C MS C (R − 1)2 = M S0 MS ≤ (12) 4R

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Observations However, the situation (12) provides the upper limit of  M S , the amount of sterile men to be discharged into the scheme, it is observed from the situation (11) that whenever the value of M S increases, A decreases and this depends on the wild male population parameters. Biologically, it implies that using futile males below this limit value reduces both the larval population and the adult pest population, which improves the plant biomass. But if we cross this specific value, the fading equilibrium will appear and stabilize.

3.2.4

Equilibrium Points Under Immigration

In this section, we evaluate the equilibrium points of the model under immigration. From (5), we get, M S = (τ −δτT )K , from Eq. (2), we get, S +m)+i((M+M S +m)+β(M+m)−βT M S ) A )C∗A I = r η A(M+Mβ(M+m)+β From (3) and (4), we get F = (η+δ ξ(C−A) T M S +δ I (M+M S +m)

and M = (1−rδ)ηMA+m Putting this value of I and F in Eq. (3), we get β(M + m)(I + i) = δ F F(M + M S + m) β(M + m)(r η A(M + M S + m)) + i((M + M S + m) − β(M + m) − βT M S ) + i(β(M + m) + βT M S + δ I (M + M S + m)) = δ F ∗ F ∗ (M + M S + m) ∗ (β(M + m) + βT M S + δ I (M + M S + m)) ⇒ β(M + m)ξ(C − A)(r η A + i(δ I + 1)) = δ F (η + δ A )C A((β + δ I )M + (βT + δ I )M S + (β + δ I )m) Putting the value of M in left-hand side, we get β(M + m)ξ(C − A)(r η A + i(δ I + 1)) = β( (1−rδ)ηMA+m + m)ξ(C − A)(r η A + i(δ I + 1)) )η2 ξr )η2 ξrC )ηξ(δ I +1) ⇒ A3 ( β(1−r ) + A2 ( β(1−r − β(1−rmδ ) + A( β(1−r )ηξδ MCi(δ I +1) + mδ M mδ M M βm(δ M +1)ξ Cr η Ci(δ I +1) ) + βm(δ M +1)ξ δM δM Now, considering right-hand side, we get δ F (η + δ A )C A((β + δ I )M + (βT + δ I )M S + (β + δ I )m) A2 β(1 − r )η2 ξr β(1 − r )η2 ξrC )+( = −A3 (( )(( + δ M δ F (η + δ A )C Cδ F (η + δ A δM β(1 − r )ηξ i(δ I + 1) (β + δ I )(1 − r )η (β + δ I )(1 − r )η )− )−( )) + δM δM δM (β + δ I )m 1 β(1 − r )ηξ Ci(δ I + 1) βm(δ M + 1)ξ Cr η − + ∗ + A( δM δM Cδ F (η + δ A ) δM βm (δ M + 1)ξ Ci(δ I + 1) 1 (βT + δ I )M S + (β + δ I )m) + ∗ δM Cδ F (η + δ A ) i(δ I + 1)R η Ri(δ I + 1) m(δ M + 1)R −η 2 ηR − − ∗R+A ( − − ) + A( − δM δM rCδ M δM rη (1 − r )η mδ M 1 βT + δ I βm (δ M + 1)ξ Ci(δ I + 1) 1 ( ∗ MS ) + + 1) − ∗ (1 − r )η δ M (1 − r )η δM Cδ F (η + δ A )

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Thus, on combining left-hand side and right-hand side, we get −A3 + S1 A2 + S2 A + S3 = 0 mδ M I +1)R M +1)R − δηM ),S2 = ( Ri(δr Iη+1) − m(δ − (1−r ( 1 + 1) − where S1 = ( ηδ MR − i(δrCδ (1−r )η )η δ M M βT +δ I (1−r )η

∗ MS ) βm (δ M +1)ξ Ci(δ I +1) δM

1 S3 = ∗ Cδ F (η+δ A) Using Descartes Rule, we get S2 < 0 β(1 − r )η2 i(δ I + 1)ξ C β(1 − r )ηi(δ I + 1)ξ C (β + δ I ) + m −( + − − δM δM δM (βT + δ I )M S − (β + δ I )m) < 0 1 ⇒ M S > (βT +δ I)

M +1)r ηξ C+M(δ M +1)i(δ I +1)ξ −(β+δ I )m ( −β(1−r )η i(δ I +1)ξ C+β(1−r )ηi(δ I +1)ξ C−m(mu − (β + δ I ) δM m) = M S1 M S1 = M S ∗ + (δ M +1)(δ I +1)rηδηξI C(1−r )(β+δ I ) If we take M S > M S1 , then S2 becomes positive. 2

3.2.5

Local Stability of P3

Now, to examine the local natue of equilibrium points, we compute the variational matrix, which leads to the following characteristic equation: λ4 + p1 λ3 + p2 λ2 + p3 λ + p4 ξF β(M + m) + βT M S where, p1 = + δI + + δF δM C M + MS + m ξF ξF β(M + m) + βT M S + (η + δ A ) + δ I + )+( + η + δ A) ∗ δI + p2 = δ F ( C M + MS + m C β(M + m) + βT M S ξF β(M + m) + βT M S + δM ( + η + δ A + δI + ) + δF δM M + MS + m C M + MS + m ξF β(M + m) + βT M S ξF + η + δ A )(δ I + ) + δ F δ M (( + η + δ A) + δI p3 = δ F (( C M + MS + m C β(M + m) + βT M S ξF β(M + m) + βT M S ) + δ M (( + η + δ A) ∗ δI + ) − δM + M + MS + m C M + MS + m β(M + m)r ηξ A β M S (I + i) A ξ(1 − ) (1 − r )η ξ(1 − ) + 2 (M + M S + m) C M + M S + m) C ξF β(M + m) + βT M S + (η + δ A ))(δ I + )+ p4 = δ M δ F ( C M + MS + m A β(M + m)r η ∗ ξ(1 − C ) β(M + m) A ) − (1 − r )η( (ξ(1 − ) δM ( M + MS + m M + MS + m C βT − β)M S (I + i) β M S (I + i) A ( − (1 − r )η ξ(1 − )(δ I + M + M S + m)2 (M + M S + m)2 C β(M + m) + βT M S ) M + MS + m By Routh–Hurwitz criterion for local stability, we can say that the model is locally stable if pi > 0, i = 1, 2, 3, 4, taking one positive and negative coefficients and comparing, we get the following:

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ξF + η + δ A δI C δ M (1−r )ηβ M S (I +i)∗ξ(1− CA ) ((M+M S +m)2 )

δM

+
M M+M S +m (M+M S +m)2 β(M+m)r η∗ξ(1− CA ) M+M S +m

(δ M (1 − r )ηβ(I + i) = β(M + m)r η)M S ∗ β(M + m)2 r η, M S
) ξ(1 − (δ I + C M + MS + m (M + M S + m)2 C M + MS + m

(1 − r )ηβ(I + i) + 2δ M δ F ( ξCF + η + δ A )(M + m) ±



(1 − r )ηβ(I + i) + 2δ M δ F ( ξCF + η + δ A )(M + m) − ηδ M δ F ( ξCF + η + δ A )(C)

2 ∗ δ M δ F ( ξCF + η + δ A )

and δM (

MS =

β(M + m) (βT − β)M S (I + i) β(M + m)r ηξ(1 − A/C) ) > (1 − r )η ∗ ξ(1 − A/C)( M + MS + m M + MS + m (M + M S + m)2

−2Mδ M + mδ M + (1 − r )η(βT − β)(I + i) ±



(δ M (−2M + m) + (1 − r )η(βT − β)(I + i))2 − 4(δ M )(M 2 + m 2 )δ M + 2Mmδ M 2δ M

M S = Max(P, Q), where P=

(1 − r )ηβ(I + i) + 2δ M δ F ( ξCF + η + δ A )(M + m) ±



(1 − r )ηβ(I + i) + 2δ M δ F ( ξCF + η + δ A )(M + m) − ηδ M δ F ( ξCF + η + δ A )(C)

2 ∗ δ M δ F ( ξCF + η + δ A )

and Q=

−2Mδ M + mδ M + (1 − r )η(βT − β)(I + i) ±



(−2Mδ M + mδ M + (1 − r )η(βT − β)(I + i))2 − 4(δ M )(M 2 + m 2 )δ M + 2Mmδ M 2δ M

Observations The relationship of sterile insect technique with immigration is very important. We get two threshold values of released sterile males and will find the maximum of both values, depends on the rate of immigration i and m, so that the internal equilibrium remains stable and positive. It may be observed that M S1 is higher than M S ∗ which, in the event of non-immigration, was the limit value for pest control. Therefore, more sterile males must be published to maintain the density of the pest population below a desirable level when pest migration occurs. Male migration may diminish the control measure’s efficiency, but there is little impact on female migration.

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It may be observed that when migration is not present, it is not feasible to eradicate pest through sterile males, but it is feasible to keep the density of pest below some required level.

4 Conclusion In this paper, we considered the stage-structured model of pest population comprising of larvae, female(before mating and mating fertilized females), and male individuals under immigration. We have proposed three cases of the SIT model, in first case we have not considered discharge of futile males and immigration, in this case if R > 1, P1 is stable else unstable and in second case release of sterile male has been considered but immigration has not been taken into the account, in this case we have found the thresholds of the system. It implies that whenever the value of M S increases, A decreases and this is determined by parameters of the wild male population, but when we reach that specific value, the equilibrium appears to be disappearing and stable. In the third case, both release of sterile male and immigration has been taken into the account, in this case M S1 is greater than M S ∗ which was the threshold value for control of pest population in case of without immigration. So larger amount of sterile male has to be discharged to have the pest density under some required level, when migration of pest is active. To make the comparison for sterile population M S , we have taken the equilibrium without immigration and with immigration which shows the clear effect of sterile insects under immigration. The function of immigrants in a population is not only crucial for the SIT, but also to assess whether local attempts to control larvae with environmental management insecticides could have an effect on adult population or whether immigration is likely to swamp them.

References 1. E.F. Knipling, Sterile-male method of population control. Science 130(3380), 902–904 (1959) 2. E.F. Knipling, Sterile insect technique as a screwworm control measure: the concept and its development, in Symposium on Eradication of the Screwworm from the United States and Mexico, Misc. Publ. Entomol. Soc. America vol. 62, ed. by O.H. Graham (College Park, MD, 1985), pp. 4–7 3. W. Costello, H. Taylor, Mathematical Models of the Sterile Male Technique of Insect Control. Lecture Notes in Biomath, vol. 35, (Springer, New York, 1975), pp. 318–359 4. K. Dietz, The effect of immigration on genetic control. Theor. Popul. Biol. 9, 58–67 (1976) 5. H.J. Barclay, Combining methods of insect pest control: modelling selection for resistance to control methos in combination, Reserv. Pop. Ecol. 34, 97–107 (1996) 6. R.E. Plant, M. Mangel, Modeling and simulation in agricultural pest management. SIAM Rev. 29, 235–261 (1987) 7. A.C. Bartlett, Insect sterility, insect genetics, and insect control, in Handbook of Pest Management in Agriculture, vol. II, ed. by D. Pimentel (CRC Press, Boca Raton, FL, 1990), pp. 279–287

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8. T. Prout, The joint effect of release of sterile males and immigration of fertilized females on a density regulated population. J. Popul. Biol. 13, 40–71 (1978) 9. L. Esteva, H.M. Yang, Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique. Math. Biosc. 198, 132–147 (2005) 10. L. Esteva, H.M. Yang, Control of dengue vector by the sterile insect technique considering logistic recruitmen. TEMA Tend. Mat. Apl. Comput. 7(2), 259–268 (2006)

Strict Practical Stability of Impulsive Differential System in Terms of Two Measures Pallvi Mahajan, Sanjay Kumar Srivastava and Rakesh Dogra

Abstract In this paper, an impulsive differential system is investigated to obtain sufficient conditions for strict practical stability. The investigations are carried out by perturbing Lyapunov function and by using the comparison principle. The stability properties are investigated in terms of two measures. Our results demonstrate that impulses do contribute to the system’s stability behaviour. An example is given to compliment on results. Keywords Impulsive differential system · Strict practical stability · Perturbed Lyapunov functions · Two measures AMS Subject Classification: 34D20

1 Introduction Stability is one of the most important feature in the qualitative theory of differential equations. In dealing with real- world problems, it is, however, possible that the system may be theoretically stable, but actually it is unstable, because the domain of attraction is very small. Practical stability is an appropriate way to overcome this problem by stabilizing the system into certain subsets of phase space. The concept of practical stability is introduced by Lakshmikantham et al. [1]. Since then, it has been investigated for various types of differential systems and is extensively studied by many researchers [2–5]. Strict practical stability is the refinement of practical stability and strict stability [6, 7], as it estimates the behaviour of a system within specified bounds, which is first presented by V. Lakshmikantham and Y. Zhang [8] for a non-impulsive differential system with delay. Shurong Sun et al. [9] P. Mahajan Inder Kumar Gujral-Punjab Technical University, Kapurthala 144601, Punjab, India P. Mahajan (B) · S. K. Srivastava · R. Dogra Beant College of Engineering and Technology, Gurdaspur 143521, Punjab, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_8

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investigated strict practical stability for a discrete hybrid system. Later on, Senlin Li et al. [10] generalized the same stability for an impulsive differential system. The approach of obtaining stability relative to two measures of stability h and h 0 , namely, h-positive definiteness and h-decrescentness; has been proved very powerful which unify a variety of stability concepts and offers a more general framework for the investigation of the stability theory rather than using standard norm [11, 12]. In this paper, we investigated sufficient criteria to obtain the strict practical stability for an impulsive differential system in terms of two measures. In order to investigate the stability properties of nonlinear differential systems, Lyapunov method is the most widely recognized tool to derive these properties. Instead of employing a single Lyapunov function to obtain the desired properties of stability, which has to satisfy rigid conditions, it becomes more advantageous to use families of Lyapunov function by perturbing it, which is first given by V. Lakshmikantham and Leela [13]. Many researchers adopted this technique of perturbing the Lyapunov function to investigate the stability of impulsive differential systems under weaker conditions [14–16]. In this contribution, we consider the (h 0 , h)- strict practical stability of an impulsive differential system. We establish sufficient conditions for the desired stability by using the concept of two measures by employing two Lyapunov-like functions and also use the comparison method. Our results prove that the system’s stability behaviour significantly depends upon impulses. We illustrate the derived result with an example.

2 Preliminaries Let R n denotes the n-dimensional Euclidean space and let R+ = [0, ∞), Consider the following impulsive differential system: 

x  = f (t, x), x = Ik (x),

t  = tk t = tk , k = 1, 2, 3 . . .

(1)

Let the following assumptions hold for system (1): (A1) 0 < t1 < t2 < . . . tk < tk+1 . . . , tk → ∞ as k → ∞; (A2) f ∈ PC[R+ × R n , R n ] is piecewise continuous in (tk−1 , tk ] × R n for x ∈ R n , lim(t,y)→(tk+ ,x) f (t, y) = f (tk+ , x) exists; (A3) Ik : R n → R n . Let x(t) = x(t; t0 , x0 ) be any solution of system (1) through (t0 , x0 ). Denote S(ρ) = {x ∈ R n : x < ρ}. Let us define the following classes of functions: K = {ω ∈ C[R+ , R+ ] : ω is strictly increasing and ω(0) = 0}

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   = h ∈ C[R+ × R n , R+ ], inf h(t, x) = 0 Now, we shall define the following class of Lyapunov function, which will be used to investigate the strict practical stability of impulsive differential system (1): Definition 1 [17]. Let V : R+ × R n → R+ . Then V belongs to the class V0 , which satisfies the following conditions: (a) V is continuous in (tk−1 , tk ] × R n and for all x ∈ R n and k ∈ N , lim(t,y)→(tk+ ,x) V (t, y) = V (tk+ , x) exists; (b) V is locally Lipschitzian with respect to x. For V ∈ V0 and (t, x) ∈ (tk−1 , tk ] × R n , define the generalized derivatives of system (1) as follows: 1 D + V (t, x) = lim+ sup {V (t + h, x + h f (t, x) − V (t, x)} h→0 h and D− V (t, x) = lim− in f h→0

1 {V (t + h, x + h f (t, x) − V (t, x)} . h

Definition 2 [17] Let h 0 , h ∈ , then V (t, x) is said to be (a) h-positive definite, if there exists a ρ > 0 and a function β ∈ K , such that h(t, x) < ρ implies β(h(t, x)) ≤ V (t, x); (b) h 0 -decrescent, if there exists a ρ > 0 and a function α ∈ K , such that h 0 (t, x) < ρ implies V (t, x)) ≤ α(h 0 (t, x)); (d) S(h, ρ) = {(t, x) ∈ R+ × R n , h(t, x) < ρ} . Definition 3 [9] Let h, h 0 ∈ , then the solution x(t) = x(t; t0 , x0 ) of the system (1) is said to be (S1) (S2) (S3) (S4)

(h 0 , h)-practically stable, if for given 0 < λ < A, we have h(t, x(t)) < A whenever h 0 (t0 , x0 ) < λ for some t0 ∈ R+ ; (h 0 , h)-uniformly practically stable, if (S1) holds for all t0 ∈ R+ ; (h 0 , h)-strictly practically stable, if (S1) holds and for every μ ≤ λ, there exists B < μ such that h(t, x(t)) > B whenever h 0 (t0 , x0 ) > μ for t ≥ t0 ; (h 0 , h)-strictly uniformly practically stable, if (S3) holds for all t0 ∈ R+ ;

In the following section, we will present two comparison principles which will be very helpful in our further investigations. Lemma 1 [10] Let V : PC[R+ × R n → R+ ] and V (t, x) is locally Lipschitzian in x. Assume that  D+ V (t, x) ≤ g1 (t, V (t, x)), t  = tk (2) + V (tk , x(tk ) + Ik (x(tk ))) ≤ ψk (V (tk , x(tk ))), t = tk , k = 1, 2, 3 . . .

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where g1 ∈ R+ × R+ → R which satisfies (A2) and ψk : R+ → R+ is nondecreasing. Let r1 (t) = r1 (t; t0 , u 0 ) be the maximal solution of the following comparison system of (1): ⎧ u  = g1 (t, u), t  = tk , ⎪ ⎨ + u(tk ) = ψk (u(tk )), (3) ⎪ ⎩ u(t0 ) = u 0 ≥ 0, k = 1, 2, 3 . . . existing for t ≥ t0 . Then, V (t0 , x0 ) ≤ u 0 implies that V (t, x(t)) ≤ r1 (t, t0 , u 0 ) for t ≥ t0 ., where x(t) = x(t; t0 , x0 ) is any solution of (1) existing for t ≥ t0 . Lemma 2 [10] Let V : PC[R+ × R n → R+ ] and V (t, x) is locally Lipschitzian in x. Assume that  D− V (t, x) ≥ g2 (t, V (t, x)), t  = tk , (4) + V (tk , x(tk ) + Ik (x(tk ))) ≥ φk (V (tk , x(tk ))), t = tk , k = 1, 2, 3 . . . where g2 ∈ R+ × R+ → R and φk : R+ → R+ is non-decreasing. Let r2 (t) = r2 (t; t0 , v0 ) be the minimal solution of the following comparison system of (1): ⎧ v  = g2 (t, v), t  = tk , ⎪ ⎨ ⎪ ⎩

v(tk+ ) = φk (v(tk )), v(t0 ) = v0 ≥ 0, k = 1, 2, 3 . . .

(5)

existing for t ≥ t0 . Then V (t0 , x0 ) ≥ v0 implies that V (t, x(t)) ≥ r2 (t; t0 , v0 ), t ≥ t0 ., where x(t) = x(t, t0 , x0 ) is any solution of (1) existing for t ≥ t0 .

3 Main Section In this section, we will obtain sufficient conditions for the strict practical stability of impulsive differential system (1) in terms of two measures. We will obtain the desired results by applying the method of perturbed Lyapunov function and comparison results. Theorem 3.1 Assume that the following conditions are fulfilled. (i) Let 0 < λ < A < ρ and h, h 0 ∈ ; (ii) Let V1 ∈ PC[R+ × S(h, ρ), R+ ], V1 (t, x) ∈ V0 , such that a1 (λ) ≤ b1 (A) for some t0 ∈ R+ , where a1 , b1 ∈ K , which satisfies b1 (h(t, x)) ≤ V1 (t, x) ≤ a1 (h 0 (t, x))

(6)

Strict Practical Stability of Impulsive Differential System …

and 

D+ V1 (t, x) ≤ 0, V1 (tk+ , x(tk )

91

t  = tk ,

+ Ik (x(tk ))) ≤ V1 (tk , x(tk ))), t = tk , k = 1, 2, 3 . . .

(7)

(iii) Let, there exist a V2 ∈ PC[R+ × S(h, ρ), R+ ], V2 (t, x) ∈ V0 , such that b2 (h 0 (t, x)) ≤ V2 (t, x) ≤ a2 (h(t, x))

(8)

where a2 , b2 ∈ K , and  V2 (tk+ , x(tk )

D− V2 (t, x) ≥ 0, t  = tk , + Ik (x(tk ))) ≥ V2 (tk , x(tk ))), k = 1, 2, 3 . . .

(9)

Then the zero solution of system (1) is (h 0 , h)-strictly practically stable. Proof For given 0 < λ < A < ρ, in order to prove the required stability, firstly we claim that h(t, x(t)) < A whenever h 0 (t0 , x0 ) < λ for some t0 ∈ R+ . If, it is not true, then there exists a solution x(t) = x(t, t0 , x0 ) of system (1) satisfying h 0 (t0 , x0 ) < λ and a t ∗ > t0 , such that tk < t ∗ ≤ tk+1 for some k satisfying h(t, x(t)) < A for t0 ≤ t ≤ tk and h(t ∗ , x(t ∗ )) ≥ A. By using (7), V1 (t, x(t)) is a decreasing function. Hence, we get b1 (A) ≤ b1 (h(t ∗ , x(t ∗ ))) ≤ V1 (t ∗ , x(t ∗ )) ≤ V1 (t0 , x(t0 )) ≤ a1 (h 0 (t0 , x(t0 )) < a1 (λ)

which contradicts condition (ii) of this Theorem. Hence, h(t, x(t)) < A whenever h 0 (t0 , x0 ) < λ for some t0 ∈ R+ holds. On the other hand, for any 0 < μ < λ, choose B < μ, such that a2 (B) ≤ b2 (μ). Now, we claim that h(t, x(t)) > B whenever h 0 (t0 , x0 ) > μ for some t ≥ t0 holds. If, it is not true, then there exists a solution x(t) = x(t; t0 , x0 ) of system (1) satisfying μ < h 0 (t0 , x0 ) < λ and a t ∗∗ > t0 , such that tk < t ∗∗ ≤ tk+1 for some k, satisfying h(t, x(t)) > B for t0 ≤ t ≤ tk and h(t ∗∗ , x(t ∗∗ )) ≤ B. Again, by using (9), we have V2 (t, x(t)) is a non- decreasing function. Consider a2 (B) = a2 (h(t ∗∗ , x(t ∗∗ ))) ≥ V2 (t ∗∗ , x(t ∗∗ )) ≥ V2 (t0 , x(t0 )) > b2 (h 0 (t0 , x(t0 )) > b2 (μ)

which is a contradiction. Hence, h(t, x(t)) > B whenever h 0 (t0 , x0 ) > μ for some t ≥ t0 holds. Therefore, system (1) is (h 0 , h)-strictly practically stable. In the next Theorem, we shall prove the same result by using the comparison method.

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Theorem 3.2 Assume that the following conditions are fulfilled. (i) Let 0 < λ < A < ρ and h, h 0 ∈ ; (ii) Let V1 ∈ PC[R+ × S(ρ), R+ ], V1 (t, x) ∈ V0 , such that b1 (h(t, x)) ≤ V1 (t, x) ≤ a1 (h 0 (t, x)), a1 , b1 ∈ K , and 

D+ V1 (t, x) ≤ g1 (t, V1 (t, x)),

(10)

t  = tk ,

V1 (tk+ , x(tk )

+ Ik (x(tk ))) ≤ ψk (V1 (tk , x(tk ))), t = tk , k = 1, 2, 3 . . . (11) where g1 : R+ × R+ → R, g1 (t, x) satisfies (A2) and ψk ∈ R+ → R+ is nondecreasing. (iii) Let there exists a V2 ∈ PC[R+ × S(ρ), R+ ], V2 (t, x) ∈ V0 , such that b2 (h 0 (t, x)) ≤ V2 (t, x) ≤ a2 (h(t, x)) where a2 , b2 ∈ K and  D− V2 (t, x) ≥ g2 (t, V2 (t, x)), V2 (tk+ , x(tk )

t  = tk ,

+ Ik (x(tk ))) ≥ φk (V2 (tk , x(tk ))), k = 1, 2, 3 . . .

(12)

(13)

where where g2 : R+ × R+ → R, g2 (t, x) satisfies (A2) and φk ∈ R+ → R+ is non-decreasing. (iv) The zero solution of comparison system (3) is practically stable with respect to (λ, A) and for every μ ≤ λ, there exists B < μ, such that for the comparison system (5), v0 > μ implies v(t) > B, t ≥ t0 , where v(t) = v(t; t0 , v0 ) is any solution of (5). Then the zero solution of system (1) is (h 0 , h)-strictly practically stable. Proof Since the zero solution of (3) is practically stable. Hence, for 0 < λ < A < ρ, we have a1 (λ) < b1 (A) and u(t) < b1 (A),

t ≥ t0 for u 0 < a1 (λ)

(14)

We claim that h(t, x(t)) < A whenever h 0 (t0 , x0 ) < λ for some t0 ∈ R+ . If, it is not true, then there exists a solution x(t) = x(t, t0 , x0 ) of system (1) satisfying h 0 (t0 , x0 ) < λ and a t  > t0 , such that tk < t  ≤ tk+1 for some k, satisfying h(t, x(t)) < A for t0 ≤ t ≤ tk and h(t  , x(t  )) ≥ A. Choose u 0 = V1 (t0 , x0 ), then by using (11) and Lemma 1, we get V1 (t, x) ≤ r1 (t; t0 , u 0 ), t0 ≤ t ≤ t  . where r1 (t; t0 , u 0 ) is the maximal solution of comparison system (3).

(15)

Strict Practical Stability of Impulsive Differential System …

93

Now, by using Eq. (10), we have V1 (t0 , x0 ) ≤ a1 (h 0 (t0 , x0 )) < a1 (λ). Also, by using Eqs. (10), (14), and (15), we get b1 (A) = b1 (h(t  , x(t  ))) ≤ V1 (t  , x(t  )) ≤ r1 (t; t0 , u 0 ) < b1 (A) which is a contradiction, hence h(t, x(t)) < A whenever h 0 (t0 , x0 ) < λ for some t0 ∈ R+ holds. On the other hand, for every μ ≤ λ, there exists B < μ, such that v0 > b2 (μ) implies v(t) > a2 (B), t ≥ t0 .

(16)

Now, we claim that h(t, x(t)) > B whenever h 0 (t0 , x0 ) > μ for some t ≥ t0 holds. If, it is not true, then there exists a solution x(t) = x(t, t0 , x0 ) of system (1), satisfying μ < h 0 (t0 , x0 ) < λ and a t  > t0 , such that tk < t  ≤ tk+1 for some k, satisfying A > h(t, x(t)) > B for t0 ≤ t ≤ tk and h(t  , x(t  )) ≤ B. Now, by using Lemma 2 and Eq. (13), we have V2 (t, x) ≥ τ (t; t0 , v0 ), t0 ≤ t ≤ t  .

(17)

where τ (t; t0 , v0 ) is the minimal solution of comparison system (5). By using Eq. (12), we have V2 (t0 , x0 ) ≥ b2 (h 0 (t0 , x0 )) > b2 (μ)

(18)

Consider, a2 (B) > a2 (h(t  , x(t  )) ≥ V2 (t  , x(t  )) ≥ τ (t; t0 , v0 ) ≥ a2 (B). Again, we get a contradiction. Hence, h(t, x(t)) > B whenever h 0 (t0 , x0 ) > μ for some t ≥ t0 holds. Therefore, the zero solution of system (1) is (h 0 , h)- strictly practically stable. In this section, we will present an example to support the proved results. Example Consider, the following impulsive differential system which represents the logistic growth model of single-species populations: ⎧ ⎪ ⎨ ⎪ ⎩

y  = r y(1 − y), Ik (y),

y = y(t0 ) = y0 > 0,

t  = tk ; t = tk , k = 1, 2, 3 . . . , t0 ≥ 0.

(19)

where r > 0 is an intrinsic birth rate constant. Set x = y − 1 to consider the strict practical stability of the equilibrium point y = 1 of system (19) by following system: ⎧ ⎪ ⎨ ⎪ ⎩

x  = −r x(x + 1), t = tk ; x = Ik (x), x(t0 ) = x0 > 0,

t = tk , k = 1, 2, 3 . . . , t0 ≥ 0.

(20)

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Proof Let ρ = 1 and choose 0 < λ < A < min {ρ, 2λ}. Let h(t, x) = x and h 0 (t, x) = x. Define V1 (x) = 21 x 2 , a1 (x) = x 2 and b1 (x) = 14 x 2 . Then, b1 (h(t, x)) ≤ V1 (t, x) ≤ a1 (h 0 (t, x)) is satisfied and D+ V1 (t, x) = −r x 2 (x + 1) ≤ 0. 2 2 2 Next, Define V2 (x) = e−x , a2 (x) = 2e−x and b2 (x) = xe−x . Clearly, b2 (h 0 (t, x)) ≤ V2 (t, x) ≤ a2 (h(t, x)) is satisfied and D+ V2 (t, x) = 2r x 2 2 (1 + x)e−x ≥ 0. Also, at t = tk , if Ik (x)(2x + Ik (x)) ≤ 0, then V1 (x + Ik (x)) ≤ V1 (x) and V2 (x + Ik (x)) ≥ V1 (x). Hence, all the conditions of Theorem 3.2 are satisfied. Therefore, the zero solution of (20) and then, y ≡ 1 of (19) is strictly practically stable.

4 Conclusion In this paper, we investigated the strict practical stability criteria for an impulsive differential system in terms of two measures. We obtained the sufficient conditions of the said stability by means of two measures h and h 0 , namely, h-positive definiteness and h-decrescentness, which was earlier investigated by Senlin Li et al. [10] by using usual norms. We have employed the perturbed Lyapunov functions to obtain the desired results. We, first proved the results analogous to Lyapunov’s original theorems and then discussed the same employing comparison principle. An example is also given to support the proved results. Acknowledgements One of us (PM) would like to thank IKGPTU for providing the online library facility.

References 1. V. Lakshmikantham, S. Leela, A.A. Martynyuk, Practical Stability of Nonlinear Systems (World Scientific, 1990) 2. Y. Zhang, J. Sun, Practical stability of impulsive functional differential equations in terms of two measurements. Comput. Math. Appl. 48, 1549–1556 (2004) 3. A.A. Soliman, On practical stability of perturbed differential systems. Appl. Math. Comput. 163, 1055–1060 (2005) 4. S. Hristova, Generalization of practical stability for delay differential equations with respect to initial time difference. AIP Conf. Proc. 1570, 313–320 (2013) 5. P. Singh, S.K. Srivastava, K. Kaur, Uniform practical stability in terms of two measures with effect of delay at the time of impulses. Nonlinear Eng. 5(2), 93–97 (2016) 6. V. Lakshmikantham, J.V. Devi, Strict stability criteria for impulsive differential systems. Nonlinear Anal., Theory, Methods Appl. 21(10), 785–794 (1993)

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7. R.P. Agarwal, S. Hristova, Strict stability in terms of two measures for impulsive differential equations with ‘supremum. Appl. Anal. 91, 1379–1392 (2012). January 8. V. Lakshmikantham, Y. Zhang, Strict practical stability of delay differential equation. Appl. Math. Comput. 122, 341–351 (2001) 9. S. Sun, W. Chen, Y. Zhao, Z. Han, Strict practical stability for discrete hybrid systems in terms of two measures. Appl. Mech. Mater. 643(2), 90–95 (2014) 10. S. Li, X. Song, A. Li, Strict practical stability of nonlinear impulsive systems by employing two lyapunov-like functions. Nonlinear Anal.: R. World Appl. 9(2), 2262–2269 (2008) 11. V. Laksmhikantham, X. Liu, Stability criteria of impulsive differential equations in terms of two measures. J. Math. Anal. Appl. 137, 591–604 (1989) 12. V. Lakshmikantham, X.Z. Liu, Perturbing families of lyapunov functions and stability in terms of two measures. J. Math. Anal. Appl. 140, 107–114 (1989) 13. V. Lakshmikantham, S. Leela, On perturbing lyapunov functions. Math. Syst. Theory 10, 85–90 (1976) 14. X. Liu, Stability analysis of impulsive system via perturbing families of lyapunov functions. Rocky Mt. J. Math. 23(2), 651–670 (1993) 15. H. Zhao, E. Feng, Stability of impulsive system by perturbing lyapunov functions. Appl. Math. Lett. 20(2), 194–198 (2007) 16. D. Stutson, A. Vatsala, Generalized practical stability results by perturbing lyapunov function. J. Appl. Math. Stoch. Anal. 9(1), 69–75 (1996) 17. V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations (World Scientific, 1989)

Free Vibration Analysis of Rigidly Fixed Axisymmetric Viscothermoelastic Cylinder Himani Mittal and D. K. Sharma

Abstract This paper represents the analysis of free vibrations of rigidly fixed, functionally graded generalized viscothermoelastic axisymmetric hollow cylinder which is considered undeformed at uniform temperature. The material of the cylinder is considered to be functionally graded according to the simple exponent law. The governing partial differential equations of motion and heat conduction have been transformed into ordinary differential equations due to time-harmonic analysis. The matrix Frobenius method of the series solution has been implemented to ordinary differential equations analytically to represent the solutions of displacement and temperature. The regular fixed boundary conditions are further solved by the use of numerical method of iteration technique with the help of MATLAB software tools. For numerical computations, we take polymethyl methacrylate material to represent natural frequencies, thermoelastic damping, frequency shift, temperature change and displacement. The behavior of frequencies, thermoelastic damping, temperature change and variation of displacement have been monitored (increase or decrease) with grading index (i.e. inhomogenous parameter). Keywords Viscothermoelasticity · Rigidly fixed vibrations · Functionally graded material · Series solution · Exponent law

1 Introduction The non-homogenous material consists of two or more different materials with changing composition across the material referred to as functionally graded materials (FGM). The theory of viscoelasticity was well established by Bland [1]. Various problems on viscothermoelastic models was utilized in vibrating solids by Biot [2], H. Mittal · D. K. Sharma (B) Department of Mathematics, School of Basic and Applied Sciences, Maharaja Agrasen University, Atal Shiksha Kunj, Baddi, Solan 174103, HP, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_9

97

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Hunter [3] and Flugge [4] to manage energy dissipation and it was observed that energy and dispersion reduces due to the produced internal friction. Heat conduction effect on Rayleigh waves for the semi-infinite elastic solid under thermal boundary conditions was introduced by Chadwick and Windle [5]. Dhaliwal and Singh [6] have described some problems in the classical and non-classical theories of thermoelasticity. Sharma [7] represented the three-dimensional transversely isotropic thermoelastic problem based on cylindrical panel in the context of coupled thermoelasticity. Keles and Tutuncu [8] studied the effect of grading index on free and forced vibrations of transversely isotropic elastic hollow cylinders and disks. Sharma et al. [9] analyzed the free vibrations in viscothermoelastic hollow sphere in which uncoupled and coupled equations are taken for first and second class vibrations, respectively. Sharma et al. [10] studied the free vibrations of the axisymmetric functionally graded thermoelastic cylinder and investigated the numerical results of thermoelastic damping and frequency shift. The analysis of stress-free axisymmetric functionally graded viscothermoelastic hollow sphere in the context of generalized thermoelasticity was presented by Sharma et al. [11]. Abbas [12] has proposed a model on generalized thermoelasticity for dual phase lag, in which he studied the effect of reinforcement on stress, temperature and displacement. Tripathi et al. [13] studied diffusion interactions in thick thermoelastic circular plate having finite thickness and infinite extent. The spheroidal and toroidal vibrations in viscothermoelastic spherical curved plates were studied by Sharma [14]. Sherief and Allam [15] had investigated electromagnetic interactions in the generalized thermoelastic two-dimensional solid cylinder. Sharma et al. [16] studied a dynamic problem of semi-infinite viscothermoelastic cylinder based on five theories of generalized thermoelasticity using Laplace and Hankel transforms. Hien and Lam [17] studied the dynamic response of functionally graded rectangular plates under moving loads in the context of coupled viscoelasticity. Wang [18] presented the dynamic stability in the Rayleigh beams in axial motion. The forced vibrations under the action of heat sources in functionally graded viscothermoelastic sphere has been studied by Sharma et al. [19]. Neuringer [20] studied the indicial equation having complex roots in series solution of Frobenius method. Propagation of waves in piezoelectric cylindrical shell was studied by Bisheh and Wu [21]. There is no problem of free vibrations which have been solved completely to obtain the results in a closed form by using different methods of differential equations available in the literature. This motivated the authors to investigate free vibrations of rigidly fixed viscothermoelastic cylinder by applying series solution of Frobenius method. The main aim of this paper is to study and investigate the free vibration analysis of rigidly fixed viscothermoelastic hollow cylinder which is subjected to the rigidly fixed boundary conditions (i.e. thermally insulated and isothermal). This problem has been modelled with the help of generalized theories of thermoelasticity i.e. Lord and Shulman (LS) [22] and Green and Lindsay (GL) [23]. The dispersion relations are derived analytically with the help of series solution and the numerical computation has been carried out with the help of polymethyl methacrylate material. Numerical results for natural frequencies, thermoelastic damping, frequency shift, temperature change and displacement (deformation) are shown graphically.

Free Vibration Analysis of Rigidly Fixed Axisymmetric …

99

1.1 Formulation of Problem We have considered a thick-walled viscothermoelastic thermally conducting hollow cylinder having its inner radius a and outer radius ξa. The cylinder is considered in undisturbed state with the uniform temperature T0 initially having domain a  r  ξa. The solid is considered axisymmetric cylinder made of functionally graded viscothermoelastic material. Here the components of displacement in cylindrical coordinates (r, θ, z) have been expressed as u θ = u z = 0 and u r = u(r, t). The basic governing equations of motion and heat conduction for the homogenous isotropic viscothermoelastic cylinder in the absence of body forces and heat sources given by Dhaliwal and Singh [6] are ∂2u ∂t 2       ∂ ∂T ∂2 ∂2 1 ∂ ∗ ∂ Kr − ρCe + t0 2 T = T0 β + t0 δ1k 2 e r ∂r ∂r ∂t ∂t ∂t ∂t σi j, j = ρ

where

  ⎫ ∂ ⎪ T⎪ σrr = (λ + 2μ )err + λ eθθ − β 1 + t1 δ2k ⎪ ⎪ ∂t ⎪ ⎪  ⎪  ⎬ ∂ ∗ ∗ ∗ ∗ T σθθ = λ err + (λ + 2μ )eθθ − β 1 + t1 δ2k ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ∂u u⎪ ⎭ , eθθ = ⎪ e = eθθ + err , err = ∂r r ∗

∗

∗

(1)

(2)

∗

(3)

here σrr and σθθ are radial and de-hoop stress components; e is dilatation; err and eθθ are the strain components; ρ is mass density; T (r, t) is temperature; u(r, t) is radial displacement; β ∗ is the viscothermoelatic coupling constant; λ∗ and μ∗ are viscoelastic parameters; Ce is specific heat; K is thermal conductivity; δi j is Kronecker delta in which k = 1 is Lord Shulemen theory and k = 2 is Green Lindsay theory. Now we consider a functionally graded material in the sense that thermal conductivity, viscothermoelasticity, density and modulus of viscoelasticity vary with radial coordinate as λ∗ = λ0

 γ  γ  γ r r r , μ∗ = μ0 , ρ∗ = ρe , a a a

K ∗ = K0

 γ  γ r r , β ∗ = β0 a a

(4) here γ indicates the degree of non-homogeneity. where    ∂ ∂ ∂ ∗ , μ0 = μe 1 + α1 , β = βe 1 + β0 , λ0 = λe 1 + α0 ∂t ∂t ∂t (3λe α0 + 2μe α1 )αT βe = (3λ + 2μ)αT , β0 = βe

(5)

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Here αT is the coefficient of linear thermal expansion of material, α0 , α1 are viscoelastic relaxation times, λe and μe are Lame’s constants Using Eqs. (3)–(5) in Eqs. (1) and (2), we get 

 2  ∂ m ∗2 ∂ ∂ m1 ∂ βe 1 + δ0 u+ 2u− 1 + β0 + ∂t ∂r 2 r ∂r r (λe + μe ) ∂t   ∂ γ ρe ∂ 2 u ∂ 1 + t1 δ2k + T = ∂t ∂r r λe + μe ∂t 2  2  ∂ ρCe ∂ K0 ∂2 m1 ∂ T T − + t + 0 T0 βe ∂r 2 r ∂r βe T0 ∂t ∂t 2    ∂u ∂ u ∂ ∂2 + t0 δ2k 2 + = 1 + β0 ∂t ∂t ∂t ∂r r

where

m 1 = γ + 1,

(6)

(7)

m ∗2 = [λ0 (γ − 1) − 2μ0 ]

2 Boundary Conditions The analysis of inhomogenous hollow cylinder is subjected to rigidly fixed, thermally insulated and isothermal boundary conditions at inner radius r = a and outer radius r = ξa. Mathematically, this gives us u(r, t) = θ = 0 u(r, t) =

∂θ =0 ∂r

at at

x = a , x = ξa x = a , x = ξa

(8) (9)

We introduce non-dimensional quantities to remove the complexity of the equations. ⎫ r u C1 t ⎪ T0 βe T0 βe2 ⎪ , x= , w= , τ= , T = ⎪ ⎪ λe + 2μe ρe Ce (λe + 2μe ) a a a ⎪ ⎪ ⎪ ∗⎪ ⎪ T C1 C1 C aω ⎪ 1 ∗ ⎪ ˆ α1 , αˆ1 = α1 , β0 = β0 ,  = θ= , αˆ0 = ⎪ T0 a a a C1 ⎬ C2 σrr σθθ C1 ⎪ ⎪ t0 ⎪ τx x = , τθθ = , δ0 = αˆ 0 + 2δ 2 (αˆ 1 − αˆ 0 ) , δ02 = 22 , τ0 = ⎪ 2 2 ⎪ a ⎪ ρe C 1 ρe C 1 C1 ⎪ ⎪ ⎪ ⎪ (λ + 2μ ) + 2μ C C C λ μ 1 1 e e e e e e⎪ ⎪  ∗ 2 2 ⎭ δ1kt0 , τ1 = t1 , ω = τ0 = , C1 = , C2 = a a K0 ρe ρe ¯ =

(10)

Substituting the above quantities from Eq. (10) in Eqs. (6) and (7), we get

Free Vibration Analysis of Rigidly Fixed Axisymmetric …  1 + δ0

∂ ∂t



101

   m 2 γ ∂2w m1 ∂ ∂ ∂2 ∂ ˆ0 ∂ w + 1 + τ + θ= + w − ¯ 1 + β δ 1 2k ∂x 2 x ∂x x2 ∂τ ∂τ ∂x x ∂x 2

(11)





∂2θ ∂2 m 1 ∂θ ∂ T  − ∗ + τ0 2 θ = + 2 ∂x x ∂x ∂τ ∂τ ¯

m 2

where

∗

1 + βˆ 0

∂ ∂τ



∂2 ∂ + τ0 2 ∂τ ∂τ



1 ∂ + w ∂x x

    ∂ ∂ γ − 1 + δˆ0 ∂τ = (1 − 2δ 2 ) 1 + α0 ∂τ

(12)

2.1 Solution of Mathematical Problem We define the cylindrical time-harmonics vibrations by Sharma et al. [11] such that ¯ iτ (w, θ)(x, τ ) = (w, ¯ θ)e

(13)

Using Eq. (13) in Eqs. (11) and (12), we get m 1 d w¯ d 2 w¯ + + 2 dx x dx



 d m2 γ ¯ i w ¯ + im + + θ=0 3 x2 dx x δ˜0

 d 1 ¯ d 2 θ¯ m 1 d θ¯ ∗ 2 3 ¯ −   τ˜0 θ + i m 4 + + θ=0 dx2 x dx dx x

(14)

(15)

where ⎫  T ∗ β˜ 0 τ˜0 γ(α˜ 0 (1 − 2δ 2 )) ¯ β˜ 0 τ˜0 ⎪ , α˜ 0 = i−1 + αˆ 0 , β˜ 0 = i−1 + βˆ 0 ⎬ − 1 , m3 = , m4 = δ0 ¯ δ˜ 0 ⎪ ⎭ τ˜1 = i−1 + τ1 δ2k , τ˜0 = i−1 + τ0 δ1k , α˜ 1 = i−1 + αˆ 1 , δ˜ 0 = i−1 + δ0 , τ˜0 = i−1 + τ0 

m2 =

(16) Introducing the transformation defined by Sharma et al. [10] as γ ¯ (w, ¯ θ)(x, τ ) = x − 2 (V, )(x, τ )

(17)

Using transformation defined by Eq. (17) in Eqs. (14) and (15), we get ⎡

 ⎤  η2 γ i d 2      im + − + 3 dx ⎢ ⎥ x2 2x δ˜ 0 ⎢  ⎥ V = 0   ⎣ 0 γ2 ⎦  1 2 + ∗ 2 τ˜0 − 4x m 4 i3 ddx − γ−2 2 2 x where

2 =

d2 dx2

+

1 d x dx

,

η2 = m 2 −

γ2 4

(18)

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3 Series Solution Since x = 0 is a regular singular point in matrix differential equation (18), so we introduce matrix Frobenius method in Eq. (18) for which there exists atleast one trivial solution takes the form as     ∞  Lk V x p+k = Mk 

(19)

k=0

Here p is an eigen value and L k and Mk are the unknown coefficients. Substituting the solution assumed from Eq. (19) in the Eq. (18) and on simplifying we get ∞  

 H1 ( p + k)x

−2

+ H2 ( p + k)x

−1

k=0

+

i δ˜ 0

0 ∗ 2 0   τ˜0

 

 Lk x p+k = 0 (20) Mk

where  0 ( p + k)2 − η 2 2 , 0 ( p + k)2 − γ4  ⎤ ⎡ γ p + k + 0 im 3 ⎢ 2 ⎥ ⎥   H2 ( p + k) = ⎢ ⎦ ⎣ ∗ 3 0 i  m 4 p + k − γ−2 2 

H1 ( p + k) =

Now equating the coefficients of the lowest power of x(i.e, x p−2 ) from Eq. (20) to zero, we get   2  p − η2 0 L 0( p j ) 2 =0 (21) M0 ( p j ) 0 p 2 − γ4 The above Eq. (21) will have a non-trivial solution, so that the roots of indicial equation are given by p1 = +η,

p2 = −η,

γ p3 = + , 2

p4 = −

γ 2

(22)

Here the roots p1 and p2 may be complex roots and the roots p3 and p4 are real roots. Due to Neuringer [20], the sufficient use of complex roots of indicial equation have two independent real solutions of Eq. (20). Further it is impossible with the development that former case is solved once rather than twice in the later one. For the choice of indicial Eq. (21), unknowns L 0 ( p j ) and M0 ( p j ) can be chosen as  1, L 0( p j ) = 0,

j = 1, 2 j = 3, 4

 ,

M0 ( p j ) =

0, 1,

j = 1, 2 j = 3, 4

(23)

Free Vibration Analysis of Rigidly Fixed Axisymmetric …

103

Now equating the next lowest power of x(i.e. x p−1 ) to zero, we get 

   L 1( p j ) L 0( p j ) H1 ( p j + 1) + H2 ( p j ) =0 M1 ( p j ) M0 ( p j ) 

(24)

On simplifying the above equation, we get 

where

1 e12 (pj) =

    1 L 1( p j ) 0 e12 ( p j ) L 0( p j ) =− 1 0 M1 ( p j ) e21 ( p j ) M0 ( p j )

im 3 ( p j + γ+2 2 ) , ( p j +1)2 −η 2

1 e21 (pj) =

(25)

i∗ 3 m 4 ( p j +( 4−γ 2 ))

Now equating like powers of the coefficient of x

2

( p j +1)2 − γ4 p+k

equal to zero, we get

   L k+2 ( p j ) L k+1 ( p j ) + H2 ( p j + k + 1) H1 ( p j + k + 2) Mk+2 ( p j ) Mk+1 ( p j )    i 0 Lk( p j ) + δ˜0 ∗ 2 = 0 ; k = 0, 1, 2.... 0   τ˜0 Mk ( p j ) 

(26)

On simplifying Eq. (26), we get following recurrence relation 

k H11 =

where k = H21

    k ( p j ) L k+1 ( p j ) L k+2 ( p j ) 0 H12 =− k Mk+2 ( p j ) Mk+1 ( p j ) H21 (pj) 0    k 0 Lk( p j ) H11 ( p j ) ; k = 0, 1, 2...... + k 0 H22 ( p j ) Mk ( p j ) i δ˜ 0 (( p j +k+2)2 −η 2 )

k , H22 =

i m 4 ( p j +k+( 4−γ 2 )) 2 ( p j +k+2)2 − γ4

∗ 2 τ˜0 2 ( p j +k+2)2 − γ4

k , H12 =

(27)

im 3 ( p j +k+( γ+2 2 )) ( p j +k+2)2 −η 2

,

3

Substituting k = 0 in Eq. (27), we get 

  2   L 2( p j ) 0 e (p ) L 0( p j ) = 11 j 2 M2 ( p j ) 0 e22 ( p j ) M0 ( p j )

2 0 1 0 where e11 ( p j ) = H12 ( p j )e21 ( p j ) − H11 (pj) , 0 H22 ( p j ) Again substituting k = 1 in Eq. (27), we get



2 0 1 e22 ( p j ) = H21 ( p j )e12 (pj) −

    3 L3 ( p j ) L0 0 e12 =− 3 M3 0 M0 e21 ( p j )

3 1 2 2 1 e12 ( p j ) = H12 ( p j )e22 ( p j ) − H11 ( p j )e12 (pj) ,

(28)

(29)

3 1 2 1 1 e21 ( p j ) = H21 ( p j )e11 ( p j ) − H22 ( p j )e21 (pj)

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Now continuing likewise, we can have [L 2k ( p j ), M2k ( p j )] has similar form as H1 ( p + k) and [L 2k+1 ( p j ), M2k+1 ( p j )] has similar form as H2 ( p + k). Thus we can write 

  2k   e (p ) 0 L 2k L0 = 11 j 2k , M2k 0 e22 ( p j ) M0



    2k+1 0 e12 ( p j ) L0 L 2k+1 = − 2k+1 M2k+1 M0 0 e21 ( p j )

(30) where

⎧ 2k 2k−2 2k−1 2k−2 2k−2 e11 = H12 ( p j )e21 ( p j ) − H11 ( p j )e11 (pj) ⎪ ⎪ ⎪ ⎨e2k = H 2k−2 ( p )e2k−1 ( p ) − H 2k−1 ( p )e2k−1 ( p ) j 12 j j 22 j 21 22 22 2k+1 2k−1 2k−1 2k−1 2k ⎪ = H ( p )e ( p ) − H ( p )e ( p e j 22 j j 12 j) ⎪ 12 12 11 ⎪ ⎩ 2k+1 2k−1 2k−1 2k−1 2k e21 = H21 ( p j )e11 ( p j ) − H22 ( p j )e21 ( p j )

(31)

From above discussion, we can proceed that     2k 0 im 3 0 e11 ( p j ) −1 ≈ O(k , ) 2k 0 i3 m 4 0 e22 (pj)     2k+1 0 e12 (pj) 0 im 3 −1 ≈ O(k ) 2k+1 i3 m 4 0 e21 (pj) 0

(32)

    2k 2k+1 0 (pj) 0 e12 e11 ( p j ) → → 0 and 2k 2k+1 0 e22 (pj) e21 (pj) 0 0 as k → ∞. This implies that the assumed sequence in Eq. (19) is uniformly and absolutely convergent having infinite radius of convergence, so that Eq. (19) can be written as        2   1 3 V 0 e12 (pj) 0 (pj) 3 0 e12 e (p ) = I− 1 x + 11 j 2 x2 − 3 x  0 0 e21 ( p j ) 0 e22 ( p j ) e21 ( p j )   L 0( p j ) p j x + ...........∞ (33) M0 ( p j ) From Eq. (32), it is clear that

General solution for Eq. (13) from Eq. (33) via Eq. (19) we get w(x, τ ) =

∞   k=0

θ(x, τ ) =

2k 2k ( p1 )x p1 + H2k e11 ( p2 )x p2 + H1k e11 γ x 2k− 2 (eiτ ) 2k+1 2k+1 H3k e12 ( p3 )x p3 +1 + H4k e12 ( p4 )x p4 +1

∞   H1k e2k+1 ( p1 )x p1 +1 + H2k e2k+1 ( p2 )x p2 +1 + 21

k=0

2k H3k e22 ( p3 )x p3

+

21 2k H4k e12 ( p4 )x p4

γ

x 2k− 2 (eiτ )

(34)

(35)

Free Vibration Analysis of Rigidly Fixed Axisymmetric …

105

∞  2k+1 2k+1  ( p1 )x p1 + H2k (R1∗ + p2 )e21 ( p2 )x p2 + 2k− γ iτ H1k (R1∗ + p1 )e21 ∂θ 2 (e = x ) 2k ( p )x p3 −1 + H (R ∗ + p )e2k ( p )x p4 −1 ∂x H3k (R2∗ + p3 )e22 3 4 22 4 4k 2 k=0

where

R1∗ = 2k +

R2∗ = 2k − γ2 .

2−γ , 2

(36)

4 Dispersion Relations The non-dimensional boundary conditions of hollow cylinder subjected to rigidly fixed thermally insulated and isothermal conditions have been applied, we get four linear algebraic equations. The system of equations lead to non-trivial solution if the coefficient of determinant equation vanishes and non-trivial solution forms the following determinant equation:  m 11  m 21  m 31  m 41

m 12 m 22 m 32 m 42

m 13 m 23 m 33 m 43

 m 14  m 24  =0 m 34  m 44 

(37)

The elements of m i j (i, j = 1, 2, 3, 4) are defined below for thermally insulated and isothermal boundary conditions for k = 0 and k > 0 in SET-1 and SET-2. SET-1: Elements for thermally insulated boundaries are Case-1: For (k = 0)   ⎫ γ γ ⎪ 1 1 ⎪ − ; m 14 = e12 m 13 = e12 ⎪ 2 2 ⎪ ⎪ ⎪ ⎪   ⎪ γ γ γ 1 γ 1−γ ⎪ ⎪ η− 2 −η− 2 0 0 1 1 ⎪ ⎪ ξ ; m 24 = e12 − ξ ; m 22 = e11 (−η)ξ ; m 23 = e12 m 21 = e11 (η)ξ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪    γ ⎬ 2−γ 2−γ 1 1 0 + η e21 (η); m 32 = − η e21 (−η); m 33 = 0; m 34 = (−γ)e22 − m 31 = 2 2 2 ⎪ ⎪ ⎪   ⎪ ⎪ γ 2−γ 2 − γ η− 2 −η− γ2 ⎪ 1 1 ⎪ ; m 42 = ;⎪ m 41 = + η e21 (η)ξ − η e21 (−η)ξ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪  ⎪ ⎪ γ ⎪ 0 −(1+γ) ⎪ ⎭ m 43 = 0; m 44 = (−γ)e22 − ξ 2 0 (η); m 11 = e11

0 (−η); m 12 = e11

(38)

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Case-2: For (k > 0)   ⎫ 2k (η); m = e2k (−η); m = e2k+1 γ ; m = e2k+1 − γ ⎪ ⎪ m 11 = e11 ⎪ 12 13 14 11 12 12 ⎪ 2 2 ⎪ ⎪ ⎪  ⎪ ⎪ γ γ γ ⎪ η+2k− 2 −η+2k− 2 2k+1 2k 2k 2k+1 ⎪ m 21 = e11 (η)ξ ξ ; m 22 = e11 (−η)ξ ; m 23 = e12 ;⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪   ⎪ ⎪ γ −γ+2k+1 2−γ ⎪ 2k+1 2k+1 ξ − ; m 31 = 2k + m 24 = e12 + η e21 (η);⎪ ⎪ ⎬ 2 2    2−γ ⎪ 2k+1 2k γ ; m = (2k − γ)e2k − γ ⎪ ⎪ m 32 = 2k + − η e21 (−η); m 33 = (2k)e22 ⎪ 34 22 ⎪ 2 2 2 ⎪ ⎪ ⎪   ⎪ γ γ⎪ 2−γ 2 − γ η+2k− 2 −η+2k− 2 ⎪ 2k+1 2k+1 ⎪ ⎪ + η e21 (η)ξ − η e21 (−η)ξ ; m 42 = 2k + m 41 = 2k + ⎪ ⎪ 2 2 ⎪ ⎪ ⎪   ⎪ ⎪ γ γ ⎪ 2k 2k−1 2k (2k−1−γ) ⎪ ξ ξ m 43 = (2k)e22 ; m 44 = (2k − γ)e22 − ⎭ 2 2

(39) SET-2: Elements for isothermal boundaries are Case-1: For (k = 0) ⎫   γ γ ⎪ 0 (η); m = e0 (−η); m = e1 1 ⎪ ; m − m 11 = e11 = e 12 13 14 ⎪ 11 12 2 12 2 ⎪ ⎪ ⎪ ⎪   ⎪ γ 1 γ 1−γ ⎪ ⎪ η− γ2 −η− γ2 0 0 1 1 ⎪ ξ ; m 24 = e12 − ξ ; m 22 = e11 (−η)ξ ; m 23 = e12 m 21 = e11 (η)ξ ⎪ ⎬ 2 2   γ γ −γ ⎪ ⎪ 1 (η); m = e1 (−η); m = e0 0 ⎪ m 31 = e21 32 33 21 22 2 ; m 34 = e22 − 2 ξ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ γ γ −γ ⎪ ⎪ η+1− γ2 −η+1− γ2 1 1 0 0 ⎭ ; m 44 = e22 − ξ ⎪ m 41 = e21 (η)ξ ; m 42 = e21 (−η)ξ ; m 43 = e22 2 2

Case-2: For (k > 0)

(40)

⎫   2k (η); m = e2k (−η); m = e2k+1 γ ; m = e2k+1 − γ ⎪ ⎪ m 11 = e11 ⎪ 12 13 14 11 12 12 2 2 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ η+2k− γ2 −η+2k− γ2 2k+1 γ 2k 2k 2k+1 ⎪ ξ ; m 22 = e11 (−η)ξ ; m 23 = e12 ;⎪ m 21 = e11 (η)ξ ⎪ ⎪ 2 ⎪ ⎪   ⎪ ⎬ γ γ 2k+1 2k+1 2k+1 2k+1−γ 2k ξ ; − ; m 31 = e21 (η); m 32 = e21 (−η); m 33 = e22 m 24 = e12 2 2 ⎪ ⎪ ⎪  ⎪ γ γ ⎪ ⎪ γ 2k−γ 2k+1 2k+1 ⎪ m 34 = e22 − ; m 41 = e21 (η)ξ η+2k+1− 2 ; m 42 = e21 (−η)ξ −η+2k+1− 2 ;⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪   ⎪ γ 2k γ 2k−γ ⎪ ⎪ 2k 2k ⎪ ⎭ m 43 = e22 ξ ; m 44 = e22 − ξ 2 2

(41)

Free Vibration Analysis of Rigidly Fixed Axisymmetric …

107

4.1 Deduction of Results If the viscous effect (α0 = 0 = α1 ) is ignored in Eqs. (5), then the analysis reduces to generalized thermoelastic cylinder. In case relaxation time is considered to be zero (t1 = t0 = 0) in Eqs. (1)–(3), the results are reduced to coupled thermoelastic cylinder. If thermal equilibrium have been introduced, i.e.( T = T = t0 = t1 = α1 = α0 = 0) in Eqs. (1)–(7) the present analysis reduces to vibrations of coupled elastic hollow cylinder and the Eqs. (34) and (35) of viscothermoelastic hollow cylinder reduces to equations of coupled elastic hollow cylinder that can be written as γ

w(x, τ ) = x − 2

∞  

 2k 2k H1k e11 ( p1 )x p1 + H2k e11 ( p2 )x p2 x 2k (eiτ )

(42)

k=0

θ(x, τ ) = x

− γ2

∞  

2k+1 H1k e21 ( p1 )x p1 +1

+

2k+1 H2k e21 ( p2 )x p2 +1

 x 2k (eiτ )

(43)

k=0

5 Numerical Results and Discussion To authenticate the analytical results, we propose some numerical techniques which have been applied with the help of MATLAB software tools for thermally insulated rigidly fixed viscothermoelastic cylinder. For numerical computations, polymethyl methacrylate material have been taken whose physical data is given by Sharma [14]: T = 0.045, ω ∗ = 1.11 × 1011 s−1 , T0 = 773 K, δ 2 = 0.333 αˆ0 = αˆ1 = 0.005, τ0 = 0.02, τ1 = 0.03, ρ = 1190 kg · m−3 K = 0.19 W · m−1 · k −1 , Ce = 1400 J · kg−1 · K−1 , αT = 77 × 10−6 · K−1 The frequency equations obtained from Eqs. (37) to (39) are complex transcendental equations due to the presence of dissipation term in heat conduction equation. Numerical computations are applied upto six decimal places by taking k = 10 to obtain natural frequency. The complex values of natural frequency () can be written as m = mR + mI , the natural frequency and dissipation factor can be written as f v = mR and D I = mI , where m is the mode number which corresponds to root of transcendental equation. The thermoelastic damping Q −1 and frequency shift s are defined below which are taken from Sharma et al. [11].    DI  Q −1 = 2  , fV

  M∗  f − fE s =  v E v  f v

(44)

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Here M ∗ represents viscothermoelastic (VTE), thermoelastic (TE), viscoelastic (VE) and E stands for elastic materials. Figures 1, 2, 3 and 4 indicate natural frequency and thermoelastic damping against mode number (m) for γ = −3.5, −1.5, 0.0, 1.5, 3.5 in case of ξ = 2.0 and ξ = 4.0, respectively. It is revealed from Figs. 1 and 3 that with increase in mode number m, the natural frequencies go on increasing. As the grading index decreases, the behaviour of natural frequencies increases. It can be inferred from Fig. 2 that initially thermoelastic damping is small upto m = 2.0 and maximum at m = 6.0, then at γ = 0.0, 1.5, 3.5 damping goes on decreasing and shows merged behaviour for γ = −3.5, −1.5 due to negative values of grading index. It is seen from Fig. 4 that damping is slow upto m = 2.0, then go on increasing with increasing value of m. Figures 5, 6 show the frequency shift against mode number (m) for homogenous and inhomogenous materials. It is revealed form Fig. 5 that frequency shift s is quite high in case of viscoelastic (VE) material as compared with viscothermoelastic (VTE) and elastic (E) case for homogenous materials. Figure 6 depicts that the frequency shift s has high variation for TE as compared to VTE and VE case for inhomogenous materials. The behaviour of Fig. 5 follows the inequality s (V E) < s (V T E) < s (T E) while Fig. 6 follows reverse inequality s (T E) > s (V T E) > s (V E) to that of Fig. 5 which clearly indicates the effect of inhomogeneity. Figures 7 and 8 have

Fig. 1 Natural frequency against mode number

Fig. 2 Thermoelastic damping against mode number

Free Vibration Analysis of Rigidly Fixed Axisymmetric …

109

Fig. 3 Natural frequency against mode number

Fig. 4 Thermoelastic damping against mode number

Fig. 5 Frequency shift against mode number

been plotted for temperature (θ)  and displacement (w) against modified thickness  ; (ξ = 0) for γ = −3.5, −1.5, 0.0, 1.5, 3.5 in case which is defined as π ∗ = x−1 ξ−1 of VTE cylinder. The temperature change is large for homogenous case γ = 0 and small for γ = 3.5 and the behaviour decreases with increase in value of π ∗ . From Fig. 8, it is noticed that the displacement (w) shows small variation in π ∗ = 0.0 and large behaviour for π ∗ = 0.1 and with increase in the value of π ∗ the variations go on decreasing. Figures 7 and 8 show that the behaviour for γ = −3.5 has large variation as compared to γ = 3.5 has small variation. This is noticed from all the figures that the

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Fig. 6 Frequency shift against mode number

Fig. 7 Variation of temperature against thickness

Fig. 8 Variation of displacement against thickness

vibrations have maximum variation at γ = −3.5 and least at γ = 3.5. This satisfies the trends for the values of γ = −3.5 > γ = −1.5 > γ = 0.0 > γ = 1.5 > γ = 3.5 which shows the effect of non-homogeneity of grading index of viscothermoelastic hollow cylinder.

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111

6 Conclusion The series solution of matrix Frobenius method is successfully applied for the analysis of free vibrations in rigidly fixed axisymmetric viscothermoelastic hollow cylinder. The dispersion relations were derived analytically and their behaviour has been analyzed numerically. The significant effects of mechanical relaxation times and thermal relaxation times have been noticed for viscothermoelastic displacement, temperature and frequencies. The continuity in case of generalized theories of thermoelasticity are more prominent than coupled theories of thermoelasticity. As the grading index increases, the variation of field functions go on decreasing. The numerical results are consistent with Keles and Tutuncu [8] in the absence of viscous and thermal effects. With the help of grading index, the deformation and change in temperature may be increased or decreased as per requirement. The inhomogeneity parameter, i.e. grading index may also be used to handle the frequency shift and thermoelastic damping to improve the quality of signals having different vibration mode numbers. Also, this index is used to optimize the energy losses.

References 1. D. Bland, The Theory of Linear Viscoelasticity (Pergamon Press, Oxford, 1960) 2. M.A. Biot, Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomen. J. Appl. Phys. 25, 1385–1391 (1954) 3. S.C. Hunter, Viscoelastic waves, Progress in Solid Mechanics (1960) 4. W. Flugge, Viscoelasticity, Blaisdell Publishing Company (Massachusetts, Toronto, London, 1967) 5. P. Chadwick, D. Windle, Propagation of Rayleigh waves along isothermal and insulated boundaries. Proc. R. Soc. A 280, 47–71 (1964) 6. R.S. Dhaliwal, A. Singh, Dynamic Coupled Thermoelasticity, Hindustan Publishing Corporation (1980) 7. J.N. Sharma, Three-dimensional vibration analysis of a homogeneous transversely isotropic thermoelastic cylindrical panel. J. Acoust. Soc. Am. 110, 254–259 (2001) 8. I. Keles, N. Tutuncu, Exact analysis of axisymmetric dynamic response of functionally graded cylinders (or disks) and spheres, J. Appl. Mech. 78 (2011), 061014-1-7 9. J.N. Sharma, D.K. Sharma, S.S. Dhaliwal, Free vibration analysis of a rigidly fixed viscothermoelastic hollow sphere. Indian J. Pure Appl. Math. 44, 559–586 (2013) 10. J.N. Sharma, P.K. Sharma, K.C. Mishra, Analysis of free vibrations in axisymmetric functionally graded thermoelastic cylinders. Acta Mech. 225, 1581–1594 (2014) 11. D.K. Sharma, J.N. Sharma, S.S. Dhaliwal, V. Walia, Vibration analysis of axisymmetric functionally graded viscothermoelastic spheres. Acta Mech. Sin. 30, 100–111 (2014) 12. I.A. Abbas, A dual phase lag model on thermoelastic interaction in an infinite Fiber-Reinforced anisotropic medium with a circular hole. Mech. Based Des. Struct. Mach. 43, 501–513 (2015) 13. J.J. Tripathi, G.D. Kedar, K.C. Deshmukh, Generalized thermoelastic diffusion in a thick circular plate including heat source. Alex. Eng. J. 55, 2241–2249 (2016) 14. D.K. Sharma, Free vibrations of homogenous isotropic viscothermoelastic spherical curved plates. Tamkang J. Sci. Eng. 19, 135–148 (2016) 15. H.H. Sherief, A.A. Allam, Electromagneto interaction in a two-dimensional generalized thermoelastic solid cylinder. Acta Mech. 228, 2041–2062 (2017)

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16. D.K. Sharma, H. Mittal, S.R. Sharma, I. Parkash, Effect of deformation on semi infinite viscothermoelastic cylinder based on five theories of generalized thermoelasticity. Math. J. Interdiscip. Sci. 6, 17–35 (2017) 17. T.D. Hien, N.N. Lam, Vibration of functionally graded plate resting on viscoelastic elastic foundation subjected to moving loads, in IOP Conference Series: Earth and Environmental Science (2018) p. 012024 18. B. Wang, Effect of rotary inertia on stability of axially accelerating viscoelastic Rayleigh beams. Appl. Math. Mech. 39, 717–732 (2018) 19. D.K. Sharma, S.R. Sharma, V. Walia, Analysis of axisymmetric functionally graded forced vibrations due to heat sources in viscothermoelastic hollow sphere using series solution, in AIP Conference Proceedings (2018), p. 030010 20. J.L. Neuringer, The Frobenius method for complex roots of the indicial equation. Int. J. Math. Educ. Sci. Technol. 9, 71–77 (1978) 21. H.K. Bisheh, N. Wu, Analysis of wave propagation characteristics in piezoelectric cylindrical composite shels reinforced with carbon nanotubes. Int. J. Mech. Sci. 145, 200–220 (2018) 22. H.W. Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967) 23. A. Green, K. Lindsay, Thermoelasticity. J. Elast. 2, 1–7 (1972)

Study on a Free Boundary Problem Arising in Porous Media Bhumika G. Choksi and Twinkle R. Singh

Abstract The present study discusses a free boundary problem arising from the steady two-dimensional seepage flow through a rectangular dam. The free boundary location, the potential velocity field, and the pressure field have been found using successive linearisation method (SLM) by solving a nonlinear partial differential equation arising as a governing equation for this problem. The SLM is a newly developed method, which is a very efficient and reliable method to handle nonlinear problems. The numerical and the graphical representation of the solution has been discussed using MATLAB under the certain valid assumption. Keywords Free boundary problems · Two-dimensional potential flows · Porous media · Seepage face · Successive linearisation method (SLM)

1 Introduction Here the problem concerned with the steady two-dimensional seepage flow in a saturated porous medium has been discussed. This problem can be formulated as a free boundary problem for an elliptic equation mathematically. Such problems arise in a wide variety of fluid mechanics applications [9]. The free boundary can be found by using any numerical methods such as finite element models [11], variational inequalities [1], etc. But here successive linearisation method (SLM) has been used to find the free boundary of the dam. For a seepage flow in a rectangular dam, the free boundary is expressed by Polubarinova–Kochina (PK) equation [17], which has been solved numerically in [10]. By using an approximate model, which provides an analytical expression for

B. G. Choksi (B) · T. R. Singh Department Applied Mathematics and Humanities, S. V. National Institute of Technology, Surat 395007, Gujarat, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_10

113

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the free boundary profile. The model generalizes for steady axisymmetric potential flow [6]. Within the range of validity of the model assumptions, the computational results show good agreement with the Polubarinova–Kochina (PK) equation.

2 Flow Problem Figure 1 shows the steady two-dimensional seepage flow through a rectangular dam. The dam is above a horizontal impermeable base and the porous medium is saturated, homogeneous, and isotropic. The x-axis represents horizontally and the z-axis represents vertically. The free boundary is denoted by h(x), the upstream flow depth by h 1 , the downstream flow depth by h 2 , the dam length by L, the seepage face by h s f = h(x = L) − h 2 . Neglecting the capillary fringe, the free boundary becomes the saturation line [2]. To find the location of the free boundary, consider the velocity field v(x, z) and the pressure field p(x, z) in the unbounded flow domain [0, L] × [0, h(x)]. The momentum and the continuity equations are expressed by Darcy’s law and solenoidality condition as [7, 12]

Fig. 1 Schematic diagram of the flow problem

Study on a Free Boundary Problem Arising in Porous Media

  p v = −k∇ z + ρg ∇ ·v =0

115

(1) (2)

where k is the constant hydraulic conductivity. Equation (1) is consistent with the irrotationality condition given as ∇ ×v =0

(3)

The boundary conditions are given by vz (x, 0) = 0

(4)

dh vz (x, h) = vx (x, h) dx p(x, h) = 0 1 p(0, z) = h 1 − z ρg 1 p(L , z) = h 2 − z; 0 ≤ z ≤ h 2 ρg p(L , z) = 0; z > h 2

(5) (6) (7) (8) (9)

where vx and vz are the horizontal and vertical velocity components respectively. Equations (4) and (5) are the kinematic boundary conditions at the base and at the free boundary, respectively; Eq. (6) is the dynamic boundary condition at the free boundary; Eq. (7) is the inflow boundary condition; Eqs. (8) and (9) are the outflow boundary conditions. To find the free boundary location and the flow rate q for the problem, the successive linearisation method (SLM) [14–16] is used. So the constant flow rate is given by h(x) vx d x q=

(10)

0

Here vx is the velocity component in the x-direction. Now the approximate velocity components can be found by using the following Picard iteration. To start the iteration, in the first approximation consider q h vz = 0

vx =

(11) (12)

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Thus Eqs. (3) and (4) are verified, while to fulfill the conditions (2) and (5) consider the following first-order term as small to be neglected by dh ≈0 dx

(13)

which defines the closure hypotheses of the first approximation. For the second approximation, consider q h vz = f 2 (x, z)

vx =

(14) (15)

where f 2 is an unknown function. By the Eqs. (2) and (4), f 2 can be written as f 2 (x, z) =

q dh z h2 d x

(16)

To verify Eq. (3) by using Eqs. (14), (15) and (16), consider the following secondorder terms as small to be neglected by 

dh dx

2 ≈ 0,

d2h ≈0 dx2

(17)

which defines the closure hypotheses of the second approximation. For the third approximation, consider q + f 3 (x, z) h q dh vz = 2 z h dx

vx =

(18) (19)

By Eqs. (3) and (10), f 3 can be written as  f3 =

     1 q d2h q dh 3 2 1 q dh 2 1 d 2 h − + − q 2 z 2 h2 d x 2 h3 d x 3 h dx 6 dx

(20)

To verify Eq. (4) by using the Eqs. (18), (19) and (20), consider the following thirdorder terms as small to be neglected by 

dh dx

3 ≈ 0,

d 3h d 2 h dh ≈ 0, ≈0 2 dx dx dx3

which defines the closure hypotheses of the third approximation.

(21)

Study on a Free Boundary Problem Arising in Porous Media

117

This iterative procedure can be extended to any required degree of accuracy. The approximate expressions found using this iterative procedure match with those obtained by using the power series expansion of the harmonic stream function [5, 18] and the Picard iteration of Cauchy–Riemann equations [13]. The proposed procedure offers the advantage of providing the closure hypotheses of the approximate velocity field. Thus, the velocity components can be expressed in an approximate form as       1 q d2h q q dh 3 2 1 q dh 2 1 d 2 h − + − q 2 z vx = + h 2 h2 d x 2 h3 d x 3 h dx 6 dx vz =

q dh z h2 d x

(22) (23)

With this, the pressure filed and the free boundary differential equations are given from the vertical and the horizontal momentum equation, respectively, as p q dh 2 =h−z− (z − h 2 ) ρg 2kh 2 d x   d2h 1 dh 2 3 3k dh = − − − dx2 h dx h q dx

(24) (25)

3 Solution of the Problem The flow rate is expressed as [2, 4] q=k

h12 − h22 2L

(26)

and the boundary conditions associated with Eq. (25) are given by h(x = 0) = h 1 d h(x = L) = 0 dx

(27) (28)

The solution of Eq. (25) subject to the above conditions (27) and (28) is obtained by using the successive linearisation method (SLM) [14–16]. Consider a solution of Eq. (25) as h = hi +

i−1 

h m ; i = 1, 2, 3, ...

m=0

By Eq. (25) and neglecting the nonlinear terms in h i , h i  and h i  , we get

(29)

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B. G. Choksi and T. R. Singh

ai−1 h i  + bi−1 h i  + ci−1 h i = φi−1

(30)

where ai−1 = q

i−1 

h m , bi−1 = 2q

m=0

⎡

and φi−1 = − ⎣2x

i−1 

h m  + 3k

m=0

i−1 



hm +

m=0

i−1 

i−1 

h m , ci−1 =

m=0

hm

 i−1 

m=0

hm

 

m=0

+

i−1 

i−1 

h m 

m=0

2 ⎤

hm



(31)

⎦

m=0

with h i (0) = 0 = h i  (L) Choose initial approximation which satisfies the boundary condition as h 0 (x) = h 1 e−x

(32)

By solving Eqs. (30) iteratively, we get each solution for h i (i ≥ 1), thus the approximate solution for h(x) is obtained as h (x) ≈

K 

h m (x)

(33)

m=0

by assuming lim h i = 0

i→∞

(34)

Now ai−1 , bi−1 , ci−1 and φi−1 of Eq. (30) are known from the previous iterations for i = 1, 2, 3, ..., so to solve Eq. (30) the Chebyshev spectral collocation method [3, 19] is used. To apply it, transform the physical region [0, L] into the region [−1,1] using mapping ξ +1 x = ; −1 ≤ ξ ≤ 1 L 2

(35)

Consider the Gauss–Lobatto collocation points [3] to define the Chebyshev nodes in [−1, 1], namely ξ j = cos

πj ; j = 0, 1, 2, ..., N N

The variable f i is approximated by the truncated Chebyshev series given as

(36)

Study on a Free Boundary Problem Arising in Porous Media

h i (ξ ) =

N 

119

h i (ξk )Tk (ξ j )

(37)

k=0

where Tk (ξ ) = cos[kcos−1 (ξ )] is called the kth Chebyshev polynomial. Derivatives of the variables at the collocation points can be given by  dr hi = Dkr j h i (ξk ) r dη k=0 N

(38)

where D = 2D, with D is the Chebyshev spectral differentiation matrix, whose entries are defined as D jk =

c j (−1) j+k ; j = k; j, k = 0, 1, 2, ..., N ; ck ξ j − ξk

Dkk =

ξk ; k = 1, 2, ..., N − 1; 2(1 − ξk2 )

D00 =

(39)

2N 2 + 1 = −D N N 6

Thus the matrix equation form is given as Ai−1 Hi = i−1

(40)

subject to h i (ξ N ) = 0,

N 

D N k h i (ξk ) = 0, h i (ξ0 ) = 0

(41)

k=0

where Ai−1 = ai−1 D 2 + bi−1 D + ci−1 Hi = [h i (ξ0 ), h i (ξ1 ), ..., h i (ξ N )]

(42) T

i−1 = [φi−1 (ξ0 ), φi−1 (ξ1 ), ..., φi−1 (ξ N )]

(43) T

(44)

where Ai−1 is an (N + 1) × (N + 1) matrix; Hi and i−1 are (N + 1) × 1 column vectors; ai−1 , bi−1 and ci−1 are (N + 1) × (N + 1) diagonal matrices and D = L2 D with Chebyshev spectral differentiation matrix D [3, 19]. Now applying the boundary conditions h i (ξ0 ) = 0 and h i (ξ N ) = 0, the solutions for h i (ξ1 ), h i (ξ2 ), ..., h i (ξ N −1 ) are obtained iteratively from solving −1 i−1 Hi = Ai−1

(45)

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4 Concluding Remarks A free boundary profile arising from the steady two-dimensional seepage flow through a rectangular dam is found from the solution (45) by the successive linearisation method (SLM) using MATLAB. Table 1 shows the numerical values of the free boundary for different distance x by the SLM and results are given in [8]. As shown in Fig. 2, within this range, the free boundary profile is deduced, which is feasible with the physical phenomenon. Also, by the comparison of the values of h by the SLM and [8], we can say that the results by the SLM are in good agreement with the results [8]. Thus, the SLM is applicable, efficient, and easy to use to handle nonlinear problems. Table 1 The value of free boundary h for different distance x by SLM and [8]

Fig. 2 Free boundary h versus distance x by SLM

x

SLM

[8]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.9919 0.9863 0.9860 0.9442 0.8463 0.7210 0.5949 0.4886 0.4096 0.3679

0.9918 0.9862 0.9858 0.9440 0.8462 0.7209 0.5948 0.4887 0.4093 0.3677

Study on a Free Boundary Problem Arising in Porous Media

121

Acknowledgements The authors are very much thankful to SVNIT and GUJCOST for support of the research work.

References 1. C. Baiocchi, Variational and quasivariational inequalities, in Applications to Free-boundary Problems (1984) 2. J. Bear, in Dynamics of Fluids in Porous Media (Courier Corporation, 2013) 3. C. Canuto, M.Y. Hussaini, A. Quarteroni, A. Thomas Jr., et al., in Spectral Methods in Fluid Dynamics (Springer Science & Business Media, 2012) 4. I. Charnyi, A rigorous derivation of Dupuit’s formula for unconfined seepage with seepage surface. Dokl. Akad. Nauk SSSR 79 (1951) 5. E. De Jager, On the origin of the Korteweg-de Vries equation (2006). arXiv:0602661 6. C. Di Nucci, A free boundary problem: steady axisymmetric potential flow. Meccanica 48(7), 1805–1810 (2013) 7. C. Di Nucci, Theoretical derivation of the conservation equations for single phase flow in porous media: a continuum approach. Meccanica 49(12), 2829–2838 (2014) 8. C. Di Nucci, A free boundary problem for fluid flow through porous media (2015). arXiv:1507.05547 9. S. Friedlander, D. Serre, in Handbook of Mathematical Fluid Dynamics (Elsevier, 2002) 10. U. Hornung, T. Krueger, Evaluation of the Polubarinova-Kochina formula for the dam problem. Water Resour. Res. 21(3), 395–398 (1985) 11. J. Istok, in Groundwater Modeling by the Finite Element Method (American Geophysical Union, Washington, DC (USA), 1989) 12. A. Lorenzi, Laminar, turbulent, and transition flow in porous sintered media. Meccanica 10(2), 75–77 (1975) 13. G. Matthew, Higher order, one-dimensional equations of potential flow in open channels. Proc. Inst. Civ. Eng. 91(2), 187–201 (1991) 14. S. Motsa, O. Makinde, S. Shateyi, Application of successive linearisation method to squeezing flow with bifurcation. Adv. Math. Phys. (2014) 15. S. Motsa, S. Shateyi, Successive linearisation analysis of unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and thermal radiation effects. Can. J. Chem. Eng. 90(5), 1323–1335 (2012) 16. S.S. Motsa, P. Sibanda, G.T. Marewo, On a new analytical method for flow between two inclined walls. Numer. Algorithms 61(3), 499–514 (2012) 17. P.Y. Polubarinova-Kochina, Theory of Groundwater Movement, translated from Russian by J. M. Roger de Wiest (Princeton Press, Princeton, NJ, 1962) 18. M. Todisco, C. Di Nucci, A. Russo Spena, On the non–linear unsteady water flow in open channels. Nuovo Cimento della Societa Italiana di Fisica B. Gen. Phys. Relativ. Astron. Math. Phys. Methods 122(3), 237–255 (2007) 19. L.N. Trefethen, Spectral Methods in MATLAB, vol. 10 (SIAM, 2000)

Effect of Habitat on Dynamic of Native and Exotic Prey–Predator Population Namita Goel, Sudipa Chauhan and Sumit Kaur Bhatia

Abstract In this paper, we have formulated a prey–predator interaction model with native species as predator and exotic species as prey. The population is also effected by the habitat. It is assumed that the prey population can invade a new environment due to the unsuccessful exploitation of the exotic prey by the native predators. The existence of the steady-state solution and boundedness of the system is obtained. Further, the local and global stability analysis of the steady-state solution is evaluated. Finally, the sensitivity analysis of the system is done based on two parameters A, i.e. (loss of habitat due to exotic prey) and r, i.e. (growth rate of Habitat). Keywords Habitat · Native species · Exotic species · Local stability · Global stability

1 Introduction The prey–predator relationship still continues to be one of the main themes in mathematical ecology due to its complex dynamic behaviour. Many prey–predator models have been studied considering different types of functional responses. It is also seen that the invasion or introduction of exotic species, in general, disrupts the tropic dynamics of native species [1–4]. Habitat loss and destruction can occur both naturally and through anthropogenic causes on interacting species. Species that are able to thrive and reproduce in a new environment are called alien, non-native or invasive. At present, it is important to study the impact of change in habitat characteristics due to its destruction on N. Goel · S. Chauhan (B) · S. K. Bhatia Department of Applied Mathematics, Amity University, Noida 201303, UP, India e-mail: [email protected] N. Goel e-mail: [email protected] S. K. Bhatia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_11

123

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the distribution and interaction of species. Many authors have studied in detail the changing habitat on survival of species and effect of exotic species on native species. It has been noted that the hydrology of Southwestern United States rivers has led to a decline in native (Popuius deltoides) species. Areas historically dominated by native have been replaced by invasive exotic (Saltcedar chinensis)and the effects of changing habitat on the survival of species [5–7]. A key factor that makes many species invasive is a lack of predators in the new environment. This is complex and results from thousands of years of evolution in a different place. Predators and prey often co-evolve in a phenomenon called the coevolutionary arms race which means that as prey evolve better defenses, predators, in turn, evolve better ways of exploiting prey. The classic example of this comes from the cheetah and antelope. Faster antelope survive better because they can better escape cheetahs. The fastest cheetahs then survive better because they can better catch the faster antelope. Neither species ultimately gains an advantage because they continually evolve in response to one another [8–10]. Some invaders can physically alter the habitat in addition to destruction. 50 beavers from Canada were relocated to Tierra del Fuego, an archipelago at the southern tip of South America, in 1946 to be hunted for their pelts. Since then, they have multiplied and now number in the hundreds of thousands. The trees in the region are not adapted to beaver activity as they are in North America, and most do not grow back after being gnawed by beavers [11, 12]. Keeping in mind the above discussion, this paper only deals with an exotic and native species living in a habitat. We begin by proposing a new model to study the effects of a habitat on exotic prey and native predator species. We have discussed the boundedness of the model and conducted local and global stability analysis on the non-trivial equilibrium point of the model. Finally, we have validated our analytic results using numerical simulations with different parametric ranges using MATLAB.

2 Biological Background In ecology, a habitat is the type of natural environment in which a particular species of organism lives. A species’ habitat is that place where it can find food, shelter, protection and mate for reproduction. In this biological system, we are explicitly focusing on exotic prey and native predator and how the habitat is effecting them very relevant example is of Blue slug which is an exotic prey when enters a new habitat negatively effect the waterweed (Habitat) or in other words habitat has a positive impact on the growth rate of Blue slug. Additionally since the native predators (Turtles) who have not evolved with prey cannot successfully exploit them. Further any kind of man-made destruction of habitat also effect the population of both Blue slug and Turtle (Fig. 1).

Effect of Habitat on Dynamic of Native and Exotic Prey–Predator Population

125

Fig. 1 Biological background

3 Mathematical Model Let H(t) represents the measure of habitat characteristic such as soil, temperature, biodiversity. N1 (t) represents the exotic prey at time t. N2 (t) represents the native predator at time t. With the above notations , the mathematical model of the system under consideration is given by following system of nonlinear ordinary differential equations: H dH = r H (1 − ) − AH N1 − B H N2 dt K

(1)

a2 N 1 N 2 d N1 = a1 (H )N1 − a2 N12 − dt 1 + αN1 + β N2

(2)

d N2 a2 N 1 N 2 = b1 (H )N2 − b2 N22 + dt 1 + αN1 + β N2

(3)

with the initial conditions as H (0) > 0, N1 (0) > 0 , N2 (0) > 0 , α = a2 h 1 , β = a2 w

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Parameters K r A B a1 (H ) a2 h1 w b1 (H ) b2

Meaning Carrying capacity Growth rate of habitat Loss due to exotic prey Loss due to native predator Specific growth rate of exotic prey depending on habitat Decay due to crowding effect of exotic prey Handling time to handle exotic prey by native predator Waiting time to searching exotic prey by native predator Specific growth rate of native predator depending on habitat Decay due to crowding effect of native predator

Here, all the parameters K, r, A, B, a1 (H ), a2 , h 1 ,w, b1 (H ), b2 are all taken to be positive constants. Case—Habitat characteristic is favourable for exotic species The function a1 (H ) is the specific growth rate coefficient of the native species and it decreases as habitat characteristics became more conducive. We assume 

a1 (0) = a10 , a1 (H ) < 0 f or H ≥ 0 and a1 (H ) ≤ a10

(4)

The function b1 (H ) is the specific growth rate coefficient of the exotic species and it increases as habitat characteristics became more conducive. We assume 

b1 (0) = b10 , b1 (H ) > 0 f or H ≥ 0 and b10 ≤ b1 (H )

(5)

4 Boundedness In this section, we will be discussing the boundedness of the system. Lemma 1 All the solutions of system (1)–(3) with the positive initial condition are uniformly bounded within the region where 5 V1 = ((H, N1 , N2 ) ∈ R+ : 0 < H ≤ K , 0 < H + N1 + N2 ≤

A1 ) θ

is a region of attraction. Proof From Eq. (1) H≤K We assume that the right-hand sides of the system (1)–(3) are smooth function of (H (t), N1 (t), N2 (t)) of t ∈ R+ .

Effect of Habitat on Dynamic of Native and Exotic Prey–Predator Population

127

d(H + N1 + N2 ) r H∗H ≤ r H∗ − − AH ∗ N1 − B H ∗ N2 + a1 (H ∗ )N1 + b1 (H ∗ )N2 dt K

The time derivative along the solution of the system (1)–(3) is dW ≤ r H ∗ + a1 (H ∗ )N1∗ + b1 (H ∗ )N2∗ − θ(H + N1 + N2 ) dt Let θ = min(H, N1 , N2 ) Thus, the above expression reduces to dW ≤ A1 − θW dt where

A1 = r H ∗ + a1 (H ∗ )N1∗ + b1 (H ∗ )N2∗ ,

therefore, the solution is W ≤

A1 θ 

This completes the proof of the lemma.

5 Basic Feasible Equilibrium Points The system has two feasible equilibrium points: 1. Boundary Equilibrium point E 1 (H ∗∗ , N1∗∗ , 0) where H ∗∗ =

K [r − AN1 ] r

provided r > AN1 N1∗∗ =

a1 (H ) a2

2. Interior Equilibrium point E 2 (H ∗ , N1∗ , N2∗ ) where H∗ =

K [r − AN1∗ − B N2∗ ] r

To find the values of N1∗ and N2∗ , we get two equations such that F(N1 , N2 ) = a1 (H )N1 − a2 N12 −

a2 N 1 N 2 1 + αN1 + β N2

(6)

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Fig. 2 Existence of interior equilibrium point

G(N1 , N2 ) = b1 (H )N2 − b2 N22 +

a2 N 1 N 2 1 + αN1 + β N2

(7)

From Eqs. (6) and (7) we can find that, (1) if N2 = 0 and F(N1 , 0) = 0 ⇒ f 1 (N1 ) = there exist N1 = C1∗ such that f 1 (C1∗ ) = 0

a1 (H ) a2

and by theory of equation

(2) if N1 = 0 and F(0, N2 ) = 0 ⇒ f 2 (N2 ) = there exist N2 = D1∗ such that f 2 (D1∗ ) = 0

a1 (H ) a2 −a1 (H )β

and by theory of equation

(3) if N2 = 0 and G(N1 , 0) = 0 ⇒ f 3 (N1 ) = there exist N1 = C2∗ such that f 3 (C2∗ ) = 0

b1 (H ) a2 −b1 (H )α

and by theory of equation

(4) if N1 = 0 and G(0, N2 ) = 0 ⇒ f 4 (N2 ) = there exist N1 = D2∗ such that f 4 (D2∗ ) = 0

b1 (H ) b2

and by theory of equation

The two Eqs. (6) and (7) intersect with each other in the positive phase plane under either of the following conditions: C2∗ < C1∗ and D1∗ < D2∗ or C2∗ > C1∗ and D1∗ > D2∗ . Which shows that E 2∗ exists and is unique if dd NN1∗ < 0 as shown in Fig. 2. 2

6 Local Stability Analysis In this section, we will study the local stability analysis of boundary and interior equilibrium points. The Jacobian corresponding to E(H, N1 , N2 ) is given by let a2 N 1 N 2 M= 1 + αN1 + β N2

Effect of Habitat on Dynamic of Native and Exotic Prey–Predator Population

129

⎤ − rKH −AH −B H  J (H, N1 , N2 ) = ⎣ a1 N1 −a2 N1 + Mα b1 (H ) − b2 N2 + Mβ ⎦  b2 N2 − Mβ b1 N2 a1 (H ) − a2 N1 − Mα The characteristic equation corresponding to Jacobian E 1 (H ∗∗ , N1∗∗ , 0) is ⎡

λ2 P + λQ + R = 0 where P= Q= and

r H ∗∗ N1∗∗ a2 >0 K

r H ∗∗ + a2 N1∗∗ > 0 K 

R = a1 (H )N1∗∗ > 0

Now, using Routh–Hurwitz Criterion, we can say that all the roots of the polynomial are either negative or with negative real parts iff, i.e. P Q − R > 0 Then E 1 is locally asymptotically stable with condition 

r H ∗∗ a1 (H )N1∗∗ r H ∗∗ a2 N1∗∗  + a2 a1 (H )N1∗∗2 > . K K The characteristic equation corresponding to Jacobian E 2 (H ∗ , N1∗ , N2∗ ) is λ3 + λ2 X + λY + Z = 0 where X=

Y =

r H∗ + N1∗ a2 + b2 N2∗ + Mβ − Mα > 0 K

r H∗ ∗ [N1 a2 + b2 N1∗ + Mβ − Mα] + a1 (H ∗ )b2 N2∗ + a2 b1 (H ∗ )N1∗ + Mαb1 (H ∗ ) K

+a1 (H ∗ )N1∗ AH ∗ + b1 (H ∗ )N2∗ B H ∗ − a1 (H ∗ )b1 (H ∗ ) − a1 (H ∗ )Mβ − b2 N2∗ Mα > 0 Z=

r H∗ ∗ [N2 a1 (H ∗ )b2 − a1 (H ∗ )b1 (H ∗ ) − a1 (H ∗ )Mβ + a2 b1 (H ∗ )N1∗ + Mαb1 (H ∗ ) − b2 MαN2∗ − αb2 N2∗ M] K 









∗

+AH ∗ [a1 (H ∗ )b2 N2∗ N1∗ + a1 (H ∗ )Mβ N1∗ + N2∗ b1 (H ∗ )b1 (H ∗ ) + b1 (H ∗ )Mβ N2∗ − b1 (H ∗ )N22 b2 ] 









+B H ∗ [a1 (H ∗ )a1 (H ∗ )N1∗ + b1 (H ∗ )a2 N2∗ N1∗ − N12 a1 (H ∗ )a2 − a1 (H ∗ )MαN1∗ − b1 (H ∗ )N2∗ Mα] > 0

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N. Goel et al.

Now, using Routh–Hurwitz Criterion, we can say that all the roots of the polynomial are either negative or with negative real parts iff, i.e. X Y − Z > 0. Then E 2 is locally asymptotically stable . In the next section, we would discuss the global stability of the interior equilibrium point.

7 Global Stability Lemma 2 E ∗ is globally asymptotically stable if 2 < a11 a22 a12

and 2 < a11 a33 a13

where a12 = A − ξ(H )c1 , a11 =

r , a22 = c1 a2 , a13 = B − η(H )c2 , a33 = c2 b2 2K

Proof We consider the positive-definite function V1 (H, N1 , N2 ) = (H − H ∗ − H ∗ ln

H N1 N2 ) + c1 [N1 − N1∗ − N1∗ ln ∗ ] + c2 [N2 − N2∗ − N2∗ ln ∗ ] H∗ N1 N2

Then, differentiating V1 with respect to t and solving, we get V1 = z 1 [

−r z 1 − Az 2 − Bz 3 ] + c1 z 2 [ξ(H )z 1 − a2 z 2 ] + c2 z 3 [η(H )z 1 − b2 z 3 ] K 

where ξ(H ) =

H = H ∗ a1 (H ), H = H ∗ 

 η(H ) =

a1 (H )−a1 (H ∗ ) , H −H ∗

b1 (H )−b1 (H ∗ ) , H −H ∗

H = H ∗ b1 (H ), H = H ∗ 

a11 z 12 a11 z 12 + a12 z 1 z 2 + a22 z 22 + + a13 z 1 z 3 + a33 z 32 ] V˙ = −[ 2 2 where a12 = A − ξ(H )c1 , a11 =

r , a22 = c1 a2 , a13 = B − η(H )c2 , a33 = c2 b2 2K

Effect of Habitat on Dynamic of Native and Exotic Prey–Predator Population

131

Now, we see that by Sylvester’s criteria under the following condition V˙ (t) is negative definite if 2 < a11 a22 (8) a12 2 < a11 a33 a13

(9) 

holds. 

Lemma 3 In addition to (4)–(5), let a1 (H ), b1 (H ) satisfy condition 0 ≤ −a1 (H ) ≤  n, b1l ≤ b1 (H ) ≤ b1m for some positive constant b1l , b1m , n . If the above (8) and (9) inequality hold then positive interior equilibrium point is globally asymptotically stable. Proof By mean value theorem |ξ(H )| ≤ n and b1l ≤ |η(H )| ≤ b1m choosing c1 and c2 such that 0 < c1
M2 > M1 > M5 > M4 For k = 3, M3 > M2 > M1 > M5 > M4 For k = 5, M3 > M2 > M1 > M5 > M4 . Here, when k varies, there is no change in ranking and the best choice also remains unchanged. Therefore, the most preferable alternative is M 3 . Table 1 Numerical values of ! M f A + , Mi

Mf

A+ ,

M1

M f A + , M2 M f A + , M3 Mf

A+ ,

M4

M f A + , M5

Table 2 Calculated numerical values of ! M f A − , Mi

M f A − , M1 Mf

A− ,

M2

M f A − , M3 M f A − , M4 Mf

A− ,

M5

! ! ! ! !

! ! ! ! !

k =1

k =3

k =5

0.8798

5.0851

21.9712

0.8121

4.2138

14.1969

0.2551

1.1535

2.7911

1.8524

29.0030

568.6678

1.2736

9.5774

54.2652

k =1 1.2813

k =3 17.0733

k =5 173.0361

1.1954

15.2920

152.8204

2.3496

117.0523

6889.4068

0.9560

39.8839

1397.0931

1.0572

19.0736

287.9358

202

R. N. Saraswat and A. Umar

Table 3 Calculated values of M K (Mi ) for k ≥ 0

k =1

k =3

k =5

M K (M1 )

0.5928

0.7705

0.8873

M K (M2 )

0.5954

0.7839

0.9149

M K (M3 )

0.9020

0.9902

0.9995

M K (M4 )

0.3404

0.5789

0.7107

M K (M5 )

0.4535

0.6657

0.8414

5.2 Comparison with the TOPSIS Method Here, we show an application of the TOPSIS Method, which was introduced by Hwang and Yoon [15] for multi-criteria decision-making. Let us consider a set P = {P1 , P2 , P3 , . . . , Pm } of m alternatives and a set T = {T1 , T2 , T3 , . . . , Tn } of n criteria. The best alternative from the set P corresponding to the set T is to be found by the decision maker. The method for solving the MCDM problem using TOPSIS method is as follows: (1) Construct a decision matrix. T1 T2 . . . z 11 z 12 . . . z 21 z 22 · · · .. .. .. . . . Pn z m1 z m2 · · · P1 P2 .. .

Tn z 1n z 2n .. . z mn

W = [w1 , w2 , . . . . . . wn ] (2) Construct the normalized decision matrix, using normalized value n i j : " # m # z i2j , i = 1, 2, . . . , m, j = 1, 2, . . . , n. n i j = z i j /$ i=1

(3) Construct the weighted normalized decision matrix using the weighted normalized value: u i j = wi n i j i = 1, 2, . . . , m, j = 1, 2, . . . , n. wi is the weight of ith attribute. (4) Find positive ideal and negative ideal solution I+ and I−, respectively, using:

New Fuzzy Divergence Measure and Its Applications …

203

    + + I + = u+ , u , . . . , u u = max ij n 1 2   i   − − I − = u− = min , u , . . . , u u ij n 1 2 i

(5) Find separation of measures using a positive ideal solution and negative ideal solution. (6) Find relative closeness to positive ideal solution of each alternative using Ci =

S− , i = 1, 2, . . . , m. S∗ + S−

(7) Rank the preference order according to the coefficient of the closeness of all alternatives.

5.3 Numerical Illustration (1) (2) (3) (4) (5)

Decision matrix is given in Table 4. Weighted normalized decision matrix is given in Table 5. Positive and Negative Ideal Solutions are given in Table 6. Separation Measures are given in Table 7. Relative closeness to positive ideal solution is given in Table 8.

Table 4 Decision matrix T1

T2

T3

T4

T5

P1

0.5

0.8

0.4

0.7

0.5

P2

0.8

0.7

0.6

0.5

0.3

P3

0.7

0.5

0.8

0.9

0.6

P4

0.6

0.3

0.9

0.2

0.4

P5

0.3

0.4

0.5

0.8

0.6

Table 5 Normalized/Weighted decision matrix T1

T2

T3

T4

T5

P1

0.3696

0.6266

0.2685

0.4383

0.4527

P2

0.5914

0.5483

0.4027

0.3131

0.2716

P3

0.5174

0.3916

0.5369

0.5636

0.5432

P4

0.4435

0.2350

0.1252

0.1252

0.3622

P5

0.2218

0.3133

0.5010

0.5010

0.5432

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Table 6 Positive and negative ideal solutions I+

0.5914

0.6266

0.6040

0.5636

0.5432

I−

0.2218

0.2350

0.2685

0.1252

0.2716

Table 7 Separation measures

S1∗ = 0.4308 S2∗ S3∗ S4∗ S5∗

Table 8 Closeness to positive ideal solution

= 0.4279 = 0.2553 = 0.6325 = 0.5574

S1− = 0.5531 S2− = 0.5366 S3− = 0.6707 S4− = 0.4121 S5− = 0.4749

C1

0.5621

C2

0.5563

C3

0.7242

C4

0.3945

C5

0.4600

(6) Now ranking the preference order according to the coefficient of closeness. Here, P3 > P1 > P2 > P5 > P4 . Here P3 is the most preferable alternative.

6 Conclusion In this paper, we have introduced a new convex function and fuzzy divergence measure and its generalization with proof of its validity. Application of new fuzzy divergence measure in multi-criteria decision-making with an illustration is also given. We applied this method to many real-world problems and found better results. A comparative study has also been provided between the proposed method and the existing method. Existing methods for solving MCDM problems are time consuming as compared to the proposed method, hence the proposed method is better than the existing methods.

References 1. R.K. Bajaj, D.S. Hooda, On some new generalized measures of fuzzy information. World Acad. Sci. Eng. Technol. 62, 747–753 (2010) 2. D. Bhandari, N.R. Pal, Some new information measures for fuzzy sets. Inf. Sci. 67(3), 209–228 (1993)

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3. P.K. Bhatia, S. Singh, A new measure of fuzzy directed divergence and its application in image segmentation. Int. J. Intell. Syst. Appl. 4, 81–89 (2013) 4. M. Emami, K. Nazari, H. Fardmanesh, Application of fuzzy TOPSIS technique for strategic management decision. J. Appl. Basic Sci. Res. 2(1), 685–689 (2012) 5. C. Ferrari, Hyperentropy and related heterogeneity divergence and information measures. Statistica 40(2), 155–168 (1980) 6. M. Ghosh, D. Das, C. Chakraborty, A.K. Roy, Automated leukocyte recognition using fuzzy divergence. Micron 41, 840–846 (2010) 7. D.S. Hooda, D. Jain, The generalized fuzzy measures of directed divergence, total ambiguity and information improvement. Investig. Math. Sci. 2, 239–260 (2012) 8. G.R. Jahanshahloo, L.F. Hosseinzadeh, M. Izadikhah, Extension of TOPSIS method for decision making problems with fuzzy data. Appl. Math. Comput. 181, 1544–1551 (2006) 9. K.C. Jain, R.N. Saraswat, A new information inequality and its application in establishing relation among various F-divergence measures. J. Appl. Math. Stat. Inform. 8(1), 17–32 (2012) 10. K.C. Jain, R.N. Saraswat, Some bounds of information divergence measure in terms of Kullback-Leibler divergence measure. Antarct. J. Math. 9(7), 613–623 (2012) 11. K.C. Jain, R.N. Saraswat, Series of information divergence measures using new F-divergences, convex properties and inequalities. Int. J. Mod. Eng. Res. 2(5), 3226–3231 (2012) 12. J.N. Kapur, Measures of Fuzzy Information (Mathematical Sciences Trust Society, New Delhi, 1997) 13. S. Kullback, R.A. Leibler, On information and sufficiency. Ann. Math. Stat. 22(1), 79–86 (1951) 14. A.D. Luca, S. Termini, A definition of non-probabilistic entropy in the setting of fuzzy set theory. Inf. Control 20(4), 301–312 (1972) 15. S. Montes, I. Couso, P. Gil, C. Bertoluzza, Divergence measure between fuzzy sets. Int. J. Approx. Reason. 30, 91–105 (2002) 16. A. Ohlan, R. Ohlan, Generalizations of Fuzzy Information Measures (Springer International Publishing, Switzerland, 2016) 17. A. Ohlan, R. Ohlan, Parametric generalized exponential fuzzy divergence measure and strategic decision making, in Generalizations of Fuzzy Information Measures, ed. by A. Ohlan, R. Ohlan (Springer International Publishing, Switzerland, 2016), pp. 53–69 18. O. Parkash, P.K. Sharma, S. Kumar, Two new measures of fuzzy divergence and their properties. SQU J. Sci. 11, 69–77 (2006) 19. C.E. Shannon, A mathematical theory of comunication. Bell Syst. Tech. J. 27(3), 379–423 (1948) 20. D. Stanujkic, N. Magdalinovic, S. Tojanovic, R. Jovanovic, Extension of ratio system part of MOORA method for solving decision making problems with interval data. Informatica 23(1), 141–154 (2012) 21. R. Verma, B.D. Sharma, On generalized exponential fuzzy entropy. World Acad. Sci. Eng. Technol. 69, 1402–1405 (2011) 22. L.A. Zadeh, Fuzzy sets. Inf. Control 8, 338–353 (1965)

An SIRS Age-Structured Model for Vector-Borne Diseases with Infective Immigrants Nisha Budhwar, Sunita Daniel and Vivek Kumar

Abstract In this paper, we develop a SIRS age-structured model with infective immigrants. We consider a fraction of the juvenile immigrants and a fraction of the adult immigrants to be infective. We calculate the equilibrium points and then check the stability of these points. The reproduction number is calculated using the NextGeneration Method. Mathematical simulation for the model is also conducted using MATLAB. It is observed that an increase in the infective immigrants does not affect the total infective persons in the population. However, there is an increase in the infective population if the rate of immigration is increased. Also, the recovered population increases as the recovery rate increases. It is seen that as the mosquito population increases due to an increase in their birth rate, the infective human population also increases. Keywords SIRS model · Infective immigrants · Equilibrium points · Stability · Numerical simulation

N. Budhwar (B) Amity School of Applied Sciences, Amity University Haryana, Gurgaon, India e-mail: [email protected] S. Daniel FORE School of Management, Qutub Institutional Area, New Delhi, India e-mail: [email protected] V. Kumar Department of Applied Mathematics, Delhi Technological University, New Delhi, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_18

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1 Introduction Vector-borne diseases are illnesses that are transmitted by vectors which include mosquitoes, ticks and fleas. These vectors can carry infective pathogens such as viruses, bacteria and protozoa, which can be transferred from one host to the other. Diseases like dengue, malaria and chikungunya are vector-borne diseases transmitted through mosquitoes. Many models for vector-borne diseases are available in the literature [1–5] and the transmission dynamics of certain of these diseases like malaria and dengue have been extensively studied [6–13]. Since these diseases spread through a carrier, we can assume that the spread of the disease from one geographical area to another is the migration of people who are carrying this disease. Hence, it can be seen that the infective immigrants are also a major cause for the spread of the disease [6, 8, 10, 11, 14, 16]. It is also observed that some of these diseases are common and prevalent among children [17, 18] and hence it is necessary to study the transmission dynamics for an age-structured population. Many age-structured population models have been studied for dengue and malaria [7, 9, 13, 15]. However, the age-structured model with infective immigrants in vector-borne diseases has not yet been studied. In this paper, we develop a SIRS age-structured model with infective immigrants among the children and the adults. In this paper, we formulate the SIRS age-structured model in the following section. The region of feasibility is found in Sect. 3. The equilibrium points and the reproduction number R0 are computed in Sects. 4 and 5 respectively. The stability of the equilibrium points is analysed in Sect. 6. We then carry out the numerical simulations in Sect. 7 and Sect. 8 is the conclusion.

2 The SIRS Model Formulation Let the population of the susceptible juveniles, infected juveniles and the recovered juveniles be denoted by X J , Y J and Z J , respectively. Similarly, let the population of the susceptible adults, infected adults and the recovered adults be denoted by X A , Y A and Z A respectively. Then N T = X J + Y J + Z J + X A + Y A + Z A is the total human population. Similarly, let X v be the susceptible mosquito population and Yv the infected mosquito population. Then Nv = X v + Yv is the total mosquito population. We now define the various parameters needed to formulate the model. Let us assume that the newly born individuals are not infected with any vector-borne diseases. Let Λ J and Λ A be the constant immigration rate of juveniles and adults respectively. We further assume that a fraction φ J of juveniles are infective and the remaining fraction (1 − φ J ) of juveniles are susceptibles. Similarly, a fraction φ A of adults J and are infective and the fraction (1 − φ A ) of adults are susceptible. Let B J = bβ NT bβ A B A = NT , where b is the average biting rate of the vector (carrier of the disease), β J is the probability of infection of susceptible juveniles per bite of the vector, β A is the

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v probability of infection of susceptible adults per bite of the vector. Similarly Bv = bβ NT where βv is the probability of infection of susceptible vectors per bite of the vector of the infected human. Let μ H be the natural death rate of the humans, μv , the natural death rate of the vector, δ the rate at which the juveniles pass into adulthood, η J (A) the rate at which recovered juveniles (adults) join the susceptible class after losing their immunity and γ J (A) the recovery rate of juveniles (adults) and λv Nv the birth rate of mosquitoes. Then the SIRS age-structured model is given by the following system of non-linear differential equations.

dXJ = (1 − φ J )Λ J − B J X J Yv − (μ H + δ)X J + η J Z J dt dY J = φ J Λ J + B J X J Yv − (μ H + γ J )Y J dt dZJ = γ J Y J − (μ H + δ)Z J − η J Z J dt

(1)

dXA = (1 − φ A )Λ A + δ X J − B A X A Yv − μ H X A + η A Z A dt dY A = φ A Λ A + B A X A Yv − (μ H + γ A )Y A dt dZA = γ AY A − μH Z A + δ Z J − η A Z A dt d Xv = λv Nv − Bv X v (Y J + Y A ) − μv X v dt dYv = Bv X v (Y J + Y A ) − μv Yv dt We find the region of feasibility of the solution of the above model.

3 Feasibility of Solution Theorem 1 The feasible solution set for model (1) given by Ω = {(X J , Y J , Z J , X A , Y A , Z A , X v , Yv ) ∈ R 8 : (X J , Y J , Z J , X A , Y A , Z A , X v , A Yv ) ≥ 0;0 ≤ N T ≤ Λ Jμ+Λ ; 0 ≤ Nv ≤ μλvv }. H is positively invariant and mathematically well-posed in the domain Ω Proof Adding the first six equations, and the last two equations of the model (1), we NH have d dt = Λ J + Λ A − μ H N T and ddtNv = λv Nv − μv Nv .

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Since ddtNT ≤ Λ J + Λ A − μ H N H and ddtNv ≤ λv Nv − μv Nv , it follows that ddtNT A ≤ 0 and ddtNv ≤ 0 if N T (t) ≥ Λ Jμ+Λ and Nv (t) ≥ λμv Nv v , respectively. H A Hence, it follows that N T (t) ≤ N T (0)e−μ H (t) + Λ Jμ+Λ [1 − e−μ H (t) ] and Nv (t) ≤ H Nv (0)e−μv (t) + λμv Nv v [1 − e−μv (t) ]. A A In particular, N T (t) ≤ Λ Jμ+Λ if N T (0) ≤ Λ Jμ+Λ and Nv (t) ≤ λμv Nv v if Nv (0) ≤ H H λv N v , respectively. So, we see that the region Ω for the model (1) is positively invariμv A ant. Further, if N T (0) > Λ Jμ+Λ and Nv (0) > λμv Nv v , then either the solution enters Ω H A is finite time or N T (t) → Λ Jμ+Λ and Nv (t) → λμv Nv v as t → ∞. Hence the theorem. H The equilibrium points are calculated in the following section.

4 Existence of Equilibrium Points In this section, we calculate the two equilibrium points.

4.1 Disease-Free Equilibrium Point The disease-free equilibrium state is one in which the population has no infected persons. It is assumed that all the persons in the population are disease free and are susceptible persons. Hence, we have φ J = φ A = 0 and also Y J = Z J = Y A = Z A = YV = 0. Using these conditions, and equating the system of equations given by (1) δΛ J J , X A = μΛHA + μ H (μ , and X v = λμv Nv v . to zero, we have X J = μΛ H +δ H +δ) Thus the disease-free equilibrium point of model (1) is given by E 1 = (X ∗J , 0, 0, X ∗A , 0, 0, X v∗ , 0) = (

Λ A (μ H + δ) + δΛ J λv Nv ΛJ , 0, 0, , 0, 0, , 0) μH + δ μ H (μ H + δ) μv

4.2 Endemic Equilibrium Point To calculate the endemic equilibrium point, we assume that the humans and vectors exist in all classes. For simplicity, we assume an SIR framework and hence η J = η A = 0. In this case, the model becomes (1 − φ J )Λ J + η J Z J (μ H + δ) + B J Yv

(2)

φ J Λ J (μ H + δ) + B J Λ J Yv + B J Yv η J Z J (μ H + γ J )(B J Yv + μ H )

(3)

XJ =

YJ =

An SIRS Age-Structured Model for Vector-Borne …

ZJ =

211

γ J φ J Λ J (μ H + δ) + γ J B J Λ J Yv + γ J B J Yv η J Z J (μ H + γ J )(μ H + δ + γ J )(B J Yv + μ H )

(4)

(1 − φ A )Λ A (B J Yv + μ H + δ) + δ(1 − φ J )Λ J (B A Yv + μ H )(B J Yv + μ H + δ)

(5)

φ A Λ A + B A X A Yv μH + γ A

(6)

γAYA + δ Z J μH

(7)

λv N v Bv (Y J + Y A ) + μv

(8)

XA =

YA =

ZA = Xv = and Yv is the solution of

c1 Yv3 + c2 Yv2 + c3 Yv + c4 = 0

(9)

where c1 = B A B J Bv μv ((μ H + γ A )Λ J + (μ H + γ J )Λ A ) + Bv B J μv μ H φ A Λ A + B A B J μ2v (μ H + γ J )(μ H + γ A ), c2 = B A Bv μv φ J Λ J (μ H + δ)(μ H + γ A ) + B A Bv φ A Λ A (μ H + δ) + B J Bv φ A Λ A (μ H + γ J ) + 1 − φ J )Λ J μv B A Bv δ(μ H + γ J ) − B A Bv Nv Λ A B J (μ H + γ J ) − Bv B J λv Nv Λ J (μ H + γ A ), c3 = Bv μ H μv φ J Λ J (μ H + δ)(μ H + γ A ) + Bv μv φ A Λ A (μ H + δ)(μ H + γ J ) + Bv μv δmu H φ A Λ A (μ H + δ) + (μ H + γ J )(μ H + γ A )μ2v μ H (μ H + δ) − B A Bv Nv λv φ J Λ J (μ H + δ)(μ H + γ A ) − B J Bv μ H λv Nv Λ J (μ H + γ A ) − φ A Λ A Bv λv Nv B J (μ H + γ J )μ H − (μ H + δ)(μ H + γ J )Bv λV Nv B A (1 − φ A )Λ A − Bv λv Nv B A δ(1 − φ J ) Λ J (μ H + γ J ) − Bv B A δλv Nv φ J Λ J (μ H + δ) and c4 = (μ H + δ)(μ H + γ J )(1 − φ A )Λ A B A μv Bv − Bv λv Nv φ J Λ J (μ H + δ)(μ H + γ A ) f − λv Nv Bv (μ H + δ)(μ H + γ J )φ A Λ A μ H . The endemic equilibrium point is the positive root of (9).

5 Basic Reproduction Number R0 In this section, we compute the reproduction number R0 which is the average number of new infections caused by one infected individual in an entirely susceptible population. We can calculate R0 using Next-Generation Method as given in [19]. We find the reproduction number R0 by considering two matrices F and V −1 where F = [ ∂ F∂i x(xj 0 ) ] and V = [ ∂ V∂i x(xj 0 ) ]. Here Fi are the new infections, Vi transfers of infections from one compartment to another and x0 is the disease-free equilibrium state.

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Then R0 is the dominant eigenvalue of the matrix F V −1 . According to model (1), the matrices F and V are given as ⎛

⎞ μh + γ J 0 0 0 μH + γ A 0 ⎠ and V = ⎝ 0 0 μv

⎞ 0 0 BJ X J 0 BA X A ⎠ F =⎝ 0 Bv X v Bv X v 0 ⎛

The inverse of V is

⎜ V −1 = ⎝

⎛

1 μh +γ J

⎛ ⎜ F V −1 = ⎝

⎞ 0 0 ⎟ ⎠

0

0 0

1 γ A +μ H

0

1 μv

0 0

0 0

BJ X J μv BA X A μv

Bv X v Bv X v μ H +γ J μ H +γ A

⎞ ⎟ ⎠

(10)

0

⎛

Thus F V −1

⎞ 00a = ⎝0 0 b⎠ cd0

Xv Xv where a = BμJ Xv J , b = B AμXv A , c = μBHv+γ and d = μBHv+γ . J A √ −1 The dominant eigenvalue √ of the matrix F V is λ = + (bd + ac). Hence we have R0 = (bd + ac). Thus,  BA X A Bv X v BJ X J + ]. R0 = [ μv (μ H + γ J ) (μ H + γ A )

6 Local Stability Analysis of the Equilibrium Points The stability of the disease-free equilibrium point and the endemic equilibrium point are given by the following two theorems in this section. Bv X v (B J X J +B A X A ) < 1. Then the diseaseμv (2μ H +γ J +γ A )+(μ H +γ J )(μ H +γ A ) Λ A (μ H +δ)+δΛ J ΛJ ( μ H +δ , 0, 0, μ H (μ H +δ) , 0, 0, λvμNvv , , 0) of (1) is locally

Theorem 2 Let R0 < 1 and free equilibrium point E 1 = asymptotically stable.

Proof Let us consider the Jacobian matrix corresponding to (1). We have,

An SIRS Age-Structured Model for Vector-Borne …

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Q1 =

213

−(μ H + δ) 0 ηJ 0 0 0 0 −(μ H + γ J ) −(η J + μ H + δ) 0 0 γJ δ 0 0 −μ H 0 δ 0 0 0 0 δ 0 0 0 0 −Bv X v 0 Bv X v 0 0 ⎤ 0 0 0 −B J X J 0 0 0 BJ X J ⎥ ⎥ 0 0 0 0 ⎥ ⎥ 0 −B A X A ⎥ 0 ηA ⎥ −(γ A + μ H ) 0 0 BA X A ⎥ ⎥ γA −(μ H + η A ) 0 0 ⎥ ⎥ 0 −μv 0 ⎦ −Bv X v 0 0 −μv Bv X v

The eigenvalues of the above matrix are −(μ H + δ), −(μ H + η J ), −(μ H + η A ), −μ H , −μv and roots of the polynomial λ3 + p1 λ2 + p2 λ + p3 . Where p1 = 2μ H + μv + γ J + γ A , p2 = (μ H + η A )(μ H + γ J + μv ) + (μ H + γ J )μv − Bv X v B J X J − Bv X v B A X A , p3 = μv (μ H + γ A )(μ H + γ J )(1 − R02 ). For the equilibrium point to be locally stable, the zeroes of the polynomial λ3 + p1 λ2 + p2 λ + p3 = 0 have to be negative. All the roots of cubic polynomial have negative real parts if and only if all the coefficients are positive and p1 p2 > p3 . This is known as the Routh–Hurwitz criterion. It is obvious that p1 > 0 and by our assumption p2 > 0. Moreover since R0 < 1, p3 > 0. Further p1 p2 − p3 = μv (μ H + η A )(μ H + η J )(1 − R02 ) + [μv (μ H + γ J )(μ H + γ A ) + (μ H + γ J )μv − Bv X v (B J X J + B A X A ) + μv (μ H + γ A )] + (μ H + γ J )2 (μv + μ H + γ A ) + (μ H + γ A )2 (μ H + γ J + μv ) > 0. Hence, the roots of the cubic polynomial are all negative, and therefore the equilibrium point is locally stable. The following theorem examines the stability the endemic equilibrium point, which is derived as the positive root of the system of equations given by (2)–(9). Theorem 3 If the endemic equilibrium point E 2 = (X J , Y J , Z J , X A , Y A , Z A , X v , Yv ) exists, then it is conditionally stable for R0 > 1. Proof The Jacobian matrix for the system of differential equations given for the SIR model is

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⎡

Q2 =

−(B J Yv + μ H + δ) 0 ⎢ B J Yv −(μ H + γ J ) ⎢ ⎢0 γJ ⎢ ⎢δ 0 ⎢ ⎢0 0 ⎢ ⎢0 0 ⎢ ⎣0 −Bv X v 0 Bv X v

0 0 −(μ H + δ) 0 0 δ 0 0

0 0 0 −(B J Yv + μ H ) 0 0 0 0 ⎤ 0 0 0 −B J X J 0 0 0 BJ X J ⎥ ⎥ ⎥ 0 0 0 0 ⎥ 0 0 0 −B A X A ⎥ ⎥ −(γ A + μ H ) 0 0 BA X A ⎥ ⎥ ⎥ γA −μ H 0 0 ⎥ ⎦ 0 −(Bv (Y J + Y A ) + μv ) 0 −Bv X v 0 Bv (Y J + Y A ) −μv Bv X v

(11)

Let a = B J Yv + μ H + δ, b = B J Yv , c = μ H + γ J , f = μ H + δ, g = B J YV + μ H , h = B A X A , i = γ A + μ H , r = μ H , j = Bv X v , k = (Bv (Y J + Y A ) + μv ), s = Bv (Y J + Y A ), m = B J X J + Yv and p = μv . The above matrix becomes ⎡

−a ⎢ b ⎢ ⎢ 0 ⎢ ⎢ δ Q2 = ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0

0 −c γJ 0 δ 0 −j j

0 0 −f 0 0 δ 0 0

0 0 0 −g 0 0 0 0

0 0 0 0 −i γA −j j

0 0 0 0 0 −r 0 0

0 0 0 0 0 0 −k s

⎤ −m m ⎥ ⎥ 0 ⎥ ⎥ −h ⎥ ⎥ h ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −p

Using the MAPLE software the eigenvalues are − f , −r , g and the roots of the fiftorder polynomial λ5 + s1 λ4 + s2 λ3 + s3 λ2 + s4 λ + s5 = 0 where s1 , s2 , s3 , s4 and s5 are constants. Since the calculation of the remaining eigenvalues are complicated, the stability of the endemic equilibrium point is shown using numerical simulation in the following section.

7 Numerical Simulation We carry out the simulations on the behaviour of the system (1) using the parameters given in [7]. We consider φ J = 0.4, φ A = 0.3, b = 0.5 day−1 , μ H = 0.00004 day−1 , β J = β A = 0.181, δ = 0.00000986 day−1 , γ J = 0.0014 day−1 , γ A = 0.0035 day−1 , Λ J = 1520, Λ A = 506, λv Nv = 500 day−1 , βv = 0.8333, μv = 0.05 day−1 , η J = η A = 0.0027 and X J (0) = 3500, Y J (0) = Z J (0) = Y A (0) = Z A (0) = 0,

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X A (0) = 6500, X v (0) = 1000, Yv (0) = 550, Nv (0) = 1550 and N T (0) = 10,000. The value of R0 for these values is 5.52. The numerical simulations were conducted using MATLAB. We calculate the endemic equilibrium point for model given by (1) for the above set of parameters. We solve the eight equations by equating them to zero. The solution thus obtained gives negative values for Y A , Z A and Yv . Hence the endemic equilibrium point does not exist for model (1). However, it is interesting to note that the solution of the set of equations given by (2)–(9) are positive and hence the endemic equilibrium point exists for the reduced model and is stable for R0 > 1. Substituting the values of the parameters in the Jacobian matrix given by (11), we get negative eigenvalues. This means that the endemic equilibrium point is locally stable for the SIR model given by the set of Eqs. (2)–(9). We analyse the SIR model given by Eqs. (2)–(9) and get the results illustrated below: Figure 1a and b are the solutions of system (1) for the juvenile population and adults population, respectively. It can be seen that both the juvenile as well as adults population (Susceptible, Infected, Recovered) are stable thus proving that the endemic equilibrium point given by (X J , Y J , Z J , X A , Y A , Z A , X v , Yv ) is stable for R0 > 1. In Fig. 2a, b, we increase the fraction of infected immigrants of juveniles as well as adults. It can be seen that there is a very slight increase in the number of infected population when the fraction of infected immigrants increases. In Fig. 3a and b, we increase the constant rate of immigration of the juvenile and adult population Λ J and Λ A , respectively. It can be seen that when we increase the recruitment rate, there is an increase in the number of infected population. In Fig. 4a, b, we increase the biting rate of the vector. It can be seen that as the biting rate increases, there is an increase in the number of infected population. It is also seen that the number of infected juveniles increase faster than the number of infected adults.

Fig. 1 Variation with respect to time in the a juvenile population as well as b adults population

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Fig. 2 Change in infective human population with time by varying the value of infective immigrants in a juvenile population and b adults population

Fig. 3 Variation in infected a juvenile population and b adults population with various values of recruitment rate

Fig. 4 Variation in a juvenile population and b adults population with increase in the biting rate of the vector (carrier)

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In Fig. 5a, b, we increase the birth rate of the vector. It can be seen that as the birth rate increases, there is an increase in the number of infected population. In these cases too, it is observed that the number of infected adults increases faster in time compared to the infected juveniles. In Figs. 6 and 7, we increase the recovery rate of the humans (juvenile and adults) population. It can be seen that as the recovery rate increases, there is an increase in the recovered population (Fig. 6a, b) and in Fig. 7a, b it is observed that there is a decrease in the infected population.

Fig. 5 Change in infective human population by varying the value of birth rate of the vector in a juvenile population and b adults population

Fig. 6 Change in recovered human population for different values of recovery rate of a juvenile population b adults population

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Fig. 7 Change in infected human population for different values of recovery rate of a juvenile population b adults population

8 Conclusion In this paper, we formulated a SIRS age-structured model for vector- borne diseases. The disease-free equilibrium point exists for the SIRS model whereas the endemic equilibrium point does not exist for the SIRS model but exists for the SIR model and is stable for R0 > 1. The reproduction number is calculated. From the numerical simulation, it is observed that the solution is stable for a SIR framework. Also an increase in the fraction of infective immigrants, be it a juvenile or an adult has very little impact on the infected juveniles (adults) population. However, if the rate of immigration of the juveniles (adults) increases, there is an increase in the infected human population. An increase in the recovery rate of the juvenile (adult) population shows an increase in the recovered juvenile (adult) population and decrease in the infected juvenile (adult) population. It was observed that an increase in the biting rate of the vector showed an increase in the infected human population. Hence, efforts must be made to reduce the vectors (carriers) in order to reduce the transmission of these diseases. Acknowledgements The infrastructure support provided by FORE School of Management, New Delhi in completing this paper is gratefully acknowledged.

References 1. A.A. Lasharia, G. Zamanb, Optimal control of a vector borne disease with horizontal transmission. Nonlinear Anal.: R. World Appl. 13, 203–212 (2012) 2. G. Kuniyoshi, P. dos Santos, Mathematical modelling of vector-borne diseases and insecticide resistance evolution. J. Venom. Anim. Toxins Incl. Trop. Dis. 23, 34 (2017) 3. H.-M. Wei, X.-Z. Li, M. Martcheva, An epidemic model of a vector-borne disease with direct transmission and time delay. J. Math. Anal. Appl. 342(2), 895–908 (2007)

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4. M. Ozair, A.A. Lashari, H. Jung, Y. Seo, B.N. Kim, Stability analysis of a vector-borne disease with variable human population. Abstr. Appl. Anal. (2013). https://doi.org/10.1155/2013/ 293293 5. N.H. Shah, J. Gupta, SEIR model and simulation for vector borne diseases. Appl. Math. 4, 13–17 (2013) 6. C. Mukandavirea, G. Musukab, G. Magombedzea, Z. Mukandavirea, Malaria model with immigration of infectives and seasonal forcing in transmission. Int. J. Appl. Math. Comput. 2(3), 1–16 (2010) 7. F. Forouzannia, A.B. Gumel, Mathematical analysis of an age structured model for malaria transmission dynamics. Math. Biosci. 247, 80–94 (2014) 8. L.N. Massawe, E.S. Massawe, O.D. Makinde, Dengue in Tanzania—vector control and vaccination. Am. J. Comput. Appl. Math. 5(2), 42–65 (2015) 9. P. Pongsumpun, I.M. Tang, Transmission of dengue hemorrhagic fever in an age structured population. Math. Comput. Model. 37, 949–961 (2003) 10. S. Olaniyi, O.S. Obabiyi, Mathematical model for malaria transmission dynamics in human and mosquito populations with nonlinear forces of infection. Int. J. Pure Appl. Math. 88(1), 125–156 (2013) 11. S. Side, M.S.M. Noorani, SEIR model for transmission of dengue fever. Int. J. Adv. Sci. Eng. Inf. Technol. 2 (2012) 12. J. Tumwiine, J.Y.T. Mugisha, L.S. Luboobi, A host-vector model for malaria with infective immigrants. J. Math. Anal. Appl. 361, 139–149 (2010) 13. J. Tumwiine, S. Luckhaus, J.Y.T. Mugisha, L.S. Luboobi, An age-structured mathematical model for the within host dynamics of malaria and the immune system. J. Math. Model. Algorithems 7, 79–97 (2008) 14. F. Brauer, P. van den Driessche, Models of transmission of diseases with immigration of infectives. Math. Biosci. 171, 143–154 (2001) 15. J.M. Addawe, J.E.C. Lope, Analysis of age-structured malaria transmission model. Philipp. Sci. Lett. 5(2) (2012) 16. M. El hia, O. Balatif, M. Rachik, J. Bouyaghroumni, Application of optimal control theory to an SEIR model with immigration of infectives. Int. J. Comput. Sci. Iss. 10(2) (2013) 17. https://www.indiatoday.in/india/north/story/children-most-vulnerable-to-vector-bornedisease-59997-2009-11-04 18. www.who.int/heca/infomaterials/vector-born.pdf 19. O. Diekmann, J.A.P. Heesterbeek, J.A.J. Metz, On the definition and the computation of the basic reproduction ratio Ro in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990)

Numerical Study of Conformable Space and Time Fractional Fokker–Planck Equation via CFDT Method Brajesh Kumar Singh and Anil Kumar

Abstract In this article, conformable fractional differential transform (CFDT) method has been successfully implemented to compute the numerical solution of space–time fractional Fokker–Planck equation with conformable fractional derivative. The computed results are compared with the existing results in the literature, and also depicted graphically for α = β = 1. The accuracy of the computed results for different values of α and β = 1 is measured in terms of L 2 error norms. The findings show that the present results agreed well with the results by various well-known methods such as Adomian decomposition method (ADM), variational iteration method (VIM), fractional variational iteration method (FVIM) and fractional reduced differential transform method (FRDTM), and so forth. The proposed results converge to the exact solutions. Keywords Conformable fractional derivative · Conformable fractional differential transform · Space–time fractional Fokker–Planck equations

1 Introduction Fractional differential equations, the generalizations of the classical differential equations, have been studied by many researchers in various fields, among them, such as fluid mechanics, plasma physics, image processing, mechanics of materials, and the other fields of science and engineering, see [1–3] and the references therein. Various types of definitions of the fractional- order derivatives, for different kinds of applications, have been defined in the literature.

B. K. Singh (B) · A. Kumar Department of Mathematics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow 226 025, UP, India e-mail: [email protected] A. Kumar e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_19

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In the past years, many analytical techniques have been developed for the study of fractional differential equations, Algorithm of homotopy asymptotic method [11], Fractional homotopy analysis transforms method [12]. Biswas et al. [4] secures dark and singular resonant optical solitons that are studied with dual-power law nonlinearity and fractional temporal evolution, considering conformable-type fractional derivatives. Reduced differential transform method [5], modified simple equation method, and the extended simplest equation method [6], several new exact solutions of the conformable space–time fractional Fokas equation in (4+1)dimensions are obtained via various developed techniques [7]. Al-Sawoor and Al-Amr used modified VIM [8] and Adomian’s decomposition method [9] to solve reaction–diffusion system with fast reversible reaction. The exact solutions of family of higher dimensional space–time fractional KdV-type equations has been reported in [10]. Kumar adopted homotopy perturbation method nonlinear waves in hyperelastic rods [13] and time fractional Ito-coupled equations [14]. A homotopy technique with Laplace transform has been adopted for the study of fractional-order multi-dimensional telegraph equation by Prakash et al. [15]. Fractional variational method has been used to solve time fractional model of coupled Burgers equation [16], fractional Bloch model arising in magnetic resonance imaging [17], and time-fractional multi-dimensional diffusion equations [18]. New iterative Sumudu transform has been implemented for solving fractional model of nonlinear Zakharov–Kuznetsov equations [19]. In recent years, the researchers started to see the weakness in most of the definitions of the fractional derivatives. In most of the definitions of the fractional derivatives, it is found that they either do not have Taylor power series expansion or unable to compute their Laplace transform. A simple and challenging definition of fractional derivative, conformable fractional derivative, was proposed by Khalil et al. [20]. This definition of fractional derivative depends on the basic limit definition of the classical derivative. Since exact solutions to conformable fractional partial differential equations are rarely available, and so, the computation of an analytical or numerical solution of conformable fractional PDE is a very challenging problem and under investigation. Based upon the conformable fractional derivative, most of the works have been done in [21–27] and the references therein. Various (analytical/numerical) methods have been proposed in many articles for computing the better approximations of conformable fractional partial differential equations (CFPDEs). For instance, the traveling wave solutions of CFPDEs using first integral method [28]. Modified Kudryashov method [24] was used to solve conformable time fractional Klein–Gordon equations with quadratic and cubic nonlinearities. Tanh method [29] is used to find analytic solutions of the conformable space–time fractional Kawahara equations. In [30], Kaplan used two reliable methods for computing the solutions of nonlinear conformable time-fractional equations. Analytical solutions of linear Navier–Stokes equation and nonlinear homogeneous, nonhomogeneous gas dynamic equations in sense of conformable space–time fractional derivatives are obtained via fractional differential transform in [46]. Now, the main goal is to compute analytical solutions of initial value system of conformable space–time fractional Fokker–Planck equation (1) below using CFDT method

Numerical Study of Conformable Space and Time Fractional Fokker–Planck …



t Tα [θ ]

223

  = −x Tβ A(x, t, θ ) +x T2β B(x, t, θ ) θ, x, t > 0,

θ (x, 0) = g(x),

(1)

0 < α, β ≤ 1.

where A(x, t, θ ) and B(x, t, θ ) are drift and diffusion coefficients, g is a smooth function. In case α = β = 1, Eq. (1) reduces to classical Fokker–Planck equation. Being a broad applications of the Fokker–Planck equations, it is studied very vigorously in many research papers [34–45]. The probabilistic solution of the fractional Fokker–Planck equation has important applications in the modeling of anomalous diffusion processes. Stochastic representation and computer simulation of fractional Fokker–Planck equation describing anomalous diffusion is given in [34]. For more details on Fokker–Planck equation, the interested readers are referred to [44, 45] and the reference therein.

2 Preliminaries This section presents some definitions and theorems on the fractional calculus theory [31–33]. Definition 1 Let α ∈ R, m − 1 < α ≤ m ∈ N , then for t > 0 (a)

Riemann–Liouville derivative of order α of g(x, t) is defined as follows: RL

(b)

Dtα g(x, t)

∂m 1 = Γ (m − α) ∂t m



t

 (t − τ )

m−α−1

g(x, τ )dτ

0

Caputo fractional derivative of g(x, t) of order α (m − 1 < α < m) is defined as follows:  t 1 α (t − τ )m−α−1 g m (τ )dτ, (2) Dt g(x, t) = Γ (m − α) 0 and Dtα [g(x, t)] =

dm [g(x, t)] dt m

whenever α = m.

Theorem 1 [46] Let α, β ∈ R with n − 1 < α < n, m − 1 < β < m where m, n ∈ N and m = n, then 

β

β

α+β

Dtα Dt u(x, t) = Dt Dtα u(x, t) = Dt Dtα Dtm u(x, t)

=

u(x, t)

Dtm Dtα u(x, t)

Due to noncommutative property of the Caputo derivative, a new kind of definition of fractional derivative, the so-called conformable fractional derivative is defined as follows:

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Definition 2 (Conformable fractional derivative) The conformable fractional derivative of order α of a function f (x, t) : R × (0, ∞) → R is defined by t Tα

f (x, t) = lim

h→0

f (x, t + ht 1−α ) − f (x, t) , h

α ∈ (0, 1].

Moreover, the conformable (left) time fractional derivative of order α of a function f (x, t) : R × (a, ∞) → R, a ≥ 0 is defined as follows: a t Tα

f (x, t) = lim

h→0

f (x, t + h(t − a)1−α ) − f (x, t) , h

α ∈ (0, 1].

The basic properties of conformable derivatives are as follows: Theorem 2 let α ∈ (0, 1], C, D any constant, the functions f (x, t), g(x, t) are α differentiable at any point (x, t) ∈ R × (0, ∞) and h(x, t) be any α— differentiable function at a point (x, t) ∈ R × (a, ∞), a ≥ 0. Then (a) (b) (c) (d) (e) (f) (g) (h)

t Tα (C

f + D g) = C t Tα f + D t Tα g, = 0, t Tα ( f g) = f t Tα g + g t Tα f , g t Tα f −gt Tα f , t Tα ( f /g) = g2 ii) t Tαa (t − a) p = p(t − a) p−α . For each p ∈ R, i) t Tα (t p ) = p t p−α , In particular, t Tαa (t − a)α = α. If f, h are differentiable with respect to t, then (ii) t Tαa h = (t − a)(1−α) ∂h , (i) T f = t 1−α ∂∂tf , ∂t  t α α 

(t−a) (t−a)α λ +x λ +x a α α e = λe . t Tα m m Let m < α ≤ m + 1 and β = α − m and if ∂∂t mh exits, then t Tαa h =t Tβa ∂∂t mh . t Tα (C)

Theorem 3 Let α ∈ (0, 1] , f (x, t) be k-times differentiable at (x, t0 ) ∈ R × (0, ∞) j such that ∂∂t jf |(x,t0 ) = 0, for each j ∈ {1, 2 . . . k − 1}. Then the k-times conformable fractional derivative of f (x, t) at (x, t0 ) is defined as follows:  t0 k−kα t Tkα f = (t − t0 )

where t Tkαt0 ≡ t Tαt0 . . .t Tαt0 .    k-times

∂k f ∂t k

 t=t0

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225

2.1 Description of CFDTM The CFDT and its properties as given in [46] are as follows: Lemma 1 Let α, β ∈ R such that 0 < α, β ≤ 1 and let the function θ (x, t) is infinitely α- differentiable at (x, t0 ) whereas ψ(x, t) is infinitely β-differentiable at (x0 , t). Then (i)

The conformable fractional power series expansion of θ (x, t) about 1 t = t0 (0 ≤ t0 ≤ t0 + α , > 0) is given as θ (x, t) =

∞ 

Θαr (x)(t − t0 )r α ,

(3)

r =0

(ii)



where Θαr (x) = r !α1 r t Trtα0 θ t=t0 is the time CFDT of θ (x, t). The expression (3) is also referred to as inversre of Θαr (x). The conformable fractional power series expansion of ψ(x, t) about 1 x = x0 (0 ≤ x0 ≤ x0 + β , > 0) is given as ψ(x, t) =

∞ 

Ψβk (t)(x − x0 )kα ,

(4)

k=0

where Ψβk (t) =

1 k!β k



x0 x Tkβ ψ

x=x0

is the CFDT of ψ(x, t). The expression (4)

is also referred to as inversre of Ψβk (t). Lemma 2 Let α ∈ R, 0 < α ≤ 1. If Θαr (x) and Ψαr (x) are time CFDT of infinitely α-differentiable functions θ (x, t) and ψ(x, t), respectively. Then (a)

The time CFDT of initial conditions in the form of integer-order derivatives of nth order partial differential equation is defined as  Θαr (x) =

(b) (c)

1 (αr )!

0,

∂rα θ

∂t r α t=t0

, r ∈ {0, 1, . . . , αn − 1}; otherwise.

If ω(x, t) = θ (x, t)ψ(x, t), then Ωαr (x) = If ω(x, t) =t Tαt0 θ (x, t), then

r i=0

Θαi (x)Ψαr −i (x)

Ωαr (x) = α(r + 1)Θαr +1 (x). (d) (e)

If ω(x, t) =x Tβx0 θ (x, t), then Ωαr (x) =x Tβx0 Θαr (x). Moreover, when   β = 1, Ωαr (x) = ∂∂x Θαr (x) . If ω(x, t) =t Tβt0 θ (x, t) and m − 1 < β ≤ m ∈ N , then

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Γ (r α + β + 1) r +β/α Θ (x). Γ (r α + β − m) α

Ωαr (x) = (f)

If ω(x, t) = h(x)(t − t0 ) p , then Ωαr (x) = h(x)δ(r − p/α) where δ(r ) = 1 if r = 0 otherwise δ(r ) = 0.

3 Analytical Study of Conformable Fractional Fokker–Planck Equation In this section, analytical solutions of conformable fractional Fokker–Planck equation are computed by using conformable fractional differential transform method. The accuracy of the computed results is measured in terms of L 2 error norms for different values of α. For this, we compute the exact square residual error in mth-order approximate analytical solution as follows: 

1

εm (θ ) =



1

  S

0

0

m 

2 θi (x, t)

dx dt

i=0

  where S [θ ] =t Tα [θ ] − −x Tβ A(x, t, θ ) +x T2β B(x, t, θ ) θ. Example 1 Consider the linear conformable space–time fractional Fokker–Planck equation ⎧  2    x θ (x, t) xθ (x, t) ⎪ ⎨ T [θ (x, t)] = T − , x, t > 0, T t α x 2β x β 12 6 ⎪ ⎩ θ (x, 0) = x 2 , 0 < α, β ≤ 1.

(5)

The time CFDT of initial value system of conformable fractional Fokker–Planck equation (5) yields the following recurrence relation: ⎧ 0 2 ⎪ ⎨ Θα (x) = x , r +1 ⎪ ⎩ Θα (x) =

1 α(r + 1)



 x T2β

   x 2 Θαr (x) xΘαr (x) −x Tβ , r ≥ 0. 12 6

(6)

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227

On solving the the recurrence relation (6), we get Θα0 (x) = x 2 , x 4−2β (4 − β) x 3−β − , 3α 2α x 4−2β (4 − β) x 5−3β (155 − 97β + 14β 2 ) x 6−4β (−24 + 26β − 9β 2 + β 3 ) − − Θα2 (x) = 24α 2 144α 2 12α 2 Θα1 (x) =

x 5−3β (20 − 13β + 2β 2 ) x 8−6β (−2 + β)2 (−96 + 116β − 43β 2 + 5β 3 ) − 432α 3 108α 3 x 6−4β (−454 + 505β − 179β 2 + 20β 3 ) − 864α 3 7−5β (11627 − 19020β + 11307β 2 − 2882β 3 + 264β 4 ) x − , 5184α 3 .. .. . .

Θα3 (x) = −

On using the inverse time CFDT, the mth-order solution of (5) is θ (x, t) = Θα0 (x) + Θα1 (x)t α + Θα2 (x)t 2α + . . . + Θαm (x)t mα .

(7)

In particular, for α = β = 1, the above solution reduces to  θ (x, t) = x 2

1 t 1+ + 2 2!

  2 t t + . . . = x 2e 2 2 t

which is the closed form of the exact solution θ (x, t) = x 2 e 2 of classical linear Fokker–Planck equation. The behavior of the solutions of the problem (5) is depicted graphically in Fig. 1a and Fig. 1b for α = β = 0.8, 1 and the two plots of the behavior for different α, β and absolute errors in different order solutions for α = β = 1 is depicted in Fig. 2a and Fig. 2b, respectively. The L 2 error in fifth- and seventh-order approximations are reported in Table 1 for α = 0.8, 0.9, 1 and β = 1. Moreover, computed absolute errors for α = β = 1 are compared with ADM, VIM, HWM, ILTM, FVIM, FRDTM solutions in Table 2. The findings show that for different values of α = 0.8, 0.9, 1, and β = 1 we get accurate results which approaches toward exact solutions. Example 2 Consider the nonlinear conformable space–time fractional Fokker– Planck equation ⎧  2  ⎪ ⎨ T [θ (x, t)] = T θ 2 (x, t) − T 4θ (x, t) − xθ (x, t) , x, t > 0, t α x 2β x β x 3 (8) ⎪ ⎩ 2 u(x, 0) = x , 0 < α, β ≤ 1.

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Fig. 1 Behavior of fifth-order solutions of the conformable space–time fractional Fokker–Planck equation (5) for a α = β = 1, b α = β = 0.8 at different time levels t ≤ 1

Fig. 2 (a) Two-dimensional plots of the fifth-order solution of conformable fractional Fokker– Planck (5) equation for different α, β and x = 0.25; (b) the absolute errors in the mth order solutions (m ∈ {3, 4, 5}) and for x = 0.5 Table 1 L 2 error norm in fifth-order approximations of Fokker–Planck equation (5) in x ∈ [0, 1] for α = 0.8, 0.9, 1 and β = 1 at different time levels t ≤ 1 x

Fifth order approximation α = 0.8

α = 0.9

Seventh order approximation α = 1.0

α = 0.8

α = 0.9

α = 1.0

0.10

1.3245500E-06 6.9731100E-07 3.9259300E-07 1.2713500E-07 5.9437600E-08 3.0094400E-08

0.20

5.2981900E-06 2.7892400E-06 1.5703700E-06 5.0853900E-07 2.3775100E-07 1.2037800E-07

0.30

1.1920900E-05 6.2757900E-06 3.5333400E-06 1.1442100E-06 5.3493900E-07 2.7085000E-07

0.40

2.1192800E-05 1.1157000E-05 6.2814900E-06 2.0341600E-06 9.5100200E-07 4.8151100E-07

0.50

3.3113700E-05 1.7432800E-05 9.8148200E-06 3.1783700E-06 1.4859400E-06 7.5236000E-07

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229

Table 2 Comparison of the absolute errors in the solution of (5) with α = β = 1 for different values of x ∈ [0, 1] at different time levels t ≤ 1 t

x

ADM

0.2

0.25

3.800E-05 3.800E-05 1.532E-04 2.000E-04 0.000E+0 0.000E+0

1.06824E-05

0.50

4.100E-05 5.000E-05 1.689E-03 8.000E-04 0.000E+0 0.000E+0

4.27295E-05

0.75

1.370E-04 1.370E-04 2.363E-03 1.700E-03 1.000E-04 1.000E-04 9.61414E-05

0.25

5.000E-05 5.000E-05 9.063E-04 8.000E-04 1.000E-04 1.000E-04 8.76724E-05

0.50

4.000E-04 4.000E-04 6.178E-03 2.900E-03 4.000E-04 4.000E-04 3.50690E-04

0.75

7.500E-04 7.500E-04 1.083E-02 6.800E-03 7.000E-04 7.000E-04 7.89051E-04

0.25

3.380E-04 3.370E-04 2.560E-03 1.500E-03 3.000E-04 3.000E-04 3.03675E-04

0.50

1.250E-03 1.250E-03 1.367E-02 6.300E-03 1.300E-03 1.300E-03 1.21470E-03

0.75

2.738E-03 2.738E-03 2.629E-02 1.410E-02 2.700E-03 2.700E-03 2.73308E-03

0.4

0.6

VIM

HWM

ILTM

FVIM

FRDTM

CFDTM

The time CFDT of initial value system of conformable fractional Fokker–Planck equation (8) yields the following recurrence relation: ⎧ 0 Θ (x) = x 2 , ⎪ ⎪ α  r  ⎪ ⎪ ⎪  ⎪ 1 ⎪ ⎨ Θαr +1 (x) = Θαi (x)Θαr −i (x) x T2β α(r + 1) i=0 ⎪   r ⎪ ⎪ ⎪  x r 4 ⎪ i r −i ⎪ ⎪ Θα (x)Θα (x) − Θα (x) , r ≥ 0. ⎩ −x Tβ x 3

(9)

i=0

On solving the recurrence relation (9), we get Θα0 (x) = x 2 ,



x 3−2β 11x β + 4(β − 4)x α



x 4−4β 316β 2 − 2186β + 3490 x β+1 + 253(β − 4)x 2β + 144 β 3 − 9β 2 + 26β − 24 x 2 2 Θα (x) = − 6α 2

Θα1 (x) = −

Θα3 (x)

11 1058β 2 − 8461β + 14540 x 5−3β =− 54α 3

3 5152β − 51415β 2 + 159779β − 155114 x 6−4β − 9α 3

7−5β 4 3000β − 34694β 3 + 144879β 2 − 259650β + 168497 2x − 9α 3

8−6β 5 4 1152 15β − 194β + 982β 3 − 2436β 2 + 2960β − 1408 x − 54α 3 .. .. . .

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B. K. Singh and A. Kumar

On using the inverse time CFDT, the mth order solution of (8) is θ (x, t) = Θα0 (x) + Θα1 (x)t α + Θα2 (x)t 2α + . . . + Θαm (x)t mα .

(10)

In particular, for α = β = 1, the above solution reduces to   t2 t3 t4 θ (x, t) = x 1 + t + + + + . . . = x 2 et 2! 3! 4! 2

which is the closed form of the exact solution θ (x, t) = x 2 et of classical linear Fokker–Planck equation. The behavior of the solutions of the problem is depicted graphically for α = β = 0.8, 1 and absolute errors in different order solutions for α = β = 1 are depicted in Fig. 3. The L 2 error in fifth- and seventh-order approximations are reported in Table 3 for α = 0.8, 0.9, 1 and β = 1. Moreover, computed absolute errors for α = β = 1 are compared with the errors obtained via ADM, VIM, HWM, ILTM, FVIM, FRDTM solutions in Table 4. The findings show that for different values of α = 0.8, 0.9, 1, and β = 1 we get accurate results which approaches towards exact solutions.

Fig. 3 Behavior of fifth-order solutions of conformable space–time fractional Fokker–Planck equation (8) for a α = β = 1, b α = β = 0.8 at different time levels t ≤ 1, and the two dimensional plots of absolute errors in mth different order solutions m ∈ {3, 4, 5} and x = 0.5 Table 3 L 2 error norm in fifth-order approximations of Fokker–Planck equation (5) in x ∈ [0, 1] for α = 0.8, 0.9, 1 and β = 1 at different time levels t ≤ 1 x Fifth order approximation Seventh order approximation α = 0.8 α = 0.9 α = 1.0 α = 0.8 α = 0.9 α = 1.0 0.10 0.20 0.30 0.40 0.50

3.73949E-04 1.49580E-03 3.36555E-03 5.98319E-03 9.34874E-03

2.21775E-04 8.87098E-04 1.99597E-03 3.54839E-03 5.54436E-03

1.38889E-04 5.55556E-04 1.25000E-03 2.22222E-03 3.47222E-03

1.62733E-05 6.50931E-05 1.46459E-04 2.60370E-04 4.06832E-04

7.60802E-06 3.04321E-05 6.84721E-05 1.21728E-04 1.90200E-04

3.85208E-06 1.54083E-05 3.46688E-05 6.16334E-05 9.63021E-05

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231

Table 4 Comparison of the present results of Fokker–Planck equation (8) with α = β = 1 for different values of x ∈ [0, 1] at different time levels t ≤ 1 t x ADM VIM HWM ILTM FVIM FRDTM CFDTM EXACT 0.2

0.4

0.06

0.25 0.50 0.75 1.00 0.25 0.50 0.75 1.00 0.25 0.50 0.75 1.00

0.0764 0.3057 0.6878 1.2227 0.0938 0.3753 0.8445 1.5013 0.0664 0.2655 0.5973 1.0620

0.0764 0.3057 0.6878 1.2227 0.0938 0.3750 0.8450 1.5013 0.0664 0.2655 0.5973 1.0620

0.0763 0.3050 0.6863 1.2200 0.0925 0.3700 0.8450 1.4800 0.0664 0.2655 0.5973 1.0618

0.0763 0.3048 0.6862 1.2190 0.0923 0.3680 0.8321 1.4780 0.0663 0.2654 0.5972 1.0617

0.0761 0.3050 0.6860 1.2190 0.0924 0.3670 0.8319 1.4770 0.0662 0.2653 0.5971 1.0617

0.0762 0.3050 0.6863 1.2200 0.0925 0.3700 0.8325 1.4800 0.0664 0.2655 0.5973 1.0618

0.0763 0.3053 0.6870 1.2213 0.0932 0.3727 0.8385 1.4907 0.0664 0.2655 0.5973 1.0618

0.0763 0.3054 0.6870 1.2200 0.0932 0.3730 0.8392 1.4918 0.0664 0.2655 0.5973 1.0618

4 Conclusion In this article, CFDT method has been adopted to compute the solution of conformable space–time fractional Fokker–Planck equation. The findings show that the present results for α = β = 1 agreed well with the results by various wellknown methods such as Adomian decomposition method (ADM), variational iteration method (VIM), fractional variational iteration method (FVIM) and fractional reduced differential transform method (FRDTM), and so forth. Also, the proposed results converge to the exact solutions for fractional values of α as well.

References 1. M. Kaplan, P. Mayeli, K. Hosseini, Exact traveling wave solutions of the Wu-Zhang system describing (1 + 1)-dimensional dispersive long wave. Opt. Quantum Electron. 49(12), 404 (2017), https://doi.org/10.1007/s11082-017-1231-0 2. S. Momani, Z. Odibat, V.S. Erturk, Generalized differential transform method for solving a space- time fractional diffusion-wave equation. Phys. Lett. A 370(5–6), 379–387 (2007). https://doi.org/10.1016/j.physleta.2007.05.083 3. H. Thabet, S. Kendre, D. Chalishajar, New analytical technique for solving a system of nonlinear fractional partial differential equations. Mathematics 25(4), 47 (2017). https://doi.org/10.3390/ math5040047 4. A. Biswas et al., Resonant optical solitons with dual-power law nonlinearity and fractional temporal evolution. Optik 165, 233–239 (2018) 5. A.F. Qasim, M.O. Al-Amr, Approximate solution of the Kersten-Krasil’shchik coupled KdvMKdV system via reduced differential transform method, Eurasian J. Sci. Eng. 4(2), 1–9 (2018) 6. O. Mohammed, Al-Amr, Shoukry El-Ganaini, new exact traveling wave solutions of the (4+1)dimensional Fokas equation. Comput. Math. Appl. 74, 1274–1287 (2017)

232

B. K. Singh and A. Kumar

7. S. El-Ganaini, M.O. Al-Amr, New abundant wave solutions of the conformable space-time fractional (4+1)-dimensional Fokas equation in water waves. Comput. Math. Appl. (2019). https://doi.org/10.1016/j.camwa.2019.03.050 8. A.J. Al-Sawoor, M.O. Al-Amr, A new modification of variational iteration method for solving reaction-diffusion system with fast reversible reaction. J. Egyptian Math. Soc. 22(3), 396–401 (2014) 9. A. Al-Sawoor, M. Al-Amr, Numerical solution of a reaction-diffusion system with fast reversible reaction by using Adomian’s decomposition method and He’s variational iteration method. Al-Rafidain J. Comput. Sci. Math. 9(2), 243–257 (2012) 10. M.O. Al-Amr, Exact solutions of a family of higher-dimensional space-time fractional KdV type equations. Comput. Sci. Inf. Technol. 8(6), 131–141 (2018) 11. Z.M. Odibat, S. Kumar, A robust computational algorithm of homotopy asymptotic method for solving systems of fractional differential equations. J. Comput. Nonlinear Dyn. https://doi. org/10.1115/1.4043617 12. M. Khader, S. Kumar, S. Abbasbandy, Fractional homotopy analysis transforms method for solving a fractional heat-like physical model. Walailak J. Sci. Technol. 13(5), 337–353 (2015) 13. S. Kumar, A new mathematical model for nonlinear wave in a hyperelastic rod and its analytic approximate solution. Walailak J. Sci. Technol. 11, 965–973 (2014) 14. S. Kumar, A new efficient algorithm to solve non-linear fractional Ito coupled system and its approximate solution. Walailak J. Sci. Technol. 11(12), 1057–1067 (2014) 15. A. Prakash, P. Veeresha, D.G. Prakasha, M. Goyal, A homotopy technique for a fractional order multi-dimensional telegraph equation via the laplace transform. Eur. Phys. J. Plus 134(19), 1–18 (2019) 16. A. Prakash, M. Kumar, K.K. Sharma, Numerical method for solving coupled Burgers equation. Appl. Math. Comput. 260, 314–320 (2015) 17. A. Prakash, M. Goyal, S. Gupta, A reliable algorithm for fractional Bloch model arising in magnetic resonance imaging, Pramana-J. Phys. 92(2), 1–10 (2019), https://doi.org/10.1007/ s12043-018-1683-1 18. A. Prakash, M. Kumar, Numerical method for solving time-fractional Multi-dimensional diffusion equations. Int. J. Comput. Sci. Math. 8(3), 257–267 (2017) 19. A. Prakash, M. Kumar, D. Baleanu, A new iterative technique for a fractional model of nonlinear Zakharov-Kuznetsov equations via Sumudu transform. Appl. Math. Comput. 334, 30–40 (2018) 20. R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. J Comput. Appl. Math. 264, 65–70 (2014), https://doi.org/10.1016/j.cam.2014.01.002 21. N. Benkhettou, S. Hassani, D.F. Torres, A conformable fractional calculus on arbitrary time scales. J. King Saud Univ. Sci. 28(1), 93–98 (2016), https://doi.org/10.1016/j.jksus.2015.05. 003 22. T. Abdeljawad, On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015). https://doi.org/10.1016/j.cam.2014.10.016, arXiv:1402.6892v1 23. K. Hosseini, A. Bekir, R. Ansari, New exact solutions of the conformable time-fractional CahnAllen and Cahn-Hilliard equations using the modified Kudryashov method. Opt. Int. J. Light Electron. Opt. 132, 203–209 (2017), https://doi.org/10.1016/j.ijleo.2016.12.032 24. K. Hosseini, P. Mayeli, R. Ansari, Modified Kudryashov method for solving the conformable time-fractional Klein-Gordon equations with quadratic and cubic nonlinearities. Opt. Int. J. Light Electron. Opt. 130, 737–742 (2017), https://doi.org/10.1016/j.ijleo.2016.10.136 25. O.S. Iyiola, O. Tasbozan, A. Kurt, Y. Çenesiz, On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion. Chaos, Solitons Fractals 94, 1–7 (2017). https://doi.org/10.1016/j.chaos.2016.11.003 26. N. Raza, S. Sial, M. Kaplan, Exact periodic and explicit solutions of higher dimen- sional equations with fractional temporal evolution. Opt. Int. J. Light Electron. Opt. 156, 628–634 (2018). https://doi.org/10.1016/j.ijleo.2017.11.107 27. E. Ünal, A. Gökdoˇgan, Solution of conformable fractional ordinary differential equations via differential transform method. Opt. Int. J. Light Electron. Opt. 128, 264–273 (2017). https:// doi.org/10.1016/j.ijleo.2016.10.031

Numerical Study of Conformable Space and Time Fractional Fokker–Planck …

233

28. M. Ekici, M. Mirzazadeh, M. Eslami, Q. Zhou, S.P. Moshokoa, A. Biswas et al., Optical soliton perturbation with fractional-temporal evolution by first integral method with conformable fractional derivatives. Opt. Int. J. Light Electron. Opt. 127(22), 10659–10669 (2016). https:// doi.org/10.1016/j.ijleo.2016.08.076 29. H.C. Yaslan, New analytic solutions of the conformable space-time fractional Kawahara equation. Opt. Int. J. Light Electron. Opt. 140, 123–126 (2017), https://doi.org/10.1016/j.ijleo.2017. 04.015 30. M. Kaplan, Applications of two reliable methods for solving a nonlinear conformable time-fractional equation. Opt. Quantum Electron. 49(9), 312 (2017), https://doi.org/10.1007/ s11082-017-1151-z 31. A.A. Kilbas, H.M. Srivastava, Trujillo JJ. Theory andapplications offractional differential equations, 204 (Elsevier Science Limited, 2006) 32. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, 198 (Academic press, 1998) 33. A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative. Open. Math. 13(1), 889–898 (2015), https://doi.org/10.1515/math-2015-0081 34. M. Magdziarz, A. Weron, K. Weron, Fractional Fokker-Planck dynamics: stochastic representation and computer simulation. Phys. Rev. E 75(1-6), 016708 (2007) 35. L. Yan, Numerical solutions of fractional Fokker-Planck equations using iterative Laplace transform method. Abstr. Appl. Anal. 2013(1-7), 465160 (2013) 36. A. Yildirim, Analytical approach to Fokker-Planck equation with space-and time-fractional derivatives by homotopy perturbation method. J. King Saud. Univ. (Science) 22, 257–264 (2010) 37. F. Liu, V. Anh, I. Turner, Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166, 209–219 (2004) 38. R. Metzler, T.F. Nonnenmacher, Space-and time fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation. Chem. Phys. 284(1–2), 67–90 (2002) 39. R. Metzler, E. Barkai, J. Klafter, Deriving fractional Fokker-Planck equations from generalised master equation. Europhys. Lett. 46(4), 431–436 (1999) 40. S.A. El-Wakil, M.A. Zahran, Fractional Fokker-Planck equation. Chaos, Solitons Fractals 11(5), 791–798 (2000) 41. V.E. Tarasov, Fokker-Planck equation for fractional systems. Int. J. Modern Phys. B 21(6), 955–967 (2007) 42. W. Deng, Finite element method for the space and time-fractional Fokker-Planck equation. SIAM J. Numer. Anal. 47(1), 204–226 (2004) 43. Z. Odibat, S. Momani, Numerical solution of Fokker-Planck equation with space and timefractional derivatives. Phys. Lett. A: Gen. Atomic Solid-State Phys. 369(5–6), 349–358 (2007) 44. A. Saravanan, N. Magesh, An efficient computational technique for solving the Fokker-Planck equation with space and time fractional derivatives. J. King Saud Univ. Sci. 28, 160–166 (2016) 45. A. Prakash, H. Kaur, Numerical solution for fractional model of Fokker-Planck equation by using q-HATM. Chaos, Solitons, Fractals 105, 99–110 (2017) 46. H. Thabet, S. Kendre, Analytical solutions for conformable space-time fractional partial differential equations via fractional differential transform. Chaos, Solitons, Fractals 109, 238–245 (2018)

Multispectral Bayer Color Image Encryption Binay Kumar Singh and Jagat Singh

Abstract A first approach for Bayer color image encryption and decryption is proposed. The original image is downsampled into three basic color components using multispectral property, then each color component is encrypted by elliptic curve cryptography followed by generalized Arnold transformation. In the first stage of encryption, on each color component separate keys of ECC are employed, and the next stage of encryption considers keys from independent parameters of the coefficient matrix of Arnold transform. The two steps of encoding apply disjoint keys for each color component in both stages, which gives a higher level of security and robustness. Simulation analysis and experimental results are performed on several test images to show the strength of the proposed technique, and a comparison is established with other proposed models. Keywords Bayer image · Elliptic curve cryptography · Elliptic curve discrete logarithmic problem · Arnold transformation

1 Introduction In today’s world with rapid development in technologies, especially telecommunication, and Internet; it is difficult to secure the transmission of valuable information and considered a big challenge among scientists and researchers. For secure transmission of data, there are several encryption and decryption techniques that exist. Furthermore, for secure image transmission, some good methods presented in [1–10]. In the past few decades, image scrambling by Arnold transformation which considers only the coefficient matrix as keys for encryption and decryption is widely been used for image encryption [1–4]. In a different perspective from the conventional method B. K. Singh (B) Department of Computer Science, Mettu University, Po Box- 318, Mettu, Ethiopia e-mail: [email protected] J. Singh Department of Mathematics, Indian Institute of Technology, Delhi, New Delhi 110016, India e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_20

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of considering the original image of square size, [1] divided the original image into multiple square regions of any sizes, then each square region is scrambled from Arnold transformation before transmission. Tang and Zhang [4] proposed solution for some of the weaknesses persist in Arnold transformation, i.e., applying the idea of randomness, and dividing the original image into multiple overlapping square size images, yielding encryption iterative times, and its order. One approach [5] designed a novel quantum circuit for Arnold transformation, and encrypted image using generalized Arnold transformation in confusion stage afterward double random phase encoding in the diffusion stage. An image encryption technique which divides the original image into multiple grids before encryption is proposed by [7]. Data encryption can also be performed based on the probability of their frequency of appearance in any standard case. For example, B. K. Singh, et al. proposed a technique for data encryption which considers the probability of data before encryption. A great deal of effort has been made toward the development of quantum computers due to its richness in computational speed. The extensive research advancing in this regard is especially for image encryption and decryption. For instance, [8] the proposed image encryption and decryption on quantum computers. For image representation, one broadly accepted method is proposed by Bryce Bayer and known as the Bayer image [11]. Based on Bayer image representation, [12] presented image authentication using double random phase encoding and photon counting. References [13, 14] introduced optical transformation in the frequency domain for demosaicking of Bayer image. J. Hardeberg [15] proposed the idea of multispectral color imaging. It was believed from a long time that three color channels are insufficient to represent high-quality imaging systems such as a museum, due to the effect of metamerism. A two-dimensional model for calculating spatial frequency response (SFR) of a Bayer color capture system, which scores for both sampling and interpolation, is presented in [16]. In this paper, a new model is designed in which the RGB color image is downsampled into Bayer image for further processing. The proposed model is based on two stages: Confusion and Diffusion. In the confusion stage, Elliptic curve cryptography will shuffle color information of each pixel in each color plane, which increases keyspace, and different keys are used on each color channel generates another difficulty of the correct arrangement of keys on each channel. While, in the diffusion stage, generalized Arnold transformation will scramble coordinates of each pixel of a partially encrypted image using key matrix or efficient matrix parameters which differs in each color plane, and also its arrangement is considered as the strength of the proposed technique. The remainder of the paper is organized as follows. In Sect. 2, a survey on the literature review is discussed. Section 3 formulated the idea of encryption and decryption in detail such that the original image is encrypted from ECC then the partially encrypted image is finally encrypted using generalized Arnold transformation. For the decryption process, the reverse procedure is applied. In Sect. 4, security analysis and experimental results are demonstrated to show the effectiveness and robustness of the proposed model. In the same section, a comparison is performed between the

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proposed model and an existing model, and a table is presented for its effectiveness. The proposed technique is concluded in Sect. 5.

2 Review of Literature Survey 2.1 Multispectral Color Image and Bayer Image Image of an object can be represented in many ways. Multispectral images are a combination of monochrome images of the same object identified through different sensors. Each of these images is referred to as a band. A well-known example is RGB (Red, Green, and Blue) color image, which contains three spectral imaging planes having a different optical wavelength, and to capture these primary colors, a separate image sensor is required. Multispectral images are highly useful in areas like medical science (MRI, XRAY, CT-SCAN), weather forecasting, remote sensing, astronomy, etc. Although, multispectral images require more space and time to process, with a constant increase in hardware, one can predict no future constraint. Bayer image was introduced by Bryce Bayer in 1976 at Kodak. Bayer filter uses color filter array (CFA) or color filter mosaic (CFM) to produce RGB images on the square grid of photosensors. For a multispectral image, by considering k as 3 (where k represents the number of monochrome images), it means three primary color components (RGB). It has been found that downsampling these components and employing Bayer image CFA, or CFM, when splitting all color channels separately, the grid filter pattern for R = 25%, G = 50%, and B = 25%.

2.2 Elliptic Curve Cryptography for Images Elliptic curve cryptography [17–22] is popular among cryptologist due to smaller key size and larger keyspace, compared to its other counterparts. It is defined as follows. An elliptic curve E is defined by Eq. (1) over finite field F p . y 2 = x 3 + ax + b

(1)

where a, b, x, y ε F p , and a, b are satisfying Eq. (2). 4a 3 + 27b2 = 0 (mod p)

(2)

A pair (x, y) is a point on the curve satisfying Eq. (1). The set of all points on elliptic curve E is denoted by E(F p ), including a point at infinity denoted as ‘∞’, and bounded by Eq. (3).

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√ √ p+1−2 p ≤ N ≤ p+1+2 p

(3)

where N is the number of points on the elliptic curve. To encrypt a message by ECC, the message must be first represented as a point on the curve. The encryption process in ECC is expressed as follows. Cm = {C1 , C2 }; such that C1 = k × P; C2 = M + k × Q. where C m is ciphertext (composed of two different points C 1 and C 2 on the same curve), M is the original message or plaintext, P is a point on E(F p ) whose order is n, the public key Q = d × P, where d is the private key chosen uniformly at random from [1, n − 1], and k is an arbitrarily selected integer. When encrypted message point C m = {C 1 , C 2 } is received, the decryption process of ECC will begin as follows to recover the original message. M = C2 − d × C1 → M + (k × d × P)−(d × k × P) → M One big factor for the widespread use of ECC in secure transmission lies in the elliptic curve discrete logarithmic problem (ECDLP). The ECDLP is given an elliptic curve E over a finite field F p , a point P ε E(F p ) of order n, and a point Q ε

, find an integer  ε [0, n − 1] such that Q =  × P. The integer  is called discrete logarithm of Q to the base P, denoted as  = logP Q. The domain parameters in ECC must be chosen carefully, in order to resist several attacks. Some well-known attacks on ECC (for finding the solution of the discrete logarithmic problem) are exhaustive search, Pohlig-Hellman, Index-calculus, Isomorphism, and Pollard-rho. However, all of these attacks will take an infeasible amount of time to extract the correct keys when the length of the keys is reasonably large. Nonetheless, an investigation done on ECDLP states that once quantum computer is far-flung; ECDLP can be solved in a feasible time. In 1994, [23] has given a polynomial time method for finding a discrete logarithm. Further, [24] proposed a k-bit instance of ECDLP on quantum computer, where k ≈ 5 k + 8 + 5log2 k. On the other hand, a k-qubit quantum computer can expeditiously factor a k-bit integer, where k ≈ 2 k, [25].

2.3 Generalized Arnold Transformation for Images Arnold transformation [26] also known as cat map was introduced by V. Arnold in the research of ergodic theory. Arnold transformation is more useful in the scrambling of image pixels. An original image, say I(i, j), where, i, j are coordinates of an original image pixel, encryption using generalized Arnold transformation will result in I’(i’, j’), where i’, j’ are new coordinates of the encrypted image.

Multispectral Bayer Color Image Encryption

     i 1 p i = mod n j q pq + 1 j

239

(4)

where, p, q are parameters of coefficient matrix multiplied with each pixel of each color component of an image, and n is the size of the image. To return back original image I(i, j), inverse generalized Arnold transformation is applied and expressed as    −1    i i 1 p mod n = j j q pq + 1

(5)

where encrypted points are multiplied by the inverse of coefficient matrix parameters.

3 Proposed Model for Image Encryption and Decryption The encryption and decryption process of the proposed technique is demonstrated in Figs. 1, and 2, respectively, and the underlying discussion is elaborated in the following sections.

Fig. 1 Diagrammatic representation of the encryption process

Fig. 2 Mapping of the decryption process

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3.1 Image Encryption Procedure The original multispectral color image I(x, y) is downsampled to generate a Bayer image I BC (x, y), having three color planes (Red, Green, Blue), shown in Eq. (6) and Eq. (7), respectively. I (x, y) → I BC (x, y)

(6)

⎧ R ⎨ I B (x, y) I BC (x, y) = I BG (x, y) ⎩ B I B (x, y)

(7)

Such that on each of these color channels an interpolation method is applied. Afterward, the first stage of encryption begins by ECC on each pixel of each color channel, which is obtained as below. For each color channel, a different key value is considered and which particular channel is encoded by which key generates another difficulty of the arrangement of keys. ECC cp (x, y) = P E[ϕ1 , ϕ2 ]

(8)

where p represents pixels, c denotes each color channel, ϕ 1 = (x 1 , y1 ), ϕ 2 = (x 2 , y2 ), and PE is partially encrypted image. Here, (x, y) is a coordinate of the original image and, {(x 1 , y1 ), (x 2 , y2 )} are corresponding coordinates of the first stage encrypted image. In other words, after the first stage of encryption, generated cipher points are denoted as C m = {ϕ 1 , ϕ 2 }. Encryption by ECC yields two cipher points ϕ 1 = k × P, and, ϕ 2 = M + k × Q, where k and d are random numbers, P and Q are two different points on the same elliptic curve. After partial encryption, generalized Arnold transformation is applied one time on both partially encrypted points, separately, where the first-point second-stage encryption can be expressed as  AT (P E[ϕ1 ]) =

0 , 1

(9)

where 0 = (x1 + η0 y1 ) mod n; 1 = (η1 x1 + (η0 η1 + 1)y1 ) mod n;

(10)

For generalized Arnold transformation η0 , η1 are keys of the coefficient matrix, and n is pixel size of the image. Similarly, for second partially encrypted point ϕ 2 , on the same stage again generalized Arnold transformation is applied given as

Multispectral Bayer Color Image Encryption

241

 AT (P E[ϕ2 ]) =

2 , 3

(11)

where 2 = (x2 + η2 y2 ) mod n; 3 = (η3 x2 + (η2 η3 + 1)y2 ) mod n;

(12)

Here, for this partially encrypted point, another set of coefficient matrix parameters η2 , η3 are keys for final encryption in generalized Arnold transform. Henceforth, final encrypted pixel points are denoted as FE = {(0 , 1 ), (2 , 3 )}.

3.2 Image Decryption Procedure When the final encrypted image is received, the reverse operation will begin. That is, in the first stage of decryption operation, inverse generalized Arnold transformation is applied on both points separately for each color component, expressed as AT −1 (P E[ϕ1 ]) =



0 , 1

(13)

where x1 = (((η0 η1 + 1)0 − η0 1 ) mod n; y1 = (−η1 0 + 1 ) mod n;

(14)

This will generate first partially decrypted point, and then second partially decrypted point in a similar manner. AT −1 (P E[ϕ2 ]) =



2 , 3

(15)

where x2 = ((η2 η3 + 1)2 − η2 3 ) mod n; y2 = (−η3 2 + 3 ) mod n;

(16)

It will yield two different partially decrypted points {(x 1 , y1 ), (x 2 , y2 )}. After that, the next stage of decryption is applied. In the next level of ECC encryption first point is multiplied by the receiver’s private key then subtracted by the second point. That means, ϕ 2 is multiplied by ‘d’ a randomly selected integer then subtracted by ϕ 1 . Now, to return back original image pixels I BC (x, y), in each color component the last

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stage of decryption using ECC will be applied over these two points. Henceforth, the final encrypted pixel points are M = ϕ2 − d × ϕ1 = M + (k × d × p) − (d × k × p) =M Therefore, M = I BC (x, y). Subsequently, each of the color components is derived.

4 Security Analysis and Experimental Results The experimental results are carried out on MATLAB 2016a software over Intel(R) Core(TM) i5-2430 M CPU. Several test images of size 512 × 512 have been taken to perform analysis. Figure 3a displays the original color Lena image, Fig. 3b shows an encrypted Lena image, while Fig. 3c presents decrypted Lena image. Here, one can see that when all parameters in both stages of decryption are applied correctly in the correct order on all channels, receiver receives the original image. In Fig. 4a, the original Lena image is obtained, while Fig. 4b shows incorrectly decrypted image. Because, ECC is applied with different keys on all color components, if in any color component keys are changed, one cannot identify the original image. Similarly, with changes in coefficient matrix parameters of generalized Arnold transform, Fig. 4c, on color channels, will reflect a different image. While Fig. 4d represents a different decoded image when keys in both stages of decryption are wrongly chosen. As all keys are correct, the original Lena image is successfully recovered, which is presented in Fig. 4e. The arrangement of color channel keys parameters in both stages of encryption is also one key. For instance, in the first stage of encryption three different color channels are encrypted with three different keys which can be arranged in 3! ways, following this, the next level of encryption

Fig. 3 a Original image; b encrypted image with all correct keys in both stages; c correctly decrypted image

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Fig. 4 a Original image; b decrypted image with different key values in ECC; c decrypted image with different coefficient matrix parameters in Arnold transformation; d decrypted image with different key values in both stages; e decrypted image with all correct parameters

using generalized Arnold transform coefficient matrix parameters on all three levels are keys, which can be separately arranged in 3! ways that itself can be arranged in 2! ways. Combining the entire proposed technique encryption keys, one can get {3! × (total number of keys in ECC) × (2! × total number of keys in coefficient matrix parameters)}. Therefore, keyspace is large and gives more effectiveness and robustness for encryption.

4.1 Statistical Analysis Statistical analysis measures the strength of the proposed technique in various manners. The following are some of the statistical analysis performed on the proposed scheme.

4.1.1

Histogram Analysis

The histogram of an image illustrates the intensity value of pixels of all color channels in a graphical format. Figure 5a shows the histogram of the original Lena of Fig. 3a,

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Fig. 5 a Histogram of Fig. 3a; b histogram of Fig. 3b; c histogram of Fig. 3c

while Fig. 5b represents the encrypted Lena of Fig. 3b. As one can see that both histograms are totally different, so it is almost impossible to extract the original image information. On the other hand, Fig. 5c represents the histogram of Fig. 3c and resembles Fig. 5a which concludes that the original image is recovered correctly.

4.1.2

Entropy Analysis

In information theory, entropy is a measure of randomness feature of information in the form of signals or images. In general, image entropy analysis H (X) of base 2 is measured by Eq. (17). H (X ) = −

N 2 −1

p(X i ) log2 p(X i )

(17)

i=0

where X represents the source of the message, p(X i ) denotes the probability of the message source X i . Table 1 demonstrates entropy analysis between original images and its corresponding encrypted images, where standard entropy value for an image is 8 bits, [9]. Table 1 Entropy analysis on some test images

Images

Entropy analysis Original images

Encrypted images

Lena

7.1125

7.8314

Barbara

7.357

7.6787

Baboon

7.2149

7.7142

Boat

7.1286

7.8451

Airplane

7.2012

7.9014

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4.2 Numerical Analysis Several numerical analyses have been performed to realize the quality of the original image to encrypted image.

4.2.1

Mean Square Error Analysis

Mean square error is calculated by Eq. (18), which is used to obtain commutative error between the original image (g) and decrypted image (g’) of size M × N. MSE =

N −1 M−1 



1

g(m∇x, n∇ y) − g  (m∇x, n∇ y) 2 M × N i=0 j=0

(18)

where ∇x and ∇ y are image pixel size. In Table 2, MSE between the original image (g) and correctly decrypted image (g’) on all color channels is calculated, separately. Besides this, the proposed model is compared with [27]. From Table 2, it is clearly evident that the proposed scheme MSE is lower than [27].

4.2.2

Structural Similarity Index (SSIM)

Structural similarity index is a method for quality assessment in terms of structural information of an object in the range of −1 to +1 to determine similarities between the reference image and encoded image. When the value is close to +1 it means both images are very similar or exact to each other, whereas when the value tends to −1 images are very distinct, [28]. SSIM is expressed in Eq. (19), where it measures three attributes of an image luminance, contrast, and structure.



2μxμx  + C1 2σ x x  + C2



SS I M(x, x ) = μx 2 + μx 2 + C1 σ x 2 + σ x 2 + C2 

(19)

where x, x  are original and correctly encoded images, respectively. Other parameters used in the calculation of SSIM of two images are: μx , μx represents mean of x, x  ; σ2x , σx 2 are variances between x, x  ; and finally, C 1 , C 2 are prespecified invariant. In Table 3, SSIM between original and decoded images on all color components on various test images is demonstrated.

246 Table 2 Mean square errors on some test images

B. K. Singh and J. Singh Images

Mean square error Split channel

Ref. [26]

Proposed model

Lena

Red

4.5817 × 10−30

3.0112 × 10−30

Green

7.5088 × 10−30

7.1462 × 10−30

Blue

9.7892 × 10−30

8.8901 × 10−30

Red

1.2020 × 10−29

1.1025 × 10−30

Green

1.2046 × 10−29

1.8945 × 10−30

Blue

8.4432 × 10−30

8.7981 × 10−30

Red

2.4168 × 10−29

2.5891 × 10−30

Green

1.8396 × 10−29

3.9783 × 10−30

Blue

7.3902 × 10−30

8.0197 × 10−30

Red

2.6061 × 10−29

3.7306 × 10−29

Green

3.2023 × 10−29

4.6018 × 10−29

Blue

1.5403 × 10−29

2.0197 × 10−29

Red

2.4385 × 10−29

5.2541 × 10−30

Green

2.3530 × 10−29

4.9871 × 10−30

Blue

5.8079 × 10−29

9.3154 × 10−30

Barbara

Baboon

Boat

Airplane

5 Conclusions In this paper, a new technique for encryption of a multispectral Bayer image in all three primary color components is presented. All components of the image are separately encrypted using elliptic curve cryptography followed by generalized Arnold transformation. Different public and private keys in the first level of encryption on each color component, so as, different values of coefficient matrix parameters in generalized Arnold transformation creates larger keyspace, and provides the proposed model higher level of security. Security analysis and experimental results showed

Multispectral Bayer Color Image Encryption Table 3 SSIM of proposed scheme on some test images

247

Images

SSIM of the proposed scheme

Lena

Red

4.1401 × 10−3

Green

3.0128 × 10−3

Blue

2.5122 × 10−3

Red

2.7841 × 10−3

Green

1.2125 × 10−4

Blue

2.1532 × 10−3

Red

−3.2145 × 10−4

Split channel

Barbara

Baboon

Green Boat

1.7819 × 10−4

Blue

−1.2315 × 10−3

Red

2.3384 × 10−3

Green Airplane

Proposed model

−3.5144 × 10−3

Blue

1.2351 × 10−3

Red

3.6587 × 10−3

Green

4.2864 × 10−4

Blue

2.3651 × 10−4

on some standard examples state that the proposed model is effective and robust for encryption and decryption of images as well as minimizes several possible attacks.

References 1. M. Li, T. Liang, Y. He, Arnold Transformation Based Image Scrambling Method (Atlantis Press, ICMT 2013, 2013), pp. 1309–1316 2. Q. Dongxu, Z. Jianchun, H. Xiaoyou, A new class of scrambling transformation and its application in the image information covering. J. Sci. China Ser. 43(3), 304–312 (2000) 3. L. Chen, D. Zhao, F. Ge, Image encryption based on singular value decomposition and Arnold transform in fractional domain. Opt. Commun. 291, 98–103 (2013) 4. Z. Tang, X. Zhang, Secure image encryption without size limitation using Arnold transform and random strategies. J Multimed. 6(2) (2011) 5. M.D. Swanson, B. Zu, H.I. Tewf, Robust data hiding for images. J. Proc. IEEE 7th Digit. Signal Process. Work. (DSP 96) 9, 37–40 (1996) 6. B.K. Singh, D.C. Mishra, M. Hanmandlu, Security of grayscale-image data using random Affine Cipher followed by discrete wavelet transformation, in International Conference on Mathematical Science 2014 (India, 2014), pp. 180–183 7. B.K. Singh, S.K. Gupta, Grid-based image encryption using RSA. Int. J. Comput. Appl. 115(1), 26–29 (2015) 8. N.R. Zhou, T.X. Hua, L.H. Gong, D.J. Pei, Q.H. Liao, Quantum image encryption based on generalized Arnold transform and double random-phase encoding. Quantum Info Process. 14, 1193–1213 (2015) 9. J.X. Chen, Z.L. Zhu, C. Fu, H. Yu, A fast image encryption scheme with a novel pixel swappingbased confusion approach. Nonlinear Dyn. 77(4), 1191–1207 (2014)

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10. B.K. Singh, A. Tsegaye, J. Singh, Probabilistic data encryption using elliptic curve cryptography and Arnold transformation, in IEEE xplore-I-SMAC 2017 (India, 2017) https://doi.org/10. 1109/i-smac.2017.8058259 11. B.E. Bayer, Color Imaging Array. US Patent No. 3971065 12. F. Yi, I. Moon, Y.H. Lee, A multispectral photon-counting double random phase encoding scheme for image authentication. Sensors 14, 8877–8894 (2014) 13. D. Tretter, C.A. Bouman, Optimal transforms for multispectral and multilayer image coding. IEEE Trans. Image Process. 4(3), 296–308 (1995) 14. E. Dubois, Frequency-domain methods for demosaicking of Bayer-sampled color image. IEEE SPL 12(12), 847–850 (2005) 15. J.Y. Hardeberg, Multispectral Color Imaging (Conexant Systems Inc., Redmond, Washington, USA, NORSIGnalet, 2001) 16. A. Golts, Y.Y. Schechner, Spatial Frequency Response of Bayer Color Image Formation. CCIT Report #868, Sept 2014 17. D. Hankerson, A. Menezes, S.A. Vanstone, Guide to Elliptic Curve Cryptography (Springer, 2004) 18. W. Stallings, Cryptography and Network Security, 4th edn. (Prentice Hall, 2006) 19. M. Rosing, Implementing Elliptic Curve Cryptography (Manning Publications, Greenwich, CT, 1999) 20. Certico, Research, Standards for Efficient Cryptography: Elliptic Curve Cryptography (Standard SEC, Certicom, 2009) 21. R.C.C. Cheng, N. Jean-baptiste, W. Luk, P.Y.K. Cheung, Customizable Elliptic Curve Cryptosystems. IEEE Trans. VLSI Syst. 13(9), 1048–1059 (2005) 22. T. ElGamal, A public key cryptosystem and a signature scheme based on discrete logarithm. IEEE Trans. Inf., Theory, IT-31, 4, 469–472 (1985) 23. P. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26, 1484–1509 (1997) 24. J. Proos, C. Zalka, Shor’s discrete logarithm quantum algorithm for elliptic curves. Quantum Inf. Comput. 3, 317–344 (2003) 25. S. Beauregard, Circuit for Shor’s algorithm using 2n + 3 qubits. Quantum Inf. Comput. 3, 175–185 (2003) 26. F.J. Dyson, H. Falk, Period of a discrete cat mapping. Am. Math. Mon. 99, 603–614 (1992) 27. Z.D. Chen, F. Ge, Image encryption based on singular value decomposition and Arnold transform in fractional domain. Opt. Commun. 2, 98–103 (2013) 28. Z. Want, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli, Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2014)

Investigation of Prospective Elementary Teachers’ Opinions About Problem Concept Danyal Soyba¸s and Sevim Sevgi

Abstract This study aims to not only determine the views of prospective elementary teacher about Polya’s four-step problem-solving process used in mathematics teaching, the problem concept and characteristics of nonroutine problems and their solutions, but also to investigate the level of knowledge about some important points concerning these concepts that prospective elementary teacher need to sufficiently know. The research sample consists of 43 prospective elementary teachers from an education faculty of a state university in Turkey. The data were obtained from the answers the prospective elementary teachers wrote on the working papers that include various styles (multiple choice, true–false judgments selection, and open-ended) of questions. Both quantitative and qualitative assessment tools were employed in the analysis of the data. Keywords Problem concept · Polya’s problem-solving processes · Nonroutine problems · The prospective elementary teacher

1 Introduction 1.1 What Is a Problem? Human beings have been faced with various situations from the moment they first came into the world and attempted to get rid of these situations which could carry negative qualities. As a result of this endeavor, human beings developed various methods with the impulse to get rid of these situations, applied to various ways and had to perform firmly against these negative situations. In this context, we can say that these problems are negative, and problem-solving strategies for various ways and D. Soyba¸s (B) · S. Sevgi Math Education Department, Faculty of Education, Erciyes University, 38039 Kayseri, Turkey e-mail: [email protected] S. Sevgi e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2020 N. Deo et al. (eds.), Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory, Springer Proceedings in Mathematics & Statistics 307, https://doi.org/10.1007/978-981-15-1157-8_21

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methods that we use to overcome this situation, and the problem-solving process that includes all this process [16, 19]. John Dewey describes the problem as everything that confuses the human mind, challenges it and makes it unclear [8]. “The making challenging to him” term has brought a strong stance against the problems. The biggest factor in exhibiting this stance is the desire to overcome the problem that appears among the people who face problems. The problem that arouses the desire to solve the person and that does not have a solution procedure but can be solved by using his/her knowledge and experience can be called a problem [16]. In a crucial point of the concept of the problem, the fact that a person calls a problem as a problem depends on the fact that the subject is an obstacle to that person reaching a certain purpose or goal. Because for people who have different lives and who have different experiences and information, every situation may not be called the same problem. A problem is confusing some people’s minds, while others may not. If a person has encountered a problem before and solved it before, that problem can no longer be a problem for the person. Then, the problem must be new and original for the person [10]. Another important point regarding the concept of the problem is the condition that people should have encountered this situation for the first time. In other words, in order for a situation or subject to be called a problem by a person, it is the first time that he encountered this situation and he should not have a solution for this situation. Three main characteristics of the problem can be revealed. These are: (1) the problem is a difficulty for the person who encountered, (2) the person needs to solve it, and (3) the person has not encountered this problem before, there is no preparation for the solution [2]. Our assessments so far have some limitations on the concept of the problem. These are, once encountered and solved, that the same situation is not a problem, a situation that is problematic for some people is not for some others, the solution does not appear suddenly and requires an effort [16]. In the most general context of a case can be counted as the problem can be counted; it can be stated that people who are uncomfortable to people and who want to solve this problem as a result of this feeling and that people should have encountered this situation for the first time [4, 5, 7]. The problems are only incomplete as a result of mental disturbance. If we explain this with an example, this is a physical problem for a person who has a disease, but this person shows us the existence of a mentally based disorder with questions and worries about his/her future life. The question you ask your friend can confuse us; on a hot day, a gum sticking to our feet as we walk down the road is a problem that we don’t want and want to get rid of it, and it is tried to be resolved by the authorities; war is a separate problem, and because people cannot find a solution to this problem (as a solution to their own right). The assignment of the teacher is a problem which stimulates the student’s mind and needs to be answered. As seen in the examples, the problem can be both mental and physical [10]. The issue of the difference between the abovementioned person and the person regarding the concept of the problem points to the fact that there is also such a definitive and semantic difference between the disciplines. In terms of the problem, mathematical problems based on four processes given at the end of the subject are thought with an understanding obtained mostly from primary school mathematics textbooks. However, the concept of the problem has a wider

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meaning than that mentioned here, and the problem does not have to be related to mathematics [2]. As with the related definition, the problem has different meanings and definitions according to the interests of the disciplines. This shows us how to generalize the problem concept. The classification of the problems in the general dimension of teaching is emerging at this point. Classifications of problems can be made with different approaches. Based on the objectives of teaching, problems can be divided into two classes as routine and nonroutine problems [2]. Researchers analyzed routine and nonroutine problems separately under another heading. The next one will be to examine the problem-solving and process of problem-solving.

1.2 Problem-Solving and Process of Problem-Solving The problem is to solve the problem situation in the problem, the discomfort of the person, the problem-solving process brings together the problem-solving process. The elimination of the difficulties encountered and the efforts to eliminate the uncertainties can be called the solution to the problem. The solution of all problems, whether mental or physical, requires a mental process and this can be called a problem-solving process [10]. Problem-solving is an action that involves a broader mental process and skills, though it is perceived as a just result [3]. This process brings with it various mental activities. In particular, there is an accepted problemsolving systematic in the solution of nonroutine problems related to daily life. There is no specific way or method used to solve all problems. If such a method would be taken away from the source of the problem [2]. This means that there is no common method or strategy used to solve all problems. However, there is an accepted problem-solving step for the completion of this process, and these steps belong to George Polya. When students encounter a problem, they often try to remember a rule to be used. This is not a good attempt. Because problem-solving does not have a rule, but it has a systematic [11]. Each problem requires a separate solution. However, it has been concluded that there are some steps in solving the mathematical problems in general [17]. Problem-solving steps: (1) understanding problem, (2) choose the appropriate strategy for problem-solving, (3) implementation of the chosen strategy, (4) evaluation of the solution. These steps, which are currently accepted, provide great convenience to individuals on the path of problem-solving and to achieve the desired goals, but they do not lead to definite success because individual factors are effective in the realization of these steps. George Polya (1887–1985) is the most widely accepted problem-solving steps of routine and nonroutine problems. Knowing these steps does not provide problem-solving, but when solving the problem, the way of working in accordance with these four steps facilitates the solution [2]. The realization of these stages is the problem-solving process itself. The most important step of this process is to understand the problem and determine the appropriate strategy for problem-solving. Especially understanding the problem is the biggest and basic step toward a solution. It is very important to understand when a problem is encountered. For a problem that the individual cannot understand, he cannot propose

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a solution, and he cannot identify any strategy and implement it. The problem-solving process according to these explanations can be explained as “To do research with controlled activities in order to reach a target that is clearly designed but not reached immediately.” [3]. Problem-solving and another important point in this process is that problem-solving will not only be realized by knowledge, but also past experiences and experiences are of great importance in this process. Problem-solving is related to past experiences of the individual (Kennedy 1980, p. 28; cited in [20]). The people who are active in this process have carried out many mental activities and will have made great progress towards gaining skills such as various classification, analysis, and interpretation. This means the development of the problem-solving abilities of these individuals.

1.3 Nonroutine Problems The problems in the teaching base are divided into two as routine and nonroutine problems. Routine problems are problems that are commonly found in textbooks such as mathematics and physics, and are known as problems (exercise). Solutions of nonroutine problems require a high level of skills such as organizing data, organizing, classifying, seeing relationships, and doing a series of activities one after the other. Various problem-solving strategies can be followed to solve routine and nonroutine problems as (making a systematic list, estimating and controlling, drawing a diagram, finding a relationship, using a variable, working backwards, making a table, etc.) (cited in [3, 11]). Nonroutine problems are very important and effective problems that can be used in the process of gaining and developing high-level skills such as interpretation, classification, and analysis. The use of these problems in educational processes is extremely beneficial. George Polya stated this as: “Failure to use nonroutine problems in educational processes is an unforgivable mistake [9].” The aim of nonroutine problems and solutions is to develop the ability to understand the logic and nature of problem-solving, to choose the appropriate strategy when a problem is encountered, to use and to interpret the results. The main purpose of instruction is problem-solving [2]. This type of mental health process and some of the skills and abilities related to everyday life and also the necessary mathematical skills to gain the ability to provide some features related to this type of problem. One of these features is the need to use more than one method and strategy to solve such problems. The truth is that there is not a single and clear way. It is necessary to determine various classifications and relationships in solving such problems. Solutions of nonroutine problems require a high level of skills such as organizing data, organizing, classifying, seeing relationships, and doing a series of activities one after the other problems (Souviney and Randall 1989; cited in [2]), the accepted problem-solving steps belong to George Polya [9]. Such problems are the bridge between real life and school. These problems are the expression of a situation encountered or encountered in real life. Therefore, these are called real-life problems [2].

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1.4 Importance of the Study The study was carried out by taking into account the fact that the learning–teaching process carried out in the first stage of primary education is a period in which students are settled in the future periods, as well as daily life and the foundation parts of education and training life. In this period, the opinions and evaluations of the students, especially the elementary teachers, on the research subjects in question and the level of the information that should be at some point is very important in order to carry out this process as desired. In this context, it is important to know how the thoughts and opinions of the elementary teachers about these concepts will be given in the first stage of this teaching–learning process. For this aim, it is necessary for the prospective elementary teachers who have an important position in this learning– teaching process to get their opinions, evaluations, and information on some points before they start into the profession, taking into account the idea that it will be useful.

2 Method 2.1 Sample The sample of the study is 43 prospective elementary teachers of a department of elementary education of Faculty of Education of a state university which is located in Turkey, there are 34 females and 9 male prospective elementary teachers in our research sample.

2.2 Instruments Prospective elementary teachers who participated in the research were administered a test with multiple-choice questions (the choices to be chosen have been prepared in a way to give information about certain levels of knowledge and opinions), right–wrong statements items and open-ended questions. In all types of questions used, questions related to the mentioned research subjects were prepared. The prepared multiplechoice questions, the statements about the right–wrong and the open-ended questions that are required to be filled, have been prepared and arranged for the purpose of obtaining the knowledge levels and opinions about the whole subjects. The test applied to the sample consisted of 4 open-ended questions, 6 right–false statements, and 3 multiple-choice questions with 4 options, which were prepared to enable the control of information and the reliability of other items. The purpose of all the questions in this test is to determine the problem concept, problem-solving concept, problem-solving processes and the properties of the problems of the nonroutine

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problems and their solutions, and to determine the extent to which some points are thought to be known.

2.3 Reliability and Validity In order to maintain the reliability of the findings, the prospective elementary teachers were randomly selected, and the questions in the instrument provided for the administration of the prospective elementary teachers were prepared according to the condition of serving the stated objectives and ensuring that the desired data could be obtained. In order to ensure the reliable implementation of the instrument, the instrument and what to do are explained to the prospective elementary teachers by pilot applications. The data obtained have been evaluated objectively and it has been tried to avoid bias which will direct the data for a certain purpose. The abovementioned processes have been meticulously carried out in order to ensure that the reliability and validity of this research are carried out in a reliable manner, and that the data can be obtained reliably and that the data that are intended to be obtained in relation to the objectives can be obtained.

3 Results After the application of the instrument to the prospective elementary teachers, an analysis of the data was obtained. For this purpose, open-ended questions in the instrument are categorized into various headings. What these headings are and which open-ended question is stated below.

3.1 What Is a Problem? The first research question of the study is “Based on my end-of-class evaluations, problem is …………………………………” The categories formed in line with the answers to the first open-ended question and the frequencies and percentages of the responses given under each category are given in Table 1. 40 of these prospective teachers have filled the right answers in this question and 3 prospective elementary teachers left this question blank. In other words, the frequencies stated in the findings are based on the answers given by 40 prospective elementary teachers. The frequencies obtained for these categories are given in Table 1. The remaining 20% ratio is 8 answers which are given in different qualities that do not match with the categories given in Table 1. These answers:

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Table 1 Descriptive of first open-ended item “Based on my end-of-class evaluations, the problem is …………………………………” Categories

f

“It is a situation that is expected and to be solved due to the problem.”

Percentage %

17

42.5

“It is the situation that needs to be solved by certain methods and strategies.”

7

17.5

“There are situations that are related to daily life and constitute an obstacle in life.”

4

10

“The person is disturbed.”

4

10

Other responses

8

20

32

80

Total

‘In life, the way to be noticed, the way to be solved.’, ‘They can be solved by knowledge and experience.’, ‘The result is unknown and the solution is process requiring.’, ‘They create confusion in the mind and have obstacles in front of the solution.’, ‘The desire of research in individuals is the effects.’, ‘There will be no solution at same situations’, ‘Situations that limit human behaviors and situations that are problematic when not solved.’.

3.2 Types of Problem “Problem in the mathematics can be divided into two categories as ………………….. and……………………..” appropriate filling of the places left blank in the item was expected from the prospective elementary teachers. 43 prospective elementary teachers responded fully to this question by taking into consideration the previous mathematics learning processes. Prospective elementary teachers gave “routine problems” and “nonroutine problems” answers to the blanks related to the question. All of the participants gave the answers, and the level of knowledge to be measured came with a 100% proportional result.

3.3 Conditions of Being Problem The third research question of the study is “In order for a subject or a situation to be considered as a problem; it must be: • ………………………………………………………………. • ………………………………………………………………. • ……………………………………………………………….” The abovementioned question was asked to write three items, at least one according to the personal evaluations of prospective elementary teachers. Responses to these items are classified into categories based on prospective elementary teachers’

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Table 2 Answers to third item (being a problem) Categories

f

Percentage %

1. To be an obstacle in front of a goal or target

12

15

2. Should create a trouble

19

23.8

3. Feeling of uncomfortable and creates uneasiness

15

18.7

4. The desire to transcend and curiosity

10

12.5

5. Can be solved by specific methods and techniques 6. Other responses

8

10

16

20

responses. Since three prospective elementary teachers did not write any items to the second question, analysis was run based on the responses of 40 prospective elementary teachers which were counted as valid answers. The total number of items obtained from prospective elementary teachers’ responses to this question is 80. The categories in this classification are given in Table 2 based on 80 answers. The answers grouped under these categories account for 80% of the total answers. The remaining 20% consists of 16 items that cannot be placed in the categories mentioned in Table 2. Examples to these answers are: ‘Make life difficult (2)’, ‘It should cause mental confusion in people (4)’, ‘It must be individual and concern only the individual (2)’, ‘Must be noticeable (1)’, ‘Must have the necessary data for the solution (1)’, ‘It should be relevant to overall. (2)’, ‘Solution requires process (1)’, ‘A final answer should be reached at the end of the solution process (3)’.

3.4 Problem-Solving Steps of Polya ‘Problem-solving steps of George Polya’s problem-solving 1-…………………………….., 2-…………………………. 3………………………..4-……………………….’. the item was asked to be filled by the prospective elementary teachers. Prospective elementary teachers’ previous math learning process considering the answer to this question posed to 7 invalid and 36 valid desired answers were obtained. Within these answers, valid and correct answers regarding the steps have been examined and indicated separately. The steps written in this question are analyzed separately and the classification of the answers is given in Table 3. The percentages of the above categorized and validated answers were determined according to the responses from the 36 prospective elementary teachers. The valid answers given to the overall problem are 83.7%.

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Table 3 Categories of problem-solving steps Steps

Categories

f

Percentage %

1.

Understanding the problem, Realizing the problem, Determining the problem, Identifying the problem

36

100

2.

Determining the appropriate strategy for the solution, Choosing a solution, Finding a method

34

94

3.

Applying the specified strategy, Reaching the result, Performing the solution, Application of solution

33

92

4.

Performing the solution, Evaluation of solution, Solution and outcome evaluation

30

83

3.5 Right/Wrong Items There are six right–false statements in this section of the instrument. Categories based on the answers to right–false statements and frequencies of them were given in Table 4. Prospective elementary teachers choose one of the options and there is no empty statement. All of the prospective elementary teachers made mistake at selecting the statement which is ‘The problem is a concept belonging merely to mathematics.’ No one of them choose the wrong option for that statement. Seven prospective elementary teachers (16.3%) marked that statement which is “There is no common method or strategy to solve all problems.” is wrong and then the remaining 36 (83.7%) of prospective elementary teachers stated that the statement is right. Four prospective elementary teachers (9.4%) marked that statement which is “The use of nonroutine problems in the teaching and learning process is a correct approach.” is wrong and then the remaining 39 (90.6%) of prospective elementary teachers stated that the statement is right. 14 (32.6%) of prospective elementary teachers marked that statement which is ‘The most important step in the solution of the problem is the phase in which the strategy is implemented.’ is wrong and then the remaining 29 (67.4%) Table 4 Descriptive of right/wrong statements Statements

Right f (%)

Wrong f (%)

1. The problem is a concept belonging merely to mathematics

0

43 (100)

2. There is no common method or strategy to solve all problems

36 (83.7%)

7 (16.3%)

3. The use of nonroutine problems in the teaching and learning process is a correct approach

39 (90.6%)

4 (9.4%)

4. The most important step in the solution of the problem is the phase in which the strategy is implemented

29 (67.4%)

14 (32.6%)

5. The definition of the problem concept does not differ between disciplines and between individuals

7 (16.3%)

36 (83.7%)

6. In nonroutine problems, the answer is clear and unique

7 (16.3%)

36 (83.7%)

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of prospective elementary teachers stated that the statement is right. 7 (16.3%) of prospective elementary teachers marked that statement which is “The definition of the problem concept does not differ between disciplines and between individuals.” is wrong and then the remaining 36 (83.7%) of prospective elementary teachers stated that the statement is right. 36 (83.7%) of prospective elementary teachers marked that statement which is ‘In nonroutine problems, the answer is clear and unique.’ is wrong and then the remaining 7 (16.3%) of prospective elementary teachers stated that the statement is right.

3.6 Multiple-Choice Items Three multiple-choice questions were directed to the prospective elementary teachers. The prospective elementary teachers answered all the questions. First multiple-choice item is: “1. Which of the following is a right statement for problems? (a) (b) (c) (d)

All problems have a mathematical content. There are certain methods or strategies used to solve each problem. Mathematical problems are the only type and solution strategies. There is no specific method or strategy for all of the mathematical problems, but there is a certain solution systematic.”

As it will be used to determine the information to be used in the unmarked, uncontrolled, and unassigned parts of this question, prospective elementary teachers’ frequencies and the percentages of options are given in Table 5. Prospective elementary teachers did not make a choice regarding A and C option. 12 (27.9%) of the class teachers nominated option B and 31 (72.1%) marked option D. Second multiple-choice item is: “Which two of the following stages constitute the two most important stages of the problem-solving process? (a) Determining the appropriate strategy for the solution—Evaluating the result and solution. (b) Applying the appropriate strategy for the solution—Evaluation of the result and solution. (c) Understanding problem—Identifying the appropriate strategy for the solution. Table 5 Distribution of multiple choice items

Options

1. Item f (%)

2. Item f (%)

3. Item f (%)

A

0

1 (2.3%)

3 (7%)

B

12 (27.9%)

4 (9.3%)

33 (76.7%)

C

0

27 (62.8%)

0

D

31 (72.1%)

11 (25.6%)

7 (16.3%)

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(d) Understanding problem—Applying the appropriate strategy for the solution.” Table 5 shows the distribution of the answers to the question. One of the prospective elementary teachers (2.3%) had the option of A, four of them (9.3%) had the option of B, 27 (62.8%) had C, and 11 (25.6%) were selected as D. Third multiple-choice item is: “Which of the following is a false statement regarding the solution of nonroutine problems? (a) The nonroutine problems have a positive effect on students’ interpretation, analysis, solution production and gaining some skills. (b) Nonroutine problems can be solved by the usual algebraic operations. (c) The most recognized system for solving nonroutine problems belongs to George Polya. (d) Solving nonroutine problems may require several different strategies and methods to be implemented.” Prospective elementary teachers’ answer to this particular item were given in Table 5. Three (7%) of prospective elementary teachers were selected option A, 33 (% 76.7) of them were selected option B, and seven (% 16.3) of them chose to mark option D.

4 Discussion and Conclusion In this study, the opinions, evaluations, and knowledge levels of the prospective elementary teachers about the problem concepts were analyzed. The first finding of the study stated that prospective elementary teachers defined the problem as a situation which has handicaps at its solving process. Problems have different components but the majority of them defined the problem in a situation and unwanted or unexpected situations. Chapman [9] revealed this finding in his research. Moreover, Özyıldırım Gümü¸s and Sahiner’s ¸ [18] study stated that prospective teachers’ expressions used for the definition of the problem were similar to this study. Prospective elementary teachers defined types of problems as routine and nonroutine problems in this study. They knew the distinction between these problem types. As stated in the literature, prospective elementary teachers chose the routine problems in their classrooms [13]. The reason for using routine problems sourced from education systems and textbooks [1, 13, 15]. Prospective elementary teachers view nonroutine problems as time-consuming, and increased teaching load at their classes so they skip these problems in their classes [12, 13]. Polya [19] defines four steps of problem-solving as understanding problems, devise a plan, carry out a plan and look back (evaluation). Prospective teachers’ percentage at each step of Polya’s problem-solving is decreasing from the first step to the last step. The prospective teachers’ attitude to check the solution is lower than the understanding of the problem. Xia and Masingila [23] stated the difficulties of prospective elementary teachers at problem-solving processes at fraction subjects.

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Their result can be overgeneralized to the other subjects since teachers have problems with the evaluation step of problem-solving. Prospective teacher confidence and sufficient experience in problem-solving are vital to model problem-solving processes in their classrooms in more convincing ways [14, 22]. Implication of this study is that prospective elementary teachers should take courses related to problem-solving steps in context and one further step should cover professional development courses related to problem-posing activities [12]. This study is restricted to only third-year prospective elementary school teachers and twelve questions about problems. The study could be improved by extending the sample to the other prospective teachers and sample size could be increased.

References 1. P.D. Artut, K. Tarım, ˙Ilkö˘gretim ö˘grencilerinin rutin olmayan sözel problemleri çözme düzeylerinin çözüm stratejilerinin ve hata türlerinin incelenmesi. Çukurova Üniversitesi Sosyal Bilimler Enstitüsü Dergisi 15(2), 39–50 (2006) 2. M. Altun, ˙Ilkö˘gretimde problem çözme. Milli E˘gitim Dergisi. Sayı 147. Sayfa 27 (2000) ˙ gretim ikinci kademede matematik ö˘gretimi (Alfa Yayıncılık, Bursa, 2002) 3. M. Altun, Ilkö˘ 4. A. Baki, Kuramdan uygulamaya matematik e˘gitimi (Derya Kitabevi, Trabzon, 2006) 5. ˙I. Bayazit, Y. Aksoy, Matematiksel problemlerin ö˘grenimi ve ö˘gretimi. ˙Ilkö˘gretimde kar¸sıla¸sılan matematiksel zorluklar ve çözüm önerileri içinde (Pegem Akademi, Ankara, 2009), pp. 287– 312 6. ˙I. Bayazit, Y. Aksoy, Matematiksel zorluklar ve çözüm önerileri (Pegem Akademi, Ankara, 2010) 7. ˙I. Bayazit, N. Koçyi˘git, Üstün zekâlı ve normal zekâlı ö˘grencilerin rutin olmayan problemler konusundaki ba¸sarılarının kar¸sıla¸stırmalı olarak incelenmesi. Abant ˙Izzet Baysal Üniversitesi E˘gitim Fakültesi Dergisi 17(3), 1172–1200 (2017) 8. Y. Baykul, P. A¸skar, Problem ve problem çözme, matematik ö˘gretimi. Anadolu Üniversitesi Yayınları No: 193. Açık ö˘gretim Fakültesi Yayınları No: 94 (1987) 9. O. Chapman, In-service teacher development in mathematical problem solving. J. Math. Teacher Educ. 2(2), 121–142 (1999) 10. S. Gelbal, Problem çözme. Hacettepe Üniversitesi, E˘gitim Fakültesi Dergisi, Sayı 6, 167–173 (1991) 11. T. Gök, ˙I. Sılay, Problem çözme stratejilerinin ö˘grenilmesinde i¸sbirlikli ö˘grenme yönteminin etkileri. Mersin Üniversitesi E˘gitim Fakültesi Dergisi, 5, Sayı 1, Sayfa 58–76 (2009) 12. C. I¸sık, T. Kar, Pre-service elementary teachers’ problem posing skills. Mehmet Akif Ersoy Üniversitesi E˘gitim Fakültesi Dergisi 23, 190–214 (2012) 13. S. Kaya, Z. Kablan, The analysis of the studies on non-routine problems. Necatibey Fac. Educ. Electron. J. Sci. Math. Educ. 12(1), 25–44 (2018). ISSN: 1307-6086 14. Y.H. Leong, J. Dindyal, T.L. Toh, K.S. Quek, E.G. Tay, S.T. Lou, Teacher education for a problem-solving curriculum in Singapore. ZDM: Int. J. Math. Educ. 43(6–7), 819–831 (2011) 15. I. Marchis, Non-routine problems in primary mathematics workbooks from Romania. Acta Didactica Napocensia 5(3), 49–56 (2012) ˙ gretimde etkinlik temelli matematik ö˘gretimi (Anı Yayıncılık, Ertem 16. S. Olkun, Z. Toluk, Ilkö˘ Matbaacılık, Ankara, 2004) 17. G. Özsoy, Problem çözme ve üstbili¸s. Ulusal Sınıf Ö˘gretmenli˘gi Kongresi Bildirileri, Cilt-II, Ankara-Gazi Üniversitesi (Kök Yayıncılık, Ankara, 2006) 18. F. Özyıldırım Gümü¸s, Y. Sahiner, ¸ The effects of teaching problem solving strategies on preservice teachers’ views about problem solving. Elem. Educ. Online 14(1), 323–332 (2015)

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19. G. Polya, How to solve it? (Çev, Feryal Halatçı, New York, 1997) (1957) ˙ 20. Y. Soylu, C. Soylu, Matematik derslerinde ba¸sarıya giden yolda problem çözmenin rolü. Inönü Üniversitesi, E˘gitim Fakültesi Dergisi, Cilt 7, Sayı 11, Sayfa: 97–111 (2006) 21. S.K. Teong, J.G. Hedberg, K.F. Ho, L.T. Lioe, Y.S.J. Tiong, K.Y. Wong, Y.P. Fang, Developing the repertoire of heuristics for mathematical problem solving: project 1. Final Technical Report for Project CRP1/04 JH (Centre for Research in Pedagogy and Practice, National Institute of Education, Nanyang Technological University, Singapore, 2009) 22. T.L. Toh, Q.K. Seng, K.S. Guan, L.Y. Hoong, T.P. Choon, H.F. Him, J. Dindyal, Infusing problem solving into mathematics content course for pre-service secondary school mathematics teachers. Math. Educ. 15(1), 98–120 (2013) 23. J. Xie, J.O. Masingila, Examining interactions between problem posing and problem solving with prospective primary teachers: a case of using fractions. Educ. Stud. Math. 96, 101–118 (2017). https://doi.org/10.1007/s10649-017-9760-9