A Closer Look of Nonlinear Reaction-diffusion Equations 9781536182576, 9781536183566, 1536182575

Table of contents :
Contents
Preface
Acknowledgments
Chapter 1
Fundamentals of Nonlinear Reaction-Diffusion Equations
1.1. Nonlinear Reaction-Diffusion System
1.2. Application of Nonlinear Reaction-Diffusion System
1.2.1. Biological Sciences
1.2.2. Chemical Sciences
1.2.3. Medical Sciences
1.2.4. Physical Sciences
1.2.5. Engineering Sciences
1.3. Some Important Nonlinear Reaction-Diffusion Equations
1.4. Various Analytical Methods for Finding the Solution of Nonlinear Equations
References
Chapter 2
Mathematical Preliminaries and Various Methods of Solving Nonlinear Differential Equations
2.1. Overview
2.2. Various Methods of Solving Nonlinear Differential Equations
2.3. Analytical Methods with Basic Explanation and Examples
2.3.1. Homotopy Perturbation Method (HPM)
2.3.1.1. Basic Idea of Homotopy Perturbation Method
2.3.1.2. Example: Polymer-Modified Ultramicroelectrodes
2.3.2. Homotopy Analysis Method (HAM)
2.3.2.1. Basic Idea of Homotopy Analysis Method
2.3.2.2. Example: Steady-State Biofilters
2.3.3. Adomian Decomposition Method
2.3.3.1. Basic Concepts of the Adomian Decomposition Method (ADM)
2.3.3.2. Example: Substrate Inhibition Kinetics in an Immobilized Enzyme
2.3.4. Variational Iteration Method (VIM)
2.3.4.1. Basic Concepts in the Variational Iteration Method
2.3.4.2. Example: Electrochemical Immobilization of Enzymes
2.3.5. Exp-Function Method
2.3.5.1. Basic Concept of Exp-Function Method
2.3.5.2. Example: Mass Transport in Heterogeneous Catalysis
2.3.6. Hyperbolic Function Method
2.3.6.1. Basic Concept of Hyperbolic Function Method
2.3.6.2. Example: Glucose Oxidase Enzyme System
2.3.7. Variational Fractal Theory
2.3.7.1. Example: Troesch’s Problem
2.3.8. Taylor Series and Padé Approximation Method
2.3.8.1. Taylor Series
2.3.8.2. Padé Approximation
2.3.8.3. Example: Electroactive Polymers
2.3.8.4. Parameter-Expanding Methods
2.3.9. Parameterized Perturbation Method
References
Chapter 3
Steady and Non-Steady State Reaction-Diffusion Equations in Plane Sheet
3.1. Introduction
3.2. Concentration of Carbon Dioxide (CO2), and Phenyl Glycidyl Ether Solution (PGE) Using Taylor’s Series and Padé Approximation
3.3. Analytical Solution One Dimensional Nonlinear Parabolic Partial Differential Equation in Chemical Sciences
References
Chapter 4
Steady and Non-Steady State Nonlinear Reaction-Diffusion in a Cylinder
4.1. Introduction
4.2. Concentration of Methanol Using HPM Method
4.3. Analytical Solution of Various Nonlinear Boundary Value Problems in Cylindrical Coordinates
References
Chapter 5
Steady and Non-Steady Nonlinear Reaction-Diffusion in a Sphere
5.1. Introduction
5.2. Concentration of Spherical Catalytic Particle in Immobilized Enzyme System
5.3. Analytical Solution of Various Nonlinear Boundary Value Problems in Spherical Coordinates
References
Chapter 6
Nonlinear Convection-Diffusion Problems
6.1. Introduction
6.2. Electrochemical Convection Diffusion Phenomena at the Rotating Disk Electrode
6.3. Nonlinear Convection Diffusion Equations and Corresponding Analytical Solutions in Various Fields of Chemical Sciences
References
Chapter 7
Numerical Methods
7.1. Introduction
7.2. Advantage of Numerical Methods
7.3. Various Numerical Methods
7.4. New Computational Methods Using Software
7.5. Analytical and Numerical (Matlab) Solutions of the Coupled Reaction and Diffusion Equations within Polymer-Modified Ultramicroelectrodes
7.6. Numerical Simulation
References
About the Authors
Index
Blank Page

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THEORETICAL AND APPLIED MATHEMATICS

A CLOSER LOOK OF NONLINEAR REACTIONDIFFUSION EQUATIONS

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THEORETICAL AND APPLIED MATHEMATICS Additional books and e-books in this series can be found on Nova’s website under the Series tab.

THEORETICAL AND APPLIED MATHEMATICS

A CLOSER LOOK OF NONLINEAR REACTIONDIFFUSION EQUATIONS

L. RAJENDRAN, R. SWAMINATHAN AND

M. CHITRA DEVI

Copyright © 2020 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470

E-mail: [email protected].

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data Names: Rajendran, Lakshmanan, author. | Swaminathan, R., 1965- author. | Chitra Devi, M., author. Title: A closer look of nonlinear reaction-diffusion equations / L. Rajendran, R. Swaminathan, M. Chitra Devi. Description: New York : Nova Science Publishers, [2020] | Series: Theoretical and applied mathematics | Includes bibliographical references and index. | Identifiers: LCCN 2020030189 (print) | LCCN 2020030190 (ebook) | ISBN 9781536182576 (hardcover) | ISBN 9781536183566 (adobe pdf) Subjects: LCSH: Reaction-diffusion equations. | Reaction-diffusion equations--Numerical solutions. Classification: LCC QA377 .R354 2020 (print) | LCC QA377 (ebook) | DDC 515/.353--dc23 LC record available at https://lccn.loc.gov/2020030189 LC ebook record available at https://lccn.loc.gov/2020030190

Published by Nova Science Publishers, Inc. † New York

Dedicated with love and regards to my family members for their sacrifices

L. Rajendran

CONTENTS Preface

ix

Acknowledgments

xi

Chapter 1 Chapter 2

Chapter 3 Chapter 4

Fundamentals of Nonlinear Reaction-Diffusion Equations

1

Mathematical Preliminaries and Various Methods of Solving Nonlinear Differential Equations

15

Steady and Non-Steady State Reaction-Diffusion Equations in Plane Sheet

63

Steady and Non-Steady State Nonlinear Reaction-Diffusion in a Cylinder

101

Steady and Non-Steady Nonlinear Reaction-Diffusion in a Sphere

123

Chapter 6

Nonlinear Convection-Diffusion Problems

151

Chapter 7

Numerical Methods

169

Chapter 5

viii

Contents

About the Authors

189

Index

193

PREFACE By using mathematical models to describe the physical, biological or chemical phenomena, one of the most common results is either a differential equation or a system of differential equations, together with the correct boundary and initial conditions. The determination and interpretation of their solution are at the base of applied mathematics. Hence the analytical and numerical study of the differential equation is very much essential for all theoretical and experimental researchers, and this book helps to develop skills in this area. Recently nonlinear differential equations were widely used to model many of the interesting and relevant phenomena found in many fields of science and technology on a mathematical basis. This problem is to inspire them in various fields such as economics, medical biology, plasma physics, particle physics, differential geometry, engineering, signal processing, electrochemistry and materials science. This book contains seven chapters and practical applications to the problems of the real world. The first chapter is specifically for those with limited mathematical background. Chapter 1 presents the introduction of nonlinear reaction-diffusion systems, various boundary conditions and examples. Real-life application of nonlinear reactiondiffusion in different fields with some important nonlinear equations is

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L. Rajendran, R. Swaminathan and M. Chitra Devi

also discussed. In Chapter 2, mathematical preliminaries and various advanced methods of solving nonlinear differential equations such as Homotopy perturbation method, variational iteration method, exponential function method etc. are described with examples. Steady and non-steady state reaction-diffusion equations in the plane sheet (Chapter 3), cylinder (Chapter 4) and spherical (Chapter 5) are analyzed. The analytical results published by various researchers in referred journals during 2007-2020 have been addressed in these Chapters 4 to 6, and this leads to conclusions and recommendations on what approaches to use on nonlinear reaction-diffusion equations. Convection-diffusion problems arise very often in applied sciences and engineering. Nonlinear convection-diffusion equations and corresponding analytical solutions in various fields of chemical sciences are discussed in Chapter 6. Numerical methods are used to provide approximate results for the nonlinear problems, and their importance is felt when it is impossible or difficult to solve a given problem analytically. Chapter 7 identifies some of the numerical methods for finding solutions to nonlinear differential equations.

ACKNOWLEDGMENTS I am very grateful to the reviewers for their valuable comments and suggestions. I take this opportunity to express my sincere gratitude to Shri J. Ramachandran, Chancellor, for his constant support, encouragement, and the excellent academic and research atmosphere provided. I wish to thank Col. Dr G. Thiruvasagam, Vice-Chancellor, AMET, University, for his various plans to develop the research activities of hardworking researchers to set new goals in interdisciplinary areas. His sound managerial principles, coupled with his future vision, have been of great help to the mathematics research community. I express my sincere thanks to Dr. M. Jayaprakashvel, Registrar, Academy of Maritime Education and Training (AMET), deemed to be University, for his constant support and encouragement. The author is also thankful to Dr. D. Rajasekar, Professor, Business School, AMET University, for the motivation. Special thanks to Marwan Abukhaled, Professor, Department of Mathematics and Statistics, American University of Sharjah, Sharjah, UAE, and Dr. Bosco Emmanuel, Scientist, CSIR CECRI, Karaikudi for his valuable suggestions. It is not out of place to acknowledge the efforts of my PhD scholars who worked with me on my CSIR, UGC and DST-SERB research

xii

L. Rajendran, R. Swaminathan and M. Chitra Devi

projects. The research work related to this field has greatly inspired me to write this book. I also express my sincere gratitude to our research scholars J. Visuvasam, R. Saravanakumar, K. Saranya, D. Gowthaman, R. Joy Salomi, R. Usha Rani, S. Vinolyn Sylvia, B. Manimegalai for their kind cooperation during the preparation of the book. I have received considerable assistance from my colleagues Dr P. Balaganesan, Dr I. Paulraj Jayasimman, in the Department of Mathematics, AMET University. I express my thanks to Dr. P. Pirabaharan, Assistant Professor, Department of Mathematics, Anna University, University College of Engineering, Dindigul. Moreover, I am especially grateful to the team of NOVA publishers for cooperation in all aspects of the production of the book. I thank my parents for their blessings. Last but not least, I thank my wife, SP. Ananthi and my daughters R. Revathy and R. Priyanka for their patience and support. I am looking forward to receiving comments and suggestions on this work from students, teachers, and researchers.

Dr. L. Rajendra

Chapter 1

FUNDAMENTALS OF NONLINEAR REACTIONDIFFUSION EQUATIONS 1.1. NONLINEAR REACTION-DIFFUSION SYSTEM Reaction-diffusion processes are the mathematical models which correspond to different physical phenomena in the applied sciences. Space and time can not be considered as independent variables in this equation. An equation of reaction-diffusion contains a term of reaction and a term of diffusion. In the system, the substances are converted into one another through chemical reactions and diffusion that allows the materials to spread across a space surface. Models of reaction-diffusion are widely used in the fields of chemistry, physics and engineering sciences. However, the model can also represent non-chemical, dynamic processes. Mathematically, the reaction-diffusion systems take the form of semi-parabolic partial differential equations. They may be described as follows in the general form (Rajendran, Kirthiga, and Laborda 2017): ∂𝑆(𝑥,𝑡) ∂𝑡

= 𝐷𝑆 ∇2 𝑆(𝑥, 𝑡) − 𝑓(𝑆, 𝑃)

(1.1)

2

L. Rajendran, R. Swaminathan and M. Chitra Devi ∂𝑃(𝑥,𝑡) ∂𝑡

= 𝐷𝑃 ∇2 𝑃(𝑥, 𝑡) + 𝑔(𝑆, 𝑃)

(1.2)

where S(x, t) (substrate) and P(x,t) (product) represents the unknown vector function, 𝐷 is diffusion coefficients, and 𝑓(𝑆, 𝑃) and 𝑔(𝑆, 𝑃) accounts for all local reactions. This is a linear or nonlinear function. This function depends upon the type and nature/order of the reactions. The first term on the right hand is called a diffusion term, and the second term is called a reaction term. ∇2 is the Laplace operator. ∇2 𝑆 = ∇2 𝑆 =

∂2 𝑆 ∂𝑥 2 ∂2 𝑆 ∂𝑥 2

(𝑝𝑙𝑎𝑛𝑎𝑟) +

∂2 𝑆

1 ∂𝑆 𝑥 ∂𝑥

(1.3)

(𝑐𝑦𝑙𝑖𝑛𝑑𝑟𝑖𝑐𝑎𝑙)

2 ∂𝑆

∇2 𝑆 = ∂𝑥 2 + 𝑥 ∂𝑥 (𝑆𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙)

(1.4)

(1.5)

Table 1.1. Various boundary conditions Boundary conditions Dirichlet Neumann

Examples 𝑆(𝑥 = 𝑙, 𝑡) = 𝑚, 𝑃(𝑥 = 𝛼, 𝑡) = 𝛽, ∀𝑥 ∈ 𝛺 𝑑𝑆 𝑑𝑛

Robin Mixed

= 𝑓(𝑥),

𝑎 𝑆(𝑥) + 𝑏

𝑑𝑃

= 𝑔(𝑥), ∀𝑥 ∈ 𝛺

𝑑𝑛 𝑑𝑆(𝑥) 𝑑𝑛

𝑆(𝑥 = 𝑙, 𝑡) = 𝑚, ∀𝑥 ∈ 𝛺 and 𝑎 𝑆(𝑥) + 𝑏

Cauchy

= 0, ∀𝑥 ∈ 𝜕𝛺 𝑑𝑆 𝑑𝑛

= 0, ∀𝑥 ∈ 𝜕𝛺

𝑆(𝑥 = 𝑙, 𝑡) = 𝑚, ∀𝑥 ∈ 𝛺 and

𝑑𝑆 𝑑𝑛

= 𝑔(𝑥), ∀𝑥 ∈ 𝛺

Eqn. No (1.6) (1.7) (1.8) (1.9) (1.10) (1.11) (1.12)

The variable x is often confined to a domain 𝛺 with boundary 𝜕𝛺, and then solutions are sought, which satisfy specific initial and boundary conditions on 𝜕𝛺. Boundary conditions are those which depends on space while the initial conditiondepends on time. Generally, the boundary conditions have the following types. There are five types

Fundamentals of Non-Linear Reaction-Diffusion Equations

3

of boundary conditions: Dirichlet, Neumann, Robin, Mixed, and Cauchy, within which Dirichlet and Neumann are predominant. In the Table 1.1, 𝑙, 𝑚, 𝛼, 𝛽, 𝑓, 𝑔 are known as scalar function and 𝑑𝑆/𝑑𝑛 is directional normal derivative. S and P are unknown functions.

1.2. APPLICATION OF NONLINEAR REACTION-DIFFUSION SYSTEM Nonlinear reaction-diffusion has been the most powerful tool for mathematical modeling of various processes in biology, geology, and physics (neutron diffusion theory) medicine (physiology, diseases, etc.), genetics, social science, finance, economics, weather prediction, astrophysics, and ecology. Pioneering models have been developed to describe some of the classical laws in physics and astronomy.

1.2.1. Biological Sciences Nonlinear differential equations describe the different biological processes (James D Murray 2002; J D Murray 2003) such as population dynamics, gene propagation, environmental invasions, predator-prey equations, and dynamics of competition. In biological sciences, the notorious nonlinear equation is the Burgers equation and the Fisher equation.

1.2.2. Chemical Sciences Nonlinear reaction-diffusion processes is applied in chemical sciences such as organic and organometallic processes, homogeneous mediated enzyme-catalyzed reactions, multi-layered enzyme-modified

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L. Rajendran, R. Swaminathan and M. Chitra Devi

electrodes and nanostructured porous film, polymer-modified ultramicroelectrodes, biosensors, biofuel cells, and bioreactors. Nonlinear reaction–diffusion systems are naturally implemented in physical and electrochemistry (Rajendran, Kirthiga, and Laborda 2017; Laborda, Henstridge, and Compton 2012; Britz 1988; Ángela Molina and Joaquín González 2016; Bieniasz 2017). This equation is obtained by combining Fick’s diffusion law with the law on chemical reaction rate. Naturally, reaction-diffusion equations occur in systems which consist of many interacting elements.

1.2.3. Medical Sciences Mathematical models are also used to describe biomedical processes. All the models are based on nonlinear PDEs, and usually, they are not linearizable. Nonlinear differential equations occur in different pharmaceutical science fields such as infectious diseases, cancer growth and blood clotting, distribution of heat sources in a human body, nerve pulse transmission, and neurosciences. An epidemic is a sudden spread of the disease in a given population to a large number of people within a limited period. Example: cholera, SARS, malaria, plague, and corona. COVID-19 is a new model (not a SIR, SEIR, and other models) in 2020. This system is based on the nonlinear rate equations of susceptible, uncovered, curable affected, genital infections infected and removed people including recovered and death people containing nonlinear terms related to the transmission period, infectious period, transmission rate, birth, and death rate, the lifetime of viruses, etc.(“Coronavirus @ Www.Who.Int,” n.d.; Zhou et al. 2020; Li et al. 2020; Huang et al. 2020; J. T. Wu, Leung, and Leung 2020).

Fundamentals of Non-Linear Reaction-Diffusion Equations

5

1.2.4. Physical Sciences Nonlinear differential equations occur in several fields of physical sciences for instance in particle physics, plasma physics, charged nanoparticles, image processes, fluid mechanics, soliton physics, quantum field theory (Leung 2009). The relevant examples are the Ricatti equation and Poisson-Boltzmann equations.

1.2.5. Engineering Sciences In many applied sciences, such as heat transfer/soil consolidation, elasticity, quantum mechanics, water flow, thermodynamics, concrete carbonation, classical and fluid mechanics, solid mechanics, and others, the problems are overseened by nonlinear partial differential equations (Zeng et al. 2014). Euler-Bernoulli beam equation is one of the essential examples of linear equations in engineering sciences. The coupled nonlinear equations of motion in heave, roll, and pitch are based on physical grounds in ship dynamics. The materials for design and implementation are composed of three components. These are the friction forces and moments, the regeneration of forces and moments and the damping of forces and moments(Skjetne, Smogeli, and Fossen 2004; Chan, Xu, and Huang 1995; Pedišić Buča and Senjanović 2006).

1.3. SOME IMPORTANT NONLINEAR REACTION-DIFFUSION EQUATIONS Nonlinear reaction-diffusion equations are one of the most common differential/partial differential equations used to model physical phenomena. Nonlinear equations describe many of the major issues in solid-state physics, optics, plasma physics, fluid mechanics, population

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L. Rajendran, R. Swaminathan and M. Chitra Devi

trends, and many others that occur in mathematical biology, engineering sciences, and other technical fields. In Table 1.2, some of the important nonlinear equations with various fields are given. Table 1.2. Examples of some important nonlinear equations Field

Equation/system of equations with their names and Reference

Diffusive wave in fluid dynamics

Burgers equation (Debnath 2012) 𝜕𝑢(𝑥,𝑡) 𝜕𝑡

Travelling waves

+ 𝑢(𝑥, 𝑡) =

𝜕𝑡

𝑑𝑡 2 𝜕𝑠

𝑑𝑢 𝑑𝑡

𝑉𝑚 𝑠

𝜕𝑥

𝑠+𝑘𝑚

− 2

𝜕2 𝑝 𝜕𝑥 2

−

= 𝐷𝑅

𝜕𝑡 𝜕𝐶𝑠 (𝑥,𝑡) 𝜕𝑡

= 𝐷𝑆

𝑑2 𝑐𝑚

𝐷𝑆

𝑑𝑥 2

𝑑2 𝑐𝑆 𝑑𝑥 2

−

−

(1.16)

𝑉𝑚 𝑠 𝑠+𝑘𝑚

𝜕2 𝐶𝑅 (𝑥,𝑡) 𝜕𝑥 2

𝜕2 𝐶𝑠 (𝑥,𝑡) 𝜕𝑥 2

+

−

𝑘𝑐𝑎𝑡𝐶𝐸 𝐾𝑀𝑆 𝐾 + 𝑀𝑀 +1 𝐶𝑠 (𝑥,𝑡) 𝐶𝑜(𝑥,𝑡)

−1 𝑐 𝑐 𝑐 𝑘𝑐𝑎𝑡 𝐾𝑚 𝑒 𝑚 𝑆 −1 (𝑐 +𝐾 )+𝑐 𝑐𝑚 𝐾𝑚 𝑆 𝑆 𝑆

=

−1 𝑐 𝑐 𝑐 𝑘𝑐𝑎𝑡𝐾𝑚 𝑒 𝑚 𝑆 −1 (𝑐 +𝐾 )+𝑐 ) 4(𝑐𝑚 𝐾𝑚 𝑆 𝑆 𝑆

(1.18)

𝜕𝑐𝑚

=

𝜕𝑡 𝜕𝑐𝑠 𝜕𝑡

Ship dynamics (Pedišić Buča and Senjanović 2006) ••

(1.17)

2 𝑘𝑐𝑎𝑡 𝐶𝐸 𝐾𝑀𝑆 𝐾 + 𝑀𝑀 +1 𝐶𝑠 (𝑥,𝑡) 𝐶𝑜(𝑥,𝑡)

Voltammetry (Saravanakumar, Ganesan, and Rajendran 2015) 𝐷𝑚

(1.15)

+ 𝑢(𝑥, 𝑡) = 0

Electrochemical mediated enzyme reactions (Kirthiga, Rajendran, and Fernandez 2018) 𝜕𝐶𝑅 (𝑥,𝑡)

(1. 19)

•

(𝐼𝑋𝑋 + 𝐽𝑋𝑋 )𝜃 + 𝐷(𝜃 ) + 𝑅(𝜃) = 𝑀(𝑡) where •

Electrostatic potential

𝜕𝑥 2

(1.14)

+ 𝑢(𝑥, 𝑡) − (𝑢(𝑥, 𝑡))2

𝜕2 𝑠

= 𝐷𝑝

𝜕𝑡

Roll motion of ships

𝜕2 𝑢(𝑥,𝑡)

− 𝜇(1 − 𝑢2 (𝑡))

= 𝐷𝑠

𝜕𝑝

Biofuel Cell

𝜕𝑥 2

Enzyme reaction (Meena and Rajendran 2010) 𝜕𝑡

Biosensor

𝜕2 𝑢(𝑥,𝑡)

Van der Pol’s equation (Wang et al. 2014) 𝑑2 𝑢(𝑡)

Amperometric biosensor

=𝑣

𝜕𝑥

Fisher–Kolmogorov equation (Rottschäfer and Wayne 2001) 𝜕𝑢(𝑥,𝑡)

Pattern dynamics

𝜕𝑢(𝑥,𝑡)

Eqn. No. (1.13)

•

•

𝑑(𝜃 ) = 𝑑1 𝜃 + 𝑓1 (𝜃 ), 𝑟(𝜃) = 𝑐1 𝜃 + 𝑐3 𝜃 3 + 𝑐5 𝜃 5 Poisson–Boltzmann equation (Duval, Town, and Van Leeuwen 2018) (1. 20) ∇2 𝜙 = 4𝜋𝑒[𝑛𝑛0 exp(𝑒𝜙/𝑘𝐵 𝑇) − 𝑛𝑝0 exp(−𝑒𝜙/ 𝑘𝐵 𝑇)]

Fundamentals of Non-Linear Reaction-Diffusion Equations

7

1.4. VARIOUS ANALYTICAL METHODS FOR FINDING THE SOLUTION OF NONLINEAR EQUATIONS The well-known linear superposition principle cannot be applied to generate new, exact solutions for nonlinear PDEs. This implies that classical methods (the Fourier method, the Laplace transformation methods, and the Green function, etc.) are not helping to solve these PDEs. Thus, constructing unique exact solutions for these equations is a non-trivial problem. But it is of fundamental importance to find accurate or approximate solutions which have a physical, chemical or biological interpretation. Recently Homotopy perturbation method (Duval, Town, and Van Leeuwen 2018; Rajendran, Kirthiga, and Laborda 2017; He 1999a; Y. Wu and He 2018; Meena and Rajendran 2010), homotopy analysis method (Ganji and Sadighi 2006; Liao 1992; 2009; Liao, and Sherif, 2004), Adomian decomposition method (Adomian 1994; Sivasankari and Rajendran 2013; Praveen, Valencia, and Rajendran 2014), Variational iteration method (He and Latifizadeh 2020; He 1999b; 2000; Rahamathunissa and Rajendran 2008), Taylor’s series method, Akbar-Ganji method (Akbari et al. 2014; Dharmalingam and Veeramuni 2019), wavelet method (Hariharan 2019) and residual method (Saunders 1985; Saranya, Mohan, and Rajendran 2020), etc. has been used to find the approximate analytical solution of nonlinear problems. The basic concept of all these methods with examples is described in the next chapter.

REFERENCES Adomian, George. 1994. “Solving Frontier Problems of Physics: The Decomposition Method.” Solving Frontier Problems of Physics: The Decomposition Method. https://doi.org/10.1007/978-94-0158289-6.

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Akbari, M. R., D. D. Ganji, A. Majidian, and A. R. Ahmadi. 2014. “Solving Nonlinear Differential Equations of Vanderpol, Rayleigh and Duffing by AGM.” Frontiers of Mechanical Engineering 9 (2): 177–90. https://doi.org/10.1007/s11465-014-0288-8. Ángela Molina, and Joaquín González. 2016. Pulse Voltammetry in Physical Electrochemistry and Electroanalysis: Theory and Application. https://doi.org/10.1007/978-3-319-21251-7. Bieniasz, L. K. 2017. “An Adaptive Huber Method for Nonlinear Systems of Volterra Integral Equations with Weakly Singular Kernels and Solutions.” Journal of Computational and Applied Mathematics 323: 136–46. https://doi.org/10.1016/j.cam.2017. 04.018. Britz, Dieter. 1988. Digital Simulation in Electrochemistry. Digital Simulation in Electrochemistry. https://doi.org/10.1007/978-3-66202549-9. Chan, H. S. Y., Z. Xu, and W. L. Huang. 1995. “Estimation of Nonlinear Damping Coefficients from Large-Amplitude Ship Rolling Motions.” Applied Ocean Research 17 (4): 217–24. https://doi.org/10.1016/0141-1187(95)00024-0. “Coronavirus @ Www.Who.Int.” n.d. Who. https://www.who.int/ health-topics/coronavirus. Debnath, Lokenath. 2012. Nonlinear Partial Differential Equations for Scientists and Engineers. Nonlinear Partial Differential Equations for Scientists and Engineers. https://doi.org/10.1007/978-0-81768265-1. Dharmalingam, K. M., and M. Veeramuni. 2019. “Akbari-Ganji’s Method (AGM) for Solving Non-Linear Reaction - Diffusion Equation in the Electroactive Polymer Film.” Journal of Electroanalytical Chemistry 844: 1–5. https://doi.org/10. 1016/j.jelechem.2019.04.061. Duval, Jérôme F. L., Raewyn M. Town, and Herman P. Van Leeuwen. 2018. “Poisson-Boltzmann Electrostatics and Ionic Partition Equilibration of Charged Nanoparticles in Aqueous Media.”

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Journal of Physical Chemistry C 122 (30): 17328–37. https://doi.org/10.1021/acs.jpcc.8b05168. Ganji, D. D., and A. Sadighi. 2006. “Application of He’s HomotopyPerturbation Method to Nonlinear Coupled Systems of ReactionDiffusion Equations.” International Journal of Nonlinear Sciences and Numerical Simulation 7 (4): 411–18. https://doi.org/10. 1515/IJNSNS.2006.7.4.411. Hariharan, G. 2019. Wavelet Solutions for Reaction–Diffusion Problems in Science and Engineering. Forum for Interdisciplinary Mathematics. Singapore: Springer Singapore. https://doi.org/10. 1007/978-981-32-9960-3. He, Ji Huan. 1999a. “Homotopy Perturbation Technique.” Computer Methods in Applied Mechanics and Engineering 178 (3–4): 257–62. https://doi.org/10.1016/S0045-7825(99)00018-3. He, Ji Huan. 1999b. “Variational Iteration Method - A Kind of NonLinear Analytical Technique: Some Examples.” International Journal of Non-Linear Mechanics 34 (4): 699–708. https://doi. org/10.1016/s0020-7462(98)00048-1. He, Ji Huan. 2000. “Variational Iteration Method for Autonomous Ordinary Differential Systems.” Applied Mathematics and Computation 114 (2–3): 115–23. https://doi.org/10.1016/S00963003(99)00104-6. He, Ji Huan, and Habibolla Latifizadeh. 2020. “A General Numerical Algorithm for Nonlinear Differential Equations by the Variational Iteration Method.” International Journal of Numerical Methods for Heat and Fluid Flow. https://doi.org/10.1108/HFF-01-2020-0029. Huang, Chaolin, Yeming Wang, Xingwang Li, Lili Ren, Jianping Zhao, Yi Hu, Li Zhang, et al. 2020. “Clinical Features of Patients Infected with 2019 Novel Coronavirus in Wuhan, China.” The Lancet 395 (10223): 497–506. https://doi.org/10.1016/S0140-6736(20)30183-5. Kirthiga, O. M., L. Rajendran, and Carlos Fernandez. 2018. “Kinetic Mechanism for Modelling of Electrochemical Mediatedenzyme Reactions and Determination of Enzyme Kinetics Parameters.”

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Russian Journal of Electrochemistry 54 (11): 783–95. https://doi. org/10.1134/S1023193518110034. Laborda, Eduardo, Martin C. Henstridge, and Richard G. Compton. 2012. “Giving Physical Insight into the Butler-Volmer Model of Electrode Kinetics: Part 2 - Nonlinear Solvation Effects on the Voltammetry of Heterogeneous Electron Transfer Processes.” Journal of Electroanalytical Chemistry 681: 96–102. https://doi.org/10.1016/j.jelechem.2012.06.008. Leung, Anthony W. 2009. Nonlinear Systems of Partial Differential Equations: Applications to Life and Physical Sciences. Nonlinear Systems of Partial Differential Equations: Applications to Life and Physical Sciences. https://doi.org/10.1142/7353. Li, Qun, Xuhua Guan, Peng Wu, Xiaoye Wang, Lei Zhou, Yeqing Tong, Ruiqi Ren, et al. 2020. “Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus-Infected Pneumonia.” New England Journal of Medicine 382 (13): 1199–1207. https://doi. org/10.1056/NEJMoa2001316. Liao, Shijun, and SA Sherif,. 2004. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Applied Mechanics Reviews. Vol. 57. https://doi.org/10.1115/1.1818689. Liao, Shijun. 1992. “Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problem,” Ph.D. Thesis. Liao, Shijun. 2009. “Notes on the Homotopy Analysis Method: Some Definitions and Theorems.” Communications in Nonlinear Science and Numerical Simulation 14 (4): 983–97. https://doi.org/10. 1016/j.cnsns.2008.04.013. Meena, A., and L. Rajendran. 2010. “Mathematical Modeling of Amperometric and Potentiometric Biosensors and System of NonLinear Equations - Homotopy Perturbation Approach.” Journal of Electroanalytical Chemistry 644 (1): 50–59. https://doi.org/10. 1016/j.jelechem.2010.03.027. Murray, J D. 2003. Mathematical Biology Biomedical Applications. Edited by J. D. Murray. Vol. 18. Interdisciplinary Applied

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Mathematics. New York, NY: Springer New York. https://doi.org/10.1007/b98869. Murray, James D. 2002. “Mathematical Biology I. An Introduction.” Interdisciplinary Applied Mathematics 17: 551. https://doi.org/ 10.1007/b98868. Pedišić Buča, Marta, and Ivo Senjanović. 2006. “Nonlinear Ship Rolling and Capsizing.” Brodogradnja 57 (4): 321–31. Praveen, T., Pedro Valencia, and L. Rajendran. 2014. “Theoretical Analysis of Intrinsic Reaction Kinetics and the Behavior of Immobilized Enzymes System for Steady-State Conditions.” Biochemical Engineering Journal 91: 129–39. https://doi.org/10. 1016/j.bej.2014.08.001. Rahamathunissa, G., and L. Rajendran. 2008. “Application of He’s Variational Iteration Method in Nonlinear Boundary Value Problems in Enzyme- Substrate Reaction Diffusion Processes: Part 1. The Steady-State Amperometric Response.” Journal of Mathematical Chemistry 44 (3): 849–61. https://doi.org/10.1007/ s10910-007-9340-9. Rajendran, L., M. Kirthiga, and E. Laborda. 2017. “Mathematical Modeling of Nonlinear Reaction–Diffusion Processes in Enzymatic Biofuel Cells.” Current Opinion in Electrochemistry 1 (1): 121–32. https://doi.org/10.1016/j.coelec.2016.11.003. Rottschäfer, V., and C. E. Wayne. 2001. “Existence and Stability of Traveling Fronts in the Extended Fisher-Kolmogorov Equation.” Journal of Differential Equations 176 (2): 532–60. https://doi.org/ 10.1006/jdeq.2000.3984. Saranya, K., V. Mohan, and L. Rajendran. 2020. “Steady-State Concentrations of Carbon Dioxide Absorbed into Phenyl Glycidyl Ether Solutions by Residual Method.” Journal of Mathematical Chemistry. https://doi.org/10.1007/s10910-020-01127-0. Saravanakumar, K., Sp Ganesan, and L. Rajendran. 2015. “Theoretical Analysis of Reaction and Diffusion Processes in a Biofuel Cell

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Electrode.” Fuel Cells 15 (3): 523–36. https://doi.org/10.1002/ fuce.201500025. Saunders, R. 1985. “Note on a Proposed Weighted Residual Method for Solving Nonlinear Differential Boundary Problems.” Applied Mathematical Modelling 9 (5): 385–86. https://doi.org/10.1016/ 0307-904X(85)90029-0. Sivasankari, M. K., and L. Rajendran. 2013. “Analytical Expression of the Concentration of Species and Effectiveness Factors in Porous Catalysts Using the Adomian Decomposition Method.” Kinetics and Catalysis 54 (1): 95–105. https://doi.org/10.1134/ S0023158413010138. Skjetne, Roger, Øyvind N. Smogeli, and Thor I. Fossen. 2004. “A Nonlinear Ship Manoeuvering Model: Identification and Adaptive Control with Experiments for a Model Ship.” Modeling, Identification and Control 25 (1): 3–27. https://doi.org/10.4173/ mic.2004.1.1. Wang, Kaier, Moira L. Steyn-Ross, D. A. Steyn-Ross, Marcus T. Wilson, Jamie W. Sleigh, and Yoichi Shiraishi. 2014. “Simulations of Pattern Dynamics for Reaction-Diffusion Systems via SIMULINK.” BMC Systems Biology 8 (1). https://doi.org/10.1186/ 1752-0509-8-45. Wu, Joseph T., Kathy Leung, and Gabriel M. Leung. 2020. “Nowcasting and Forecasting the Potential Domestic and International Spread of the 2019-NCoV Outbreak Originating in Wuhan, China: A Modelling Study.” The Lancet 395 (10225): 689– 97. https://doi.org/10.1016/S0140-6736(20)30260-9. Wu, Yue, and Ji Huan He. 2018. “Homotopy Perturbation Method for Nonlinear Oscillators with Coordinate-Dependent Mass.” Results in Physics 10: 270–71. https://doi.org/10.1016/j.rinp.2018.06.015. Zeng, Fanhai, Fawang Liu, Changpin Li, Kevin Burrage, Ian Turner, and V. Anh. 2014. “A Crank-Nicolson Adi Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-

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Diffusion Equation.” SIAM Journal on Numerical Analysis 52 (6): 2599–2622. https://doi.org/10.1137/130934192. Zhou, Peng, Xing Lou Yang, Xian Guang Wang, Ben Hu, Lei Zhang, Wei Zhang, Hao Rui Si, et al. 2020. “A Pneumonia Outbreak Associated with a New Coronavirus of Probable Bat Origin.” Nature 579 (7798): 270–73. https://doi.org/10.1038/s41586-0202012-7.

Chapter 2

MATHEMATICAL PRELIMINARIES AND VARIOUS METHODS OF SOLVING NONLINEAR DIFFERENTIAL EQUATIONS 2.1. OVERVIEW The nonlinear differential equations can describe most of the physical processes that arise in mathematical physics/chemistry and engineering. The problems occur in various fields of applied mathematics, physics, and engineering, including particle physics, astrophysics, condensed-matter physics, fluid dynamics, optics, solid mechanics, plasma physics quantum field theory and electrochemistry, etc. (Debnath 2012; Wazwaz 2009; Miller and Ross 1993; Saha Ray 2015; Kilbas, Srivastava, and Trujillo 2006; Gorenflo and Mainardi 1997). Some assumptions in the nonlinear problems deviate from the real solutions; thus, they may vary from the actual physical behavior. By overcoming those limitations, physically accurate or approximate solutions can be obtained to know the normal behavior of physical systems.

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L. Rajendran, R. Swaminathan and M. Chitra Devi

2.2. VARIOUS METHODS OF SOLVING NONLINEAR DIFFERENTIAL EQUATIONS The approximate solutions for strongly nonlinear differential equations depend upon (i) one dimensional or multidimensional, (ii) steady-state or non-steady-state, (iii) migration effects, (iv) system of ODE/PDEs coupled by the homogenous kinetics/ heterogeneous kinetics terms and (v) hydrodynamics related to fast/slow diffusion. Over the last few decades, different methods have been reported to obtain the analytical solutions such as extended tanh (El-Wakil and Abdou 2007), Jacobi-elliptic function expansion (Inc and Ergüt 2005), F-expansion, and the First integral (Z. Feng 2002), Tanh-sech (Malfliet 1992) and sine-cosine method (Mitchell 1980). Some of the powerful analytical methods which are applied to solve the nonlinear problems recently are listed below. In the next section, the basic concept of the following method is discussed. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Homotopy Perturbation Method (HPM) Homotopy Analysis Method (HAM) Adomian Decomposition Method (ADM) Variational Iteration Method (VIM) Exp-Function Method Hyperbolic Function Method Variational Fractal Theory Taylor Series and Padé Approximation Method Parameter-Expanding Methods Parameterized Perturbation Method, etc.

Mathematical Preliminaries and Various Methods …

17

2.3. ANALYTICAL METHODS WITH BASIC EXPLANATION AND EXAMPLES In this section basic concept of the analytical methods mentioned above are explained with simple examples.

2.3.1. Homotopy Perturbation Method (HPM) The homotopy perturbation method is an effective and useful method for finding solutions to nonlinear equations. This method was first introduced by He in 1998 (J. H. He 1999; 2000b).

2.3.1.1. Basic Idea of Homotopy Perturbation Method Let’s consider the following function to explain the method 𝐴(𝑤) − 𝑓(𝑟) = 0, 𝑟 ∈ Ω

(2.1)

The given boundary conditions is 𝜕𝑤

𝐵 (𝑤, 𝜕𝑛 ) = 0, 𝑟 ∈ Γ

(2.2)

where 𝐴 and 𝐵 are a general differential and a boundary operator. 𝑓 is a given analytical function and 𝛺 is the boundary of the domain, The operator 𝐴 can be split into a linear part 𝐿 and a nonlinear part 𝑁. Eqn. (2.1) can therefore, be written as 𝐿(𝑤) + 𝑁(𝑤) − 𝑓(𝑟) = 0

(2.3)

We construct a homotopy 𝑧(𝑟, 𝑝): 𝛺 × [0,1] → 𝑅 using the homotopy technique as follows:

18

L. Rajendran, R. Swaminathan and M. Chitra Devi 𝐻(𝑧, 𝑝) = (1 − 𝑝)[𝐿(𝑧) − 𝐿(𝑤0 )] + 𝑝[𝐴(𝑧) − 𝑓(𝑟)] = 0

where 𝑝 ∈ [0,1], 𝑟 ∈ 𝛺

(2.4)

or 𝐻(𝑧, 𝑝) = 𝐿(𝑧) − 𝐿(𝑤0 ) + 𝑝𝐿(𝑤0 ) + 𝑝[𝑁(𝑧) − 𝑓(𝑟)] = 0

(2.5)

where 𝑝 ∈ [0,1] is an embedding parameter, and 𝑤0 is an initial approximation of Eqn. (2.1), which satisfies the boundary conditions. Obviously, from Eqns. (2.4) and (2.5), we will have 𝐻(𝑧, 0) = 𝐿(𝑧) − 𝐿(𝑤0 ) = 0

(2.6)

𝐻(𝑧, 1) = 𝐴(𝑧) − 𝑓(𝑟) = 0

(2.7)

The changing process of p from zero to unity is 𝑧(𝑟, 𝑝) from 𝑤0 to 𝑤(𝑟). We can first use the embedding parameter p as a small parameter according to the HPM. Assume that the solutions Eqn. (2.4) or (2.5) can be presented as a power series in 𝑝: 𝑧 = 𝑧0 + 𝑝𝑧1 + 𝑝2 𝑧2 + ⋯

(2.8)

Setting 𝑝 = 1 results in the approximate solution of Eqn. (2.8) 𝑤 = 𝑙𝑖𝑚 𝑧 = 𝑧0 + 𝑧1 + 𝑧2 + ⋯ 𝑝→1

(2.9)

The combination of the perturbation method and the Homotopy method is called the HPM, which eliminates the disadvantages of traditional methods of perturbation while retaining all its advantages.

Mathematical Preliminaries and Various Methods …

19

2.3.1.2. Example: Polymer-Modified Ultramicroelectrodes The following second-order nonlinear equation represents the amperometric response for conducting polymer-modified ultramicroelectrodes (Rebouillat, Lyons, and Flynn 1999). 𝑑2 𝑈 𝑑𝑋 2 𝑑2 𝑉 𝑑𝑋 2

− 𝛾𝑆 𝑈 𝑉 = 0

(2.10)

− 𝛾𝐸 𝑈 𝑉 = 0

(2.11)

where 𝑈 and 𝑉 represent the normalized concentration of substrate and product, 𝑋 denotes the normalized distance from the electrode (Senthamarai and Rajendran 2010). The diffusion and reaction parameter are 𝛾𝐸 and 𝛾𝑆 . The boundary conditions are as follows: at 𝑋 = 0, 𝑈 ′ = 0, 𝑉 = 1

(2.12)

at 𝑋 = 1, 𝑈 = 1, 𝑉′ = 0

(2.13)

Adding Eqns. (2.10) and (2.11) and integrating the resulting equation twice we get, 𝑈

𝑉 = 𝛾𝐸 (𝛾 + 𝐴𝑋 + 𝐵)

(2.14)

𝑆

where 𝐴 and 𝐵 are constants that can be obtained by use of the boundary conditions. Homotopy has been developed to evaluate the solution of Eqn. (2.10) as follows: 𝑑2 𝑈

𝑑2 𝑈

𝑈

(1 − 𝑝) [𝑑𝑋 2 ] + 𝑝 [𝑑𝑋 2 − 𝛾𝑆 𝛾𝐸 𝑈 (𝛾 + 𝐴𝑋 + 𝐵)] = 0 𝑆

The approximate solutions of Eqn. (2.10) is

(2.15)

20

L. Rajendran, R. Swaminathan and M. Chitra Devi 𝑈 = 𝑈0 + 𝑝 𝑈1 + 𝑝2 𝑈2 + 𝑝3 𝑈3 + ⋯

(2.16)

Substituting Eqn. (2.16) into Eqn. (2.15) and comparing the coefficients of like powers of 𝑝 (1 − 𝑝)[(𝑈0 + 𝑝 𝑈1 + 𝑝2 𝑈2 +. . . )″ ] + (𝑈0 + 𝑝 𝑈1 + 𝑝2 𝑈2 +. . . )″ 𝑝[ ]=0 −𝛾𝐸 (𝑈0 + 𝑝 𝑈1 + 𝑝2 𝑈2 +. . . )2 2 −𝛾𝑆 𝛾𝐸 (𝑈0 + 𝑝 𝑈1 + 𝑝 𝑈2 +. . . )(𝐴𝑋 + 𝐵)

(2.17)

𝑝0 : 𝑈0 ″ = 0

(2.18) 𝑈

𝑝1 : 𝑈1 ″ − 𝛾𝑆 𝛾𝐸 𝑈0 (𝛾 0 + 𝐴𝑋 + 𝐵) = 0

(2.19)

𝑆

The boundary conditions of the above equations are as follows: at 𝑋 = 0, 𝑈0′ = 0 and 𝑈𝑖′ = 0

(2.20)

at 𝑋 = 1, 𝑈0 = 1 and 𝑈𝑖 = 0 here 𝑖 ≥ 1

(2.21)

Finding solution of the Eqns. (2.18) and (2.19), using Eqns. (2.20) and (2.21) yields. 𝑈0 (𝑋) = 1 𝑈1 (𝑋) =

(2.22)

𝛾𝐸 𝛾𝑆 6

𝛾𝐸 𝛾𝑆

𝐴𝑋 3 + (

2

𝐵+

𝛾𝐸 2

) 𝑋2 −

𝛾𝐸 2

−

𝛾𝐸 𝛾𝑆 6

𝐴−

𝛾𝐸 𝛾𝑆 2

𝐵 (2.23)

According to the HPM, we can conclude that 𝑈(𝑋) = 𝑙𝑖𝑚𝑈(𝑋) = 𝑈0 + 𝑈1 + 𝑈2 𝑝→1

(2.24)

Mathematical Preliminaries and Various Methods … 𝛾𝐸 𝛾𝑆

𝑈(𝑋) = 1 −

2

𝐵−

𝛾𝐸 2

−

𝛾𝐸 𝛾𝑆 6

𝛾𝐸 𝛾𝑆

𝐴+(

2

𝐵+

𝛾𝐸 2

) 𝑋2 +

𝛾𝐸 𝛾𝑆 6

21 𝐴𝑋 3 (2.25)

𝑈(𝑋)

𝑉(𝑋) = 𝛾𝐸 (

𝛾𝑆

) + 𝛾𝐸 (𝐴𝑋 + 𝐵)

(2.26)

By using the boundary conditions (2.12) and (2.13), we can find the constants 𝐴 and 𝐵 as follows: 12

𝐴 = −12+𝛾 2 ; 𝐵 = 𝐸

−12𝛾𝐸 +12𝛾𝑆 +𝛾𝐸 3 +6𝛾𝐸 𝛾𝑆 (−12+𝛾𝐸 2 )𝛾𝐸 𝛾𝑆

(2.27)

2.3.2. Homotopy Analysis Method (HAM) The homotopy analysis method is a semi-analytical procedure used to solve nonlinear differential equations. The HAM was first developed by Liao Shijun of Shanghai Jiaotong University in his Ph.D. thesis (S. Liao 1992) in 1992 and further modified (S. J. Liao 1999) to introduce a non-zero auxiliary parameter in 1997. The convergence-control parameter is a non-physical variable that provides an easy way to check and implement the convergence of a series of solutions.

2.3.2.1. Basic Idea of Homotopy Analysis Method Consider the following differential equation: 𝑁[𝑢(𝑡)] = 0

(2.28)

where, 𝛮 is a nonlinear operator, 𝑡 denotes an independent variable, 𝑢(𝑡) is an unknown function. We disregard, for convenience, all boundary or initial conditions that can be viewed in a similar way. Liao constructed the zero-order deformation equation by generalizing the traditional homotopy approach as:

22

L. Rajendran, R. Swaminathan and M. Chitra Devi (1 − 𝑝)𝐿[𝜙(𝑡; 𝑝) − 𝑢0 (𝑡)] = 𝑝ℎ𝐻(𝑡)𝑁[𝜙(𝑡; 𝑝)]

(2.29)

where 𝑝 ∈ [0,1] is the embedding parameter, ℎ ≠ 0 is a nonzero auxiliary parameter, H(t) ≠ 0 is an auxiliary function, 𝐿 is an auxiliary linear operator, 𝑢0 (𝑡) is an initial guess of 𝑢(𝑡) and 𝜙(𝑡: 𝑝) is an unknown function. It is essential, that one has great freedom to select auxiliary unknowns in HAM. Obviously, when 𝑝 = 0 and 𝑝 = 1, it holds: 𝜙(𝑡; 0) = 𝑢0 (𝑡) and 𝜙(𝑡; 1) = 𝑢(𝑡)

(2.30)

respectively. Thus, as 𝑝 increases from 0 to 1, the solution 𝜙(𝑡; 𝑝) varies from the initial guess 𝑢0 (𝑡) to the solution 𝑢(𝑡). Expanding 𝜙(𝑡; 𝑝) in Taylor series with respect to 𝑝, we have: 𝑚 𝜙(𝑡; 𝑝) = 𝑢0 (𝑡) + ∑+∞ 𝑚=1 𝑢𝑚 (𝑡)𝑝

(2.31)

where 𝑢𝑚 (𝑡) =

1 𝜕𝑚 𝜙(𝑡;𝑝) 𝑚!

𝜕𝑝𝑚

|𝑝=0

(2.32)

The series (2.31) converges at 𝑝 = 1 then we have: 𝑢(𝑡) = 𝑢0 (𝑡) + ∑+∞ 𝑚=1 𝑢𝑚 (𝑡)

(2.33)

Define the vector →

𝑢𝑛 = {𝑢0 , 𝑢1 , . . . , 𝑢𝑛 }

(2.34)

Mathematical Preliminaries and Various Methods …

23

Differentiating Eqn. (2.29) for 𝑚 times with respect to the embedding parameter 𝑝, and then setting 𝑝 = 0 and finally dividing them by 𝑚!, we obtain 𝑚𝑡ℎ -order deformation equation as: →

𝐿[𝑢𝑚 − 𝜒𝑚 𝑢𝑚−1 ] = ℎ𝐻(𝑡)ℜ𝑚 (𝑢𝑚−1 )

(2.35)

where →

1

ℜ𝑚 (𝑢𝑚−1 ) = (𝑚−1)!

𝜕𝑚−1 𝑁[𝜙(𝑡;𝑝)] 𝜕𝑝𝑚−1

|𝑝=0

(2.36)

and 𝜒𝑚 = {

𝑎𝑛𝑑 0, 𝑚 ≤ 1, 𝑎𝑛𝑑 1, 𝑚 > 1.

(2.37)

applying 𝐿−1 on both side of Eqn. (2.35), we get 𝑢𝑚 (𝑡) = 𝜒𝑚 𝑢𝑚−1 (𝑡) + ℎ𝐿−1 [𝐻(𝑡)ℜ𝑚 (𝑢 ⃗ 𝑚−1 )]

(2.38)

In this way, it is easily to obtain 𝑢𝑚 for 𝑚 ≥ 1, at 𝑀𝑡ℎ order, we have 𝑢(𝑡) = ∑𝑀 𝑚=0 𝑢𝑚 (𝑡)

(2.39)

when 𝑀 → +∞, we get an accurate approximation of the original equation (2.28). For the convergence of the above method we refer the reader to Liao (Liao, and Sherif, 2004). If equation (2.28) admits a unique solution, then the unique solution will be made. If equation (2.28) does not have a particular solution, the HAM would be presenting a solution among several (possible) solutions. HPM is special cases of HAM. When we set ℎ = −1 we will

24

L. Rajendran, R. Swaminathan and M. Chitra Devi

get the same solutions for all the problems by above methods (Van Gorder 2015).

2.3.2.2. Example: Steady-State Biofilters The mathematical model of steady-state biofiltration is based on the system of nonlinear reaction/diffusion equations contains a nonlinear term related to Monod kinetics (Zarook and Shaikh 1997). 𝑑 2 𝐶1 𝑑𝑋 2

−

𝑑 2 𝐶10 𝑑𝑋 2

𝜑2 𝐶1 1+𝑀𝐶1

=0

(2.40)

𝜑2 𝜃𝐶

− 1+𝑀𝐶1 = 0

(2.41)

1

where 𝐶1 and 𝐶10 denotes the dimensionless concentration of VOC and oxygen, X is the dimensionless position in the biolayer. 𝜑 2 represents the Thiele modulus, 𝜃, 𝑀, 𝐿 𝑎𝑛𝑑 𝑁 are dimensionless constants. The boundary conditions are (M. Sivasankari and Rajendran 2013) 𝐴𝑡 𝑋 = 0; 𝐶1 = 1 𝑎𝑛𝑑 𝐶10 = 1 and 𝑎𝑡 𝑋 = 1:

𝑑𝐶1 𝑑𝑋

=

𝑑𝐶10 𝑑𝑋

=0

(2.42)

Using the relationship between Eqns. (2.40) - (2.41) and using boundary conditions (2.42) we obtain the relation between the concentrations as follows 𝐶10 = 1 − 𝜃(1 − 𝐶1 )

(2.43)

To solve Eqn. (2.40) by means of HAM, we first construct the zeroth-order deformation equation by taking 𝐻(𝑡) = 1: We construct the Homotopy as follows:

Mathematical Preliminaries and Various Methods … 𝑑 2 𝐶1

(1 − 𝑝) [

𝑑𝑋 2

𝑑2 𝐶

1 −1

] = 𝑝ℎ [ 𝑑𝑋 21 − 𝜑 2 (𝑀 + 𝐶 ) ]

25 (2.44)

1

The approximate solutions of Eqn. (2.44) as follows: 𝐶1 = 𝐶1 0 + 𝑝𝐶11 + 𝑝2 𝐶1 2 + ⋯

(2.45)

The rearranged boundary conditions are as follows 𝐶1 0 (0) = 1 and 𝐶1 0,𝑥 (1) = 0

(2.46)

𝐶1 𝑖 (0) = 0 and 𝐶1 𝑖,𝑥 (1) = 0, 𝑖 = 1,2

(2.47)

Substituting Eqn. (2.45) in Eqn. (2.44) and equating the like powers of p, we get 𝑝0 :

𝑝1 :

𝑑 2 𝐶1 0 𝑑𝑋 2

𝑑 2 𝐶1 1 𝑑𝑋 2

=0

=

𝑑 2 𝐶1 0 𝑑𝑋 2

(2.48)

(ℎ + 1) − ℎ [𝜑 2 (𝑀 +

1 𝐶1 0

−1

) ]

(2.49)

Solving the Eqns. (2.48) and (2.49), we can find the following results: 𝐶1 0 (𝑋) = 1

(2.50) 𝑥−2

𝐶11 (𝑋) = −ℎ𝜑 2 (2(1+𝑀)) 𝑥

(2.51)

According to HAM, we can conclude that 𝑥−2

𝐶(𝑋) = lim 𝐶1 (𝑋) = 𝐶1 0 + 𝐶11 = 1 − ℎ𝜑 2 (2(1+𝑀)) 𝑥 𝑝→1

(2.52)

26

L. Rajendran, R. Swaminathan and M. Chitra Devi

2.3.3. Adomian Decomposition Method George Adomian (1923–1996) developed an influential approach to answering nonlinear equations in the 1980s. The method is called the Adomian decomposition method (ADM) (George Adomian 1994). This technique is based upon functional equation representation as a series of functions. Every term in the series is obtained from a polynomial generated by an expansion of the power series.

2.3.3.1. Basic Concepts of the Adomian Decomposition Method (ADM) Consider the nonlinear differential equation 𝑦 ” + 𝑁(𝑦) = 𝑔(𝑥)

(2.53)

with boundary conditions 𝑦(0) = 𝐴, 𝑦 ′ (𝑏) = 𝐵

(2.54)

where 𝑁(𝑦) is a nonlinear function, 𝑔(𝑥) is the known function and 𝐴, 𝐵, 𝑏 are given constants. As below we also provide the differential operator 𝐿=

𝑑2 𝑑𝑥 2

(2.55)

So, Eqn. (2.53) can be written as 𝐿(𝑦) = 𝑔(𝑥) − 𝑁(𝑦)

(2.56)

The inverse operator 𝐿−1 is therefore considered as a two-fold integral operator, as below

Mathematical Preliminaries and Various Methods … 𝑥

𝑥

𝐿−1 (. ) = ∫0 ∫𝑏 (. ) 𝑑𝑥𝑑𝑥

27 (2.57)

Applying the inverse operator 𝐿−1on both sides of Eqn. (2.56) yields 𝑦(𝑥) = 𝐿−1 (𝑔(𝑥)) − 𝐿−1 (𝑁(𝑦)) + 𝑦′(𝑏)(𝑥 − 0) + 𝑦(0)

(2.58)

Using the boundary conditions Eqn. (2.54), Eqn. (2.58) becomes 𝑦(𝑥) = 𝐿−1 (𝑔(𝑥)) − 𝐿−1 (𝑁(𝑦)) + 𝐵𝑥 + 𝐴

(2.59)

The Adomian decomposition method introduce the solution 𝑦(𝑥) and the nonlinear function 𝑁(𝑦) by infinite series 𝑦(𝑥) = ∑∞ 𝑛=0 𝑦𝑛 (𝑥)

(2.60)

𝑁(𝑦) = ∑∞ 𝑛=0 𝐴𝑛

(2.61)

and

where the components 𝑦𝑛 (𝑥) of the solution 𝑦(𝑥) are recurrently calculated and the Adomian polynomials 𝐴𝑛 of 𝑁(𝑦) are evaluated using the formula 1 𝑑𝑛

𝑛 𝐴𝑛 (𝑥) = 𝑛! 𝑑𝜆𝑛 𝑁(∑∞ 𝑛=0 𝜆 𝑦𝑛 )𝜆=0

(2.62)

which gives 𝐴0 = 𝑁(𝑦0 ), 𝐴1 = 𝑁 ′ (𝑦0 )𝑦1 , 1

𝐴2 = 𝑁′(𝑦0 )𝑦2 + 2 𝑁”(𝑦0 )𝑦12 , 1

𝐴3 = 𝑁′(𝑦0 )𝑦3 + 𝑁”(𝑦0 )𝑦1 𝑦2 + 3! 𝑁′”(𝑦0 )𝑦13 , ⋮

(2.63)

28

L. Rajendran, R. Swaminathan and M. Chitra Devi By replacing Eqns. (2.60) and (2.61) in Eqn. (2.59) gives ∞ −1 −1 ∑∞ 𝑛=0 𝑦𝑛 = 𝐿 (𝑔(𝑥)) − 𝐿 (∑𝑛=0 𝐴𝑛 ) + 𝐵𝑥 + 𝐴

(2.64)

Formerly linking the terms in the linear system of Eqn. (2.63) provides the recurrent relation 𝑦0 = 𝐿−1 (𝑔(𝑥)) + 𝐵𝑥 + 𝐴, 𝑦𝑛+1 = −𝐿−1 (𝐴𝑛 ), 𝑛 ≥ 0

(2.65)

which gives 𝑦0 = 𝐿−1 (𝑔(𝑥)) + 𝐵𝑥 + 𝐴, 𝑦1 = −𝐿−1 (𝐴0 ), 𝑦2 = −𝐿−1 (𝐴1 ), 𝑦3 = −𝐿−1 (𝐴2 ), ⋮

(2.66)

From Eqn. (2.63) and (2.66), we can regulate the components of 𝑦𝑛 (𝑥), and hence the solution of 𝑦𝑛 (𝑥) in Eqn. (2.59) can be directly attained.

2.3.3.2. Example: Substrate Inhibition Kinetics in an Immobilized Enzyme A two-parameter model for an immobilized enzyme for the substrate inhibition kinetics is given as follows (Malík and Štefuca 2002): 𝑑2 𝑐 𝑑𝑥 2

− 𝛷2

𝑐 1+𝑐+

𝑐2 𝛽

=0

(2.67)

where 𝑐 is the normalized substrate concentration, 𝑥, 𝛷 and 𝛽 stands for normalized particle radial coordinate, Thiele modulus and the kinetic

Mathematical Preliminaries and Various Methods …

29

parameter respectively(Margret PonRani, Rajendran, and Eswaran 2011). The dimensionless boundary conditions are 𝑑𝑐

𝑥 = 0, 𝑑𝑥 = 0

(2.68)

𝑥 = 1, 𝑐 = 𝑚

(2.69)

Our goal is to apply the ADM (G. Adomian 1976; G. Adomian and Adomian 1984) to the model. Initially we rewrite the Eqn. (2.67) in operator form as 𝐿(𝑐) = 𝛷2 𝑁(𝑐)

(2.70)

where 𝑑2

𝐿 = 𝑑𝑥 2 and 𝑁(𝑐) =

𝑐 1+𝑐+

𝑐2 𝛽

(2.71)

Put on the inverse operator 𝐿−1 on each sides of the Eqn. (2.70) yields 𝑐(𝑥) = 𝑎𝑥 + 𝑏 + 𝛷2 𝐿−1 (𝑁(𝑐))

(2.72)

where a, b are an integrating constants. We let 𝑐(𝑥) = ∑∞ 𝑛=0 𝑐𝑛 (𝑥)

(2.73)

According to Adomian decomposition method, it is assumed that the nonlinear term 𝑁(𝑐) can be expressed as an infinite series in terms of the Adomian polynomials 𝐴𝑛 as given below

30

L. Rajendran, R. Swaminathan and M. Chitra Devi 𝑁(𝑐) = ∑∞ 𝑛=0 𝐴𝑛 (𝑥)

(2.74)

where the Adomian polynomials 𝐴𝑛 of 𝑁(𝑐) are evaluated using the formula 1 𝑑𝑛

𝑛 𝐴𝑛 = 𝑛! 𝑑𝜆𝑛 𝑁(∑∞ 𝑛=0(𝜆 𝑐𝑛 ))|

𝜆=0

(2.75)

where the hypothetical parameter 𝜆 ∈ [0,1]. Substituting Eqns. (2.73) and (2.74) in (2.72) gives 2 −1 ∑∞ ∑∞ 𝑛=0 𝑐𝑛 (𝑥) = 𝑎𝑥 + 𝑏 + 𝛷 𝐿 𝑛=0 𝐴𝑛

(2.76)

Let us classify the zeroth element as 𝑐0 (𝑥) = 𝑎𝑥 + 𝑏

(2.77)

The boundary conditions are transformed as follows 𝑑𝑐0

|

= 0, 𝑐0 (1) = 𝑚

(2.78)

|

= 0, 𝑐𝑖 (1) = 0; 𝑖 = 1,2,3, …

(2.79)

𝑑𝑥 𝑥=0 𝑑𝑐𝑖

𝑑𝑥 𝑥=0

Solving the Eqn. (2.77) and by means of the boundary conditions Eqns. (2.78) and (2.79), we get 𝑐0 = 𝑚

(2.80)

The recurrence formula is attained by associating the terms in the Eqn. (2.76): 𝑐0 (𝑥) = 𝑎𝑥 + 𝑏 = 𝑚

(2.81)

Mathematical Preliminaries and Various Methods … 𝑐𝑛+1 (𝑥) = 𝛷2 𝐿−1 (𝐴𝑛 ) ; n ≥ 0

31 (2.82)

The resultant Adomian polynomials of 𝑐0 , 𝑐1 , 𝑐2 , 𝑐3 , … , 𝑐𝑛 are denoted by 𝐴𝑛 . We can find the 𝐴𝑛 first and the corresponding 𝑐𝑛+1 as follows: 𝐴0 = 𝑁(𝑐0 ) =

𝑐0 𝑐 2 1+𝑐0 + 0 𝛽

=

𝑚 1+𝑚+

(2.83)

𝑚2 𝛽

To obtain 𝑐1 we substitute Eqn. (2.83) in Eqn. (2.82) then, we get 𝑐1 (𝑥) = 𝛷2 𝐿−1 (𝐴0 ) =

𝛷 2 𝑚(𝑥 2 −1)

(2.84)

2

2 (1+𝑚+𝑚 ) 𝛽

Now, substituting 𝑛 = 1 in Eqn. (2.75) 𝐴1 =

𝑑 𝑑𝜆

𝑁(𝑐0 + 𝜆𝑐1 )| =

𝛷 2 𝛽 2 𝑚(𝛽−𝑚2 )(𝑥 2 −1)

(2.85)

(𝛽+𝛽𝑚+𝑚2 )3

2

To obtain 𝑐2 we substitute Eqn. (2.85) in Eqn. (2.82). Then, we get 𝑐2 (𝑥) = 𝛷2 𝐿−1 (𝐴1 ) =

𝑚(𝛽−𝑚2 )

𝛷2 𝛽2 2

𝑥4

(

(𝛽+𝛽𝑚+𝑚2 )3 12

−

𝑥2 2

+

5 12

)

(2.86)

The remaining polynomials can be generated easily. 𝑐(𝑥) = 𝑐0 (𝑥) + 𝑐1 (𝑥) + 𝑐2 (𝑥) + ⋯ 𝑐(𝑥) =

𝛷 2 𝛽𝑚 2

(𝛽 2 −𝛽𝑚2 )

𝑥4

((𝛽+𝛽𝑚+𝑚2)3 (12 −

𝑥2 2

(2.87) 5

(𝑥 2 −1)

+ 12) + (𝛽+𝛽𝑚+𝑚2)) + 𝑚 (2.88)

32

L. Rajendran, R. Swaminathan and M. Chitra Devi

2.3.4. Variational Iteration Method (VIM) The variational iteration method (VIM) introduced by He in 1999 in (J. H. He and Wu 2006a), will be used to analyze the linear and nonlinear wave equation, and wave-like equation in bounded and unbounded domains. The variational iteration method (VIM) has been a familiar semi‐analytical method for finding the solution of nonlinear partial differential equations. The most significant advantage of the approach is its simplicity and ability to solve nonlinear equations rapidly. By this method, if such a solution exists, one can find the successive convergent approximations to the exact solution of the differential equations.

2.3.4.1. Basic Concepts in the Variational Iteration Method The following nonlinear partial differential equation is considered to demonstrate the basic concepts of the variational iteration method (VIM): 𝐿 [𝑢(𝑥)] + 𝑁 [𝑢(𝑥)] = 𝑔(𝑥)

(2.89)

where L is a linear operator, N is a nonlinear operator, and g(x) is a given continuous function (J. H. He and Wu 2006a). According to the Variational iteration method, we can construct a correct functional as follows: 𝑥

𝑢𝑛+1 (𝑥) = 𝑢𝑛 (𝑥) + ∫0 𝜆[𝐿(𝑢𝑛 (𝜁)) + 𝑁(𝑢𝑛 (𝜁)) − 𝑔(𝜁)] 𝑑𝜁

(2.90)

where λ is a general Lagrange multiplier [24, 29] which can be identified optimally via variational theory, 𝑢𝑛 is the nth approximate solution, and 𝑢𝑛 denotes a restricted variation, i.e., 𝛿𝑢𝑛 = 0.

Mathematical Preliminaries and Various Methods …

33

2.3.4.2. Example: Electrochemical Immobilization of Enzymes Amperometric electrodes with thin, immobilized layers containing redox enzymes are finding increasing application in analytical chemistry and biosensors. Consider the mathematical model of glucose oxidase entrapped within a redox hydrogel film or immobilized within a polyelectrolyte membrane. The corresponding system of steady-state nonlinear differential equations in this model are given as follows (Philip N. Bartlett and Eastwick-Field 1993): 𝑑 2 𝑢(𝜒) 𝑑𝜒2 𝑑 2 𝑣(𝜒) 𝑑𝜒2 𝑑 2 𝑤(𝜒) 𝑑𝜒2

− 𝜅𝑢(𝜒)𝑣(𝜒) = 0

(2.91)

− 𝜅𝛾𝑢(𝜒)𝑣(𝜒) = 0

(2.92)

+ 𝜅𝑢(𝜒)𝑣(𝜒) = 0

(2.93)

The boundary conditions for the above set of equations can be communicated as 𝑑𝑢 𝑑𝜒

𝑑𝑣

= −1, 𝑑𝜒 = 0, 𝑤 = 0 at 𝜒 = 0

𝑢 = 0, 𝑣 = 1, 𝑤 = 0 at 𝜒 = 1

(2.94) (2.95)

By using the relation between the Eqns. (2.91) and (2.92) and using boundary conditions Eqns. (2.94) and (2.95). The relation between 𝑢 and 𝑣 is obtained as follows 𝑢(𝜒) =

𝑣(𝜒)−1 𝛾

+ (1 − 𝜒)

(2.96)

Similarly, the upcoming result is attained by adding Eqn. (2.92) and Eqn. (2.93), using boundary conditions Eqns. (2.94) and (2.95)

34

L. Rajendran, R. Swaminathan and M. Chitra Devi 1

𝑤(𝜒) = 𝛾 (−𝑣(𝜒) + 𝛼𝜒 + 𝛽)

(2.97)

Let 𝑣0 be the initial approximations chosen as follows: 𝑣0 = 1 + 𝑎 − 𝑎𝜒 2

(2.98)

Substituting (2.96) into Eqn. (2.92)we get 𝑑 2 𝑣(𝜒) 𝑑𝜒2

− 𝜅(𝑣 2 (𝜒) − 𝑣(𝜒)) − 𝜅𝛾(1 − 𝜒)𝑣(𝜒) = 0

(2.99)

The following approximation is obtained using the variational iteration method defined above 𝑥

𝑣𝑛+1 (𝜒) = 𝑣𝑛 (𝜒) + ∫0 𝜆[𝑣 ″ (𝜉) − 𝜅(𝑣 2 (𝜉) − 𝑣(𝜉)) − 𝜅𝛾(1 − 𝜉)𝑣(𝜉)] 𝑑𝜉

(2.100)

The Lagrange multiplier for the above system can be founded as (Rahamathunissa et al. 2011) 𝜆 =𝜉−𝜒

(2.101) 𝑥

𝑣𝑛+1 (𝜒) = 𝑣𝑛 (𝜒) + ∫0 (𝜉 − 𝜒)[𝑣 ″ (𝜉) − 𝜅(𝑣 2 (𝜉) − 𝑣(𝜉)) − 𝜅𝛾(1 − 𝜉)𝑣(𝜉)] 𝑑𝜉

(2.102)

Substituting the values of 𝑣0 and n = 0 in in the above iteration formula, we get 𝑥

𝑣1 (𝜒) = 𝑣0 (𝜒) + ∫0 (𝜉 − 𝜒)[𝑣 ″ (𝜉) − 𝜅(𝑣0 2 (𝜉) − 𝑣0 (𝜉)) − 𝜅𝛾(1 − 𝜉)𝑣0 (𝜉)] 𝑑𝜉

(2.103)

Mathematical Preliminaries and Various Methods …

35

𝑥

𝑣1 (𝜒) = 1 + 𝑎 − 𝑎𝜒 2 + ∫0 (𝜉 − 𝜒) [

−𝜅((1 + 𝑎 − 𝑎𝜒 2 )2 − 1 + 𝑎 − 𝑎𝜒 2 ) ] 𝑑𝜉 −𝜅𝛾(1 − 𝜉)(1 + 𝑎 − 𝑎𝜒 2 ) − 2𝑎 𝜅+𝜅𝛾

𝑣1 (𝜒) = 1 + 𝑎 + (( 𝜅+𝜅𝛾

((

12

)𝑎 +

𝜅𝑎 2 6

) 𝜒4 +

2

𝜅𝛾𝑎 20

)𝑎 +

𝜅𝑎 2 2

𝜅𝑎 2

𝜒5 +

30

+

𝜅𝛾

1+𝑎

) 𝜒2 − ( 2

6

𝜒6

) 𝜅𝛾𝜒 3 − (2.104)

𝑣(𝜒) ≈ 𝑣1 (𝜒)

(2.105)

where the value of 𝑎 is obtained by using boundary condition as follows 𝑣(1) = 1

(2.106) 𝜅+𝜅𝛾

1 + 𝑎 + ((

2

𝜅+𝜅𝛾

− ((

𝑎+

5𝑘𝑎 12

+

12

3𝑘𝛾𝑎 20

)𝑎 +

)𝑎 +

+

𝜅𝑎 2 2

𝜅𝑎 2 6

11𝑘𝑎 2 30

+

𝜅𝛾 2

𝜅𝛾𝑎

)+

=−

20

𝑘𝛾 3

1+𝑎

)−( +

6

𝜅𝑎 2

) 𝜅𝛾 }=1

(2.107)

30

(2.108)

The value of 𝑎 can be obtained from the above equation for particular value of 𝑘 and 𝛾.

2.3.5. Exp-Function Method He and Wu (J. H. He and Wu 2006b) first suggested this approach and were successfully applied to obtain the solitary and periodic solutions of partial nonlinear differential equations. Furthermore, several researchers used this approach to handle various other problems such as stochastic equations (Dai and Zhang 2009), a system of the

36

L. Rajendran, R. Swaminathan and M. Chitra Devi

partial differential equations (Misirli and Gurefe 2010), highdimensional nonlinear evaluation equation (Boz and Bekir 2008), differential equation (Bekir 2010), and long-wave nonlinear dispersive equation (Zhang, Tong, and Wang 2009).

2.3.5.1. Basic Concept of Exp-Function Method Let us consider the following nonlinear partial differential equation to understand the exp-function method briefly (J. H. He and Wu 2006b; J. H. He and Abdou 2007; (Benn)Wu and He 2008; Bekir and Boz 2009; Biazar and Ayati 2009) 𝐹(𝑢, 𝑢𝑥 , 𝑢𝑡 , 𝑢𝑥𝑥 , 𝑢𝑥𝑡 , 𝑢𝑡𝑡 , . . . ) = 0

(2.109)

Here, subscripts indicate partial differentiation with respect to indicated variables in the subscript. Using travelling wave transformation 𝑢 = 𝑢(𝜂), 𝜂 = 𝑘𝑥 + 𝜔𝑡 (where 𝑘 and 𝜔 are constants) in Eqn. (2.109), it is possible to transform the partial differential equation into a nonlinear ordinary differential equation 𝐹(𝑢, 𝑢′ , 𝑢″ , . . . ) = 0

(2.110)

where the prime indicates derivative with respect to 𝜂. Using exp-function method, the solution of ordinary differential Eqn. (2.110) can be expressed as 𝑢(𝜂) =

𝑛𝜂 ∑𝑑 𝑛=−𝑐 𝑎𝑛 𝑒 𝑞

∑𝑛=−𝑝 𝑏𝑛 𝑒 𝑛𝜂

(2.111)

where 𝑐, 𝑑, 𝑝 and 𝑞 are positive integers yet to be determined, and 𝑎𝑛 and 𝑏𝑛 are unknown constants. The values of c and p are obtained by balancing exp-functions of the linear term of the lowest order in Eqn. (2.110) with the lowest-order nonlinear term. Similarly, balancing the exp-functions of the highest-

Mathematical Preliminaries and Various Methods …

37

order linear term in Eqn. (2.110) with the highest-order nonlinear term, values of d and q can be obtained.

2.3.5.2. Example: Mass Transport in Heterogeneous Catalysis The mass transport equation in heterogeneous catalysts is (Hayes et al. 2007) 𝑑2 𝑈 𝑑𝑋 2

𝛼2𝑈𝑝

− (1+𝑀𝑈)𝑚 = 0

(2.112)

where 𝑈 is the dimensionless concentration of substrate (M. K. Sivasankari and Rajendran 2013). The boundary conditions are 𝑑𝑈

= 0 when 𝑋 = 0

(2.113)

𝑈 = 1 when 𝑋 = 1

(2.114)

𝑑𝑋

Assume approximate trial solution of Eqn. (2.112) is in the following form: 𝑈(𝑋) = A𝑒 𝑏𝑋 + 𝐵𝑒 −𝑏𝑋

(2.115)

where 𝐴, 𝐵 and 𝑏 are constants. These constants can be obtained using boundary condition (Eqn. (2.113) and (2.114)) as follows: 𝐴 = 𝐵, 𝐵 = 𝑒 𝑏 + 𝑒 −𝑏

(2.116)

Replacing these constants into Eqn. (2.115) we find 𝑈(𝑋) =

𝑒 𝑏𝑋 +𝑒 −𝑏𝑋 𝑒 𝑏 +𝑒 −𝑏

where 𝑏 is the constant coefficient.

(2.117)

38

L. Rajendran, R. Swaminathan and M. Chitra Devi Now to find the value of 𝑏, consider Eqn. (2.112) as follows 𝑑2 𝑈

𝛼2𝑈𝑝

𝑓(𝑋) = 𝑑𝑋 2 − (1+𝑀𝑈)𝑚 = 0

(2.118)

Substituting the value of Eqn. (2.117) in Eqn. (2.118) 𝑓(𝑋) = 𝑏 2

𝑒 𝑏𝑋 +𝑒 −𝑏𝑋 𝑒 𝑏 +𝑒 −𝑏

𝑝

−

𝛼 2 (𝑒 𝑏𝑋 +𝑒 −𝑏𝑋 )

𝑚

(𝑀(𝑒 𝑏𝑋 +𝑒 −𝑏𝑋 )+𝑒 𝑏 +𝑒 −𝑏 )

=0

(2.119)

Now, substituting 𝑋 = 1 in above equation after differentiation, we get

𝑓 ′ (𝑋 = 1) =

𝛼2 𝑝 𝛼2 𝑚𝑀 − ) (M+1)𝑚 (M+1)𝑚+1 2b (𝑒 +1)

𝑏(𝑒 2b −1)(−𝑏2 +

=0

(2.120)

After simplification of Eqn. (2.120) we get 𝑝

𝑚𝑀

𝑏 = 𝛼√(M+1)𝑚 − (M+1)𝑚+1

(2.121)

Hence the approximate analytical solution of the Eqns. (2.112)(2.114) using modified hyperbolic function method is given as follows 𝑈(𝑋) =

𝑒 𝑏𝑋 +𝑒 −𝑏𝑋 𝑒 𝑏 +𝑒 −𝑏 𝑝

𝑚𝑀

where 𝑏 = 𝛼√(M+1)𝑚 − (M+1)𝑚+1

(2.122)

2.3.6. Hyperbolic Function Method This method was first proposed by Devi et al. (Devi, Pirabaharan, Rajendran, et al. 2020; Visuvasam, Meena, and Rajendran 2020; Devi,

Mathematical Preliminaries and Various Methods …

39

Pirabaharan, Abukhaled, et al. 2020) and was successfully applied to obtain the approximate analytical solutions of nonlinear differential equations. Hyperbolic function method, is a special case of the expfunction method. Further, this method was used by many researchers for handling various nonlinear differential equations.

2.3.6.1. Basic Concept of Hyperbolic Function Method The differential equation of 𝑝𝑘 which is a function of 𝑢 and their derivatives are considered as follows (Devi, Pirabaharan, Rajendran, et al. 2020): 𝑝𝑘 : 𝑓(𝑢𝑘 , 𝑢𝑘′ , 𝑢𝑘″ , 𝑎𝑘 , 𝑏𝑘 ) = 0

(2.123)

where 𝑝𝑘 represents a nonlinear differential equation, 𝑢𝑘 = 𝑢𝑘 (𝑥, 𝑎𝑘 , 𝑏𝑘 , … )∀𝑘 = 1 … 𝑛in which 𝑎𝑘 , 𝑏𝑘 are given parameters in the differential equation, 𝑘 = 1,2, … , 𝑛 and 𝑥 ∈ [𝐿, 𝑈]. The boudary conditions are 𝑢𝑘 (𝑥) = 𝑢𝑘𝐿0 or 𝑢𝑘′ (𝑥) = 𝑢𝑘𝐿1 𝑎𝑡 𝑥 = 𝐿

(2.124)

𝑢𝑘 (𝑥) = 𝑢𝑘𝑈0 or 𝑢𝑘′ (𝑥) = 𝑢𝑘𝑈1 𝑎𝑡 𝑥 = 𝑈

(2.125)

Assume that the trail solution of Eqn.(2.123) is a hyperbolic function of the form. 𝑢𝑘 (𝑥) = 𝐴𝑘 𝑐𝑜𝑠ℎ(𝑏𝑥) + 𝐵𝑘 𝑠𝑖𝑛ℎ(𝑏𝑥)

(2.126)

The constant coefficients 𝐴𝑘 and 𝐵𝑘 are obtained by solving the equations using the boundary conditions Eqns.(2.124) and (2.125) as follows: 𝑢𝑘 (𝐿) = 𝐴𝑘 𝑐𝑜𝑠ℎ(𝑏𝐿) + 𝐵𝑘 𝑠𝑖𝑛ℎ(𝑏𝐿) = 𝑢𝐿0 ,

40

L. Rajendran, R. Swaminathan and M. Chitra Devi

or 𝑢𝑘′ (𝐿) = 𝑚(𝐴𝑘 𝑐𝑜𝑠ℎ(𝑏𝐿) + 𝐵𝑘 𝑠𝑖𝑛ℎ(𝑏𝐿)) = 𝑢𝐿1 ,

(2.127)

𝑢𝑘 (𝑈) = 𝐴𝑘 𝑐𝑜𝑠ℎ(𝑏𝑈) + 𝐵𝑘 𝑠𝑖𝑛ℎ(𝑏𝑈) = 𝑢𝑈0 , or 𝑢𝑘′ (𝑈) = 𝑚(𝐴𝑘 𝑐𝑜𝑠ℎ(𝑏𝑈) + 𝐵𝑘 𝑠𝑖𝑛ℎ(𝑏𝑈)) = 𝑢𝑈1 ,

(2.128)

The unknown parameter 𝑏can obtain by substituting the Eqn. (2.126) in the original differential Eqn. (2.123) by giving specific value for 𝑥 = 𝐾 where 𝐿 ≤ 𝐾 ≤ 𝑈. 𝑝𝑘 : 𝑓(𝑢𝑘 (𝐾, 𝐴𝑘 , 𝐵𝑘 , 𝑏), 𝑢𝑘′ (𝐾, 𝐴𝑘 , 𝐵𝑘 , 𝑏), 𝑢𝑘″ (𝐾, 𝐴𝑘 , 𝐵𝑘 , 𝑏)) = 0 (2.129) The unknown parameter 𝑏 is obtained by solving the above equation.

2.3.6.2. Example: Glucose Oxidase Enzyme System The system of steady-state nonlinear equations in glucose oxidase enzyme (P. N. Bartlett and Whitaker 1987) is in the following dimensionless form (Shanmugarajan et al. 2011). 𝑑 2 𝑠(𝑥) 𝑑𝑥 2 𝑑 2 𝑏(𝑥) 𝑑𝑥 2

𝑘𝑠(𝑥)

− 1+𝛼𝑠(𝑥) = 0 𝜂𝑠(𝑥)

+ 1+𝛼𝑠(𝑥) = 0

where s and b are the concentration substrate and mediator.

(2.130)

(2.131)

Mathematical Preliminaries and Various Methods … 𝑑𝑠 𝑑𝑥

= 0 when 𝑥 = 0; 𝑠 = 𝐾𝑠∞ when 𝑥 = 1

𝑏 = 0 when 𝑥 = 0, 𝑥 = 1

41 (2.132) (2.133)

Let the probable trial solution of Eqn. (2.130) in the following form: 𝑠(𝑥) = A 𝑐𝑜𝑠ℎ(𝑚𝑥) + 𝐵 𝑠𝑖𝑛ℎ(𝑚𝑥)

(2.134)

where the constants 𝐴, 𝐵 and m are obtained by using boundary condition (Eqns. (2.132) and (2.133)): 𝐴 = 𝐾𝑠∞ sech(𝑚), 𝐵 = 0

(2.135)

Substituting the value of constants into Eqn. (2.134) we find 𝑠(𝑥) = 𝑠𝑒𝑐ℎ(𝑚) 𝑐𝑜𝑠ℎ(𝑚𝑥)

(2.136)

To find the value of the constant m, reconsider Eqn. (2.130) as follows 𝑑2 𝑠

𝑘𝑠

𝑓(𝑥) = 𝑑𝑥 2 − 1+𝛼𝑠 = 0

(2.137)

Substituting the value of (2.136) in (2.137) 𝑘 𝑠𝑒𝑐ℎ(𝑚) 𝑐𝑜𝑠ℎ(𝑚𝑥)

𝑓(𝑥) = 𝑚2 𝑠𝑒𝑐ℎ(𝑚) 𝑐𝑜𝑠ℎ(𝑚𝑥) − 1+𝛼 𝑠𝑒𝑐ℎ(𝑚) 𝑐𝑜𝑠ℎ(𝑚𝑥) = 0 (2.138) Now, substituting 𝑥 = 1 in above equation, we get 𝑘

𝑓(𝑥 = 1) = 𝑚2 − 1+𝛼 = 0

(2.139)

42

L. Rajendran, R. Swaminathan and M. Chitra Devi 𝑘

𝑚 = √1+𝛼

(2.140)

Similarly, by assuming the probable trial solution of Eqn. (2.131) in the following form 𝑏(𝑥) = 1 + ( A1 𝑐𝑜𝑠ℎ(𝑙𝑥) + 𝐵1 𝑠𝑖𝑛ℎ(𝑙𝑥))

(2.141)

And repeating the steps from Eqns. (2.135)-(2.140) we get 𝑏(𝑥) = 1 − 𝑐𝑜𝑠ℎ(𝑙𝑥) −

1−𝑐𝑜𝑠ℎ(𝑙) 𝑠𝑖𝑛ℎ(𝑙)

𝑠𝑖𝑛ℎ(𝑙𝑥)

(2.142)

where 𝜂

𝑙 = √1+𝛼

(2.143)

Hence the approximate analytical solution of the Eqns. (2.130)(2.133) using modified hyperbolic function method is given as follows 𝑠(𝑥) = 𝑠𝑒𝑐ℎ (√

𝑘

𝑘

) 𝑐𝑜𝑠ℎ (√1+𝛼 𝑥) 1+𝛼

𝜂

𝑏(𝑥) = 1 − 𝑐𝑜𝑠ℎ (√1+𝛼 𝑥) −

(2.144)

𝜂 ) 1+𝛼

1−𝑐𝑜𝑠ℎ(√

𝜂 𝑠𝑖𝑛ℎ(√ ) 1+𝛼

𝜂

𝑠𝑖𝑛ℎ (√1+𝛼 𝑥) (2.145)

2.3.7. Variational Fractal Theory The most interesting features of the variational fractal theory method are its extreme simplicity and concise forms of variational functionals for a wide range of nonlinear problems.

Mathematical Preliminaries and Various Methods …

43

2.3.7.1. Example: Troesch’s Problem Troesch’s problem was first described and solved by Weibel (Weibel 1958). It has become a widely used test problem and has been solved many times, (Roberts and Shipman 1976) by using a shooting method (Troesch 1976), Laplace transform technique (Khuri 2003) and a modified homotopy perturbation technique (X. Feng, Mei, and He 2007). The differential equation is 𝑦 ″ (𝑡) = 𝜆 𝑠𝑖𝑛ℎ(𝜆𝑦(𝑡)) , 𝑡 ∈ [0,1]

(2.146)

The boundary conditions are 𝑦(0) = 0 𝑎𝑛𝑑 𝑦(1) = 1

(2.147)

Considering the Eqn. (2.146), a fractal modification has to be adopted: 𝑑

dy

( ) = 𝜆 𝑠𝑖𝑛ℎ(𝜆𝑦(𝑡)) , 𝑡 ∈ [0,1]

dt𝛼 dt𝛼

(2.148)

dy

where dt is the fractal derivative (J. H. He 2018; Li et al. 2019) 𝛼

dy dt𝛼

(𝑡0 ) = 𝛤(1 + 𝛼) 𝑙𝑖𝑚

𝑡→𝑡0 →𝛥𝑡

𝑦(𝑡)−𝑦(𝑡0 ) (𝑡−𝑡0 )𝛼

(2.149)

Using the two-scale transform (Ain and He 2019; Ji-Huan and FeiYu 2019) 𝑠 = 𝑡𝛼

(2.150)

one can change Eqn. (2.148) approximately into the following one 𝑦 ″ (𝑠) = 𝜆 𝑠𝑖𝑛ℎ(𝜆𝑦(𝑠)), 𝑦(0) = 0 𝑎𝑛𝑑 𝑦(1) = 1

(2.151)

44

L. Rajendran, R. Swaminathan and M. Chitra Devi

Higher order derivatives of y is obtain by differentiating Eqn. (2.151) with respect to s 𝑦 (3) (𝑠) = 𝜆2 𝑐𝑜𝑠ℎ(𝜆𝑦(𝑠)) 𝑦 ′ (𝑠)

(2.152) 2

𝑦 (4) (𝑠) = 𝜆3 𝑠𝑖𝑛ℎ(𝜆𝑦(𝑠)) (𝑦 ′ (𝑠)) + 𝜆2 𝑐𝑜𝑠ℎ(𝜆𝑦(𝑠)) 𝑦 ′′ (𝑠) (2.153) 3

𝑦 (5) (𝑠) = 𝜆4 𝑐𝑜𝑠ℎ(𝜆𝑦(𝑠)) (𝑦 ′ (𝑠)) + 3𝜆3 𝑠𝑖𝑛ℎ(𝜆𝑦(𝑠)) 𝑦 ′ (𝑠)𝑦 ″ (𝑠) + 𝜆2 𝑐𝑜𝑠ℎ(𝜆𝑦(𝑠)) 𝑦 ‴ (𝑠)

(2.154)

We assume that 𝑦 ′ (0) = 𝜂

(2.155)

where the unknown constant 𝜂 to be determined further. Setting 𝑠 = 0 and using the initial conditions 𝑦(0) = 0 , 𝑦 ′ (0) = 𝜂 in Eqns. (2.151) and Eqn. (2.152)-(2.154), we have 𝑦 ″ (𝑠) = 0

(2.156)

𝑦 ‴ (𝑠) = 𝜆2 𝜂

(2.157)

𝑦 (4) (𝑠) = 0

(2.158)

𝑦 (5) (𝑠) = 𝜆4 𝜂(𝜂2 + 1)

(2.159)

The Taylor series solution is 𝑠2

𝑠3

𝑠4

𝑦(𝑠) = 𝑦(0) + 𝑠𝑦 ′ (0) + 2! 𝑦 ″ (0) + 3! 𝑦 ‴ (0) + 4! 𝑦 (4) (0) + 𝑠5

𝑠3

𝑠5

𝑦 (5) (0) = 0 + 𝑠𝜂 + 0 + 3! 𝜆2 𝜂 + 0 + 5! 𝜆2 𝜂(𝜂2 + 1) 5!

(2.160)

Mathematical Preliminaries and Various Methods …

45

Using the boundary condition (1) = 1 , we obtain 𝑠3

𝑠5

𝑦(𝑠)|𝑠=1 = 𝑠𝜂 + 3! 𝜆2 𝜂 + 5! 𝜆2 𝜂(𝜂2 + 1)| 1 5!

2

1

𝑠=1

= 𝜂 + 3! 𝜆2 𝜂 +

2

𝜆 𝜂(𝜂 + 1) = 1

(2.161)

Substituting particular value of 𝜆 in Eqn. (2.161), 𝜂 can be identified simply. Therefore one obtain the approximate solution as follows 𝑦(𝑡) = 𝑡 𝛼 𝜂 +

𝑡 3𝛼 2 𝜆 𝜂 3!

+0+

𝑡 5𝛼 5!

𝜆2 𝜂(𝜂2 + 1)

(2.162)

If the approximate solution is required in higher order, without any difficulty one can carry on the solution process defined above.

2.3.8. Taylor Series and Padé Approximation Method 2.3.8.1. Taylor Series Taylor series is a description of a function as an infinite number of terms, determined at a single point from the values of the function’s derivatives. This subject was first formulated by the Scottish mathematician James Gregory and formally introduced by the English mathematician Brook Taylor in 1715. If the Taylor series is centered at zero, then that series is also called a Maclaurin series, after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. 2.3.8.2. Padé Approximation The Padé approximants are a special type of rational approximation. This technique was invented by Henri Padé around 1890 but goes back to Georg Frobenius, who introduced the concept and characteristics of

46

L. Rajendran, R. Swaminathan and M. Chitra Devi

rational approximations. The Padé approximant also gives the function a better estimate than truncating its Taylor series, and it can also operate where the Taylor series does not converge. Padé approximants are widely used in computer calculations for these purposes. Given a function f and two integers 𝑚 ≥ 0 and 𝑛 ≥ 1, the Padé approximant of an order [𝑚/𝑛] is the rational function 𝑅(𝜒) =

𝑎0 +𝑎1 𝜒+𝑎2 𝜒2 +⋯+𝑎𝑚 𝜒𝑚 1+𝑏1 𝜒+𝑏2 𝜒2 +⋯+𝑏𝑛 𝜒𝑛

(2.163)

which agrees with 𝑓(𝑥) to the highest possible order, which amounts to 𝑓(0) = 𝑅(0) 𝑓 ′ (0) = 𝑅′ (0) 𝑓 ″ (0) = 𝑅″ (0) ⋮ 𝑓 (𝑚+𝑛) (0) = 𝑅(𝑚+𝑛) (0)

(2.164)

Equivalently, if R(x) is expanded in a Maclaurin series (Taylor series at 0), its first 𝑚 + 𝑛 terms would cancel the first 𝑚 + 𝑛 terms of 𝑓(𝑥), and as such 𝑓(𝑥) − 𝑅(𝜒) = 𝑐𝑚+𝑛+1 𝜒 𝑚+𝑛+1 + 𝑐𝑚+𝑛+2 𝜒 𝑚+𝑛+2

(2.165)

The Padé approximant is unique for given 𝑚 and 𝑛, that is, the coefficients 𝑎0 , 𝑎1 … 𝑎𝑚 , 𝑏1 , 𝑏2 … 𝑏𝑛 can be uniquely determined. The Padé approximant defined above is also denoted as [𝑚/𝑛]𝑓(𝑥).

2.3.8.3. Example: Electroactive Polymers The substrate concentration in the electroactive polymer film (Lyons et al. 1996) is

Mathematical Preliminaries and Various Methods … 𝑑2 𝑆 𝑑𝜒2

𝛾𝑆

− 1+𝛼𝑆 = 0

47 (2.166)

where 𝑆 represent the concentration of substrate and 𝜒 denotes the normalized distance. The diffusion and reaction parameter are 𝛾 and 𝛼. The boundary conditions are as follows: at 𝜒 = 0, 𝑆 ′ = 0 and at 𝜒 = 1, 𝑆 = 1

(2.167)

Consider the Maclaurin series (Taylor’s series at 𝜒 = 0) for dimensionless concentration of substrate, 𝑆(𝜒). 𝜒

𝜒2

𝜒3

𝑆(𝜒) = 𝑆(0) + 𝑆′(0) 1! + 𝑆 (2) (0) 2! + 𝑆 (3) (0) 3! + 𝜒4

𝑆 (4) (0) 4! +. . . . . . ..

(2.168)

Let us consider, 𝑆(0) = 𝑎 where 𝑎is constant. From the boundary conditions (Eqn. (2.167)), we have 𝑆′(0) = 0. Setting χ = 0 and using the boundary condition 𝑆′(0) = 0 in Eqns. (2.166) and the continuous derivatives of the Eqns. (2.166), we have 𝛾𝑎

𝑆 ″ (0) = 1+𝛼𝑎

(2.169)

𝑆 (3) (0) = 0

(2.170) 𝛾𝑎

𝑆 (4) (0) = (1+𝛼𝑎)3

(2.171)

𝑆 (5) (0) = 0

(2.172)

5

𝑆

(6) (0)

=

𝛾 2 𝑎(1−5𝛼𝑎−𝑎 2 )

𝑆 (7) (0) = 0

(1+𝛼𝑎)5

(2.173) (2.174)

48

L. Rajendran, R. Swaminathan and M. Chitra Devi 𝑆 (8) (0) =

𝛾 3 𝑎(1−𝑎2 −35𝛼𝑎+90𝛼2 𝑎 2 ) (1+𝛼𝑎)7

(2.175)

Consider, 𝑆(𝜒) ≈ 𝑐0 + 𝑐1 𝜒 + 𝑐2 𝜒 2 + ⋯ + 𝑐7 𝜒 7 + 𝑐8 𝜒 8

(2.176)

where, 𝑐0 = 𝑎

(2.177)

𝑐1 = 0

(2.178) 𝛾𝑎

𝑐2 = 2(1+𝛼𝑎)

(2.179)

𝑐3 = 0

(2.180) 𝛾𝑎

𝑐4 = 48(1+𝛼𝑎)3

(2.181)

𝑐5 = 0

(2.182)

5

𝑐6 =

𝛾 2 𝑎(1−5𝛼𝑎−𝑎 2 ) 34560(1+𝛼𝑎)5

𝑐7 = 0 𝑐8 =

(2.183) (2.184)

𝛾 3 𝑎(1−𝑎2 −35𝛼𝑎+90𝛼2 𝑎 2 ) 1393459200(1+𝛼𝑎)7

(2.185)

The Padé approximants is a rational fraction and is equal to Maclaurin series as follows:

Mathematical Preliminaries and Various Methods … 𝑆(𝜒) =

𝑎0 +𝑎1 𝜒+𝑎2 𝜒2 +𝑎3 𝜒3 +𝑎4 𝜒4 1+𝑏1 𝜒+𝑏2 𝜒2 +𝑏3 𝜒3 +𝑏4 𝜒4

49 (2.186)

Substitute Eqn. (2.176) instead of 𝑆(𝜒), we get 𝑐0 + 𝑐1 𝜒 + 𝑐2 𝜒 2 + 𝑐3 𝜒 3 + 𝑐4 𝜒 4 𝑎0 +𝑎1 𝜒+𝑎2 𝜒2 +𝑎3 𝜒3 +𝑎4 𝜒4 5 6 7 8 } = 1+𝑏1 𝜒+𝑏2 𝜒2 +𝑏3 𝜒3 +𝑏4 𝜒4 +𝑐5 𝜒 + 𝑐6 𝜒 + 𝑐7 𝜒 + 𝑐8 𝜒

(2.187)

𝑐0 + (𝑐1 + 𝑐0 𝑏1 )𝜒 + (𝑐2 + 𝑐1 𝑏1 + 𝑐0 𝑏2 )𝜒 2 + (𝑐3 + 𝑐2 𝑏1 + 𝑐1 𝑏2 + 𝑐0 𝑏3 )𝜒 3 + ⇒ = 𝑎0 + 𝑎1 𝜒 + ⋮ +(𝑐8 + 𝑐7 𝑏1 + 𝑐6 𝑏2 + 𝑐5 𝑏3 + 𝑐4 𝑏4 )𝜒 8 } 2 𝑎2 𝜒 + 𝑎3 𝜒 3 + 𝑎4 𝜒 4 (2.188)

Equating the coefficients of 𝜒 0 , 𝜒, 𝜒 2 , 𝜒 3 , 𝜒 4 , 𝜒 5 , 𝜒 6 , 𝜒 7 , 𝜒 8 , we get 𝑐0 = 𝑎0

(2.189)

𝑐1 + 𝑐0 𝑏 = 𝑎1

(2.190)

𝑐2 + 𝑐1 𝑏1 + 𝑐0 𝑏2 = 𝑎2

(2.191)

𝑐3 + 𝑐2 𝑏1 + 𝑐1 𝑏2 + 𝑐0 𝑏3 = 𝑎3

(2.192)

𝑐5 + 𝑐4 𝑏1 + 𝑐3 𝑏2 + 𝑐2 𝑏3 + 𝑐1 𝑏4 = 0

(2.193)

𝑐6 + 𝑐5 𝑏1 + 𝑐4 𝑏2 + 𝑐3 𝑏3 + 𝑐2 𝑏4 = 0

(2.194)

𝑐7 + 𝑐6 𝑏1 + 𝑐5 𝑏2 + 𝑐4 𝑏3 + 𝑐3 𝑏4 = 0

(2.195)

𝑐8 + 𝑐7 𝑏1 + 𝑐6 𝑏2 + 𝑐5 𝑏3 + 𝑐4 𝑏4 = 0

(2.196)

50

L. Rajendran, R. Swaminathan and M. Chitra Devi Since 𝑐1 = 0, 𝑐3 = 0, 𝑐5 = 0, 𝑐7 = 0, solving equations (2.193)-

(2.196), we get 𝑏1 = 𝑏3 = 0,𝑏2 =

𝑐8 𝑐2 −𝑐6 𝑐4 𝑐4 2 −𝑐6 𝑐2

𝑐8 𝑐2 −𝑐6 𝑐4

𝑎0 = 𝑎, 𝑎1 = 0, 𝑎2 = 𝑐2 + 𝑎 ( 𝑐 𝑐 −𝑐 𝑐

𝑐4 2 −𝑐6 𝑐2

and 𝑏4 =

(𝑐6 2 −𝑐4 𝑐8 ) 𝑐2 2 −𝑐2 𝑐6

. Hence,

) , 𝑎3 = 0, 𝑎4 = 𝑐4 +

𝑐 2 −𝑐 𝑐

𝑐2 ( 𝑐8 22−𝑐 6𝑐 4) + 𝑎 (𝑐62−𝑐4𝑐8). 4

6 2

2

2 6

Substituting the values of 𝑎𝑖 , ∀𝑖 =0 to 4 and 𝑏𝑗 , ∀𝑗 =1 to 4 in Eqn. (2.187), we get

𝑆(𝜒) =

𝑐 𝑐 −𝑐 𝑐 𝑐 𝑐 −𝑐 𝑐 𝑐 2 −𝑐 𝑐 𝑎+(𝑐2 +𝑎( 8 22 6 4 ))𝜒2 +(𝑐4 +𝑐2 ( 8 22 6 4 )+𝑎( 6 2 4 8 ))𝜒4 𝑐4 −𝑐6 𝑐2 𝑐4 −𝑐6 𝑐2 𝑐2 −𝑐2 𝑐6 𝑐 𝑐 −𝑐 𝑐 𝑐 2 −𝑐 𝑐 1+( 8 22 6 4 )𝜒2 +( 62 4 8 )𝜒4 𝑐4 −𝑐6 𝑐2 𝑐2 −𝑐2 𝑐6

(2.197)

In order to find the unknown constant “𝑎” we have to substitute the Eqns. (2.177)-(2.185) in Eqn. (2.197), and using the boundary condition 𝑆(1) = 1 and substitute 𝛼 = 0.1, 𝛽 = 0.02, 𝜙𝑠 2 = 0.1, we get the unknown constant 𝑎 = 0.598. Hence, 𝑆(𝜒) =

0.598+0.1878𝜒2 +0.19𝜒4 1−0.1187×10−2 𝜒2 +0.417×10−9 𝜒4

(2.198)

2.3.8.4. Parameter-Expanding Methods Consider the Duffing equation with 5th order nonlinearity 𝑑2 𝑢

𝑎 𝑑𝑥 2 − 𝜀𝑢5 = 0

(2.199)

𝑢′(0) = 0, 𝑢(0) = 𝐴

(2.200)

Re-write Eqn. (2.199) in the form 𝑑2 𝑢

𝑎 𝑑𝑥 2 + 0 𝑢 − 𝜀𝑢5 = 0

(2.201)

Mathematical Preliminaries and Various Methods …

51

Applying the modified Lindstedt-Poincare method, suppose that the solution of Eqn. (2.201), the constant a and 0, can be expressed in the forms 𝑢 = 𝑢0 + 𝜀𝑢1 + 𝜀 2 𝑢2 + ⋯. 0 = 𝜔2 + 𝜀𝑐1 + 𝜀𝑐2 + ⋯. 𝑎 = 1 + 𝜀𝑚1 + 𝜀𝑚2 +. . ..

(2.202)

Substituting Eqns. (2.202) into Eqn. (2.201) and associating the coefficients of like powers of 𝜀 yields the following equations (1 + 𝜀𝑚1 + 𝜀𝑚2 + ⋯ )(𝑢0 + 𝜀𝑢1 + 𝜀 2 𝑢2 + ⋯ . )″ }=0 −𝜀(𝑢0 + 𝜀𝑢1 + 𝜀 2 𝑢2 +. . . . )5 2 2 +(𝜔 + 𝜀𝑐1 + 𝜀𝑐2 +. . . . )(𝑢0 + 𝜀𝑢1 + 𝜀 𝑢2 +. . . . )

(2.203)

𝜀 0 : 𝑢0′′ + 𝜔2 𝑢0 = 0, 𝑢0′ (0) = 0, 𝑢0 (0) = 𝐴

(2.204)

𝜀 1 : 𝑢1′′ + 𝜔2 𝑢1 + 𝑚1 𝑢0′′ + 𝑎1 𝑢0 + 𝑢0 5 = 0, with 𝑢0′ (0) = 0, 𝑢0 (0) = 0

(2.205)

solving Eqn. (2.204) results in 𝑢0 = 𝐴 𝑐𝑜𝑠(𝜔𝑥)

(2.206)

substituting above value in Eqn. (2.205), yields 𝑢1′′ + 𝜔2 𝑢1 − (𝑚1 𝜔2 − 𝑐1 )𝐴 𝑐𝑜𝑠(𝜔𝑥) + 𝐴5 10 𝑐𝑜𝑠(𝜔𝑥) + 5 𝑐𝑜𝑠(3𝜔𝑥) ( )=0 16 + 𝑐𝑜𝑠(5𝜔𝑥)

(2.207)

52

L. Rajendran, R. Swaminathan and M. Chitra Devi 10𝐴4

𝑢1′′ + 𝜔2 𝑢1 − (

16

𝐴5

+ 𝑚1 𝜔2 − 𝑐1 ) 𝐴 𝑐𝑜𝑠(𝜔𝑥) + 16 (5 𝑐𝑜𝑠(3𝜔𝑥) +

𝑐𝑜𝑠(5𝜔𝑥)) = 0

(2.208)

avoiding the presence of a secular term needs 10𝐴4 16

+ 𝑚1 𝜔2 − 𝑐1 = 0

(2.209)

from Eqn. (2.202), Taking the first-order approximate solution for, we have 0 = 𝜔2 + 𝜀𝑐1 𝑎 = 1 + 𝜀𝑚1

(2.210)

from Eqns. (2.209) and (2.210), we get 𝜔=√

10𝜀𝐴4 16𝑎

(2.211)

2.3.9. Parameterized Perturbation Method He’s (J. He 1999; J. H. He 2000a) applied a parameterized perturbation method to find approximate solutions for nonlinear differential-difference equations. Simple but traditional examples are used to demonstrate the validity of the process and its great prospective. The highly accurate results reveal that the technique is beneficial and straightforward. 𝑢 = 𝜀𝑣 + 𝑏

(2.212)

where 𝜀 is the introduced perturbation parameter, b is a constant. We consider Duffing equation with 3rd order nonlinearity, which reads

Mathematical Preliminaries and Various Methods … 𝑢″ + 𝑢 + 𝛼𝑢3 = 0; 𝑢(0) = 𝐴, 𝑢0′ (0) = 0

53 (2.213)

Substituting Eqn. (2.212) into Eqn. (2.213) results in 𝜀𝑣 ″ + (𝜀𝑣 + 𝑏) + 𝛼(𝜀𝑣 + 𝑏)3 = 0; with 𝑣(0) =

𝐴−𝑏 𝜀

, 𝑣 ′ (0) = 0

𝑏

𝑣 ″ + 𝑣 + 𝜀 + 𝛼 (𝜀 2 𝑣 3 + 3𝑏𝜀𝑣 2 + 3𝑏 2 𝑣 +

(2.214) 𝑏3 𝜀

)=0

(2.215)

Setting 𝑏 = 0 for simplicity, we have the following equation with an artificial parameter 𝜀:

{

𝑣 ″ + 𝑣 + 𝛼𝜀 2 𝑣 3 = 0 𝐴 𝑣(0) = 𝜀 , 𝑣 ′ (0) = 0

(2.216)

Assume that the solution of Eqn. (2.216) can be written in the form 𝑣 = 𝑣0 + 𝜀𝑣1 + 𝜀 2 𝑣2 + ⋯ (2.217) Unlike the traditional perturbation methods, we keep 𝑣0 (0) = 𝑣(0), 𝑎𝑛𝑑 ∑𝑖=1 𝑣𝑖 (0) = 0

(2.218)

Substituting Eqn. (2.216) in Eqn. (2.217) (𝑣0 + 𝜀𝑣1 + 𝜀 2 𝑣2 )″ + 𝑣0 + 𝜀𝑣1 + 𝜀 2 𝑣2 + 𝛼𝜀 2 (𝑣0 + 𝜀𝑣1 + 𝜀 2 𝑣2 )3 = 0 (2.219) Associating coefficients of like powers of 𝜀 produces the following equations

54

L. Rajendran, R. Swaminathan and M. Chitra Devi 𝐴

𝑣0″ + 𝑣0 = 0; 𝑣0 (0) = 𝜀 , 𝑣0′ (0) = 0

(2.220)

𝑣1″ + 𝑣1 = 0; 𝑣1 (0) = 0, 𝑣1′ (0) = 0

(2.221)

𝑣2″ + 𝑣2 + 𝛼𝑣03 = 0; 𝑣2 (0) = 0, 𝑣2′ (0) = 0

(2.222)

Solving Eqns. ((2.220)-(2.222)), we get 𝑣0 =

𝐴 𝑐𝑜𝑠(𝑥)

(2.223)

𝜀

𝑣1 = 0 𝑣2 = −

(2.224) 𝐴3 𝑐𝑜𝑠(𝑥) 16𝜀 3

𝑠𝑖𝑛(𝑥) (6𝑥 + 𝑠𝑖𝑛(2𝑥))

(2.225)

The second-order approximate solution of Eqn. (2.217) is 𝑣 = 𝑣0 + 𝜀𝑣1 + 𝜀 2 𝑣2 =

𝐴 𝑐𝑜𝑠(𝑥) 𝜀

𝐴3

− 16𝜀3 𝑠𝑖𝑛(𝑥) (6𝑥 + 𝑠𝑖𝑛(2𝑥)) (2.226)

The approximate solution of the original Eqn. (2.213) is: 𝐴3

𝑢 = 𝜀𝑣 = 𝜀(𝑣0 + 𝜀𝑣1 + 𝜀 2 𝑣2 ) = 𝐴 𝑐𝑜𝑠(𝑥) − 16 𝑠𝑖𝑛(𝑥) (6𝑥 + 𝑠𝑖𝑛(2𝑥))

(2.227)

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Method.” Applied Mathematics and Computation 216 (9): 2623–27. https://doi.org/10.1016/j.amc.2010.03.105. Mitchell, A. R. 1980. The Finite Difference Method in Partial Differential Equations.” Rahamathunissa, G., P. Manisankar, L. Rajendran, and K. Venugopal. 2011. “Modeling of Non-linear Boundary Value Problems in Enzyme-Catalyzed Reaction Diffusion Processes.” Journal of Mathematical Chemistry 49 (2): 457–74. https://doi.org/10.1007/ s10910-010-9752-9. Rebouillat, Serge, Michael E. G. Lyons, and Andrew Flynn. 1999. “Heterogeneous Redox Catalysis at Conducting Polymer Ultramicroelectrodes.” The Analyst 124 (11): 1635–44. https://doi. org/10.1039/a905973c. Roberts, S. M, and J. S. Shipman. 1976. “On the Closed Form Solution of Troesch’s Problem.” Journal of Computational Physics 21 (3): 291–304. https://doi.org/10.1016/0021-9991(76)90026-7. Saha Ray, Santanu. 2015. Fractional Calculus with Applications for Nuclear Reactor Dynamics. Fractional Calculus with Applications for Nuclear Reactor Dynamics. Taylor and Francis group. https://doi.org/10.1201/b18684. Senthamarai, R., and L. Rajendran. 2010. “System of Coupled NonLinear Reaction Diffusion Processes at Conducting PolymerModified Ultramicroelectrodes.” Electrochimica Acta 55 (9): 3223– 35. https://doi.org/10.1016/j.electacta.2010.01.013. Shanmugarajan, Anitha, Subbiah Alwarappan, Subramaniam Somasundaram, and Rajendran Lakshmanan. 2011. “Analytical Solution of Amperometric Enzymatic Reactions Based on Homotopy Perturbation Method.” Electrochimica Acta 56 (9): 3345–52. https://doi.org/10.1016/j.electacta.2011.01.014. Sivasankari, M. K., and L. Rajendran. 2013. “Analytical Expression of the Concentration of Species and Effectiveness Factors in Porous Catalysts Using the Adomian Decomposition Method.” Kinetics

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and Catalysis 54 (1): 95–105. https://doi.org/10.1134/S00231584 13010138. Sivasankari, Mayakkannan, and Lakshmanan Rajendran. 2013. “Analytical Expressions of Concentration of VOC and Oxygen in Steady-State in Biofilteration Model.” Applied Mathematics 04 (02): 314–25. https://doi.org/10.4236/am.2013.42048. Troesch, B. A. 1976. “A Simple Approach to a Sensitive Two-Point Boundary Value Problem.” Journal of Computational Physics 21 (3): 279–90. https://doi.org/10.1016/0021-9991(76)90025-5. Visuvasam, J., A. Meena, and L. Rajendran. 2020. “New Analytical Method for Solving Non-linear Equation in Rotating Disk Electrodes for Second-Order ECE Reactions.” Journal of Electroanalytical Chemistry, March, 114106. https://doi.org/10. 1016/j.jelechem.2020.114106. Wazwaz, Abdul-Majid. 2009. Partial Differential Equations and Solitary Waves Theory. Non-linear Physical Science. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/ 978-3-642-00251-9. Weibel, E. S. 1958. “Confinement of a Plasma Column by Radiation Pressure.” In Plasma in a Magnetic Field, 60. Zarook, S. M., and A. A. Shaikh. 1997. “Analysis and Comparison of Biofilter Models.” Chemical Engineering Journal 65 (1): 55–61. https://doi.org/10.1016/S1385-8947(96)03101-4. Zhang, Sheng, Jing Lin Tong, and Wei Wang. 2009. “Exp-Function Method for a Non-linear Ordinary Differential Equation and New Exact Solutions of the Dispersive Long Wave Equations.” Computers and Mathematics with Applications 58 (11–12): 2294– 99. https://doi.org/10.1016/j.camwa.2009.03.020.

Chapter 3

STEADY AND NON-STEADY STATE REACTION-DIFFUSION EQUATIONS IN PLANE SHEET 3.1. INTRODUCTION Reaction-diffusion equations for the density/concentration/heat of a substance/population in one spatial dimension are considered, where the diffusion coefficient, as well as creation and annihilation terms, are monomials. Diffusion to this surface is effectively planar (the effects of the edges are negligible), hence the system of nonlinear onedimensional reaction-diffusions are given as follows: 𝜕𝑎 𝜕𝑡 𝜕𝑏 𝜕𝑡 𝜕𝑐 𝜕𝑡

=𝐷

𝜕2 𝑎

− 𝑓(𝑎, 𝑏, 𝑐, 𝑡)

(3.1)

= 𝐷 𝜕𝑥 2 + 𝑔(𝑎, 𝑏, 𝑐, 𝑡)

(3.2)

𝜕𝑥 2 𝜕2 𝑏

𝜕2 𝑐

= 𝐷 𝜕𝑥 2 + ℎ(𝑎, 𝑏, 𝑐, 𝑡)

(3.3)

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L. Rajendran, R. Swaminathan and M. Chitra Devi

where 𝑎 = 𝑎(𝑥, 𝑡), 𝑏 = 𝑏(𝑥, 𝑡), 𝑐 = 𝑐(𝑥, 𝑡) are independent state variables and describes density/concentration/ heat of a substance/ population at position 𝑥 ∈ 𝛺 ⊂ 𝑅 𝑛 at time 𝑡 (Ω is a open set). Thus the first term to the right describes the diffusion, including D as the diffusion coefficient. The second term is a smooth function 𝑓, 𝑔, ℎ: 𝑅 → 𝑅 and describes processes that are changing (birth, death, chemical reaction), not just spatially diffuse. Sometimes it is also possible that the second term (reaction term) depends not only on 𝑎 , 𝑏, 𝑐, but also on the first derivative of 𝑎, 𝑏, 𝑐, or explicitly on x.

3.2. CONCENTRATION OF CARBON DIOXIDE (CO2), AND PHENYL GLYCIDYL ETHER SOLUTION (PGE) USING TAYLOR’S SERIES AND PADÉ APPROXIMATION The mass balance equation between the reaction CO2 and phenyl glycidyl ether (PGE) to form the 5-membered cyclic carbonate is given as follows (Choe et al. 2010; Subramaniam, Krishnaperumal, and Lakshmanan 2012): 𝐷𝐴 𝐷𝐵

𝑑 2 𝐶𝐴 𝑑𝑧 2 𝑑 2 𝐶𝐵 𝑑𝑧 2

− 𝑟𝐴 = 0

(3.4)

− 𝑟𝐴 = 0

(3.5)

The reaction rate of CO2 under the condition of a steady state is presented as follows: 𝑟𝐴 =

𝐶𝐵 𝑆𝑡 𝐶 1 1 + + 𝐵 𝑘1 𝐾1 𝑘3 𝐶𝐴 𝑘3 𝐶𝐴

(3.6)

Steady and Non-Steady State Reaction-Diffusion Equations …

65

where 𝑧 is the distance, 𝑧𝐿 is the film thickness, 𝐷𝐴 , 𝐷𝐵 are the diffusivity of CO2 and PGE respectively. Consider the following set of dimensionless variables: 𝑧

𝐶

𝐶

𝜒 = 𝑧 , 𝑃 = 𝐶 𝐴 , 𝑆 = 𝐶 𝐵 , 𝛼1 = 𝐿

𝐶𝐴𝑖 𝐾1 𝑘3

𝐴𝑖

, 𝛽2 =

𝑘1

𝐵0

𝐶𝐵0 𝐾1 𝑘3 𝑘1

,𝑘 =

𝑧𝐿 2 𝐶𝐵0 𝑆𝑡 𝐾1 𝑘3 𝐷𝐴

, 𝛼2 =

𝑧𝐿 2 𝐶𝐴𝑖 𝑆𝑡 𝐾1 𝑘3

𝐶𝐴 𝐶𝐴𝐿

𝐷𝐵

, 𝛽1 = (3.7)

where 𝑃 and 𝑆 represent dimensionless concentrations, and 𝑥 is the distance parameter. We get the dimensionless nonlinear equations as follows: 𝑑2 𝑃 𝑑𝜒2 𝑑2 𝑆 𝑑𝜒2

𝛼 𝑆𝑃

− 1+𝛽 1𝑃+𝛽

2𝑆

1

−

𝛼2 𝑆𝑃 1+𝛽1 𝑃+𝛽2 𝑆

=0

(3.8)

=0

(3.9)

The boundary conditions are 𝑑𝑆 𝑑𝑥

= 0, 𝑃 = 1 at 𝜒 = 0

𝑆 = 1, 𝑃 = 𝑘 at 𝜒 = 1

(3.10) (3.11)

From Eqns. (3.8)-(3.9), we have the following relation 𝑃″ (𝑥) 𝛼1

−

𝑆 ″ (𝑥) 𝛼2

=0

(3.12)

Solving above equation and using the boundary conditions (Eqns. (3.10)-(3.11)) we obtain, approximation of the PGE concentration as follows

66

L. Rajendran, R. Swaminathan and M. Chitra Devi 𝛼

𝑃(𝑥) = 1 + (𝑘 − 1)𝜒 + 𝛼1 ((𝑆(0) − 1)𝜒 − 𝑆(0) + 𝑆(𝑥)) 2

(3.13)

Taylor’s series and Pade approximation method is used to solve nonlinear differential Eqns. (3.8)-(3.11). The basic principle of this method is described in chapter II. A detailed derivation of CO2 from the nonlinear Eqn. (3.8) is described below. The Maclaurin series (Taylor’s series at χ = 0) for the concentration of the substrate is given as follows. 𝑑𝑞 𝑆

𝑆(𝜒) = ∑𝑟𝑞=0 (𝑑𝜒𝑞 |

𝜒𝑞

𝜒=0

) 𝑞! = ∑𝑟𝑞=0 𝐴𝑞

𝜒𝑞

(3.14)

𝑞!

where 𝑑𝑞 𝑆

|

𝑑𝜒𝑞 𝜒=0

= 𝐴𝑞

(3.15)

Now assume that 𝑆(0) = 𝐴0 = 𝑚 where 𝑚is constant. From the boundary conditions, we get 𝐴1 = 𝑆′(0) = 1. Now from Eqn. (3.9) and using 𝑃(𝜒 = 0) = 1 we get 𝑑2 𝑆

𝐴2 = 𝑆 ″ (0) = 𝑑𝜒2|

𝛼 𝑆(𝜒=0)𝑃(𝜒=0)

𝜒=0

= 1+𝛽 2𝑃(𝜒=0)+𝛽

2 𝑆(𝜒=0)

1

=

𝛼1 𝑚 1+𝛽1 +𝛽2 𝑚

(3.16)

Then differentiating Eqn. (3.10) continuously and putting 𝜒 = 0 in the equations results in 𝐴3 = 𝑆 ‴ (0) 𝛼 𝛼2 𝑚(𝑘−1+ 1(𝑚−1))

=

𝛼2

1+𝛽1 +𝛽2 𝑚

(1 − 1+𝛽

𝛽1

1 +𝛽2 𝑚

𝛼 𝑚

𝐴4 = 𝑆 (𝐼𝑉) (0) = 1+𝛽 1+𝛽 1

2𝑚

)

(3.17) (3.18)

Steady and Non-Steady State Reaction-Diffusion Equations …

67

⋮ 𝑑𝑞 𝑆

|

𝑑𝜒𝑞 𝜒=0

= 𝐴𝑞 where 𝑞 = 0 𝑡𝑜 𝑟

(3.19)

Consider, 𝑆(𝜒) ≈ 𝑐0 + 𝑐1 𝜒 + 𝑐2 𝜒 2 + 𝑐3 𝜒 3 + 𝑐4 𝜒 4 + 𝑐5 𝜒 5 + 𝑐6 𝜒 6 + 𝑐7 𝜒 7 + 𝑐8 𝜒 8 + 𝑐9 𝜒 9 + 𝑐10 𝜒10 (3.20)

where, 𝑐𝑞 =

𝐴𝑞

(3.21)

𝑞!

The Padé approximants is a rational fraction and is equal to Maclaurin series as follows: 𝑆(𝜒) =

𝑎0 +𝑎1 𝜒+𝑎2 𝜒2 +𝑎3 𝜒3 +𝑎4 𝜒4 +𝑎5 𝜒5 1+𝑏1 𝜒+𝑏2 𝜒2 +𝑏3 𝜒3 +𝑏4 𝜒4 +𝑏5 𝜒5

(3.22)

Substitute Eqn. (3.16) instead of 𝑆(𝜒), we get 𝑐0 + 𝑐1 𝜒 + 𝑐2 𝜒 2 + 𝑐3 𝜒 3 + 𝑐4 𝜒 4 + 𝑐5 𝜒 5 + }= 𝑐6 𝜒 6 + 𝑐7 𝜒 7 + 𝑐8 𝜒 8 + 𝑐9 𝜒 9 + 𝑐10 𝜒10 𝑎0 +𝑎1 𝜒+𝑎2 𝜒2 +𝑎3 𝜒3 +𝑎4 𝜒4 +𝑎5 𝜒5 1+𝑏1 𝜒+𝑏2 𝜒2 +𝑏3 𝜒3 +𝑏4 𝜒4 +𝑏5 𝜒5

(3.23)

𝑐0 + (𝑐1 + 𝑐0 𝑏1 )𝜒 + (𝑐2 + 𝑐1 𝑏1 + 𝑐0 𝑏2 )𝜒 2 } = 𝑎0 + 𝑎1 𝜒 + 𝑎2 𝜒 2 + 𝑎3 𝜒 3 + 𝑎4 𝜒 4 + +(𝑐3 + 𝑐2 𝑏1 + 𝑐1 𝑏2 + 𝑐0 𝑏3 )𝜒 3 + ⋯ 𝑎5 𝜒 5

(3.24)

Equating the coefficients of 𝜒 0 , 𝜒, 𝜒 2 , 𝜒 3 , 𝜒 4 , 𝜒 5 , 𝜒 6 , 𝜒 7 , 𝜒 8 , 𝜒 9 , 𝜒10 , we get 𝑐0 = 𝑎0

(3.25)

68

L. Rajendran, R. Swaminathan and M. Chitra Devi 𝑐1 + 𝑐0 𝑏 = 𝑎1

(3.26)

𝑐2 + 𝑐1 𝑏1 + 𝑐0 𝑏2 = 𝑎2

(3.27)

𝑐3 + 𝑐2 𝑏1 + 𝑐1 𝑏2 + 𝑐0 𝑏3 = 𝑎3

(3.28)

𝑐4 + 𝑐3 𝑏1 + 𝑐2 𝑏2 + 𝑐1 𝑏3 + 𝑐0 𝑏4 = 𝑎4

(3.29)

𝑐5 + 𝑐4 𝑏1 + 𝑐3 𝑏2 + 𝑐2 𝑏3 + 𝑐1 𝑏4 + 𝑐0 𝑏5 = 𝑎5

(3.30)

𝑐6 + 𝑐5 𝑏1 + 𝑐4 𝑏2 + 𝑐3 𝑏3 + 𝑐2 𝑏4 + 𝑐1 𝑏5 = 0

(3.31)

𝑐7 + 𝑐6 𝑏1 + 𝑐5 𝑏2 + 𝑐4 𝑏3 + 𝑐3 𝑏4 + 𝑐2 𝑏5 = 0

(3.32)

𝑐8 + 𝑐7 𝑏1 + 𝑐6 𝑏2 + 𝑐5 𝑏3 + 𝑐4 𝑏4 + 𝑐3 𝑏5 = 0

(3.33)

𝑐9 + 𝑐8 𝑏1 + 𝑐7 𝑏2 + 𝑐6 𝑏3 + 𝑐5 𝑏4 + 𝑐4 𝑏5 = 0

(3.34)

𝑐10 + 𝑐9 𝑏1 + 𝑐8 𝑏2 + 𝑐7 𝑏3 + 𝑐6 𝑏4 + 𝑐5 𝑏5 = 0

(3.35)

From Eqns. (3.25)-(3.30) we get 𝑎𝑖 , 𝑖 = 0. .5, Solving Eqns. (3.31)(3.35), we get 𝑏𝑖 , 𝑖 = 0. .5. Hence, 𝑎𝑖 𝑎𝑛𝑑 𝑏𝑖 , 𝑖 = 0 to 5 is a function of 𝛼1 , 𝛼2 , 𝛽1 , 𝛽2 , 𝑘 and 𝑚.Using the boundary condition 𝑆(1) = 1 we can find the unknown constant “𝑚.” For example using this method, the analytical expression for the concentrations can be obtained from the above expression for the parameters 𝛼2 = 𝛽1 =1, 𝛼1 = 𝛽2 =0, k = 10 as follows: 1+0.25𝜒 2 +0.375𝜒 3 -0.8334𝜒 4 +2.3156𝜒 5 -6.9045𝜒 6 𝑆(𝜒) = ( )𝑚 +22.1313𝜒 7 -74.5689𝜒 8 +260.7134𝜒 9 -937.8758𝜒 10 (3.36)

Steady and Non-Steady State Reaction-Diffusion Equations …

69

Substituting the Pade approximant coefficients values 𝑎𝑗 , ∀𝑗 = 1𝑡𝑜4 and 𝑏𝑗 , ∀𝑗 = 1to 4 in Eqn. (3.36), we get 𝑆(𝜒) =

(1+8.2059𝜒+20.9598𝜒2 +17.4302𝜒3 +5.5593𝜒4 +6.0805𝜒5 )𝑚 1+8.2059𝜒+20.7098𝜒2 +15.0037𝜒3 −1.8621𝜒4 −0.9139𝜒5

(3.37)

Now, substituting 𝜒 = 1 in Eqn. (3.37) , we get 𝑆(1) =

(1+8.2059+20.9598+17.4302+5.5593+6.0805)𝑚 1+8.2059+20.7098+15.0037−1.8621−0.9139

= 1.40557 𝑚

(3.38)

Since 𝑆(1) = 1, the above equation yields 𝑆(0) = 𝑚 = 0.7114. Substituting this value in the Eqn. (3.37)we get 𝑆(𝜒) =

0.7114+5.8376𝜒+14.9108𝜒2 +12.3998𝜒3 +3.9548𝜒4 +4.3256𝜒5 1+8.2059𝜒+20.7098𝜒2 +15.0037𝜒3 −1.8621𝜒4 −0.9139𝜒5

(3.39)

Figure 3.1. Normalized concentration of CO2 for various values of parameter 𝑘 is plotted using Eqn. (3.22). The key to the graph (solid line) represents the Eqn. (3.22) and (* symbol) represents the numerical simulation.

70

L. Rajendran, R. Swaminathan and M. Chitra Devi

Figure 3.2. Normalized concentration of CO2 for various values of parameter 𝛼1 is plotted using Eqn. (3.22). The key to the graph (solid line) represents the Eqn. (3.22) and (* symbol) represents the numerical simulation.

Figure 3.3. Normalized concentration of CO2 for various values of parameter 𝛼2 is plotted using Eqn. (3.22). The key to the graph (solid line) represents the Eqn. (3.22) and (* symbol) represents the numerical simulation.

Steady and Non-Steady State Reaction-Diffusion Equations …

71

Figure 3.4. Normalized concentration of CO2 for various values of parameter 𝛽1 is plotted using Eqn. (3.22). The key to the graph (solid line) represents the Eqn. (3.22) and (* symbol) represents the numerical simulation.

Figure 3.5. Normalized concentration of CO2 for various values of parameter 𝛽2 is plotted using Eqn. (3.22). The key to the graph (solid line) represents the Eqn. (3.22) and (* symbol) represents the numerical simulation.

72

L. Rajendran, R. Swaminathan and M. Chitra Devi

Thus at 𝜒 = 0 the error estimation between analytical and numerical experimental results is given as follows (for 𝛼2 = 𝛽1 =1,𝛼1 = 𝛽2 =0,k = 10) Error percentage = | 100 = 0.0985

Numerical-Analytical

0.7107−0.7114

Numerical

0.7107

| × 100 = |

|× (3.40)

The error percentage will be decreased successively by increasing the number of terms in Taylor’s series and Pade approximation. Our analytical result for the concentration of CO2 is compared with simulation results (Matlab) for various parameter values in Figures 3.13.5. A agreeable result is noted.

3.3. ANALYTICAL SOLUTION ONE DIMENSIONAL NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATION IN CHEMICAL SCIENCES In this section, we classify typical reactions for the reactiondiffusion equations, which may appear as terms of “reaction.” If the diffusion is ignored (i.e., spatial effects), they are ODEs. The reaction term depends upon various real-life problems such as population predator-prey, competition, symbiosis, chemical reactions. The examples of the above types are given in ref (Kuttler, Christina 2011). Unusually for a chemical reaction, this term depends upon reversible, irreversible, Michalis-Menten, Non-Michael-Menten kinetics, etc. Various methods for solving these nonlinear equations are discussed with examples in the previous chapter-2. In Table 3.1 and 3.2, the recent contribution to mathematical modeling of one dimensional nonsteady and steady state reaction-diffusion equations with various reaction mechanisms and enzymatic scheme with corresponding nonlinear equations and its solutions are given respectively.

1

S. No

The boundary conditions are: when 𝑡 = 0 and 0 ≤ 𝑥 ≤ 𝑑 𝑠 = 0, 𝑝 = 0 when 𝑡 > 0 and 𝑥 = 0 𝑠 ′ = 0, 𝑝 = 0 when 𝑥 = 𝑑 𝑠 = 𝑠0 , 𝑝 = 0

Experimental Nonlinear differential Analytical techniques and equations with initial and techniques enzymatic scheme boundary conditions 𝜕𝑠 𝜕2 𝑠 𝑉 𝑠 (Meena and Amperometric HPM = 𝐷𝑠 2 − 𝑚 𝜕𝑡 𝜕𝑥 𝑠+𝑘𝑚 Rajendran 2010b) 𝐸 + 𝑆 ↔ [𝐸𝑆] → 𝐸 + 2 𝜕𝑝 𝜕 𝑝 𝑉 𝑠 = 𝐷𝑝 2 + 𝑚 𝑃 𝜕𝑡 𝜕𝑥 𝑠+𝑘𝑚

Reference

∞

) + (1 −

1 𝑐𝑜𝑠ℎ(𝜎)

)𝑋 −

∑∞ 𝑛=1 { 𝑛

𝑠𝑖𝑛(𝑛𝜋𝑋)

1 𝑐𝑜𝑠ℎ(𝜎)

𝑛+1 ) + 2𝜋 ∑∞ {(𝑛 + 1/ 𝑛=0(−1)

𝑥 𝑑

,𝑇 =

𝑑2

𝐷𝑠 𝑡

,𝑆 =

𝑠 𝑠0

,𝑃 = √𝑘 , 𝑚 = (𝑛 + 1/2)2𝜋 2 + 𝜎.

𝜇

𝑋=

𝑠0

𝑝

; 𝑟=

𝑠𝑒𝑐(√𝑛2 𝜋 2 − 𝜎 2 ) 𝑒𝑥𝑝(−𝑛2 𝜋 2 𝑇)] where, 𝐷𝑠

𝐷𝑝

𝜕𝑥 𝑋=0

𝜕𝑃

,𝐺 = [ ]

𝑛 2)𝑚−1/2 𝑐𝑜𝑡 √𝑚} 𝑒𝑥𝑝(−𝑚𝑇) + 2 ∑∞ 𝑛=1[(−1) −

𝐺(𝑇) = (1 −

𝑠 𝑀

𝑉 𝑑2

,𝜎 = √ 𝑚 = 𝐷 𝐾

[(−1)𝑛+1 + 𝑠𝑒𝑐(√𝑛2 𝜋 2 − 𝜎 2 )]} × 𝑒𝑥𝑝(−𝑛2 𝜋 2 𝑇)

The current response for saturated catalytic kinetics

𝜋

2

𝑐𝑜𝑠(√𝑚) 𝑠𝑖𝑛(𝑋√𝑚) 𝑛 (𝑛+1/2) [ ]} + 2𝜋 ∑∞ 𝑛=0(−1) { 𝑚 + 𝑐𝑜𝑠((𝑛 + 1/2)𝜋𝑋) − 𝑐𝑜𝑠(𝑋√𝑚)

𝑐𝑜𝑠ℎ(𝜎)

𝑛=0 1−𝑐𝑜𝑠ℎ(𝜎𝑋)

(𝑛 + 1/2) 𝑐𝑜𝑠[(𝑛 + 1/2)] 𝜋𝑋 𝑐𝑜𝑠ℎ(𝜎𝑋) } 𝑒𝑥𝑝(−𝑚𝑇) − 2𝜋 ∑(−1)𝑛 { 𝑐𝑜𝑠ℎ(𝜎) 𝑚

𝑃(𝑋, 𝑇) = (

=

𝑆(𝑋, 𝑇)

Expressions for concentration and current

Table 3.1. Recent contribution to mathematical modelling of one dimensional non-steady state reactiondiffusion equations

3

2

S. No

𝐶𝐵 → Products

𝑘1

(Eswari and Cyclic-Voltammetry Rajendran 2011a) 𝐸𝐴 + 𝑒 ↔ 𝐵

= 0, 𝜕𝑥

𝜕𝑝

=0

= 𝐷𝐵

= 𝐷𝐴 𝜕𝑥 2

𝜕𝑥 2 𝜕2 [𝐵]

𝜕2 [𝐴]

− 𝑘1 [𝐵]

The initial and boundary conditions are [𝐴] → [𝐴]𝑏𝑢𝑙𝑘 , [𝐵] = 0 at 𝑡=0 [𝐴] → [𝐴]𝑏𝑢𝑙𝑘 , [𝐵] → 0 at 𝑥→∞

𝜕𝑡

𝜕𝑡 𝜕 [𝐵]

𝜕 [𝐴]

when 𝑡 > 0 and 𝑥 = 𝑑 𝑠(𝑑, 𝑡) = 𝑠0 , 𝑝(𝑑, 𝑡) = 0

𝜕𝑥

𝜕𝑠

The boundary conditions are: when 𝑡 = 0 and 0 ≤ 𝑥 ≤ 𝑑 𝑠(𝑥, 0) = 0, 𝑠(𝑑, 0) = 𝑠0 , 𝑝(𝑥, 0) = 0 when 𝑡 > 0 and 𝑥 = 0

Laplace Transformation

Experimental Nonlinear differential Analytical techniques and equations with initial and techniques enzymatic scheme boundary conditions 𝜕𝑠 𝜕2 𝑠 𝑉 𝑠 (Meena and Potentiometric HPM = 𝐷𝑠 2 − 𝑚 𝜕𝑡 𝜕𝑥 𝑠+𝑘𝑚 Rajendran 2010b) 𝐸 + 𝑆 ↔ [𝐸𝑆] → 𝐸 + 2 𝜕𝑝 𝜕 𝑝 𝑉 𝑠 = 𝐷𝑝 2 + 𝑚 𝑃 𝜕𝑡 𝜕𝑥 𝑠+𝑘𝑚

Reference

∞

𝑛=0

∞

𝑛=0

𝑥 𝑑

,𝑇 =

𝑑2

𝐷𝑠 𝑡

,𝑆 =

𝑠0

𝑠

,𝑃 =

2𝑛−1

𝑠0

𝑝

; 𝑟=

𝐷𝑠

𝐷𝑝

]+

𝐻

𝑘𝑓

𝐻3

2 𝑋𝑒 𝐻𝑋𝑒 𝐻 𝜏

2

[

𝐻

𝑘𝑓

2𝐻 5

)−

[

𝐻2

𝐾1 𝑘𝑓 −𝑋

2√𝜏

𝑋

2

−

𝑋

1 𝐻3

−

−

𝑒

𝐻2

1

−

𝑋

2𝐻 4

]+

2𝐻 4

𝑋𝑒 𝐻𝑋

+

2𝐻 5

3𝑒 𝐻𝑋

]−

2

𝑋

𝐻2

𝜏

[

𝜏 𝐻2

𝑋2

+

2

+

2𝐻 2 2𝐻 2

𝑋2

[

𝑋

[

]

𝑋 𝐻3

]

𝐻

𝜏

−

𝐻

𝑘

𝜇

=√ ,

+ +

𝑋2

2𝐻

−

𝐻3 −𝑋

+

2

𝐾1 𝑘𝑓 𝑋 2

2𝐻 2 𝐾1 𝑘𝑓

+

2√𝜏

)] −

[

2 𝐾1 𝑘𝑓

2√𝜏

𝑋

2

2 𝐾1 𝑘𝑓

)] + 𝑒𝑟𝑓𝑐 ( 𝑋𝑒 𝐻𝑋

2√𝜏

𝑒𝑟𝑓𝑐 (𝐻√𝜏 +

𝐻3 𝐻𝑋 𝐻 2 𝜏

− [𝑒

𝐾1 𝑘𝑓 3𝑒 𝐻𝑋

2

[𝑒 𝐻𝑋 𝑒 𝐻 𝜏 𝑒𝑟𝑓𝑐 (𝐻√𝜏 +

]−

𝑏 = 𝑒𝑟𝑓𝑐 (

𝐻2

2 𝑒 𝐻𝑋 𝑒𝐻 𝜏

𝑎 =1+

𝐷𝑠 𝐾𝑀

,𝜎 = √

)−

𝜕𝑥 𝑋=0

𝜕𝑃

,𝐺 = [ ]

} 𝑒𝑥𝑝(−(𝑛 − 1)2 𝜋 2 𝑇)

𝑐𝑜𝑠((𝑛−1/2)𝜋𝑋)

𝑚 = (𝑛 + 1/2)2 𝜋 2 + 𝜎. For small 𝐾1, the concentration of a and b becomes

𝑋=

Where

𝜋

𝑛 + ∑∞ 𝑛=0(−1) {

4

(𝑛 + 1/2) 𝑐𝑜𝑠[(𝑛 + 1/2)] 𝜋𝑋 } 𝑒𝑥𝑝(−𝑚𝑇) − 2𝜋 ∑(−1)𝑛 { 𝑚

𝑉𝑚 𝑑2

(𝑛 + 1/2) 𝑐𝑜𝑠[(𝑛 + 1/2)] 𝜋𝑋 𝑐𝑜𝑠ℎ(𝜎𝑋) } 𝑒𝑥𝑝(−𝑚𝑇) − 2𝜋 ∑(−1)𝑛 { 𝑐𝑜𝑠ℎ(𝜎) 𝑚

𝑃(𝑋, 𝑇) 1 − 𝑐𝑜𝑠ℎ(𝜎𝑋) ) =( 𝑐𝑜𝑠ℎ(𝜎)

=

𝑆(𝑋, 𝑇)

Expressions for concentration and current

Table 3.1. (Continued)

S. Reference No

Experimental techniques and enzymatic scheme

𝑅𝑇

𝐸 = 𝐸1 + 𝑣𝑡 for (𝑡 < 𝑡𝑠𝑤𝑖𝑡𝑐ℎ ) 𝐸 = 𝐸1 + 𝑣𝑡𝑠𝑤𝑖𝑡𝑐ℎ − 𝑣(𝑡 − 𝑡𝑠𝑤𝑖𝑡𝑐ℎ ) for (𝑡 > 𝑡𝑠𝑤𝑖𝑡𝑐ℎ )

𝐸𝑓0 (𝐴/𝐵)]

) [𝐵]

) [𝐴] −

𝑅𝑇 (1−𝛼)𝐹𝑊

𝛼𝐹𝑊

where 𝑊 = [𝐸 −

𝑘 0 exp (−

𝑘 0 exp (−

𝜕𝑥

Nonlinear differential Analytical equations with initial and techniques boundary conditions at 𝑥 = 0 𝜕[𝐴] 𝜕[𝐵] 𝐷𝐴 = −𝐷𝐵 𝜕𝑥 𝜕𝑥 𝜕[𝐴] 𝐷𝐴 [ ] = 2

− [ 2 2𝑘𝑓

2

𝑘 𝑋 𝑘 𝜏 𝑋𝑒 𝑓 𝑒 𝑓

−

𝑘𝑓 2

]

𝑋 2√𝜏

𝑋 2√𝜏

)+

2

+

𝜏=𝑡

0.28217(𝑘𝑏 + √𝐾1 )2 𝜏 −3⁄2 𝑘𝑓2 𝐾1

=

𝐹𝐸 𝑘1 𝑅𝑇 ,𝐾 = 𝑅𝑇 1 𝑣𝐹

𝐹𝑣 𝐹𝑣 𝑘 0 𝑅𝑇𝐷 [𝐴] [𝐵] , 𝑋 = 𝑥√ ,𝛬 = √ ,𝑎 = ,𝑏 = ,𝛩 𝑅𝑇 𝑅𝑇𝐷 𝐷 𝐹𝑣 [𝐴]𝑏𝑢𝑙𝑘 [𝐴]𝑏𝑢𝑙𝑘

where

𝜓 = 0.56433𝜏 −1⁄2 −

− 1.12867𝑘𝑓 (𝑘𝑏 + 𝑘𝑓 )√𝜏 + 𝑘𝑓 (𝑘𝑏 + 𝑘𝑓 )2 𝜏 The current when 𝜏 is large

2𝐾1

2

2 𝑒 𝑘𝑓 𝜏 𝑒 −√𝐾1 𝑋 𝑋𝑘𝑓

𝜓 = 𝑘𝑓 + 0.376228[(𝑘𝑏 + 𝑘𝑓 )𝐾1 − 2(𝑘𝑏 + 𝑘𝑓 )3 − 𝐾1 𝑘𝑓 ]𝜏 −1⁄2

𝐾1

𝑘 𝜏 𝑘𝑓 𝑘𝑏 𝑒 −√𝐾1 𝑋𝜏 𝑒 𝑓

) + 𝑒𝑟𝑓𝑐 (

𝑒𝑟𝑓𝑐(𝑘𝑓 √𝜏)] −

The current when 𝜏 is small

2𝜏 𝑘𝑓

3 2𝑘𝑓

𝑘 𝑋 𝑘 𝜏 3𝑒 𝑓 𝑒 𝑓

2𝑘𝑓 𝑒 𝑘𝑓𝑋 𝑒 𝑘𝑓𝜏 −

𝑏 = 𝑘𝑓 𝑒 −√𝐾1 𝑋 √1⁄𝐾1 [𝑒

√𝐾1

𝑘𝑓 𝑘𝑏

𝑘 𝑋 𝑘2 𝜏 𝑒 𝑓 𝑒 𝑓 𝜏

𝑎 = 1 + 𝑒 𝑘𝑓𝑋 𝑒 𝑘𝑓 𝜏 𝑒𝑟𝑓𝑐 (𝑘𝑓 √𝜏 +

2

For large 𝐾1, the concentration of a and b becomes

Expressions for concentration and current

Reference

(Eswari and Rajendran 2011b)

S. No

4

𝑅𝑇

) [𝐵]

) [𝐴] −

𝑅𝑇 (1−𝛼)𝐹𝑊

𝛼𝐹𝑊

𝐸 = 𝐸1 + 𝑣𝑡 (𝑡 < 𝑡𝑠𝑤𝑖𝑡𝑐ℎ ) 𝐸 = 𝐸1 + 𝑣𝑡𝑠𝑤𝑖𝑡𝑐ℎ − 𝑣(𝑡 − 𝑡𝑠𝑤𝑖𝑡𝑐ℎ ) for (𝑡 > 𝑡𝑠𝑤𝑖𝑡𝑐ℎ )

where 𝑊 = [𝐸 − 𝐸𝑓0 (𝐴/𝐵)]

𝑘 0 exp (−

0

𝑘 exp (−

𝜕𝑥

Experimental Nonlinear differential Analytical techniques and equations with initial and techniques enzymatic scheme boundary conditions Cyclic Voltammetry 𝜕 [𝐴] = 𝐷 𝜕2 [𝐴] HPM 𝐴 𝜕𝑥 2 𝜕𝑡 𝐸: 𝐴 + 𝑒 ↔ 𝐵 2 𝜕 [𝐵] 𝜕 [𝐵] = 𝐷𝐵 2 − 𝑘1 [𝐵]2 𝐶2 : 𝐵 + 𝜕𝑡 𝜕𝑥 𝑘1 The initial and boundary 𝐵 → Products conditions are [𝐴] → [𝐴]𝑏𝑢𝑙𝑘 , [𝐵] = 0 at 𝑡=0 [𝐴] → [𝐴]𝑏𝑢𝑙𝑘 , [𝐵] → 0 at 𝑥→∞ at 𝑥 = 0 𝜕[𝐴] 𝜕[𝐵] 𝐷𝐴 = −𝐷𝐵 𝜕𝑥 𝜕𝑥 𝜕[𝐴] 𝐷𝐴 [ ] = +

 𝐻 [𝑒

2𝜏

7/2

𝑒𝑟𝑓𝑐 (𝐻√𝜏 +

2√𝜏

𝑋

𝑋 2√𝜏

)−

+

𝐻3

−

𝐻4

2𝑘 𝐾1 𝑘𝑓 𝑏

𝐻3

2𝑘 𝑋 2 0.1667𝐾1 𝑘𝑓 𝑏

𝑋

−

𝐻4

𝐻5

𝑋 2√𝜏

] 𝜏2

)+

𝐻3

2 0.667𝐾1 𝑘𝑓

−

𝐻2

−

+

2𝑋 2 0.334𝐾1 𝑘𝑓

𝐻3

2𝑘 𝑋 0.2508𝐾1 𝑘𝑓 𝑏

)] + [ ] 𝜏 + 0.7523 [ 2𝑘 0.1667𝐾1 𝑘𝑓 𝑏 𝐻3 2 0.1667𝐾1 𝑘𝑓 𝐻2

+

2𝑘 2𝐾1 𝑘𝑓 𝑏

𝑋 2√𝜏

]𝜏 + [

)] + 𝑒𝑟𝑓𝑐 ( 2𝑘 2𝐾1 𝑘𝑓 𝑏 𝐻5

2√𝜏

[𝑒 𝐻𝑋 𝑒 𝐻 𝜏 𝑒𝑟𝑓𝑐 (𝐻√𝜏 +

] 𝜏 3/2 + [ −

+

𝐻

2𝑘 𝑋 𝐾1 𝑘𝑓 𝑏

)−

2

2√𝜏

𝑋

𝑘𝑓

] 𝜏2

] 𝜏 3/2 − 0.167 [

2𝑘 𝑋 0.333𝐾1 𝑘𝑓 𝑏

𝐻4

2 𝐾1 𝑘𝑓

2

2 𝑘 −1.667𝐾 𝑘 2 𝑘 )𝑋 (0.66𝐾1 𝑘𝑓 1 𝑓 𝑏 𝑏 𝐻4

[𝑒 𝐻𝑋 𝑒 𝐻 𝜏 𝑒𝑟𝑓𝑐 (𝐻√𝜏 +

2𝐻 2 𝐾1 𝑘𝑏 𝑘𝑓 𝐻5

+

𝐻

𝑘𝑓

𝑏(𝑋, 𝜏) = 𝑒𝑟𝑓𝑐 (

2𝑘 0.7524𝐾1 𝑘𝑓 𝑏 𝐻4

2𝑘 𝑋 2 −0.167𝐾1 𝑘𝑓 𝑏 𝐻3

𝑎(𝑋, 𝜏) = 1 +

The approximation of concentration a and b when 𝜏 is large

[

7

)] − 0.056𝐾1 𝑘𝑓2 𝜏 3 +

)] + 𝑒𝑟𝑓𝑐 (

𝑒𝑟𝑓𝑐(𝐻√𝜏) + 0.056𝐾1 𝑘𝑓2 𝑘𝑏 𝜏 3 − 0.0859𝐾1 𝑘𝑓2 𝑘𝑏 𝐻𝜏 2 +

𝜏

𝑋

2√𝜏 4

−3𝐾1 𝑘𝑓2 𝑘𝑏 𝐻2 + 1.667𝐾1 𝑘𝑓3 𝑘𝑏 𝐻 ] 𝜏4 0.0417 [ +1.667𝐾1𝑘𝑓2𝑘𝑏2 𝐻 + 3.333𝐾1 𝑘𝑓2 𝑘𝑏 𝐻2

𝜓 = 𝑘𝑓 𝑒 𝐻

𝑒

0.0287𝐾1𝑘𝑓2 𝑘𝑏 ]

)

+

2√𝜏

𝑋

The normalized current

[0.057𝐾1 𝑘𝑓2 𝐻

𝑏(𝑋, 𝜏) = 𝑒𝑟𝑓𝑐 (

0.0417𝐾1 𝑘𝑓2 𝑘𝑏 𝐻𝜏 𝑘𝑓 𝐻𝑋 𝐻 2 𝜏

2

[𝑒 𝐻𝑋 𝑒 𝐻 𝜏 𝑒𝑟𝑓𝑐 (𝐻√𝜏 +

𝐻 7/2

𝑘𝑓

0.0287𝐾1 𝑘𝑓2 𝑘𝑏 𝜏

𝑎(𝑋, 𝜏) = 1 +

The approximation of concentration a and b when 𝜏 is small

Expressions for concentration and current

Table 3.1. (Continued)

Reference

(Indira and Rajendran 2012)

S. No

5

Nonlinear differential Analytical equations with initial and techniques boundary conditions

=𝐷 𝜕𝑥 2

𝜕2 𝑐

𝜕𝑥

+ 𝑘𝑓 𝑎 − 𝑘𝑏 𝑏𝑐 =

𝜕𝑥

𝜕𝑎

= 0, 𝑏 = 0, 𝜕𝑥

𝜕𝑐

=0

𝑛𝐹𝐴𝐷

|

𝜕𝑥 𝑥=1

𝜕𝑏

𝑥 → ∞, 𝑎 → 𝑎0 , 𝑏 → 𝑏0 , 𝑐 → 𝑐0 The current equation is 𝑖 =

𝑥 = 𝑙,

0 The boundary conditions are 𝑡 = 0, 𝑎 = 𝑎0 , 𝑏 = 𝑏0 , 𝑐 = 𝑐0

𝜕𝑡

𝜕𝑐

0

𝜕𝑡

Chronoamperometric 𝜕𝑎 = 𝐷 𝜕2 𝑎 − 𝑘 𝑎 + 𝑘 𝑏𝑐 = HPM 𝑓 𝑏 𝜕𝑡 𝜕𝑥 2 𝐴↔𝐵+𝐶 0 − 𝐵 ± 𝑒 → 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠 𝜕𝑏 𝜕2 𝑏 = 𝐷 2 + 𝑘𝑓 𝑎 − 𝑘𝑏 𝑏𝑐 =

Experimental techniques and enzymatic scheme

1

𝑒

(𝑋−1)2 − 4𝑇

]} −

𝜋

1

𝑒

𝐻4

2𝑘 𝐾1 𝑘𝑓 𝑏

] 𝜏+

]} +

)+

𝑇

𝜋

2

[2√ 𝑒

𝑘5

(𝑋−1)2 − 4𝑇

𝑇 𝜋 √𝑇

+ 𝑘3 √

0.56419

2√𝑇

2√𝑇

2√𝑇

) 𝑒𝑟𝑓𝑐 (

2√𝑇

)−

𝑋−1

)] (𝑋 − 1)

2√𝑇

𝑋−1

𝑋−1

) + √𝑇(𝑋 − 1) [(

2√𝑇

𝑋−1

)] (𝑋 −

)−

𝑋−1

)] (𝑋 − 1)

𝑋−1

) 𝑒𝑟𝑓𝑐 (

− (𝑋 − 1)𝑒𝑟𝑓𝑐 (

− (𝑋 − 1)𝑒𝑟𝑓𝑐 (

2√𝑇

𝜋

𝑇

[2√ 𝑒 𝑋−1

2

𝑘3

{𝑇𝑒𝑟𝑓𝑐 (

2√𝑇

(𝑋−1)2 − 4𝑇

− (𝑋 − 1)𝑒𝑟𝑓𝑐 (

2√𝑇

𝑋−1

Where 𝑘𝑓 𝑙 2 𝑎 𝑏 𝑐 𝑥 𝐷𝑡 𝑘𝑏 𝑙 2𝑏0 𝑐0 𝑢 = , 𝑣 = , 𝑐 = , 𝑋 = , 𝑇 = 2 , 𝑘1 = , 𝑘2 = , 𝑘3 𝑎0 𝑏0 𝑐0 𝑙 𝑙 𝐷 𝐷𝑎0 2 2 2 𝑘𝑓 𝑙 𝑎0 𝑘𝑓 𝑙 𝑎0 𝑘𝑏 𝑙 𝑐0 𝑘𝑏 𝑙 2𝑏0 = , 𝑘4 = , 𝑘5 = , 𝑘6 = 𝐷𝑏0 𝐷 𝐷𝑐0 𝐷

𝜓=

] 𝜏 1/2 − [

) + √𝑇(𝑋 − 1) [(

The analytical expression for current

√𝜋

(𝑋−1)2 − 4𝑇

2

𝑘5

2√𝑇

𝑋−1

𝑘1𝑅𝑇 𝑣𝐹

(𝑋−1)2 − 4𝑇

𝑋−1

𝑇

[2√ 𝑒

2

1)𝑤(𝑋, 𝑇) = 1 +

=

{𝑇𝑒𝑟𝑓𝑐 (

𝑘1

2

𝑘1

𝑣(𝑋, 𝑇) = 4 − 𝑒𝑟𝑓𝑐 (

√𝜋

𝐻5

2𝑘 2.2573𝐾1 𝑘𝑓 𝑏

𝐹𝑣 𝐹𝑣 𝑘 0 𝑅𝑇𝐷 [𝐴] [𝐵] 𝐹𝐸 √ , 𝑋 = 𝑥√ ,𝛬 = ,𝑎 = ,𝑏 = ,𝛩 = ,𝐾 𝑅𝑇 𝑅𝑇𝐷 𝐷 𝐹𝑣 [𝐴]𝑏𝑢𝑙𝑘 [𝐴]𝑏𝑢𝑙𝑘 𝑅𝑇 1

𝐻3

] 𝜏 3/2

𝑒𝑟𝑓𝑐(𝐻√𝜏) + [

𝑢(𝑋, 𝑇) = 1 −

𝜏=𝑡

where

0.2508 [

2𝜏

2𝑘 𝐾1 𝑘𝑓 𝑏

𝜓 = 𝑘𝑓 𝑒 𝐻

The normalized current is

Expressions for concentration and current

Reference

(Kirthiga, Rajendran, and Fernandez 2018)

(Rahamathunissa and Rajendran 2008a)

S. No

6

7 𝐾2

𝑜

−

𝜕𝑥

𝜕𝐶𝑠

(𝐸 − 𝐸0 )}

𝐷𝑠 𝜕𝜒

− 2

𝜕2 𝑆

𝜕𝑥

𝜕𝐶𝑅

1+𝑆/𝐾𝑀

𝐾𝑆

𝑗 = 𝑛𝑒 𝐹𝐷𝑅 ( 𝑥=0

=0

)

The current density𝑗(𝜇𝐴/ 𝑐𝑚2 ) is given by

𝑅𝑇

𝑛𝑒 𝐹

)=0

Where 𝜂 = {

𝑒 𝜂 ]−1 , (

when 𝑡 = 0, 𝑥 ≥ 0 𝑎𝑛𝑑 𝑡 > 0, 𝑥 → ∞ 𝐶𝑅 = 𝐶𝑅 ∞ , 𝐶𝑠 = 𝐶𝑠 ∞ when 𝑡 > 0, 𝑥 = 0 𝐶𝑅 = 𝐶𝑅 ∞ [1 +

𝜕𝑥 2

𝜕2 𝐶𝑠(𝑥,𝑡)

𝐾𝑀𝑆 𝐾 + 𝑀𝑀 +1 𝐶𝑠 (𝑥,𝑡) 𝐶𝑜(𝑥,𝑡)

𝜕𝑡 𝑘𝑐𝑎𝑡 𝐶𝐸

= 𝐷𝑆

𝐸 + 𝑆 ↔ 𝐸𝑆 → 𝐸 + S(0, χ) = 𝑆 (χ) (χ ∈ Ω) 0 𝑃

𝐾𝑀

Amperometric

𝑆

𝜕𝐶𝑠 (𝑥,𝑡)

Experimental Nonlinear differential techniques and equations with initial and enzymatic scheme boundary conditions Cyclic voltammetry 𝜕𝐶𝑅 (𝑥,𝑡) = 𝐷 𝜕2 𝐶𝑅(𝑥,𝑡) + 𝑅 𝜕𝑡 𝜕𝑥 2 𝑅 ↔ 𝑂 + 𝑒− 2𝑘𝑐𝑎𝑡 𝐶𝐸 𝐾𝑀𝑆 𝐾 2𝑂 + 𝑆 → 2𝑅 + 𝑃 + 𝑀𝑀 +1 𝐶 (𝑥,𝑡) 𝐶 (𝑥,𝑡)

VIM

HPM& Danckwerts’ Method

Analytical techniques

𝐾𝑀𝑀

𝑎=

𝑒𝜂

𝑗 ∞ 𝑛𝑒 𝐹𝐶𝑅 √𝛼1 𝐷𝑅

∞ 𝐶𝑅

2𝑘𝑐𝑎𝑡 𝐶𝐸

√𝜋𝜏

𝑒 −𝛼′.𝜏

]

, 𝑋 = 𝑥√

= 𝑎 [√𝛼′𝑒𝑟𝑓(√𝛼′𝜏) +

, 𝛼1 =

2√𝜏

𝛼1 𝐷𝑅

2√𝜏

, 𝜏 = 𝛼1 𝑡, 𝛽 =

𝐶𝑆∞

𝐾𝑀𝑆

,𝛾 =

2𝛼

𝐾𝑥 2

+

𝑎𝛼2

(1+𝑎𝛼)

𝐾𝑎𝛼 2+𝑎𝛼)2

𝑙𝑛 [

𝑢=

𝑠 𝑘 𝑠∞

where

𝐾 2𝛼

𝐷𝑠

𝑘 𝐿2

= 𝜙2 , 𝑥 =

𝐿

𝜒

𝑥 2 , when 𝛼𝑢 ≫ 1

,𝐾 =

𝑢(𝑥) = 𝑎 +

,𝛼 =

𝐾𝑀

𝑘 𝑠∞

𝐿2

𝐷𝑠 𝑡

.

√2𝐾 √𝐾𝑎𝛼 𝑥 tan−1 [ 𝑥] 𝛼3/2 √𝑎 √2+𝑎𝛼)

,𝜏 =

𝑥 2 + 1] − 𝑢(𝑥) = 𝑎 cosh(√𝐾𝑥) when 𝛼𝑢 ≪ 1

𝑢(𝑥) = 𝑎 +

In this case 𝜓 = 𝑎/√𝜋𝜏. This is the Cottrell equation of current for planar electrode.

𝜓𝑠𝑠 = 𝑎√𝛼′ at (𝜏 → ∞) When 𝐾𝑀𝑆 or 𝐾𝑀𝑀 are very large (𝛽 or 𝛾 is large) 𝛼′ = 0.

𝜓=

1 (𝛽𝑎+𝛾+𝑎)

)

, 𝛼′ =

2√𝜏

𝑋

The current density

(1+𝑒 𝜂 )

where

∞ 𝐶𝑅

𝑋−2𝜏√𝛼′

) + 1] + 2𝑒 −𝜏𝛼′ − 2𝑒 −√𝛼′𝑋 𝑒 −√𝛼′𝑋 [𝑒𝑟𝑓 ( 𝑎 2√𝜏 }+ = 𝑢(𝑋, 𝜏) = − { 𝑋+2𝜏√𝛼′ 𝑋 2 ) − 1] − 2𝑒 −𝜏𝛼′ 𝑒𝑟𝑓 ( ) +𝑒 √𝛼′𝑋 [𝑒𝑟𝑓 (

𝑎𝑒 −𝛼′𝜏 𝑒𝑟𝑓𝑐 (

∞ 𝐶𝑅

𝐶𝑜

When 𝜏 is very small.

Expressions for concentration and current

Table 3.1. (Continued)

Reference

(Rahamathunissa and Rajendran 2008b)

S. No

8

𝐶 →𝐶

′

′ 𝑘𝐸

[𝑃𝐶 ′ ] → 𝑃 + 𝐶 ′

𝑘𝑐

𝑆 + 𝐶 → [𝑆𝐶] → [𝑃𝐶 ′ ]

𝐾𝑀

Experimental techniques and enzymatic scheme Amperometric

s(L, t) = κs ∞

∂x x=0

Nonlinear differential Analytical equations with initial and techniques boundary conditions ∂s(x,t) ∂2 s(x,t) k s(x,t) = Ds − s(x,t) Danckwerts’ ∂t ∂x2 1+ K M Method, Boundary conditions: Variable s(x, 0) = 0 Separable ∂s Method [ ] =0 𝜋

4𝐾 2

1 2 2

𝜋2

2𝛼𝐾2 1 2

1 1 2

−

𝐾 2𝛼

u=

3𝛼

𝐾2

− 2𝐾 ∑∞ 𝑛=1 𝐴𝑛

(2𝑛−1)𝜋

2

1 2

𝑒𝑥𝑝 [− {(𝑛 − ) 𝜋 2 } 𝜏] s x Ds t κs ∞ k L2 ,χ = ,τ = 2 ,α = ,K = ∞ κs L L KM Ds

where

𝑦(𝜏) = 𝐾 −

(−1)𝑛

(𝜒 2 − 1) − ∑∞ 𝑛=1 𝐴𝑛 𝑐𝑜𝑠[(𝑛 − 1/2)𝜋𝜒]𝑒𝑥𝑝 [− {(𝑛 −

The transient current

2

) 𝜋 2 } 𝜏]

1 2

𝑢(𝜒, 𝜏) = 1 +

𝑦(𝑠𝑡𝑒𝑎𝑑𝑦 𝑠𝑡𝑎𝑡𝑒) = 𝛼√𝐾 tanh(√𝐾) Concentration for saturated (zero order) catalytic kinetics when 𝛼𝑢 ≫ 1

}

(𝑛+ ) [(𝑛+ ) 𝜋2 +𝐾] 2 2

2

} 𝑒𝑥𝑝 [− {(𝑛 + ) 𝜋 2 + 𝐾} 𝜏]}

1 2 𝑒𝑥𝑝[−{(𝑛+2) 𝜋2 +𝐾}𝜏] 𝑛 1 2 (𝑛+ ) 𝜋2 +𝐾 2

∑∞ 𝑛=0

2𝛼𝐾 ∑∞ 𝑛=0(−1) {

𝑦 = 𝛼𝐾 −

The transient normalized current is

(𝑛+ ) 𝜋2 +𝐾

1 2

1

1 2

{cos[(𝑛 + ) 𝜋𝜒]} −

(−1)𝑛+1 (2𝑛+1)[(𝑛+ ) 𝜋 2 +𝐾] 2

(𝑛+1/2)cos[(𝑛+1/𝜋𝜒]

∑∞ 𝑛=0

𝑛 {2𝜋 ∑∞ 𝑛=0(−1) {

𝑢(𝜒, 𝜏) = 1 +

Concentration for unsaturated (first order) catalytic kinetics is: when 𝛼𝑢 ≪ 1

Expressions for concentration and current

Reference

(Manimozhi, Subbiah, and Rajendran 2010)

(Kirthiga and Rajendran 2015)

S. No

9

10

𝜕𝑡

𝐾𝑚 +𝑆(𝑋,𝑇)

𝑠 𝜕𝜒2

𝑆

𝐾𝑚 +𝑆(𝑋,𝑇)

𝑆(𝑋,𝑇)

= 𝐷𝑝 𝜕𝑋 2

+

𝜕𝑆 𝜕𝑋

= 0, 𝑃 =

𝑗 = 𝑛𝑒 𝐹𝐷𝑝 ( 𝜕𝑋 𝑋=0

)

𝜕𝑃(𝑋,𝑇)

0 𝑇 ≥ 0, 𝑋 = 𝑑: 𝑆 = 𝑆0 , 𝑃 = 0 Current density j (A/cm)

𝑇 > 0, 𝑋 = 0:

The initial and boundary conditions are: 𝑇 = 0,0 ≤ 𝑋 < 𝑑: 𝑆 = 0, 𝑃 = 0

𝑉𝑚

𝜕𝑇

𝜕𝑃(𝑋,𝑇)

𝜕2 𝑃(𝑋,𝑇)

𝑆2

1+𝐾 +𝐾 𝐾 𝑀 𝑖 𝑀

HAM

HPM

Nonlinear differential Analytical equations with initial and techniques boundary conditions 𝜕𝑆 𝜕2 𝑆 𝐾𝑆 VIM =𝐷 −

𝐸 + 𝑃 Boundary conditions: 𝑡 = 0, 𝑆 = 0 𝐸𝑆 + 𝑆 ↔ 𝐸𝑆3 𝜒 = 0, 𝑆′ = 0 𝜒 = 0, S = 1 𝜕𝑆(𝑋,𝑇) 𝜕2 𝑆(𝑋,𝑇) Amperometric = 𝐷𝑠 − 𝜕𝑇 𝜕𝑋 2 𝐸 + 𝑆 ↔ 𝐸𝑆 → 𝐸 + 𝑆(𝑋,𝑇) 𝑉𝑚 𝑃

𝑘𝑐 𝑘 𝑆 →𝑘1−1 𝐸𝑆 →

Experimental techniques and enzymatic scheme Amperometric 𝐸+ 2

𝐾−𝑎)

2

− 𝑎[𝛼(1 − 𝑎) + 𝛽−𝑎)2 , 𝐵 =

𝑠 𝑘 𝑠∞

𝜋

𝑐𝑜𝑠(√[𝑎/(𝑟−1)])

]𝑒

+ 𝑎𝑟 𝑡 𝑟−1

1

1

𝑟

[

] + 𝑎𝜋 ∑∞ 𝑛=0 𝑟 𝑐𝑜𝑠ℎ(√𝑎)

𝐿2

𝐷𝑠 𝑡

,𝛽 =

[𝑥 +

2

𝐾

𝐾𝑖 𝐾𝑀

𝐾

𝑛

[𝑥 4 +

(−1)𝑛 (1+2𝑛)

[

[

𝑠𝑖𝑛(2√[𝑎/(𝑟−1)])

+

24 𝐾2

]

+ 𝑐𝑜𝑠[ (1 +

+

𝑚=𝑎+

where 4

𝜋 2 (2𝑛+1)2

and 𝑎 =

1+𝑘

𝜇

Dimensionless current density becomes for the case of steady state conditions (𝑡 → ∞), 1 1 𝐺=[ − ] 𝑟 𝑟 𝑐𝑜𝑠ℎ( √𝜇/(1 + 𝑘))

+

𝑠𝑖𝑛(√[𝑎/(𝑟−1)])

(−1)𝑛 (1+2𝑛) 𝑐𝑜𝑡(√𝑚/𝑟) −𝑚𝑡 𝑒 √𝑚𝑟[𝑎𝑟−𝑚(𝑟−1)]

−

𝑠𝑖𝑛(√[𝑎/(𝑟−1)]𝑥)

−

𝑠𝑖𝑛(√(𝑚/𝑟))

𝑠𝑖𝑛(√(𝑚/𝑟)(𝑥−1)) 𝑠𝑖𝑛[𝑛𝜋(𝑥−1)]

𝑚[𝑎𝑟−𝑚(𝑟−1)]

𝐾

12𝑥 2

, 𝐴 + 𝐵 + 𝐶 = 𝑎.

]−

15

−𝛽𝑎 3

2

𝑘 𝑠∞

,𝐶 =

𝑛[𝑎𝑟−𝑛2 𝜋 2 𝑟(𝑟−1)] 𝑐𝑜𝑠ℎ(√𝑎−𝑛2 𝜋 2 𝑟)

(−1)𝑛

+ 𝑎𝜋 ∑∞ 𝑛=0

(−1)𝑛 −𝑠𝑒𝑐(√𝑎−𝑛2 𝜋 2 ) −𝑛2 𝜋 2 𝑟𝑡 2𝑎 ∑∞ 𝑛=1 [ 𝑎𝑟−𝑛2 𝜋 2 𝑟(𝑟−1) ] 𝑒

𝑟

𝐺=[ −

,𝜏 =

𝐾

(𝑟−1) 2 𝑠𝑖𝑛(√[𝑎/(𝑟−1)](𝑥−1))

∑∞ 𝑛=1

𝐾𝑀

𝑘 𝑠∞

Dimensionless current density is given by

−

−𝑛2 𝜋2 𝑟𝑡

𝑐𝑜𝑠(√[𝑎/(𝑟−1)]𝑥)

𝑠𝑖𝑛( 𝑛𝜋𝑥)] 𝑒

2

2𝑎

𝑟 𝑐𝑜𝑠ℎ(√𝑎)

𝑛=0 1−𝑥−𝑐𝑜𝑠ℎ(√𝑎𝑥)

2𝑛) 𝑥]] 𝑒 −𝑚𝑡 −

𝑟

𝑥

,𝛼 =

−

𝑚

(−1)𝑛 2𝑛 + 1 (2𝑛 + 1) 𝑐𝑜𝑠 ( 𝜋𝑥) 𝑒 −𝑚𝑡 𝑚 2

𝐿

𝜒

−𝜋∑

∞

= 𝜙2, 𝑥 =

𝑐𝑜𝑠ℎ( √𝑎)

𝑣(𝑥, 𝑡) = + 𝜋

𝐷𝑠

𝑘 𝐿2

𝑐𝑜𝑠ℎ( √𝑎𝑥)

,𝐾 =

𝑢(𝑥, 𝑡) =

𝑢=

Where

𝐾

1

12

𝑎𝐾−2𝑎 2 [𝛼+2𝛽−𝑎)]

𝑢(𝑥) = 1 − 𝑎 + 𝑎𝑥 + 2𝐶1 cosh √𝐾 𝑥 −

𝐴=

𝑢(𝑥) = 𝑢1 (𝑥) = 1 − 𝑎 + 𝐴𝑥 2 + 𝐵𝑥 4 + 𝐶𝑥 6 ,

Expressions for concentration and current

Table 3.1. (Continued)

Reference

(Rasi, Rajendran, and Subbiah 2015)

S. No

11

Experimental techniques and enzymatic scheme Voltammetry

0 𝐶𝑃 𝐹 0 )] (𝐸−𝐸𝑃𝑄 𝑅𝑇

𝑖 = 𝐹𝑆𝐷𝑃 ( 𝜕𝑥

𝜕[𝑄]

) 𝑥=0

𝑥 = ∞, 𝜕[𝑄]/𝜕𝑥 = 0 The current flowing through the electrode is

1+𝑒𝑥𝑝[

Nonlinear differential Analytical equations with initial and techniques boundary conditions 𝜕2 [𝑄] NHPM and 𝐷𝑃 2 − 𝜕𝑥 Laplace 0 𝐶𝐸 𝜕[𝑄] 1 1 1 1 = transform 𝜕𝑡 +𝑘 +𝑘 +𝑘 [𝑄] 0 𝑘1 𝐶𝑆 1,2 2,2 2 technique 𝑡 = 0, 𝑥 ≥ 0and𝑥 = ∞, [𝑄] = 0 𝑥 = 0, 𝑡 ≥ 0: [𝑄] = 1+𝑒

+𝜎

1+𝑒−𝜉 2√ 𝑧 −𝜉

[

𝑒𝑟𝑓 (

𝑒𝑟𝑓𝑐 ( 2√𝜏

1+𝑒−𝜉 +𝜎

1 (1+𝑒 −𝜉 )√𝜋𝜏

=

𝜋𝜏

√𝜋𝜏

𝜏

√1+𝜎𝑒𝑟𝑓(√1+𝜎)+𝑒

𝜏 ) 1+𝜎

+𝜎

𝑖

=

1 𝑘1 𝐶𝑆0

) , 𝜏 = 𝑘2 𝐶𝐸0𝑡

0 𝐶𝑃

[𝑄]

,𝑞 =

0 𝑘2 𝐶𝐸

𝐷𝑃

𝜏

𝜋𝜏

+𝜎

when(𝜉 → ∞)

−(

𝑅𝑇

𝐹

𝜏 ) 1+𝜎 ]

,𝜉 = −

(1+𝑒 −𝜉 )[√1+𝜎𝑒𝑟𝑓(√1+𝜎)+𝑒

√(1+𝑒

Where 𝑧 = 𝑥√

𝑖pl

−𝜉

(1+𝑒−𝜉 )𝜏 ) 1+𝑒−𝜉 +𝜎

−(

)

1 𝑘2,2

0 (𝐸 − 𝐸𝑃𝑄 ), 𝜎 = 𝑘2 𝐶𝑃0 (

(1+𝑒−𝜉)𝜏 −( ) −𝜉)𝜋𝜏 (1+𝑒−𝜉)𝜏 )+𝑒 1+𝑒−𝜉+𝜎 𝑒𝑟𝑓(√ 1+𝑒−𝜉 +𝜎 1+𝑒−𝜉+𝜎

The normalized current is

𝐶 𝜙pl

−𝜉

−(

1+𝑒

)+

(1+𝑒 )𝜋𝜏 (1+𝑒 )𝜏 (√ −𝜉 𝑒𝑟𝑓 (√ −𝜉 ) + 𝑒

The plateau current is

𝜙𝐶 =

]

1+𝑒

)+1

2√𝜏

1+𝑒−𝜉+𝜎

1+𝑒−𝜉 2√ 𝜏−𝑧

1+𝑒−𝜉 2√ 𝜏+𝑧

2(1+𝑒 −𝜉 )

1+𝑒−𝜉 −√ 𝑧 1+𝑒−𝜉 +𝜎

The catalytic current response is

𝑒

𝑞(𝑧, 𝜏) =

𝑒

Expressions for concentration and current

+

1 𝑘1,2

+

12

S. No

𝑘𝑐

𝐵 + 2𝑒 − → 𝐶

𝑘

𝐸2 + 𝐴 → 𝐸1 + 𝐵

𝑃

Experimental techniques and enzymatic scheme (Swaminathan et al. 𝐸1 + 𝑘𝑐𝑎𝑡 2019) 𝑘 𝑆 →𝑘1−1 𝐸1 𝑆 → 𝐸2 +

Reference

𝜕𝑥 2

𝜕2 𝑏(𝑥,𝑡)

𝜕𝑥 2

−

𝜕𝑠 𝜕𝑥

= 0;

𝜕𝑠 𝜕𝑥 ∞

= 0,

𝑥 = 𝐿; 𝑠 = 𝑘𝑠 𝑠 , 𝑏 = 𝑘𝑏 𝑏 ∞

𝑥 = 0;

The initial and boundary conditions are: 𝑡 = 0; 𝑠 = 𝑘𝑠 ∞ ; 𝑏 = 0,

𝐾𝑚 +𝑠(𝑥,𝑡)

𝑘𝑐𝑎𝑡 𝑒𝑇𝑠(𝑥,𝑡)

= 𝐷𝐵

𝑘𝑏(𝑥, 𝑡) +

𝜕𝑡

𝑠(𝑥,𝑡)

𝑆

𝐾𝑚 +𝑠(𝑥,𝑡)

𝜕𝑏(𝑥,𝑡)

𝑘𝑐𝑎𝑡

𝜕𝑡

Nonlinear differential Analytical equations with initial and techniques boundary conditions 𝜕𝑠(𝑥,𝑡) 𝜕2 𝑠(𝑥,𝑡) HAM =𝐷 −

𝛾

𝑐𝑜𝑠ℎ(√ 𝜉)

𝛾

𝑐𝑜𝑠ℎ(√ 𝜉𝑥)

𝑐𝑜𝑠ℎ(√𝑎)

𝑐𝑜𝑠ℎ(√𝑎𝜒)

+

+ 𝛼𝑎

𝑥

𝛾

𝛾

𝑐𝑜𝑠ℎ(√ 𝜉) 2

−

[(2𝑛+1)2 𝜋2 𝜉+4𝛾]

𝐾𝑀

,𝛽 =

𝐿2

𝐷𝑆 𝑡

𝑘𝑠 𝑠 ∞

;𝑇 =

𝛼=

𝐷𝑆

𝐷𝐵

)+

𝜏

𝐾𝑀

𝑘𝑏 𝑏∞

,𝛾 =

𝐷𝑆

𝑘𝐿2

,𝜑 =

,𝑢 =

𝐷𝑆 𝐾𝑀

𝑘𝑐𝑎𝑡 𝑒𝑇𝐿2

1+𝛼

𝜑

𝑘𝑠 𝑠 ∞

𝑠

;𝑣 =

𝑘𝑏 𝑏∞

𝑏

𝐿

𝑥

;𝜒 = ;𝜉 =

64𝜋𝛼𝑎 2 (2𝑛 + 1)𝑒 −[(2𝑛+1)2𝜋2𝜉+4𝛾]4 , + 4𝛾][(2𝑛 + 1)2 𝜋 2 𝜉 − 4(𝐴 − 𝛾)][(2𝑛 + 1)2 𝜋 2 (𝜉 − 1) − 4(𝑎 − 𝛾)]

𝑚 = 4𝑎 + 𝜋 2 (2𝑛 + 1)2 “𝑎𝑛𝑑 “ 𝑎 =

(𝛽[(2𝑛 +

1)2 𝜋 2 𝜉

(𝜋𝜉𝛽(2𝑛+1)[(2𝑛+1)2 𝜋2 +4𝑎][(2𝑛+1)2 𝜋2 (𝜉−1)−4(𝑎−𝛾)]

−

𝑐𝑜𝑠ℎ(√𝑎)

𝑐𝑜𝑠ℎ(√𝑎𝜒)

𝜏 − [(2𝑛+1)2 𝜋2 𝜉+4𝛾]( ) 4

𝜏 −[(2𝑛+1)2𝜋2 +4𝑎]( ) 4 64𝛼𝑎2 𝑒

where µ𝐴𝑛 (𝜏) =

4𝜋𝜉(2𝑛+1)𝑒

2𝑛+1

(−1)𝑛 𝑐𝑜𝑠( 2 𝜋𝑥) −𝑚𝑡/4 𝑒 𝑚 (2𝑛+1) 𝑐𝑜𝑠ℎ(√ 𝜉𝑥)

(

∑∞ 𝑛=0

𝛽(𝜉𝑎−𝛾)

𝜋

16𝑎

𝑛 ∑∞ 𝑛=0(−1) µ𝐴𝑛 (𝜏)𝑐𝑜𝑠 ((2𝑛 + 1)𝜋 )

𝑣(𝜒, 𝜏) =

𝑢(𝜒, 𝜏) =

Expressions for concentration and current

Table 3.1. (Continued)

(Thiagarajan et Mediated al. 2011) bioelectrocatalysis 𝑆+ 𝑘𝑐𝑎𝑡

𝑀𝑜𝑥 ↔ 𝑆𝑀𝑜𝑥 → 𝑃 + 𝑀𝑟𝑒𝑑

𝑘𝑀

𝑃2

2

𝑃 2 1 𝐸2

1

𝑆1 + 𝑆2 →

𝑆1 →

𝐸1

(Praveen and Rajendran 2014)

Experimental techniques and enzymatic scheme Amperometric

1

S. Reference No

𝑑𝑥 2

𝑑𝑥 2 𝑑2 𝑃2,𝑒

𝑑2 𝑃1,𝑒

+

+

=0

𝑘21 𝑆1,𝑒 +𝑘22 𝑆2,𝑒

2 𝑘21 𝑘22 𝑒2 𝑆1,𝑒 𝑆2,𝑒

𝑘1 𝑒1 𝑆1,𝑒

|

𝑑𝑥 𝑥=0

𝑑𝑐𝑒

= 0, 𝑃1,𝑒 (𝑥 = 0) =

=0

−

𝐾𝑐𝑎𝑡 [𝐸] 1+𝐾𝑀 ⁄[𝑀𝑜𝑥 ]

=0

𝑛𝐹𝐴

𝑖

= −𝐷𝑀 ( 𝑑𝑋 𝑋=0

)

𝑑[𝑀𝑜𝑥 ]

The boundary conditions are [𝑀𝑜𝑥 ]𝑋=0 = [𝑀𝑟𝑒𝑑 ]∗ [𝑀𝑜𝑥 ]𝑋=𝛿 = 0 The current equation is given by

𝐷𝑀

𝑑2 [𝑀𝑜𝑥 ] 𝑑𝑋 2

1,2 𝑓𝑜𝑟 𝑥 = 𝑑𝑒 + 𝑑𝑑

0, 𝑐 = 𝑆1, 𝑆2 , 𝑃2 𝑆𝑖,𝑒 (𝑥) = 𝑆𝑖0 , 𝑃𝑖,𝑒 (𝑥) = 0, 𝑖 =

𝐷𝐶𝑒

Boundary conditions:

𝐷𝑃2,𝑒

𝐷𝑃1,𝑒

HPM

Nonlinear differential equations Analytical with initial and boundary techniques conditions NHPM 𝑑 2𝑆1,𝑒 𝐷𝑆1,𝑒 − 𝑘1𝑒1 𝑆1,𝑒 𝑑𝑥 2 𝑘21 𝑘22 𝑒2 𝑆1,𝑒 𝑆2,𝑒 − =0 𝑘21𝑆1,𝑒 + 𝑘22 𝑆2,𝑒 2 𝑑 𝑆2,𝑒 𝑘21 𝑘22 𝑒2𝑆1,𝑒 𝑆2,𝑒 𝐷𝑆2,𝑒 − 𝑑𝑥 2 𝑘21 𝑆1,𝑒 + 𝑘22 𝑆2,𝑒 =0

and 𝛽 =

𝑘21

=

𝐷𝑆1,𝑒

𝑑2 𝑘22 𝑒2

𝑆10

𝑆1,𝑒

𝐷𝑆2,𝑒

𝐷𝑃1,𝑒

𝑆10

𝑃1,𝑒

𝐾𝑀 (𝑒 4√𝛽𝛿 −2𝑒 2√𝛽𝛿 +1)

+

= (𝐴 − 𝐵)√𝛽 + [𝑀𝑟𝑒𝑑 ]∗ √𝛽 − [𝑀𝑟𝑒𝑑 ]∗ 3𝐾𝑀 (𝑒 4√𝛽𝛿 −2𝑒 2√𝛽𝛿 +1)

2([𝑀𝑟𝑒𝑑 ]∗ )2 (𝑒 4√𝛽𝛿 −1)√𝛽

𝑛𝐹𝐴𝐷𝑀 [𝑀𝑟𝑒𝑑 ]∗

𝑖𝛿

The current is

([𝑀𝑟𝑒𝑑

]∗ )2 𝑒 2√𝛽𝑋 [𝑒 −4√𝛽(𝑋−𝛿) +1+6𝑒 −2√𝛽(𝑋−𝛿) ]

𝑒 √𝛽𝛿 𝑠𝑖𝑛ℎ(√𝛽𝛿)

𝑠𝑖𝑛ℎ(√𝛽𝛿)

𝑠𝑖𝑛ℎ(√𝛽𝑋)

𝐷𝑃2,𝑒

𝑑2 𝑘1 𝑒1

,𝑦 = ,𝛼 =

𝑆10

𝑃2,𝑒

]−

, 𝑃2𝑁 = , 𝛾𝑃2 =

, 𝑃1𝑁 = 𝑑2 𝑘1 𝑒1

𝑆20

𝑆2,𝑒

, 𝛾𝑃1 =

, 𝑆2𝑁 = 𝑑2 𝑘22 𝑒2

, 𝑆1𝑁 =

2𝐴2 𝑐𝑜𝑠ℎ(𝐴)

𝛾𝑃1 [𝑐𝑜𝑠ℎ(𝐴)−1]

, 𝛾𝑆3 =

𝛼+𝛽

𝛾𝑆3

𝑦=0

,𝐵 = √

, 𝛾𝑆2 =

𝑘22

𝐷𝑆1,𝑒

𝑑2 𝑘1 𝑒1

𝛼+𝛽

𝑑𝑦

𝑑𝑃1𝑁

[𝑀𝑜𝑥 ] = +𝐴𝑒 √𝛽𝑋 + 𝐵𝑒 −√𝛽𝑋 + [𝑀𝑟𝑒𝑑 ]∗𝑒 √𝛽𝑋 − [𝑀𝑟𝑒𝑑 ]∗𝑒 √𝛽𝛿 [

𝑆20

𝑆10

𝑑

=

𝛾𝑆1 (𝛼+𝛽)+𝛾𝑆2

, 𝛾𝑆1 =

𝐴=√ 𝑥

𝑖𝑑 𝑛𝑒 𝐹𝐴𝐷𝑃1,𝑒 𝑆10

where

𝜓=

|

𝑐𝑜𝑠ℎ(𝐵)

𝛾𝑆3

]

2𝐴2 𝑐𝑜𝑠ℎ(𝐴) 𝑐𝑜𝑠ℎ(𝐵𝑦)

[1 −

𝛾𝑃2

𝑐𝑜𝑠ℎ(𝐵) 𝛾𝑃1 [1−𝑐𝑜𝑠ℎ(𝐴𝑦)+𝑦 𝑐𝑜𝑠ℎ(𝐴)−𝑦]

𝑐𝑜𝑠ℎ(𝐴) 𝑐𝑜𝑠ℎ(𝐵𝑦)

𝑐𝑜𝑠ℎ(𝐴𝑦)

The Current Density

𝑃2𝑁 (𝑦) =

𝑃1𝑁 (𝑦) =

𝑆2𝑁 (𝑦) =

𝑆1𝑁 (𝑦) =

Expressions for concentration and current

Table 3.2. Recent contribution to mathematical modelling of steady-state differential equations arises in planar electrode

−

𝐾𝑐𝑎𝑡 [𝐸]

𝐴→𝐵

1+𝐾𝑀 ⁄[𝑀𝑜𝑥 ]

=0

𝑋=𝛿

|

=0

𝑑𝑥 2

𝑑2 𝑏

+

𝑑𝑋 𝑋=0

)

𝑘𝑐𝑎𝑡 𝑠+𝐾𝑀 𝑘𝑎+𝑘𝑎𝑠

𝑘𝑐𝑎𝑡 𝑘𝑎𝑠(𝑒∑ )

= −𝐷𝑀 (

𝑑[𝑀𝑜𝑥 ]

=0

The boundary conditions are 𝑑𝑠 𝑥 = 0; = 0; 𝑏 = 0 𝑑𝑥

𝐷𝐵

𝑛𝐹𝐴

𝑖

The current equation is given by

𝑑𝑋

𝑑𝑀𝑜𝑥

Boundary conditions are [𝑀𝑜𝑥 ]𝑋=𝛿 = 0

𝑑𝑋 2

HPM

𝑑2𝑠 (Shanmugarajan Amperometric HPM 𝐷𝑆 2 𝐾𝑀 𝑘𝑐𝑎𝑡 et al. 2011) 𝑑𝑥 𝑆 + 𝐸1 ↔ [𝐸1 𝑆] → 𝑃 𝑘𝑐𝑎𝑡 𝑘𝑎𝑠(𝑒∑ ) + 𝐸2 − =0 𝑘𝑐𝑎𝑡 𝑠 + 𝐾𝑀 𝑘𝑎 + 𝑘𝑎𝑠

𝑑2 [𝑀𝑜𝑥 ] 𝑒 √𝛽𝛿

𝑟𝑒𝑑 ]

,𝛿 = √

∗

2√𝛽𝛿𝑒 3√𝛽𝛿

2

∗

−2𝑒 √𝛽𝛿 )

3 𝑐𝑜𝑠ℎ(√𝛽𝛿)

2√𝛽𝛿𝑒 2√𝛽𝛿 2

−

2

2

6 𝑐𝑜𝑠ℎ(√𝛽𝛿)

(𝑒 √𝛽𝛿 −2𝑒2√𝛽𝛿 )

−

3 𝑐𝑜𝑠ℎ(√𝛽𝛿)

2

(2𝑒 2√𝛽𝛿 −𝑒 3√𝛽𝛿 )

+

3

6 𝑐𝑜𝑠ℎ(√𝛽𝛿)

3

(2𝑒 3√𝛽𝛿 −𝑒 2√𝛽𝛿 𝑠𝑖𝑛ℎ(2√𝛽𝛿))

3𝐾𝑀 (𝑒 √𝛽𝑘𝛿 − 1) (𝑒 √𝛽𝑘𝛿 + 1)

3

and

3𝐾𝑀 (𝑒 √𝛽𝛿 −1) (𝑒 √𝛽𝛿 +1)

3

3 𝑐𝑜𝑠ℎ(√𝛽𝛿)

|

(4𝑒 √𝛽𝑋 −

| )

𝐺 𝑠𝑖𝑛ℎ(√𝛽𝑋)

𝐾𝑀

[𝑀𝑟𝑒𝑑 ]∗

𝛼√𝛽𝛿𝑒 2√𝛽𝛿 𝑠𝑖𝑛ℎ(2√𝛽𝛿)

− 1) +

𝐾𝑀

[𝑀𝑟𝑒𝑑 ]∗

([𝑀𝑟𝑒𝑑 ]∗ )2 [𝑒 3√𝛽𝛿 −7𝑒 2√𝛽𝛿 −𝑒 √𝛽𝛿 −1]

−

=−

3 𝑐𝑜𝑠ℎ(√𝛽𝛿)

(

[𝑀𝑟𝑒𝑑] 𝐾𝑀

2

3 𝑐𝑜𝑠ℎ(√𝛽𝛿)

𝑒 √𝛽𝛿 (3+

2𝐷𝑀 [𝑀𝑟𝑒𝑑 ]∗ ,𝐴 𝑘𝑐𝑎𝑡 [𝐸]

𝐾𝑀

|[𝑀

|√𝛽𝛿 (

6𝐾𝑀 𝑐𝑜𝑠ℎ(√𝛽𝛿)

[3 𝑠𝑖𝑛ℎ(√𝛽𝑋) +

−2

3 𝑐𝑜𝑠ℎ(√𝛽𝛿)

([𝑀𝑟𝑒𝑑 ]∗ )2 (𝑒 √𝑘𝛿 ) [𝑒 3√𝛽𝛿 + 7𝑒 2√𝛽𝛿 + 𝑒 2√𝛽𝛿 − 1]

𝐾𝑀 𝐷𝑀

𝐾𝑐𝑎𝑡 [𝐸]

=

−

  l   2 x    3  cosh   cosh   X K    X K   cosh(x / X K )  ( Ks  ) s s      Ks  cosh(l / X K ) 6 cosh(l / X K ) 3   x   2l    cosh X  3  cosh X    K    K   

𝐺=

𝐵=

𝛽=

where

𝑖𝛿 𝑛𝐹𝐴𝐷𝑀 [𝑀𝑟𝑒𝑑 ]∗

The current is

3𝐾𝑀

[𝑀𝑟𝑒𝑑 ]∗ (𝑒 √𝛽𝑋 −𝑒 2√𝛽𝑋 )

[𝑀𝑟𝑒𝑑 ]∗ 𝑒 2√𝛽𝛿 [4𝑒 √𝛽𝑋 −𝑐𝑜𝑠ℎ(2√𝛽𝑋)−3]

= 𝑒 √𝛽𝑋 +

𝑒 2√𝛽𝑋 − 3)] +

[𝑀𝑟𝑒𝑑 ]∗

[𝑀𝑜𝑥 ]

Analytical Expressions for concentration and current techniques

4

𝑘𝑐𝑎𝑡

𝑀𝑜𝑥 ↔ 𝑆𝑀𝑜𝑥 → 𝑃 + 𝑀𝑟𝑒𝑑

𝑘𝑀

𝐷𝑀

Nonlinear differential equations with initial and boundary conditions

(Thiagarajan et al. 2011)

Experimental techniques and enzymatic scheme Mediated bioelectrocatalysis 𝑆+

3

S. Reference No

Table 3.2. (Continued)

Amperometric and Potentiometric 𝐸 + 𝑆 ↔ [𝐸𝑆] → 𝐸 + 𝑃

5

(Meena and Rajendran 2010b)

Experimental techniques and enzymatic scheme

S. Reference No

𝑑𝑥 2

𝑑2 𝑝

𝑑𝑥 2

𝑑2 𝑠

+

− 𝑠+𝑘𝑚

𝑉𝑚 𝑠

𝑠+𝑘𝑚

𝑉𝑚 𝑠

=0

=0

The boundary conditions are i. When 𝑥 = 0 ; 𝑠′ = 0, 𝑝 = 0 ii. When 𝑥 = 𝑑 ; 𝑠 = 𝑠0, 𝑝 = 0

𝐷𝑝

𝐷𝑠

𝑥 = 𝑙; 𝑠 = 𝐾𝑠∞ ; 𝑏 = 0

Nonlinear differential equations with initial and boundary conditions

HPM

𝑏𝐷𝐵 𝐷𝑆 𝐾𝑠∞

≈

1

)

)𝑋

,𝜂 =

𝐷𝐵 𝐾𝑀

𝑘𝑐𝑎𝑡 𝑒∑

Vm d 2  Ds K M

𝑙

𝑋𝐾

2𝑙

𝑋𝐾

𝑙

 k

𝑋𝐾

2𝑙

𝑋𝐾

[3 − 2 𝑐𝑜𝑠ℎ ( ) − 𝑐𝑜𝑠ℎ ( )]

𝛼𝑋𝐾 (𝐾𝑠∞ ) 6𝑙 𝑐𝑜𝑠ℎ(𝑙/𝑋𝐾)3

Dp Dt x s p  P  X  , T  s2 , S  , P  ,r  ,G    ,  d s0 p0 Ds d  X  X 0

𝑐𝑜𝑠ℎ(𝜎)

1

𝐷𝑆 𝐾𝑀

𝑘𝑐𝑎𝑡 (𝑒∑ )

𝑐𝑜𝑠ℎ(𝜎)

,𝜅 =

) + (1 −

,𝑚 = (𝑛 + 1/2)2 𝜋 2 + 𝜎.

Where

𝐺(𝑇) = (1 −

The current

𝑐𝑜𝑠ℎ(𝜎)

1−𝑐𝑜𝑠ℎ(𝜎𝑋)

𝑐𝑜𝑠ℎ(𝜎𝑋) 𝑐𝑜𝑠ℎ(𝜎)

𝑃(𝑋) = (

𝑆(𝑋) =

𝐾𝑀 𝑘𝑎

𝑋𝐾

𝑙

2𝑙

𝑋𝐾

[2 𝑠𝑖𝑛ℎ ( ) − {3 − 𝑐𝑜𝑠ℎ ( )} 𝑡𝑎𝑛ℎ ( )]

𝛼(𝐾𝑠∞ ) 6 𝑐𝑜𝑠ℎ(𝑙/𝑋𝐾 )2

[𝑙 − 𝑠𝑒𝑐 ℎ ( )] +

𝑘𝑐𝑎𝑡 +𝑘𝑎

𝑙

𝑋𝐾

𝑋𝐾 = √1/𝜅, 𝛼 =

𝐷𝐵 (𝐾𝑠∞ )

𝑗 𝐵 𝑋𝐾

1

𝑋𝐾

≈ 𝑡𝑎𝑛ℎ ( ) −

𝑗 𝑆 𝑋𝐾 𝐷𝑆 (𝐾𝑠∞ )

where

𝜓𝐵 =

𝜓𝑆 =

≈

𝑥 [(1 − 𝑐𝑜𝑠ℎ( 𝑥/𝑋𝐾 )) − (1 − 𝑐𝑜𝑠ℎ( 𝑙/𝑋𝐾 )) ] 𝑙 𝑐𝑜𝑠ℎ( 𝑙/𝑋𝐾 ) 3 − 𝑐𝑜𝑠ℎ( 2𝑙/𝑋𝐾 ))[𝑐𝑜𝑠ℎ( 𝑥/𝑋𝐾 ) − 1] − 𝑐𝑜𝑠ℎ( 𝑙/𝑋𝐾 )[1 − 𝑐𝑜𝑠ℎ( 2𝑥/𝑋𝐾 )] } 𝛼(𝐾𝑠∞ ) { 𝑥 −[2 𝑐𝑜𝑠ℎ( 𝑙/𝑋𝐾 ) + 𝑐𝑜𝑠ℎ( 2𝑙/𝑋𝐾 ) − 3] 𝑙 + 3 6 𝑐𝑜𝑠ℎ( 𝑙/𝑋𝐾 ) The expression of the normalized current becomes

𝜑𝐵 =

Analytical Expressions for concentration and current techniques

6

S. No

𝑑𝑥 𝑙𝑗𝑆

𝐽𝑜𝑏𝑠 =

∞

𝐷𝐴 𝐾𝐴 [𝐵∑ ]

𝑙𝑗𝑜𝑏𝑠 ∞

𝑑𝜒 𝜒=0

𝑑𝑎

Mediator

= −( )

𝑑𝜒 𝜒=0

𝑑𝑎

=

=0𝑥 =

𝛾 𝑑𝜒 𝜒=1

𝜂 𝑑𝑠

= ( )

−( )

𝐷𝐴 𝐾𝐴 [𝐵∑ ]

𝑑𝜒 𝜒=1

( )

𝑑𝑎

𝐽𝑆 =

𝑑𝑥

𝑑[𝐴]

= 0, [𝐴] = [𝑆]𝜀

Flux of the substrate

0,

𝑑[𝑆]

𝑥 = 𝑙, [𝑆] = [𝑆]∞ 𝐾𝑆 , , 𝑊2 =

𝛾

𝜂

𝑎𝜀

(𝜅𝛾)2 𝑎𝜀 3 2(𝑒 2𝜅 +1)𝜂

𝛾

𝜂

𝑊1 𝛾 𝑐𝑜𝑠ℎ 𝜅 2𝜂

, 𝑊8 = [𝑊1 (3 − 𝑐𝑜𝑠ℎ 2 𝜅) + 𝑊2 ]

, 𝑊4 = 1 − 2𝑒 𝜅 + 𝜅 + 𝜂, 𝑊5 = (1 + − 𝜅 + 𝜂), 𝑊7 =

, 𝑊3 =

2𝜅 (1

𝑐𝑜𝑠ℎ 𝜅

𝜇(2 + 𝜇), 𝑊6 = 𝑒

6 𝑐𝑜𝑠ℎ3 𝜅

𝛾𝑎𝜀 2 (1+𝜇)

𝐽𝑆 = 𝐽𝑜𝑏𝑠 = (𝑊8 𝑠𝑖𝑛ℎ 𝜅 − 2𝑊7 𝑠𝑖𝑛ℎ 2 𝜅 − 12𝑊7 𝜅)

𝑒

2𝜅 )𝜂

𝑊1 =

𝐾𝑀

Experimental Nonlinear differential equations with Analytical Expressions for concentration and current techniques and initial and boundary conditions techniques enzymatic scheme (Loghambal and Amperometric HPM 𝑑 2 [𝑆] 𝑠 = 1 − 𝑊8 (1 + 𝜅 𝑠𝑖𝑛ℎ 𝜅) + 𝑊7 (1 + 2𝜅 𝑠𝑖𝑛ℎ 2 𝜅) + 6𝑊7 𝜅 2 + [𝑊8 𝜅 𝑠𝑖𝑛ℎ 𝜅 − 𝐷𝑆 𝑒− 2 Rajendran 𝑑𝑥 2𝑊7 𝜅 𝑠𝑖𝑛ℎ 2 𝜅]𝜒 − 6𝜅 2 𝑊7 𝜒 2 + 𝑊8 𝑐𝑜𝑠ℎ(𝜅(1 − 𝜒)) − 𝑊7 𝑐𝑜𝑠ℎ(2𝜅(1 − 𝜒)) 𝐵→ 𝐴 𝑘𝐴 𝑘𝑐𝑎𝑡 [𝐴][𝑆][𝐸𝛴 ] 𝑑 2[𝐴] 2010) 𝑘𝐴 𝑎 = 𝑊1 𝑐𝑜𝑠ℎ 𝜅 [𝑐𝑜𝑠ℎ(2𝜅(1 − 𝜒)) − 3] + 𝑐𝑜𝑠ℎ(𝜅(1 − 𝜒)) [𝑊2 + 𝑊1 (3 − 𝑐𝑜𝑠ℎ 2 𝜅)] − 𝐴 + 𝐸2 → 𝐵 + = 𝑘 [𝐴](𝐾 + [𝑆]) + 𝑘 [𝑆] 𝐷𝐴 𝑑𝑥 2 𝐴 𝑚 𝑐𝑎𝑡 𝑊3 (𝑊4 + 𝑊5 + 𝑊6 )(2𝜒 − 𝜒 2) 𝐸1 𝑘𝐴 𝑘𝑐𝑎𝑡 [𝐴][𝑆][𝐸𝛴 ] Where, 𝑘𝐸 = 𝑘𝐴 [𝐴](𝐾𝑚 + [𝑆]) + 𝑘𝑐𝑎𝑡 [𝑆] 𝐸1 + 𝑆 → 𝐸2 + 1/2 𝑘𝐴 𝐾𝐴 [𝐵∑ ] 𝐾𝑀 𝑘𝐴 [𝐸∑ ] [𝐴] [𝑆] 𝑥 𝐷 𝑘 𝐾 ∞ 𝑎 = [𝐵 ] , 𝑠 = [𝑆] , 𝜒 = , 𝜅 = 𝑙 ( ) ,𝜂 = 𝑆 𝐴 𝑀,𝛾 = ,𝜇 = [𝐸 ] [𝐸 ] [𝐸 ] = + 𝑃 𝛴 1 2 𝐾𝐴 ∑ 𝐾𝑆 ∞ 𝑙 𝐷𝐴 𝐷𝐴 𝑘𝑐𝑎𝑡 𝑘𝑐𝑎𝑡 𝐾𝑆 [𝑆]∞ ∞ The boundary conditions are 𝐾𝑆 [𝑆]∞

Reference

Table 3.2. (Continued)

Reference

(Indira and Rajendran 2011)

S. No

7

𝑃𝑃𝑂

𝑂2 → 𝑃2 +

𝐻2 𝑂

2

1

= 0, [𝑆1 ] + [𝑆2 ] = [𝑆1]∞ , [𝑃2] = 0

when 𝑋 = 𝐿 + 𝛿 [𝑆1 ] = [𝑆1]∞ , [𝑆2 ] = 0, [𝑃2 ] = 0

𝑑𝑋

𝑑[𝑆1 ]

when 𝑋 = 0

2𝜒

2 𝜇

𝜇1 =

𝛬1 2

𝐿2

Where

1)]

, 𝜇2 =

𝐿2 𝛬2 2

1

1

, 𝛼1 =

𝐾1

[𝑆1 ]∞

−

, 𝛼2 =

𝐾2

[𝑆1 ]∞

+

+ 2)) −

]

𝑠𝑖𝑛ℎ(2√𝜇2 𝑚) 𝑐𝑜𝑠ℎ(√𝜇2 𝑚) 𝑐𝑜𝑠ℎ(2√𝜇2 𝑚) − ( + 3 2 𝑠𝑖𝑛ℎ(√𝜇2 𝑚) 3

𝑐𝑜𝑠ℎ(√𝜇1 𝑚) 2

𝑐𝑜𝑠ℎ(2√𝜇1 𝑚)−3

×(

𝛼1 6 𝑐𝑜𝑠ℎ2 (√𝜇1 𝑚)

3 𝑠𝑖𝑛ℎ(√𝜇2 𝑚)

) [

2

𝑐𝑜𝑠ℎ(√𝜇1 𝑚)

𝑐𝑜𝑠ℎ(√𝜇1 𝑚)

(1 − (1 −

𝑠𝑖𝑛ℎ(√𝜇2 𝑚)

𝜇2 𝑐𝑜𝑠ℎ(√𝜇2 𝑚)

𝛼2 √𝜇2 × 𝑠𝑖𝑛ℎ2 (√𝜇2 𝑚)

𝐼=√

) − 2(𝜇1 + 𝜇2 )

𝜇 𝑠𝑖𝑛ℎ( 𝜇 𝑚)

(√𝜇1 +√𝜇2 )2 𝑠𝑖𝑛ℎ((√𝜇1 −√𝜇2 )𝜒+√𝜇2 𝑚)−(√𝜇1 −√𝜇2 )2 𝑠𝑖𝑛ℎ((√𝜇1 +√𝜇2 )𝜒−√𝜇2 𝑚) ) 𝑠𝑖𝑛ℎ(√𝜇2 𝑚)

𝑐𝑜𝑠ℎ(√𝜇1𝜒) 𝑠𝑖𝑛ℎ(√𝜇2(𝑚 − 𝜒)) 1 − (1 − ) 𝑐𝑜𝑠ℎ(√𝜇1 𝑚) 𝑠𝑖𝑛ℎ(√𝜇2 𝑚) 𝑐𝑜𝑠ℎ(√𝜇1𝑚) (𝜇1𝛼2 + 𝜇2 𝛼1) 1 − (1 − ) 2 𝑐𝑜𝑠ℎ(√𝜇1𝑚) (𝜇1 − 𝜇2 )2 𝑐𝑜𝑠ℎ(√𝜇1𝑚)

√ 1 + ((𝜇1 + 𝜇2 ) − √ 1 √ 2 𝑠𝑖𝑛ℎ(√𝜇2 𝑚) [ 𝑚 The dimensionless current

×

(

𝑤 =1−

Experimental Nonlinear differential equations with Analytical Expressions for concentration and current techniques and initial and boundary conditions techniques enzymatic scheme [𝑆1 ] 𝑐𝑜𝑠ℎ(√𝜇1 𝜒) 𝛼1 𝑐𝑜𝑠ℎ(√𝜇1 𝜒) 𝑐𝑜𝑠ℎ(2√𝜇1 𝑚) 𝑐𝑜𝑠ℎ(2√𝜇1 𝜒) 𝑑2 [𝑆1 ] Amperometric HPM 𝑢= + [ ×( − 1) − + 1] − 2 =0 𝑐𝑜𝑠ℎ(√𝜇1 𝑚) 2 𝑐𝑜𝑠ℎ2 (√𝜇1 𝑚) 𝑐𝑜𝑠ℎ(√𝜇1 𝑚) 3 3 𝑑𝑋 2 𝛬1 (1+([𝑆1 ]⁄𝐾1 )) 𝑃𝑃𝑂 𝑆1 + 𝑂2 → 𝑃2 2 [𝑆 ] 𝑠𝑖𝑛ℎ( 𝜇 (𝑚−𝜒)) 1 𝛼 𝑐𝑜𝑠ℎ(2 𝜇 𝑚)−3 𝛼2 √ 2 √ 1 1 [𝑆2 ] 𝑑 2 𝑣 = [1 − − × + 2] + × − 2 =0 𝑠𝑖𝑛ℎ(√𝜇2 𝑚) 𝑐𝑜𝑠ℎ(√𝜇1 𝑚) 6 𝑐𝑜𝑠ℎ2 (√𝜇1 𝑚) 𝑐𝑜𝑠ℎ(√𝜇1 𝑚) 𝑠𝑖𝑛ℎ2 (√𝜇2 𝑚) + 𝐻2𝑂 𝑃2 + 2𝑒 − 𝑑𝑋 2 𝛬2 (1 + ([𝑆2]⁄𝐾2 )) 2 𝑘𝑟 1 𝑠𝑖𝑛ℎ(√𝜇2 (𝑚−𝜒)) 𝑐𝑜𝑠ℎ(2√𝜇2 𝑚) 2 𝑠𝑖𝑛ℎ(√𝜇2 𝜒) 𝑐𝑜𝑠ℎ(2√𝜇2 (𝑚−𝜒)) (1 − ) [ ×( + 1) + − − + 2𝐻 + ↔ 𝑆2 𝑑2 [𝑃2 ] 𝑑2 [𝑆1 ] 𝑑2 [𝑆2 ] 𝑐𝑜𝑠ℎ(√𝜇1 𝑚) 2 𝑠𝑖𝑛ℎ(√𝜇2 𝑚) 3 3 𝑠𝑖𝑛ℎ(√𝜇2 𝑚) 6 𝑘0 + + =0 𝑑𝑋 2 𝑑𝑋 2 𝑑𝑋 2 1 ] 𝑆2 + The boundary conditions are: 2

9

8

(PonRani and Rajendran 2012) 𝑘2

=

𝑑𝑦 2 𝑘2 [𝐸𝑇 ]

𝑑2 [𝑂2 ]

𝑑𝑦 2

𝑑2 [𝑆]

=

𝑘2 [𝐸𝑇 ] (𝛽𝑆 /[𝑆]+𝛽𝑂 /[𝑂2 ])+1

𝐷𝑃 𝑑𝑥 2

𝑑2 𝐺̄

− 𝑟 𝐾𝑚 +𝐺̄

2 𝑣𝑚 𝐺̄

𝑦 = 0: [𝑂2 ] = 0;

=0

𝑑𝑦

𝑑[𝑆]

=0

Boundary conditions are: at𝑦 = 𝑑: [𝑂2 ] = [𝑂2 ]𝑏 = 𝐾𝑂 [𝑂2 ]∞ [𝑆] = [𝑆]𝑏 = 𝐾𝑆 [𝑆]∞

𝐷𝑆

(𝛽𝑆 /[𝑆]+𝛽𝑂 /[𝑂2 ])+1

𝐷𝑂

Nonlinear differential equations with initial and boundary conditions

𝑑𝑧 𝑧=1

𝐽𝑖𝑗 = ( )

𝑑𝐶

𝐺 + 𝐸 ↔ 𝑋 → 𝐹 Boundary conditions are 𝑘−1 𝑘−2 𝐺̄ = (𝐺0 − 𝐺𝑒 )𝛼1, 𝑥 = 0 +𝐸 𝐺̄ = (𝐺0 − 𝐺𝑒 )𝛼1, 𝑥 = 𝑙𝑝

𝐾1

Amperometric

𝐸𝑟𝑒𝑑 + 𝑂2 → 𝐸𝑂𝑋 + 𝐻2𝑂2

𝑘3

Experimental techniques and enzymatic scheme (Loghambal Amperometric and 𝐸𝑂𝑋 𝑘𝑀 𝑘2 Rajendran + 𝑆 ↔ 𝐸𝑆 → 𝐸𝑟𝑒𝑑 𝑘−1 2011) +𝑃

S. Reference No

HPM

HPM

)

𝑑𝑋 𝑋=0

𝑑𝐹𝑂

= 1 − 𝑤2 [𝑤1(𝑤1 + 2𝐵𝑆 ) + 2𝐵𝑆 (𝑤1 + 𝐵𝑆 )(𝑙𝑛( 𝐵𝑆 ) − 𝑙𝑛( 𝑤1 + 𝐵𝑆 ))]

+𝑆𝑖𝑛ℎ(2𝜑𝑝 𝑧)[2𝛽𝛼1𝛼2 − 2𝛽𝛼12𝐵 − 2𝛽𝛼1𝛼2 𝐵 2 + 2𝛽𝛼12𝐵 3 ] −6𝛽𝛼1𝛼2 𝐴𝐵 + 6𝛽𝛼12𝐴𝐵 2 − 3𝛽𝛼1𝐴 + 12𝛼1𝛼2𝛽𝐴 6𝛼1 𝛽 − 4𝛽𝛼1𝛼2𝐵 2 + 18𝛽𝛼1𝛼2 𝐵 − 6𝛽𝛼12 𝐵 2 +𝑆𝑖𝑛ℎ(𝜑𝑝 𝑧) [ ] +4𝛽𝛼12 𝐵 3 − 2𝛽𝛼1𝐵 − 6𝛼1𝐵 3 − 6𝛼2 2 2 2𝛽𝛼1𝐴 − 6𝛼1𝐴 + 12𝛼1𝛼2𝛽𝐴 − 4𝛽𝛼1 𝐴𝐵 +𝐶𝑜𝑠ℎ(𝜑𝑝 𝑧) [ ] { +4𝛽𝛼1 𝛼2𝐴𝐵 + 6𝛼1𝐴𝐵 2 𝐶= 6𝐴(𝐵 2 − 1)

+𝑆𝑖𝑛ℎ(2𝜑𝑝 )𝑆𝑖𝑛ℎ(𝜑𝑝 𝑧)[2𝛽𝛼1𝛼2𝐴 − 2𝛽𝛼12𝐴𝐵]

Normalized parameters are 𝐵𝑂 = [𝑂2 ]𝑏 /𝛽𝑂 , 𝐵𝑆 = [𝑆𝑏 ]/𝛽𝑆 , 𝐹𝑂 = [𝑂2 ]/[𝑂2 ]𝑏 , 𝐹𝑆 = [𝑆]/[𝑆]𝑏 , 𝑋 = 𝑦/𝑑 The analytical expression of the concentration is 2𝛽𝛼1𝛼2𝐴𝐵𝐶𝑜𝑠ℎ(2𝜑𝑝 𝑧) + 6𝛼2𝑆𝑖𝑛ℎ(𝜑𝑝 𝑧)𝐶𝑜𝑠ℎ2(𝜑𝑝 𝑧) − 2𝛽𝛼12 𝐴𝐵 2 𝐶𝑜𝑠ℎ(2𝜑𝑝 𝑧) +𝐶𝑜𝑠ℎ(2𝜑𝑝 )𝑆𝑖𝑛ℎ(𝜑𝑝 𝑧)[2𝛽𝛼12 𝐵 2 − 2𝛽𝛼1𝛼2 𝐵 − 𝛽𝛼1]

𝐽=(

𝜑 2 𝐵𝑆 2 (1+𝐵 )3 2𝐵𝑂 𝑆

The dimensionless flux is

𝑤1 = 𝐵𝑂 (1 + 𝐵𝑆 ), 𝑤2 =

}

𝐹𝑂 = 𝑤2[2𝐵𝑆2 [𝑙𝑛( 𝐵𝑆 ) − 𝑙𝑛( 𝑤1𝑋 + 𝐵𝑆 )] + {𝑤2−1 + 2𝑤1𝐵𝑆 [𝑙𝑛( 𝑤1 + 𝐵𝑆 ) − 𝑙𝑛( 𝑤1𝑋 + 𝐵𝑆 )] + 2𝐵𝑆2 [𝑙𝑛( 𝑤1 + 𝐵𝑆 ) − 𝑙𝑛( 𝐵𝑆 )] − 𝑤12}𝑋 + 𝑤12𝑋2 ] 𝐹𝑆 = 1 − 𝑤2[𝑤12 + 2𝑤1𝐵𝑆 [1 − 𝑙𝑛( 𝑤1 + 𝐵𝑆 ) + 𝑙𝑛( 𝐵𝑆 )] + 2𝐵𝑆2 [𝑙𝑛( 𝑤1𝑋 + 𝐵𝑆 ) − 𝑙𝑛( 𝑤1 + 𝐵𝑆 )] − 2𝑤1𝐵𝑆 [1 − 𝑙𝑛( 𝑤1𝑋 + 𝐵𝑆 ) + 𝑙𝑛( 𝐵𝑆 )]𝑋 − 𝑤12𝑋 2 ] where

Analytical Expressions for concentration and current techniques

Table 3.2. (Continued)

𝑑𝑥 2

𝑑𝑥 2 𝑑2 𝑦

− 𝑘𝑏𝑦 = 0

− 𝑘𝑏𝑦 = 0

= 𝐷𝑦

′ 𝑘𝐸 𝑦0

𝑑𝑏

𝑑𝑥 0

−𝑏𝐿′ = 𝐷𝑒

−𝑘 ′′ 𝑏𝐿 𝑦𝑠

𝑑𝑥 𝐿

𝑥 = 𝐿 ; y = 𝐾𝑦𝑠 ; ( ) =

𝑦0′

𝑥 = 0 ; 𝑏 = 𝑏0 ; ( ) =

𝑑𝑦

Boundary conditions are

𝐷𝑌

𝐷𝑒

𝑑2 𝑏

Experimental Nonlinear differential techniques and equations with initial and enzymatic scheme boundary conditions

10 (Meena and chemically Rajendran modified 2011) electrode

S. Reference No

HPM

𝐺̄ (𝐺0 −𝐺𝑒)

,𝑧 =

𝑥 𝑙𝑝

,𝛽 =

𝐶̄0 𝐾𝑚

, 𝜑𝑝2 =

2𝑣 2𝑙𝑝 𝑚

𝑟𝐷𝑝 𝐾𝑚

𝐴 = 𝑆𝑖𝑛ℎ(𝜑𝑝 ) and 𝐵 = 𝐶𝑜𝑠ℎ(𝜑𝑝 )

𝜇𝐿 = 𝑋𝐿 / 𝐿, 𝜇0 = 𝑋0 / 𝐿, 𝑢 = 𝑏 / 𝑏0 , 𝑣 = 𝑦/𝐾𝑦𝑠 , 𝑢𝐿 = 𝑏𝐿 ⁄𝑏0 ,𝑣0 = 𝑦0 ⁄𝐾𝑦𝑠 , 𝜆𝐵 = k’’ 𝑦𝑠 𝐿/𝐷𝑒 , 𝜆𝑌 = k′𝐵 𝐿/𝐷𝑦

{+𝑆𝑖𝑛ℎ(2𝜑𝑝 )[2𝐴𝛼1𝛽𝜑𝑝 + 6𝛼12 𝛽𝐴𝐵 2 𝜑𝑝 ] } 𝑢 = 1 + 𝐴𝜒 + 𝐵𝑚𝜒 2 + 𝐶𝑚𝜒 3 − 𝐷𝜒 4 𝑣 = 1 + 𝐴1 + 𝐵1 𝜒 + 𝐵𝑟𝜒 2 + 𝐶𝑟𝜒 3 − 𝐷𝜒 4 where 𝑚 = 1/𝜇02 𝑣0 ,𝑟 = 1/𝜇𝐿2 𝑢𝐿 and 𝐴 = −0.16666𝑚𝜆𝐵 𝑢𝐿 𝜆𝑌 𝑣0 + 0.5𝑚𝜆𝑌 𝑣0 + 0.5𝑚𝜆𝐵 𝑢𝐿 − 𝜆𝐵 𝑢𝐿 − 𝑚𝐵 = 0.5(1 − 𝜆𝑌 𝑣0 ), 𝐶 = 0.16666(−𝜆𝐵 𝑢𝐿 + 𝜆𝐵 𝑢𝐿 𝜆𝑌 𝑣0 + 𝜆𝑌 𝑣0 ) , 𝐷 = 0.08333𝑚𝜆𝐵 𝜆𝑌 𝑢𝐿 𝑣0 𝐴1 = −𝜆𝑌 𝑣0 − 0.08333𝑟𝜆𝐵 𝜆𝑌 𝑢𝐿 𝑣0 + 0.33333𝑟𝜆𝑌 𝑣0 + 0.166666𝑟𝜆𝐵 𝑢𝐿 − 0.5𝑟 , 𝐵1 = 𝜆𝑌 𝑣0,𝜒 = 𝑥 / 𝐿,

+6𝛼2 𝐵 3𝜑𝑝 − 6𝛼2 𝐵 − 6𝛼1 𝐴2𝜑𝑝 + 6𝛼1𝐵 2𝜑𝑝 − 4𝛼1 𝛼2𝛽𝐵 3 𝜑𝑝 + 18𝛼1𝛼2 𝛽𝐵 2𝜑𝑝 + 2𝛼1 𝛽𝐴2𝜑𝑝 −12𝛼1 𝛼2𝛽𝐴2𝜑𝑝 − 6𝛼12 𝛽𝐵 3𝜑𝑝 + 4𝛼12 𝛽𝐵 4𝜑𝑝 − 4𝛼12 𝛽𝐴2𝐵 2𝜑𝑝 + 4𝛼1 𝛼2𝛽𝐴2𝐵𝜑𝑝 0.1666 2 4 2 2 [ 2 ] −2𝛼1 𝛽𝐵 𝜑𝑝 + 3𝛼1𝛽𝐵𝜑𝑝 − 6𝛼1𝐵 𝜑𝑝 + 6𝛼1𝐴 𝐵 𝜑𝑝 − 12𝛼1𝛼2 𝛽𝐵𝜑𝑝 𝐴(𝐵 −1) 2 3 +𝐶𝑜𝑠ℎ(2𝜑𝑝 )[6𝛼1 𝛽𝐵 𝜑𝑝 − 𝛼1𝛽𝐵𝜑𝑝 + 4𝛼1 𝛼2𝛽𝜑𝑝 − 4𝛼12 𝛽𝐵𝜑𝑝 ]

The dimensionless current is 𝐽𝑖𝑗 =

Where 𝐶 =

Analytical Expressions for concentration and current techniques

Experimental techniques Nonlinear differential and enzymatic scheme equations with initial and boundary conditions

𝑘𝐴

𝐴𝐻 ⇔′ 𝐴− + 𝐻 +

𝑘𝑍 𝑘𝐴

𝑍𝐻 ⇔′ 𝑍− + 𝐻 +

𝑘𝑍

𝑘𝑝

𝑃ℎ 𝐻 ⇔′ 𝑃ℎ− + 𝐻 −

𝑘𝑝

𝑑𝑥 2

𝑑𝑥 2 𝑑2 [𝐻 + ]

𝑑𝑥 2 𝑑2 [𝐴− ]

𝑑𝑥 2 𝑑2 [𝐴𝐻]

𝑑2 [𝑍 − ]

𝑑𝑥 2

𝑑𝑥 2 𝑑2 [𝑍𝐻]

𝑑𝑥 𝑑2 [𝑃ℎ− ]

, 𝑣 (𝑘2

𝑘 𝑘3 [𝐸]𝑡 , 𝑘𝑚 2 +𝑘3 ) 𝑚𝑎𝑥

+ 𝑟𝑃ℎ 𝐻 + 𝑟𝑍𝐻 +

+ 𝑟𝐴𝐻 = 0

− 𝑟𝐴𝐻 = 0

+ 𝑟𝑍𝐻 = 0

+ 𝑅 − 𝑟𝑍𝐻 = 0

+ 𝑟𝑃ℎ 𝐻 = 0

= 0 for 𝐶𝑖 =

=

𝐶𝑖 (𝐿) = 𝐶𝑖 𝑏 𝑓𝑜𝑟 𝐶𝑖 = [𝑆], [𝐴] 𝑇 , [𝐻 + ] 𝑇 [𝑃ℎ ]𝑇 (𝐿) = [𝑍] 𝑇 (𝐿) = 0

[𝑆], [𝑃ℎ ] 𝑇 , [𝑍]𝑇 , [𝐴] 𝑇 , [𝐻 + ] 𝑇

𝑑𝑥 𝑥=0

|

(𝑘2 +𝑘3 ) 𝑘1 𝑑𝐶𝑖

(𝑘2 +𝑘−1 ) 𝑘3

[𝑆]+𝑘𝑚

𝑣𝑚𝑎𝑥

𝑟𝐴𝐻 = 0 Where 𝑅=

𝐷

𝐷

𝐷

𝐷

𝐷

𝐷

𝐸 𝑑2 [𝑆] 11 (Meena and Potentiometric𝑆 ↔ 𝑃ℎ 𝐻 + 𝐷 2 − 𝑅 = 0 𝑑𝑥 𝐻2 𝑂 Rajendran 2010a) 𝑑2 [𝑃ℎ 𝐻] 𝑍𝐻 𝐷 + 𝑅 − 𝑟𝑃ℎ 𝐻 = 0 2

S. Reference No HPM 𝑐𝑜𝑠ℎ 𝑎

[𝑆]𝑏 𝑐𝑜𝑠ℎ(𝑎𝑥)

𝑏 𝑏

[1 −

𝑥

𝐿

; 𝐶𝑖 = [𝑆], [𝑃ℎ ] 𝑇 , [𝑍] 𝑇 , [𝐴]𝑇 , [𝐻 + ]𝑇 , 𝑥 = , 𝑎2 =

𝐶𝑖 𝑘𝑚

𝐷𝑘𝑚

𝐿2 𝑉𝑚𝑎𝑥

[𝑃ℎ ]𝑇 = [𝑃ℎ 𝐻] + [𝑃ℎ− ], [𝑍]𝑇 = [𝑍𝐻] + [𝑍 − ], [𝐴] 𝑇 = [𝐴𝐻] + [𝐴− ], [𝐻 + ]𝑇 = [𝐻 + ] + [𝑃ℎ 𝐻] + [𝑍𝐻] + [𝐴𝐻]

𝐶𝑖 =

𝑏

] [𝑐𝑜𝑠ℎ( 𝑎𝑥) − 𝑐𝑜𝑠ℎ 𝑎]

[𝐴]𝑇 = [𝐴]𝑇 Where the dimensionless variable defined as

𝑏

3

𝑐𝑜𝑠ℎ(2𝑎𝑥)

[𝐻 + ]𝑇 = 2 (−[𝑆] + [𝑆] ) + [𝐻 + ]𝑇

[𝑍] 𝑇 = −[𝑆] + [𝑆]

𝑏

([𝑆] )2

(𝑐𝑜𝑠ℎ 𝑎)2 𝑏

−

[𝑃ℎ ]𝑇 = −[𝑆] + [𝑆]

[𝑆] =

Analytical Expressions for concentration and current techniques

Table 3.2. (Continued)

𝑘𝑐𝑎𝑡

+𝑃

𝑘𝑜𝑓𝑓

𝑆 + 𝐸1 ↔ [𝐸𝑆] → 𝐸

𝑘𝑜𝑥

Michaelis-Menten Kinetics

14 (Renuga Devi, Sevukaperumal, and Rajendran 201)

𝐸+𝑃

voltammetry 𝑆 + 𝐸𝑟𝑒𝑑 ↔ 𝐸𝑆 → 𝑃 + 𝐸𝑂𝑋 𝑀𝑟𝑒𝑑 + 𝐸𝑂𝑋 ↔ 𝐸𝑀 → 𝑀𝑂𝑋 + 𝐸𝑟𝑒𝑑 𝑀𝑂𝑋 + 𝑒 − ↔ 𝑀𝑟𝑒𝑑

𝐸𝑆 + 𝑆 ↔ 𝐸𝑆3

𝐸+

𝑘𝑐 𝑘 𝑆 →𝑘1−1 𝐸𝑆 →

𝑠 𝜕𝜒2

1+

𝑆 𝑆2 + 𝐾𝑀 𝐾𝑖 𝐾𝑀

−

2ℎ2 𝐴 𝛼1 +𝐴

=0

𝑑𝑥

Boundary conditions: 𝑑𝐴 𝑥 = 0, 𝑑𝑥 = 0 (symmetry condition) 𝑑𝐴 𝑥 = 1, = 𝑆ℎ (1 − 𝐴) 𝑑𝑥 where 𝑆ℎ = (𝑘𝑚 𝐿/𝐷)

𝑑𝑥 2

𝑑2 𝐴

Boundary conditions: t = 0, S= 0 𝜕𝑆 𝜒 = 0, =0 𝜕𝜒 𝜒 = 0, S = 1 𝑑 2𝑐𝑚 𝐷𝑚 𝑑𝑥 2 −1 𝑘𝑐𝑎𝑡 𝐾𝑚 𝑐𝑒 𝑐𝑚 𝑐𝑆 = −1 (𝑐 + 𝐾 ) + 𝑐 𝑐𝑚 𝐾𝑚 𝑆 𝑆 𝑆 𝑑 2 𝑐𝑆 𝐷𝑆 𝑑𝑥 2 −1 𝑘𝑐𝑎𝑡 𝐾𝑚 𝑐𝑒 𝑐𝑚𝑐𝑆 = −1 (𝑐 + 𝐾 ) + 𝑐 ) 4(𝑐𝑚 𝐾𝑚 𝑆 𝑆 𝑆 𝑑𝑐𝑆 𝑥 = 0: =0 𝑑𝑥 𝑑𝑐 𝑥 = 𝜙: 𝑐𝑆 = 𝑐𝑆0, 𝑚 = 0

𝜕𝑡

Experimental techniques Nonlinear differential and enzymatic scheme equations with initial and boundary conditions 𝜕𝑆 𝜕2 𝑆 𝐾𝑆 Amperometric =𝐷 −

13 (Saravanakumar, Rajendran, and Sangaranarayanan 2015)

12 (Manimozhi, Subbiah, and Rajendran 2010)

S. Reference No

ADM

HPM

HPM

VIM 2

𝐾−𝑎) 2

− 𝑎[𝛼(1 − 𝑎) + 𝛽−𝑎)2 ], 𝐵 =

,𝐾 = ∞ 𝐷𝑠

𝑘 𝐿2

[

1 𝐾𝑚

+

𝐾𝑆 𝐾𝑚 𝑐𝑆0

+

𝐾𝑚 𝑐𝑆0

+

]

]

0 𝑒𝑥𝑝(−𝑉 ) 𝑐𝑚 𝑁 −1

0 𝑒𝑥𝑝(−𝑉 ) 𝑐𝑚 𝑁

2 𝑐𝑜𝑠ℎ(𝑉𝑁 )

+

𝐾𝑆

,𝜏 =

2

×[

(𝛼1 +1)

[

2

𝑚(ℎ)𝛼1 3𝑙(ℎ)

+

𝑙(ℎ) = − [

ℎ2 𝛼1 +1

+

2ℎ2

] 𝑎𝑛𝑑𝑚(ℎ) = 𝑆ℎ (𝛼1 +1)

ℎ2 𝛼1 +1

12

5𝑚(ℎ)

where ℎ = (𝑉𝑚 /2𝐷𝑆0 )1/2𝐿, 𝐴 = 𝑆/𝑆0 , 𝛼1 = 𝐾𝑚 /𝑆0 and

𝐴(𝑥) = 1 + 𝑙(ℎ) + 𝑚(ℎ)𝑥 2 − 2

−

2

𝑙(ℎ)𝑥 2

−

12

𝑚(ℎ)𝑥 4

]

2𝑖0 𝑐𝑜𝑠ℎ(𝑉𝑁 )

0 𝑛𝐹𝐷𝑚 𝜙𝑐𝑚

𝐾𝑖 𝐾𝑀

+

24 𝐾2

+ 𝜙𝑥 −

𝐾

12𝑥 2

]

,𝐴+𝐵+𝐶 =𝑎

4

[𝑥 +

15 𝐾 𝑘 𝑠∞

𝑛

−𝛽𝑎3

(𝜙 2 − 𝑥 2 )

−1

𝐿

]

]−

,𝛽= 2

𝐾 𝐷𝑠 𝑡

[𝑥 +

2 𝑐𝑜𝑠ℎ(𝑉𝑁 )

𝐾𝑀

𝑘 𝑠∞

𝐾

0 𝑒𝑥𝑝(−𝑉 ) 𝑐𝑚 𝑁 −1 2 𝑐𝑜𝑠ℎ(𝑉𝑁 )

+

,𝛼 =

𝐾𝑚 𝑐𝑆0

𝐾𝑆

𝐿

𝜒

−

2

,𝐶 =

The analytical expression of concentration of the substrate

𝐾𝑚

−𝑛𝐹𝑘𝑐𝑎𝑡 𝑐𝑒 𝜙

[

1

8𝐷𝑚 𝐾𝑚 𝐾𝑚

𝑘𝑐𝑎𝑡 𝑐𝑒

+

= 𝜙 2, 𝑥 =

0 𝑒𝑥𝑝(−𝑉 ) 𝑐𝑚 𝑘 𝑐 1 𝑁 − 𝑐𝑎𝑡 𝑒 [ 2 𝑐𝑜𝑠ℎ(𝑉𝑁 ) 𝐷𝑚 𝐾𝑚 𝐾𝑚

𝑘𝑠

𝑠

] 𝑐𝑚 (𝑥) = 𝑐𝑆0 −

𝑖=

2

𝑥2

𝑐𝑚 (𝑥) =

Where 𝑢 =

𝐾

1

12 𝑚

𝑎𝐾−2𝑎2 [𝛼+2𝛽−𝑎)]

𝑢(𝑥) = 1 − 𝑎 + 𝑎𝑥 + 2𝐶1 cosh √𝐾 𝑥 −

𝐴=

𝑢(𝑥) = 𝑢1 (𝑥) = 1 − 𝑎 + 𝐴𝑥 2 + 𝐵𝑥 4 + 𝐶𝑥 6

Analytical Expressions for concentration and current techniques

15 (Shunmugham and Rajendran 2013)

S. Reference No

𝑘3

𝐸red + 𝑀𝑒𝑑ox → 𝐸OX + 𝑀𝑒𝑑red

𝑘4

𝐸red + 𝑂2 → 𝐸OX + 𝐻2𝑜2

−1

dy2

+ 1)

−1

𝛽𝑆 [𝑆]

+

= 𝑑𝑦

𝑑[𝑆]

= 𝑑𝑦

𝑑[𝑀𝑒𝑑𝑟𝑒𝑑 ]

=0

At the electrode, y = d [MedOX ] = [MedOX]𝑏 = 𝐾𝑂 [MedOX]∞ , [𝑆] = [𝑆]𝑏 = 𝐾𝑆 [𝑆]∞ , [Medred ] =0

𝑑𝑦

𝑑[𝑀𝑒𝑑𝑂𝑋 ]

where (𝛽𝑆 = (𝑘−1 + 𝑘2)/𝑘1 and 𝛽𝑂 = 𝑘2 /𝑘4 ). At the far wall, y = 0

[MedOX ]

dy2

+

= − 𝑘2[𝐸𝑇 ] (

−1

𝑑2 [Medred ]

𝛽𝑂

𝐷𝑀

[MedOX ]

𝛽𝑆

[𝑆]

= 𝑘2 [𝐸𝑇 ] (

+ 1)

𝑑2 [𝑆]

𝛽𝑂

𝐷𝑆

+ 1)

Experimental techniques Nonlinear differential and enzymatic scheme equations with initial and boundary conditions Amperometric 𝑑 2[MedOX ] 𝐷𝑀 𝑘1 𝑘2 dy2 𝐸Ox + 𝑆 ↔ 𝐸𝑆 → 𝐸Red 𝑘−1 𝛽𝑆 𝛽𝑂 = 𝑘2[𝐸𝑇 ] ( + +𝑃 [𝑆] [MedOX ] ADM

2 𝜑𝑂 𝐵𝑂 𝐵𝑆 (𝐵𝑆 +𝐵𝑂 𝜇𝑆 ) . 12(𝐵𝑂 +𝐵𝑆 +𝐵𝑂 𝐵𝑆 )2

𝐹𝑂 = [MedOX ]⁄[MedOX]𝑏 , ⥂ 𝐹𝑆 = [𝑆]⁄[𝑆]𝑏 , 𝐹𝑅 = [Medred ]⁄[Medred ]𝑏 , 𝜒 = 𝑦⁄𝑑 , 𝐵𝑂 = [MedOX]𝑏 ⁄𝛽𝑂 , 𝐵𝑆 = [𝑆]𝑏 ⁄𝛽𝑆 , 𝜑𝑂2 = 𝑑 2𝑘2 [𝐸𝑇 ]⁄𝐷𝑀 [MedOX ]𝑏 and 𝜇𝑆 = 𝐷𝑀 [MedOX]𝑏 ⁄𝐷𝑆 [𝑆]𝑏 the dimensionless current, 𝑑𝐹𝑆 𝜑𝑂2 𝐵𝑂 𝐵𝑆 (1 − 4𝑤1) 𝐼 = −( ) = 𝑑𝜒 𝜒=1 (𝐵𝑂 + 𝐵𝑆 + 𝐵𝑂 𝐵𝑆 )

Where 𝑤1 =

𝐹𝑆 (𝜒) = 𝜇𝑆 (𝐹𝑂 (𝜒) − 1) + 1; 𝐹𝑅 (𝜒) = 1 − 𝐹𝑂 (𝜒) 𝜑𝑂2 𝐵𝑂 𝐵𝑆 [5𝑤1 − 1 + (1 − 6𝑤1 )𝜒 2 + 𝑤1𝜒 4] 𝐹𝑂 (𝜒) = 1 + 2(𝐵𝑂 + 𝐵𝑆 + 𝐵𝑂 𝐵𝑆 )

Analytical Expressions for concentration and current techniques

Table 3.2. (Continued)

𝐹 0 )] (𝐸−𝐸𝑃𝑄 𝑅𝑇

𝐶𝑃0

, 𝜕𝑥

𝜕[𝑆]

=0

=0

𝑑𝑥 2 2

𝑑2 𝑀𝑜𝑥

−

𝑘𝑐𝑎𝑡 𝐸𝑆𝑀𝑜𝑥

𝑥=0

)

𝐾𝑆 𝑀𝑜𝑥 +𝐾𝑀 𝑆+𝑆𝑀𝑜𝑥

𝜕𝑥

𝜕[𝑄]

=0

𝑑𝑥

) 𝑑𝑥

= 0, 𝑆 = 𝑆0

(1+𝑒

𝑖(𝑆) = 2𝐹𝑗(𝑆) = 2𝐹𝐷𝑆

|

𝑑𝑥 𝑥=𝑙

𝑑𝑆

The surface flux (j) and current density (i)

𝑥 = 𝑙:

𝑑𝑀𝑜𝑥

𝑑 𝑆 𝑘𝑐𝑎𝑡 𝐸𝑆𝑀𝑜𝑥 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑑𝑒 𝐷 − 2𝑀𝑟𝑒𝑑 → 2𝑀𝑜𝑥 + 2𝑒 − 𝑆 𝑑𝑥 2 2(𝐾𝑆 𝑀𝑜𝑥 + 𝐾𝑀 𝑆 + 𝑆𝑀𝑜𝑥 ) G denote glucose =0 𝑀𝑜 𝑑𝑆 𝑔 denote gluconolactone 𝑥 = 0: 𝑀𝑜𝑥 = , =0 −𝜀

𝐷𝑀

𝑖 = 𝐹𝑆𝐷𝑃 (

𝑥 = ∞, 𝜕[𝑄]/𝜕𝑥 = 0 The current flowing through the electrode is

[𝑄] =

−

1 1 1 1 + + + 𝑘1 [𝑆] 𝑘1,2 𝑘2,2 𝑘2 [𝑄] 𝐶𝐸0 1 1 1 1 + + + 𝑘1 [𝑆] 𝑘1,2 𝑘2,2 𝑘2 [𝑄]

1+𝑒𝑥𝑝[

𝑑𝑥 2

x  0,

𝐷𝑆

𝑑2 [𝑆]

𝑃 𝑑𝑥 2

Experimental techniques and Nonlinear differential equations enzymatic scheme with initial and boundary conditions 𝑑2 [𝑄] 𝐶𝐸0 Voltammetry 𝐷 − =0

17 (Muthuramalingam and oxidation of glucose 𝐺𝑂𝑥 Lakshmanan 2016) 𝐺 + 2𝑀𝑜𝑥 → 𝑔 + 2𝑀𝑟𝑒𝑑

16 (Rasi, Rajendran, and Subbiah 2015)

S. Reference No

ADM

NHPM and Laplace transform technique 2(1+𝑒 −𝜉 )

1

𝑒

1+𝑒−𝜉

1+𝑒−𝜉+𝜎

−√

𝑧

(2 + 𝑒

1

1 √1+𝜎

𝑖

=

√1+𝜎

}𝑋 + {

2

𝐿

𝑦

𝜈𝐿

𝑆

𝜅4 𝑚𝜀 (𝜇𝜂+𝜎𝑚𝜀 2 )

𝜅4 𝜇𝜂𝑚𝜀

𝜅4 𝑚𝜀

𝑙

𝜇

𝑙

𝑥

𝑘𝑐𝑎𝑡 𝐸 𝐷𝑀 𝑀𝑜

3𝜂

}

𝜅2 𝑚𝜀

𝜅2 𝑚𝜀

𝑀𝑜

,𝜎 =

𝑆𝑜

𝐾𝑆

,𝜂 =

𝐷𝑀 𝑀𝑜

,𝑌 =

} 𝑋4

} 𝑋2 −

2𝐷𝑆 𝑆𝑜

24𝜂2 [(𝜎+1)𝑚𝜀 +𝜇]3

𝜅4 𝑚𝜀 (𝜇𝜂+𝜎𝑚𝜀 2 )

−

2[(𝜎+1)𝑚𝜀 +𝜇]

2𝜂[(𝜎+1)𝑚𝜀 +𝜇]

𝐾𝑀

𝜎𝑚𝜀 2

,𝜇 =

{ − 𝜂2 [(𝜎+1)𝑚𝜀 +𝜇]3 3

+{

}𝑋 + {

3

24𝜂2 [(𝜎+1)𝑚𝜀 +𝜇]3 6𝜂2 [(𝜎+1)𝑚𝜀 +𝜇]3

+

} 𝑋4 𝜅4 𝑚𝜀 (3𝜇𝜂+5𝜎𝑚𝜀 2 )

24𝜂2 [(𝜎+1)𝑚𝜀 +𝜇]3

),

}𝑋 + {

𝑧

3𝜂2 [(𝜎+1)𝑚𝜀 +𝜇]3

, 𝑋 = , 𝜅 = 𝑙√

−

𝑎𝐿2

𝑆𝑜

,𝛿 =

,𝑠 = 𝐷𝑆

𝑀𝑜

𝑀𝑟𝑒𝑑

, 𝑃𝑒 =

𝑚=

𝜅2 𝑚𝜀 𝜂[(𝜎+1)𝑚𝜀 +𝜇]

where

𝜓=

+

𝜅4 𝑚𝜀 (𝜇𝜂+𝜎𝑚𝜀 2 )

the dimensionless current is

4𝜂2 [(𝜎+1)𝑚𝜀 +𝜇]3

𝜅4 𝜎𝑚𝜀 3

𝜅2 𝑚𝜀

} 𝑋3 + { 2𝜂[(𝜎+1)+𝜇]

6𝜂[(𝜎+1)𝑚𝜀 +𝜇]3

𝑠(𝑋) = 1 −

{

𝜅4 𝑚𝜀 𝜇

[(𝜎+1)𝑚𝜀 +𝜇]

𝜅2 𝑚𝜀

√(1+𝑒 −𝜉 )(1+𝑒 −𝜉 +𝜎)

𝑚(𝑋) = 𝑚𝜀 + {

𝑖pl

1+𝑒−𝜉

1+𝑒−𝜉+𝜎

2√

The normalized steady state current is

𝐶 𝜙pl =

,

√(1+𝑒−𝜉 )(1+𝑒 −𝜉 +𝜎)

The plateau current

𝜙𝐶 =

The catalytic current response is

𝑞(𝑧) =

Analytical Expressions for concentration and current techniques

20 (Saravanakumar, Ganesan, and Rajendran 2015)

Amperometric

19 (Rahamathunissa and Rajendran 2008a)

𝑘2

𝑘3

𝐸red + 2MOx → 2MRed + 𝐸OX

𝑘−1

𝐸Ox + 𝑆 ↔ 𝐸𝑆 → 𝐸Red + 𝑃

𝑘1

Ping-pong mechanism

𝐸 + 𝑆 ↔ 𝐸𝑆 → 𝐸 + 𝑃

𝐾2

Amperometric

18 (Varadha-rajan and Rajendran 2011)

𝐾𝑀

Experimental techniques and enzymatic scheme

S. Reference No

+

𝛬

𝐾𝑆

1

=0

,𝛼 = 2

1

1+𝛼𝑆

=0

,𝛬 = [ 𝑘𝑐𝑎𝑡 (𝐸)𝑇

𝐷𝐾𝑆

]

1 2

= 𝐷𝑠

= 𝐷𝑠

𝜕2 𝑆

𝜕𝜒2

𝜕2 𝑆

𝜕𝜒2

−

− 1+𝑆/𝐾𝑀

𝐾𝑆

𝐾𝑀 +𝑆

𝑘2 𝐸0 𝑆

= 𝐷𝑀 𝜕𝑥 2

𝜕2 𝐶𝑀𝑅 (𝑥,𝑡)

− 2𝑣𝐸𝑛𝑧

𝑥 = 𝑙, 𝐶𝑆 = 𝐶𝑆𝑏 ; 𝜕𝑥

𝜕𝐶𝑀𝑅

=0

𝑡 = 0, 𝐶𝑆 = 0; 𝐶𝑀𝑅 = 𝐶𝑀𝑇 − 𝑀0∗

𝜕𝑡

𝜕𝐶𝑀𝑅 (𝑥,𝑡)

𝜕𝐶𝑆 (𝑥, 𝑡) 𝜕 2 𝐶𝑆 (𝑥, 𝑡) = 𝐷𝑆 − 𝑣𝐸𝑛𝑧 𝜕𝑡 𝜕𝑥 2

S(0, χ) =𝑆0 (χ) (χ ∈ Ω)

𝜕𝑡

𝜕𝑆

𝜕𝑡

𝜕𝑆

Using the following boundary conditions: 𝑆 = 𝑆∞ , 𝑃 = 𝑃∞ = 0 when 𝑥 ≥ 𝐿 + 𝛿 𝑆 = 0, 𝑃 = 0 when 𝑥 = 0,

𝛾=

where

𝑑𝑥 2

1+𝛼𝑆 𝛾𝑆

𝛾𝑆

𝑆 𝑆∞

[ 𝛿

𝛿

𝐿

𝛿

𝐿

𝛿

𝐿

𝛿 3

12𝛾𝐸 (1 − 𝛼1 )𝑋 2 − 2𝛾𝐸 2 (1 + ) 𝑋 3 + 𝛾𝐸 2 𝑋 4

𝐿

𝛿 3

𝛿

𝐿

(𝛾𝐸 2 (1 + ) + 12𝛼1 𝛾𝐸 (1 + ) − 12𝛾𝐸 (1 + )) 𝑋 +

,𝑉 = 𝑃∞

𝑃 𝐿

𝑥

, ⥂ 𝑋 = , 𝛾𝐸 = 𝛾𝐿2 , 𝛼1 =⥂ 𝛼𝑆∞

(12𝛾𝐸 (1 + ) − 12𝛾𝐸 𝛼1 (1 + ) − 𝛾𝐸 2 (1 + ) ) 𝑋 + 𝐿 𝐿 𝐿 [ ] 24 𝛿 12𝛾𝐸 (𝛼1 − 1)𝑋 2 − 𝛾𝐸 2 𝑋 4 + 2𝛾𝐸 2 (1 + ) 𝑋 3 1

24

1

]

2𝛼

𝐾𝑥 2

+

𝑎𝛼2

(1+𝑎𝛼)

𝑙𝑛 [

𝐾𝑎𝛼 2+𝑎𝛼)2

𝑥 2 + 1] − 𝐾 2𝛼

𝑥2

∞

when 𝛼𝑢 ≫ 1

𝑐𝑜𝑠ℎ(√𝜙2 𝑙)

√2𝐾 √𝐾𝑎𝛼 𝑥 tan−1 [ 𝑥] 𝛼3/2 √𝑎 √2+𝑎𝛼)

−

𝐶𝑀𝑇

𝑀0∗

]}

𝛿𝑀12 𝐷𝑀 𝑁2

𝑙

2𝑛+1 −𝑁2 𝑡 ∑∞ [ 𝑒 𝑛=1(−1)

2𝐶𝑀𝑇

𝑀0∗ 𝑁2 ] 𝑐𝑜𝑠(𝑀1 (𝑥 − 𝑙)) 𝐶𝑀𝑇

𝑛=1

∞

2 𝑒 −𝑁2𝑡 + ∑(−1)𝑛+1 [𝛿𝑀12 𝐷𝑀 𝑙 𝑀1 𝑁2

𝑖(𝑡) = 𝑛𝐹𝐴𝑟𝑝 𝐷𝑀 {𝐶𝑀𝑇 𝛿√𝜙2 𝑡𝑎𝑛ℎ(√𝜙2 𝑙) −

𝐶𝑀𝑇

=

u(x) = a +

𝑢(𝑥) = 𝑎 cosh(√𝐾𝑥) when 𝛼𝑢 ≪ 1

𝑢(𝑥) = 𝑎 +

−

The dimensionless current is given by 𝜓 = 𝐼𝑎 / 2𝐹𝐴𝐷𝑆∞ = (𝑑𝑈 ⁄𝑑𝑋 )𝑋=0 = 0.5𝛾𝐸 (1 + 𝛿 ⁄𝐿) − 0.5𝛾𝐸 𝛼1 (1 + 𝛿 ⁄𝐿) − 𝛾𝐸2 (1 + 𝛿 ⁄𝐿)3

𝑈=

where

𝑉(𝑋) =

𝑈(𝑋) = 1 +

Expressions for concentration and current

NHPM, 𝐶𝑆 (𝑥, 𝑡) 𝑐𝑜𝑠ℎ(√𝜙1 𝑥) 2𝐷𝑆 𝑀1 −𝑁 𝑡 = + ∑(−1)𝑛+1 𝑒 1 𝑐𝑜𝑠(𝑀1 𝑥) 𝐶𝑆𝑏 𝑙 𝑁1 Laplace 𝑐𝑜𝑠ℎ(√𝜙1 𝑙) 𝑛=1 Transformation 𝐶𝑀𝑜 (𝑥, 𝑡) 𝛿 𝑐𝑜𝑠ℎ (√𝜙2 (𝑥 − 𝑙))

VIM

HPM

−

𝑑2 𝑆 𝑑𝑥 2 𝑑2 𝑃

Analytical techniques

Nonlinear differential equations with initial and boundary conditions

Table 3.2. (Continued)

Steady and Non-Steady State Reaction-Diffusion Equations …

95

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Electrode.” Fuel Cells 15 (3): 523–36. https://doi.org/10.1002/fuce. 201500025. Saravanakumar, K, L Rajendran, and M V Sangaranarayanan. 2015. “Current-Potential Response and Concentration Profiles of Redox Polymer-Mediated Enzyme Catalysis in Biofuel Cells - Estimation of Michaelis-Menten Constants.” Chemical Physics Letters 621: 117–23. https://doi.org/10.1016/j.cplett.2014.12.030. Shanmugarajan, Anitha, Subbiah Alwarappan, Subramaniam Somasundaram, and Rajendran Lakshmanan. 2011. “Analytical Solution of Amperometric Enzymatic Reactions Based on Homotopy Perturbation Method.” Electrochimica Acta 56 (9): 3345–52. https://doi.org/10.1016/j.electacta.2011.01.014. Shunmugham, Loghambal, and L Rajendran. 2013. “Analytical Expressions for Steady-State Concentrations of Substrate and Oxidized and Reduced Mediator in an Amperometric Biosensor.” International Journal of Electrochemistry 2013 (7): 1–12. https://doi.org/10.1155/2013/812856. Subramaniam, Muthukaruppan, Indira Krishnaperumal, and Rajendran Lakshmanan. 2012. “Theoretical Analysis of Mass Transfer with Chemical Reaction Using Absorption of Carbon Dioxide into Phenyl Glycidyl Ether Solution.” Applied Mathematics 03 (10): 1179–86. https://doi.org/10.4236/am.2012.310172. Swaminathan, Rajagopal, Kothandapani Venugopal, Muthuramalingam Rasi, Marwan Abukhaled, and Lakshmanan Rajendran. 2019. “Analytical Expressions for the Concentration and Current In the Reduction of Hydrogen Peroxide at a Metal-Dispersed Conducting Polymer Film.” Química Nova. https://doi.org/10.21577/01004042.20170454. Thiagarajan, S., A. Meena, S. Anitha, and L Rajendran. 2011. “Analytical Expression of the Steady-State Catalytic Current of Mediated Bioelectrocatalysis and the Application of He’s Homotopy Perturbation Method.” Journal of Mathematical

Steady and Non-Steady State Reaction-Diffusion Equations …

99

Chemistry 49 (8): 1727–40. https://doi.org/10.1007/s10910-0119854-z. Varadharajan, Govindhan, and Lakshmanan Rajendran. 2011. “Analytical Solution of the Concentration and Current in the Electoenzymatic Processes Involved in a PPO-Rotating-DiskBioelectrode.” Natural Science 03 (01): 1–8. https://doi. org/10.4236/ns.2011.31001.

Chapter 4

STEADY AND NON-STEADY STATE NONLINEAR REACTION-DIFFUSION IN A CYLINDER 4.1. INTRODUCTION We consider a long circular cylinder in which diffusion is everywhere radial. In this case, concentration is a function of radius r and time t only, and the system of nonlinear reaction-diffusion equation becomes 𝜕2 𝐶

𝐷𝑆 [ 𝜕𝑟2𝑆 +

1 𝜕𝐶𝑆

𝜕2 𝐶

1 𝜕𝐶𝑃

𝐷𝑃 [ 𝜕𝑟2𝑃 +

𝑟 𝜕𝑟

𝑟 𝜕𝑟

] − 𝑓(𝐶𝑆 , 𝐶𝑃 ) = ] − 𝑔(𝐶𝑆 , 𝐶𝑃 ) =

𝜕𝐶𝑆 𝜕𝑡 𝜕𝐶𝑃 𝜕𝑡

(4.1)

(4.2)

where 𝑓(𝐶𝑆 , 𝐶𝑃 ) and 𝑔(𝐶𝑆 , 𝐶𝑃 ) are nonlinear functions and these depends the order and rate of the reaction and enzyme kinetics. For the steady-state conditions the above equation becomes

102

L. Rajendran, R. Swaminathan and M. Chitra Devi 𝑑2 𝐶

𝐷𝑆 [ 𝑑𝑟2𝑆 +

1 𝑑𝐶𝑆

𝑑2 𝐶

1 𝑑𝐶𝑃

𝐷𝑃 [ 𝑑𝑟2𝑃 +

𝑟 𝑑𝑟

𝑟 𝑑𝑟

] − f(𝐶𝑆 , 𝐶𝑃 ) = 0 ] − 𝑔(𝐶𝑆 , 𝐶𝑃 ) = 0

(4.3)

(4.4)

If the nonlinear tern 𝑓(𝐶𝑆 , 𝐶𝑃 ) = 0 and 𝑔(𝐶𝑆 , 𝐶𝑃 ) = 0 𝑑2 𝐶

𝐷𝑆 [ 2𝑆 + 𝑑𝑟

1 𝑑𝐶𝑆 𝑟 𝑑𝑟

]=0

(4.5)

Now the solution of the linear steady state equation becomes 𝐶𝑆 (𝑟) = 𝑐1 𝑙𝑜𝑔( 𝑟) + 𝑐2

(4.6)

If the nonlinear tern 𝑓(𝐶𝑆 , 𝐶𝑃 ) = 𝑘 , the equation becomes 𝑑2 𝐶

[ 𝑑𝑟2𝑆 +

1 𝑑𝐶𝑆 𝑟 𝑑𝑟

]−𝑘 = 0

(4.7)

Now the solution becomes 𝐶𝑆 (𝑟) = 𝑐1 𝑙𝑜𝑔( 𝑟) + 𝑐2 + 𝑘𝑟 2 /4

(4.8)

If the nonlinear tern 𝑓(𝐶𝑆 , 𝐶𝑃 ) = 𝑘𝐶𝑆 , the equation becomes 𝑑2 𝐶

[ 𝑑𝑟2𝑆 +

1 𝑑𝐶𝑆 𝑟 𝑑𝑟

] − k C𝑆 = 0

(4.9)

Now the solution becomes 𝐶𝑆 (𝑟) = 𝑐1 𝐽0 (𝑖√𝑘𝑟) + 𝑐2 𝑌0 (−𝑖√𝑘𝑟)

(4.10)

The constant 𝑐1 𝑎𝑛𝑑 𝑐2 can be obtained for the boundary conditions.

Steady and Non-Steady State Nonlinear Reaction-Diffusion … 103 For non-steady-state conditions the equation is 𝜕2 𝐶

1 𝜕𝐶𝑆

𝐷 [ 2𝑆 + 𝜕𝑟

𝑟

] = 𝜕𝑟

𝜕𝐶𝑆

(4.11)

𝜕𝑡

The solution of the above equation becomes (Carslaw and Jaeger 1959) 𝐶𝑆 (𝑟, 𝑡) =

2 𝑎2

∑∞𝑛=1 𝑒𝑥𝑝( − 𝐷𝛼𝑛 𝑡)

𝑎 𝐽0 (𝑟𝛼𝑛 ) ∫ 𝑟𝑓(𝑟) 𝐽0 (𝑟𝛼𝑛 )𝑑𝑟 𝐽02 01 (𝑎𝛼𝑛 ) 0

(4.12)

4.2. CONCENTRATION OF METHANOL USING HPM METHOD Methanol concentration within a biofilm phase of a biofilter bed under steady-state condition can be written as follows (Krailas and Pham 2002): 𝑑2 𝐶

𝐷𝑆 [ 𝑑𝑟2𝑆 +

1 𝑑𝐶𝑆 𝑟 𝑑𝑟

] − 𝑟𝑚 𝐾

𝐶𝑆

𝑚

+ 𝐶𝑆

=0

(4.13)

where rm is the maximum rate of reaction per unit biofilm volume and Km is the Michaelis-Menten constant per unit biofilter volume. The boundary conditions are 𝐶𝑆 = 𝑑𝐶𝑆 𝑑𝑟

𝐶𝑔 𝑚

at 𝑟 = 𝑅𝐶 + 𝛿

= 0 at 𝑟 = 𝑅𝐶

(4.14) (4.15)

104

L. Rajendran, R. Swaminathan and M. Chitra Devi

where 𝐶𝑔 the concentration in the gas phase, 𝑅𝐶 is the particle radius, 𝛿 the biofilm thickness, 𝑚 is the distribution coefficient between gas phase and biofilm phase. Consider the following normalized parameters (Varadharajan and Rajendran 2011): 𝑢=

𝐶𝑔

𝑟

,𝑅 = 𝑅 ,k = 𝑚 𝐶

𝐾𝑚 𝐶𝑔

, 𝛾2 =

𝑟𝑚 𝑅𝐶 2

(4.16)

𝐷𝑆 𝐶𝑔

Eqn. (4.13) is transformed into dimensionless form using above 𝑑2 𝑢 𝑑𝑅2

+

1 𝑑𝑢 𝑅 𝑑𝑅

−

𝛾2𝑢

(4.17)

𝑘+𝑢

The boundary conditions are represented as follows: 𝑑𝑢 𝑑𝑅

= 0 when 𝑅 = 1

(4.18)

1

and 𝑢 = 𝑚 when 𝑅 = 1 + 𝛿 ⁄𝑅𝑐 = 𝐿

(4.19)

We aim to solve the above equation using NHPM by considering the steps below. Homotopy is constructed to find the solution of Eqn. (4.17) is as follows (1 − 𝑝) [(𝑘 + 𝑢|𝑅=𝐿 ) (

𝑑2 𝑢

1 𝑑𝑢

𝑑2 𝑢

1 𝑑𝑢

+ 𝑅 𝑑𝑅)] + 𝑝 [(𝑘 + 𝑢) (𝑑𝑅2 + 𝑅 𝑑𝑅) − 𝑑𝑅2

𝛾 2 𝑢] = 0

(4.20)

The approximate solution of Eqn. (4.17) is 𝑢 = 𝑢0 + 𝑝 𝑢 1 + 𝑝2 𝑢2 +. ..

(4.21)

Steady and Non-Steady State Nonlinear Reaction-Diffusion … 105 And the initial approximations are as follows: 1

𝛿

𝑢0 = 𝑚 ; 𝑢𝑖 = 0; ∀𝑖 = 1,2,3, … at 𝑅 = 1 + 𝑅

(4.22)

𝑐

𝑑𝑢𝑖 𝑑𝑅

= 0 ; ∀𝑖 = 0,1,2, … at 𝑅 = 1

(4.23)

Substituting Eqn. (4.21) in Eqn. (4.20) and comparing the coefficients of like powers of p: 𝑝0 :

𝑑 2 𝑢0 𝑑𝑅2

+

1 𝑑𝑢0 𝑅 𝑑𝑅

=0

(4.24)

𝑑2 𝑢

1

1 𝑑𝑢0 )− 𝑅 𝑑𝑅

𝑝 :

𝑑2 𝑢

1 𝑑𝑢0

) − ( 𝑑𝑅20 + 𝑅 𝑑𝑅

)) + (𝑘 + 𝑢0 ) ( 𝑑𝑅20 + 𝑑𝑅

𝛾 2 𝑢0 = 0

(4.25) 𝑑2 𝑢

1

2

𝑑2 𝑢

1 𝑑𝑢1

𝑝1 : (𝑘 + 𝑚) (( 𝑑𝑅21 + 𝑅

𝑑 2 𝑢1

(𝑘 + 𝑢0 ) (

𝑑𝑅2

1 𝑑𝑢1

+𝑅

𝑑2 𝑢

1 𝑑𝑢2

(𝑘 + 𝑚) (( 𝑑𝑅22 + 𝑅 𝑑𝑅

1 𝑑𝑢1

) − ( 𝑑𝑅21 + 𝑅 𝑑𝑅

𝑑𝑅

)) +

𝑑 2 𝑢0

2

}=0 ) 𝑑𝑅

1 𝑑𝑢0

) − 𝛾 𝑢1 + 𝑢1 ( 𝑑𝑅2 + 𝑅

(4.26) Solving the Eqns. (4.24) and using the boundary conditions (4.22) and (4.23) we can obtain the following results 𝑑 2 𝑢0 𝑑𝑅2

1 𝑑𝑢0

+𝑅

𝑑𝑅

1

= 0 ⇒ 𝑢0 = 𝑚

(4.27)

On simplifying Eqns. (4.25) and solving using boundary conditions (4.22) and (4.23)we have 𝑑 2 𝑢1 𝑑𝑅2

1 𝑑𝑢1

+𝑅

𝑑𝑅

𝛾2𝑢

− (𝑘+𝑢0 ) = 0 ⟹ 0

𝑑 2 𝑢1 𝑑𝑅2

1 𝑑𝑢1

+𝑅

𝑑𝑅

=

𝛾2 1 𝑚

𝑚(𝑘+ )

(4.28)

106

L. Rajendran, R. Swaminathan and M. Chitra Devi 𝛾2

𝑢1 (𝑅) = 4(1+𝑘𝑚) (𝑅2 − 2 𝑙𝑜𝑔(𝑅) − (𝐿2 − 2 𝑙𝑜𝑔(𝐿)))

(4.29)

On simplifying Eqns. (4.26) and solving 𝑑2 𝑢

1 𝑑𝑢2

( 𝑑𝑅22 + 𝑅

)− 𝑑𝑅

𝛾2 𝑘+

1 𝑚

𝑢1 = 0

𝑅2 + 8 𝑙𝑜𝑔(𝐿) − 4𝐿2 2 )𝑅 −8 𝑙𝑜𝑔(𝑅) + 8 𝛾4𝑚 𝑢2 (𝑅) = 64(1+𝑘𝑚)2 +12 𝑙𝑜𝑔(𝐿) − 8𝐿2 (𝑙𝑜𝑔(𝐿) + 1) +3𝐿4 + 16 𝑙𝑜𝑔(𝐿)2 ( +4 𝑙𝑜𝑔(𝑅) (2𝐿2 − 3 − 4 𝑙𝑜𝑔(𝐿)))

(4.30)

(

(4.31)

According to the HPM, we have 𝑢 (𝑅) = lim 𝑢 ( 𝑅 ) = 𝑢0 + 𝑢1 + 𝑢2 + ⋯

(4.32)

𝑝→1

Using Eqns. (4.27),(4.29) and (4.31) in Eqn. (4.32), we obtain the final result as given in Eqn. (4.33). 𝑅2 − 2 𝑙𝑜𝑔(𝑅) )+ −(𝐿2 − 2 𝑙𝑜𝑔(𝐿)) 𝑅2 + 8 𝑙𝑜𝑔(𝐿) − 4𝐿2 2 ( )𝑅 −8 𝑙𝑜𝑔(𝑅) + 8 𝛾4𝑚 +12 𝑙𝑜𝑔(𝐿) − 8𝐿2 (𝑙𝑜𝑔(𝐿) + 1) ∀𝑘 64(1+𝑘𝑚)2 +3𝐿4 + 16 𝑙𝑜𝑔(𝐿)2 ( +4 𝑙𝑜𝑔(𝑅) (2𝐿2 − 3 − 4 𝑙𝑜𝑔(𝐿))) 1

𝛾2

𝑢(𝑅) = 𝑚 + 4(1+𝑘𝑚) (

(4.33)

Steady and Non-Steady State Nonlinear Reaction-Diffusion … 107 Limiting Case 1: Unsaturated (First-Order) Catalytic Kinetic The kinetic behavior of concentration methanol depends on two factors 𝑘 and. When 𝑢 k or k / u is small (first-order reaction) the Eqn. (4.17) becomes 𝑑2 𝑢 𝑑𝑅2

1 𝑑𝑢

+ 𝑅 𝑑𝑅 = 𝛾 2

(4.40)

Now the solution becomes as follows: 𝑢(𝑅) =

𝛾2 4

4

(𝑅2 − 2 𝑙𝑜𝑔(𝑅) − (𝐿2 − 2 𝑙𝑜𝑔(𝐿) − 𝑚))

(4.41)

The Eqns. (4.33), (4.39) and (4.41) satisfies the boundary conditions (4.18) and (4.19). This equation represents the new approximate analytical expressions for the substrate concentration 𝑢(𝑅) for all values of parameters 𝑘, 𝛾, 𝑚 and L. To find the accuracy of the

Figure 4.1. Plot of the normalized methanol concentration 𝑢(𝑅) versus normalized distance 𝑅, for the fixed values of 𝑘 = 1, 𝑚 = 1, 𝐿 = 3 and for various values of 𝛾. The solid curves are plotted using Eqn. (4.33) and the symbol (∗) represents numerical simulation.

Steady and Non-Steady State Nonlinear Reaction-Diffusion … 109

Figure 4.2. Plot of the normalized methanol concentration 𝑢(𝑅) versus normalized distance 𝑅, for the fixed values of 𝛾 = 1, 𝑚 = 1, 𝐿 = 3 and for various values of 𝑘 . The solid curves are plotted using Eqn. (4.33) and the symbol (∗) represents numerical simulation.

Figure 4.3. Plot of the normalized methanol 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝑢(𝑅) versus normalized distance 𝑅, for 𝑘 = 1, 𝑚 = 1, 𝐿 = 3 and for various values of 𝛾 . The solid curves are plotted using Eqn. (4.39) and the symbol (∗) represents numerical simulation.

110

L. Rajendran, R. Swaminathan and M. Chitra Devi

Figure 4.4. Plot of the normalized methanol concentration 𝑢(𝑅) versus normalized distance 𝑅, for the fixed values of γ = 1, 𝑚 = 1, 𝐿 = 3 and for various values of 𝑘 . The solid curves are plotted using Eqn. (4.39) and the symbol (∗) represents numerical simulation.

Figure 4.5. Plot of the normalized methanol concentration 𝑢(𝑅) versus ormalized distance 𝑅, for 𝑘 = 1, 𝑚 = 1, 𝐿 =3 and for various values of γ . The solid curves are plotted using Eqn. (4.41) and the symbol (∗) represents numerical simulation.

Steady and Non-Steady State Nonlinear Reaction-Diffusion … 111 analytical solution, it is compared with numerical solution (Matlab) for various values of parameter𝛾and 𝑘 in Figures 4.1-4.5. A satisfactory agreement is noted.

4.3. ANALYTICAL SOLUTION OF VARIOUS NONLINEAR BOUNDARY VALUE PROBLEMS IN CYLINDRICAL COORDINATES The Cartesian coordinate system provides a simple way to describe where points in space are located. However, certain surfaces can be challenging to model with the Cartesian system. There are two different ways of defining the position of space points, both of which are based on polar coordinate extensions. As the name suggests, cylindrical coordinates are useful for handling cylinder-shaped problems. Similarly, spherical coordinates are helpful when dealing with sphererelated issues. Microelectrodes are one of the essential tools in electrochemical research (Le, Kätelhön, and Compton 2020). Microcylinder electrodes, mostly in the form of microwire electrodes, are commonly used by microelectrodes (Molina et al. 2015; Chen, Lin, and Compton 2018) because of their low cost and ease of manufacturing process. Cylindrical electrodes, one micro dimension (the radius), and one macro size (the length) play an essential role in diverse research areas, including environmental sensors (Privett, Shin, and Schoenfisch 2010), medical diagnosis(Harfield, Batchelor-McAuley, and Compton 2012), and nano-impact studies (Li, Batchelor-McAuley, and Compton 2019). Various theoretical studies (Aoki et al. 1985) have modeled the current responses of infinitely long cylindrical electrodes. Tables 4.1-4.2 provides a recent contribution to the mathematical methods of nonsteady and steady state reaction- diffusion equations in cylindrical coordinate systems.

1

( 𝜕𝑟

𝜕𝑐𝐵

) 𝑟=0

=0

Experimental Nonlinear differential techniques equations with initial and and enzymatic boundary conditions scheme (Rahamathunissa, Chronoamperometric 𝜕𝑐𝐵 = 𝜕2 𝑐𝐵 + 1 𝜕𝑐𝐵 − 𝐾𝑐 𝐵 𝜕𝜏 𝜕𝑟 2 𝑟 𝜕𝑟 Basha, and 𝐴 ± 𝑒− → 𝐵 Initial and boundary conditions 𝑘 Rajendran 2007) 𝐵 + 𝑍 → 𝐴 + are products 𝜏 = 0: 𝑐𝐵 = 0 (EC’ reaction) 𝑟 → ∞: 𝑐𝐵 = 0 𝑐𝐵 = 𝑐𝐴∗ on the electrode surface

S. Reference No

Expressions for concentration and current or flux

1 ) 𝑑𝑢] 𝑙𝑛[(1.4986𝑢1/2 + 5.2945] + 𝑒 −𝐾𝑢 [0.5642𝜏 −1/2 𝑒𝑥𝑝( − 0.1772𝜏 1/2 )

+

1 𝑙𝑛[(1.4986𝑢1/2 +5.2945]

when K = 0

+

1 ] for all K 𝑙𝑛[(1.4986𝑢1/2 + 5.2945] 𝜑(𝜏) = 0.5642𝜏 −1/2 𝑒𝑥𝑝( − 0.1772𝜏1/2 ) +

0

𝜑 = [𝐾 ∫ 𝑒 −𝐾𝑢 (0.5642𝑢1/2 𝑒𝑥𝑝( − 0.1772𝑢1/2 )

Shifting The transient limiting current for first-order EC’ reaction 𝐼(𝜏)𝑎 1 1 1 Formula, 𝜑≡ 𝑒 (−𝐾𝜏) + (√𝐾 − ) 𝑒𝑟𝑓(√𝐾𝜏) ∗𝐴 = 2 + 4𝑛𝐹𝐷𝐴 𝑐𝐴 8√𝐾 √𝜋𝜏 Danckwerts’ The current for all time for all values of reaction Method 𝜏

Analytical techniques

Table 4.1. Recent contribution to mathematical modelling of non-steady state reaction-diffusion equations in cylinder

2

𝜕𝑧

𝜕𝑐𝐵

) 𝑧=0

=0

𝐼 ∗𝑏 4𝑛𝐹𝐷𝐴 𝑐𝐴

𝜋

=

1 𝜕𝑐 ± ∫𝑎/𝑏 [ 𝐵 ] 𝑟𝑑𝑟 2 𝜕𝑧 𝑧=0

𝜑≡

The normalized current or flux is given by

(

Experimental Nonlinear differential techniques equations with initial and and enzymatic boundary conditions scheme (Rajendran 2006) Chronoamperometric 𝜕𝑐𝐵 = 𝜕2 𝑐𝐵 + 1 𝜕𝑐𝐵 + 𝜕2 𝑐𝐵 − 𝜕𝑇 𝜕𝑟 2 𝑟 𝜕𝑟 𝜕𝑧 2 𝐴 ± 𝑒− → 𝐵 𝐾𝑐𝐵 𝑘 𝐵+𝑍 →𝐴+ Initial and boundary conditions products are 𝑇 = 0: 𝑐𝐵 = 0 𝑟 → ∞: 𝑐𝐵 = 0 𝑐𝐵 = 𝑐𝐴∗ ontheelectrodesurface

S. Reference No

Expressions for concentration and current or flux

−1/2

𝜏

𝐷𝐵 𝑡

4

𝑏0 −𝑎1

𝑎0 −𝑏1

|,𝛾 =

𝑏+𝑎 2(𝑏−𝑎)

√𝜋𝛾 2𝛾+1

𝜋𝛾 2𝛾+1

𝑙0 4𝑏

, 𝑏1 =

𝑎0 , 𝑎1 , 𝑏0 , 𝑏1 , 𝐴1 , 𝐵1. 𝐶1 , 𝐷1 for different values of 𝛾is given in (Rajendran 2006).

( I 0 / b ) empirical,

, 𝐴1 = 𝑎1 , 𝐵1 = 𝑎0 , 𝐶1 = 𝑏0 − 𝑎1 , 𝐷1 =

=

, 𝑏0 = 𝑏 𝑒𝑚𝑝𝑖𝑟𝑖𝑐𝑎𝑙

𝑙

, ( 0)

, 𝑎1 = (𝛾+1/2) 𝑙𝑛(32𝛾)

2𝜋2 𝛾

, 𝑎0 =

The numerical values of (𝐼0/𝑏 ) exact,

|

𝑙𝑛(32(𝛾11/2)+𝑒𝑥𝑝(𝜋2 /4)

4𝜋2 𝛾/(2𝛾+1)

=

𝑇(2𝛾+1)2

𝑏 𝑒𝑥𝑎𝑐𝑡

𝑙

= , ( 0)

(𝑏−𝑎)2

(2𝛾+1)𝑏02 𝜋3/2

𝜏=

0.39115𝜏 −1/2 − 𝐾𝜏)] Dimensionless current for ring electrode (𝛾 > 0.5): 𝜑(𝜏) = 𝐴1 + 𝐵1 𝜏 −1/2 + 𝐶1 𝑒𝑥𝑝( − 𝐷1 𝜏 −1/2 ), ∀𝜏when K = 0 where

𝜏

0.2146[𝐾 ∫0 𝑒𝑥𝑝( − 0.39115𝑢−1/2 − 𝐾𝑢)𝑑𝑢 + 𝑒𝑥𝑝( −

𝜑(𝜏) = 0.7854 + 0.4431[√𝜋𝐾𝑒𝑟𝑓(√𝐾𝜏) + 𝑒 −𝐾𝜏 𝜏 −1/2 ] +

𝐷1 𝑢 − 𝐾𝑢)𝑑𝑢 + 𝑒𝑥𝑝( − 𝐷1 𝜏 − 𝐾𝜏)]∀𝜏 Dimensionless current for disc electrode (𝛾 = 0.5)

−1/2

𝜑(𝜏) = 𝐴1 + 𝐵1 [√𝜋𝐾𝑒𝑟𝑓(√𝐾𝜏) + 𝑒 −𝐾𝜏 𝜏 −1/2 ] + 𝐶1 [𝐾 ∫0 𝑒𝑥𝑝( −

𝜑(𝜏) = 𝑏0 + 𝑏1 [√𝜋𝐾𝑒𝑟𝑓(√𝐾𝜏) + 𝑒 −𝐾𝜏 𝜏 −1/2 ], 𝜏 → ∞

Danckwerts’ The transient current for EC reaction for ring electrode Method 𝜑(𝜏) = 𝑎1 + 𝑎0 [√𝜋𝐾𝑒𝑟𝑓(√𝐾𝜏) + 𝑒 −𝐾𝜏 𝜏 −1/2 ], 𝜏 → 0

Analytical techniques

3

products

Experimental techniques and enzymatic scheme (Rahamathunissa, Chronoamperometric Basha, and 𝐴 ± 𝑒− → 𝐵 𝑘 Rajendran 200 𝐵+𝑍 →𝐴+

S. Reference No

( 𝜕𝑟

𝜕𝑐𝐵

) 𝑟=0

=0

√𝜋𝑢

+

1

) 𝑑𝑢] + ]

𝑙𝑛[(4𝑒 −𝛾 𝑢)1/1 +𝑒 5/3 ]

𝑙𝑛[(4𝑒 −𝛾 𝑢)1/1 +𝑒 5/3 ]

1

+

+

𝜑(𝜏) =

√𝜋𝜏

𝑒𝑥𝑝[(−√𝜋𝜏/10)

+

𝑙𝑛[(4𝑒 −𝛾 𝜏)1/2 +𝑒 5/3 ]

1

The current for an EC reaction when K = 0,

] 𝑙𝑛[(1.4986𝑢1/2 +5.2945]

1

1 ) 𝑑𝑢] 𝑙𝑛[(1.4986𝑢1/2 + 5.2945]

+𝑒 −𝐾𝑢 [0.5642𝜏 −1/2 𝑒𝑥𝑝( − 0.1772𝜏 1/2 ) +

0

𝜑 = [𝐾 ∫ 𝑒 −𝐾𝑢 (0.5642𝑢1/2 𝑒𝑥𝑝( − 0.1772𝑢1/2 )

𝜏

Substituting the numerical values of the constants we obtain

𝑒 −𝐾𝑢 [

𝑒 −√𝜋𝑢/10

√𝜋𝑢

𝑒 −√𝜋𝑢/10

𝜑 = [𝐾 ∫0 𝑒 −𝐾𝑢 (

𝜏

Danckwerts’ 𝜑 ≡ 𝐼(𝜏)𝑎 = 1 + 1 𝑒 (−𝐾𝜏) + (√𝐾 − 1 ) 𝑒𝑟𝑓(√𝐾𝜏) ∗𝐴 4𝑛𝐹𝐷𝐴 𝑐𝐴 2 8√𝐾 √𝜋𝜏 expression The current for all values of reaction rate for an EC reaction

 2 c B 1 c B c   Kc B  B r r  r 2 Initial and Boundary conditions are 𝑡 = 0, 𝑐𝐵 = 0 𝑟 → ∞: 𝑐𝐵 = 0 𝑟 = 0, 𝑐𝐵 = 𝑐𝐴∗

Expressions for concentration and current or flux

Analytical techniques

Nonlinear differential equations with initial and boundary conditions

Table 4.1. (Continued)

(Margret Enzyme flow PonRani, calorimetry Rajendran, and Eswaran 2011)

(Venugopal, Eswari, and Rajendran 2011)

2

3

Amperometric 𝑂2 + 2𝑐 → 2𝑜 − 𝑞 + 2𝐻2 𝑂 𝑜 − 𝑞 + 2𝐻 + + 2𝑒 − → 𝑐 where 𝑐 denotes catechol 𝑞 denotes quinone

(Devi et al. 2020)

Experimental techniques and enzymatic scheme Enzyme flow calorimetry

1

S. Reference No + 𝑟 𝑑𝑟

1 𝑑𝑐𝑠

−

𝑉𝑚 𝑐𝑠 𝑐2

𝑖

𝐷𝑒 (𝐾𝑚 +𝑐𝑠 +𝐾𝑠 )

+ 𝑟 𝑑𝑟

− 𝑐2

𝑖

𝐷𝑒 (𝐾𝑚 +𝑐𝑠 +𝐾𝑠 )

𝑉𝑚 𝑐𝑠

(𝑟

(𝑟

=0

𝑑𝑟

𝑑𝑟 𝑑𝑐𝑄

)+

)− 𝑐𝐶 +𝐾𝑀

𝑐𝐶 +𝐾𝑀 𝑘𝑐𝑎𝑡 𝑐𝐸 𝑐𝐶

𝑘𝑐𝑎𝑡 𝑐𝐸 𝑐𝐶

=0

=0

= 0,𝑟 = 𝑅𝑝 , 𝑐𝑆 = 𝑐𝑆0

𝑑𝑐𝐶

𝑑𝑟

𝑟 = 𝑟1; 𝐶𝐶 = 𝐶𝐶∗, 𝐶𝑄 = 0

Boundary conditions are: 𝑟 = 𝑟0; 𝐶𝐶 = 𝐶𝐶∗, 𝐶𝑄 = 0

𝑟 𝑑𝑟

𝑟 𝑑𝑟 𝐷𝑄 𝑑

𝐷𝐶 𝑑

𝑟 = 0,

𝑑𝑐𝑠

=0

= 0, 𝑟 = 𝑅𝑝 , 𝑐𝑆 = 𝑐𝑆0

1 𝑑𝑐𝑠

𝑑𝑟

𝑑𝑐𝑠

Boundary conditions are:

𝑑𝑟 2

𝑑2 𝑐𝑠

𝑟 = 0,

Boundary conditions are:

𝑑𝑟 2

𝑑2 𝑐𝑠

Nonlinear differential equations with initial and boundary conditions

HPM

HPM

MHFM

Analytical techniques

𝑐𝑜𝑠ℎ(𝑏)

𝑐𝑜𝑠ℎ(𝑏𝑥)

𝑐𝑠 𝐾𝑚

,𝑥 = 𝑅𝑝

𝑐𝐶∗

𝑐𝑄 (𝑟)

𝑛𝐹𝐿𝐷𝐶 𝑐𝐶∗

𝐼

𝜓=

=[

=

𝐼

𝐷𝐶

0

]

]

𝑘𝑚

𝐾𝑖

𝐾𝑚

𝑐𝑆0

,𝑐0 =

,𝑐0 =

𝐾𝑚

𝑐𝑆0

𝛾𝑆

𝛾𝐸

𝑑𝑄

𝑑𝑅 𝑅=1

= 2𝜋 [

𝑐𝐶∗

𝑐𝐶

0

]

;𝑅=

2(1+𝛼)

𝑟

𝑐𝐶∗

𝑐𝑄

𝛾𝑆 (1+𝑟1 )−2𝛾𝑆

;𝑄 =

𝑟0

𝑟

;𝛼=

𝐾𝑀

𝑐𝐶∗

; 𝛾𝐸 =

2𝜋𝜒𝑟0 {𝐾1 (𝜒𝑟0 )[𝐼0 (𝜒𝑟1 )−𝐼0 (𝜒𝑟0 )]−𝐼1 (𝜒𝑟0 )[𝐾0 (𝜒𝑟0 )−𝐾0 (𝜒𝑟1 )]} [𝐾0 (𝜒𝑟0 )𝐼0 (𝜒𝑟1 )−𝐾0 (𝜒𝑟1 )𝐼0 (𝜒𝑟0 )]

2(1+𝛼)

𝑟 𝑟 −𝛾𝑆 𝑅 2 +𝛾𝑆 (1+𝑟1 )𝑅−𝛾𝑆 (𝑟1 ) 0 0

= 2𝜋 ( )

= 𝑛𝐹𝐿𝐷𝑄 𝑐𝐶∗

;

𝑘𝑐𝑎𝑡 𝑐𝐸 𝑟02 𝐷𝑄 𝐷𝑄 𝐾𝑀

𝑟

,𝛽 =

2(1+𝛼)

0

𝐾𝑖 𝑘𝑚

𝛾𝐸 𝑅 2 −𝛾𝐸 (1+𝑟1 )𝑅+𝛾𝐸 (𝑟1 )

𝐾𝑚 𝐷𝑒

𝑉𝑚

,𝛽 =

𝑐𝐶∗

𝑟

, 𝛷 = 𝑅𝑝 √

(𝑥 − 1)

Where 𝜒 2 = 𝑘𝑐𝑎𝑡 𝑐𝐸 ⁄𝐷𝐶 𝐾𝑀 ,𝐶 =

𝑗=

𝑉𝑚 𝐾𝑚 𝐷𝑒

𝑐𝐶 (𝑟)

=1+[

𝐾𝑚

𝑟

2(𝛽+𝛽𝑐0 +𝑐02 )

𝑐𝑠

2

, 𝛷 = 𝑅𝑝 √

The sensor response 𝑗 is

Q(R) =

C(R) =

Where

𝑐=

𝑟 𝑅𝑝

𝛷 2 𝛽𝑐0

,𝑥 =

𝑐(𝑥) = 𝑐0 +

and 𝑐 =

𝐷𝐶 𝐾𝑀

𝛷2 𝑐2

; 𝛾𝑆 =

1+𝑐0 + 𝛽0

𝑘𝑐𝑎𝑡 𝑐𝐸 𝑟02

where the parameter b can be obtained from the equation 𝑏2 + 𝑏 𝑡𝑎𝑛ℎ(𝑏) =

𝑐(𝑥) = 𝑐0

Expressions for concentration and current or flux

Table 4.2. Recent contribution to mathematical modelling of steady- state differential equations arises in cylindrical electrode

(Rajendran 2006)

(Eswari and Rajendran 2011)

4

5

S. Reference No

Amperometric 𝑂2 + 2𝑐 → 2𝑜 − 𝑞 + 2𝐻2 𝑂 𝑜 − 𝑞 + 2𝐻 + + 2𝑒 − → 𝑐 where c denotes catechol q denotes quinone

𝐵 + 𝑍 → 𝐴 + products

𝑘

Experimental techniques and enzymatic scheme Chronoamperometric 𝐴 ± 𝑒− → 𝐵 + 𝜕𝑧 2

𝜕2 𝑐𝐵

− 𝐾𝑐𝐵 = 0

)

𝜕𝑧 𝑧=0

𝜕𝑐𝐵

=0

𝐼

=

(𝑟

(𝑟 𝑑𝑟

)+

)−

𝑑𝑟 𝑑𝑐𝑄

𝑑𝑐𝐶

𝑐𝐶 +𝐾𝑀

𝑐𝐶 +𝐾𝑀 𝑘𝑐𝑎𝑡 𝑐𝐸 𝑐𝐶

𝑘𝑐𝑎𝑡 𝑐𝐸 𝑐𝐶

=0

=0

The normalized current is 𝜓 = 𝐼 ⁄𝑛𝐹𝐿 𝐷𝑄 𝑐𝐶∗

𝑟 = 𝑟1; 𝐶𝐶 = 𝐶𝐶∗, 𝐶𝑄 = 0

Boundary conditions are: 𝑟 = 𝑟0; 𝐶𝐶 = 𝐶𝐶∗, 𝐶𝑄 = 0

𝑟 𝑑𝑟

𝑟 𝑑𝑟 𝐷𝑄 𝑑

𝐷𝐶 𝑑

𝜋

∗𝑏 4𝑛𝐹𝐷𝐴 𝑐𝐴 1 𝜕𝑐𝐵 ± ∫𝑎/𝑏 [ ] 𝑟𝑑𝑟 2 𝜕𝑧 𝑧=0

𝜑≡

The normalized current or flux is given by

(

VIM

Laplace transform

𝑟 𝜕𝑟

1 𝜕𝑐𝐵

Boundary conditions are: 𝑟 → ∞: 𝑐𝐵 = 0 𝑐𝐵 (𝑟, 0) = 1ontheelectrodesurface

+

𝜕2 𝑐𝐵 𝜕𝑟 2

Analytical techniques

Nonlinear differential equations with initial and boundary conditions 1

∗𝑏 4𝑛𝐹𝐷𝐴 𝑐𝐴

∗𝑏 4𝑛𝐹𝐷𝐴 𝑐𝐴 1

∗𝑏 4𝑛𝐹𝐷𝐴 𝑐𝐴 1

= 𝐴 + 𝐵𝐾 1/2 + 𝐶 𝑒𝑥𝑝( − 𝐷𝐾 1/2 )∀𝐾

= 𝑏0 + 𝑏1 √𝜋𝐾 + 𝑎1 + 𝑂(𝐾), 𝐾 → 0

= 𝑎0 √𝜋𝐾 + 𝑎1 + 𝑂(𝐾 −1/2), 𝐾 → ∞

2(𝑏−𝑎)

𝑏+𝑎

1

=

2𝛾+1

𝜋𝛾

√𝜋(𝑎0 −𝑏1 ) 𝑏0 −𝑎1

=

𝐶

(𝜋𝛾/2𝛾+1)−((2𝛾+1)(𝑙0

𝜋𝛾

− (2𝛾+1),

, 𝑏1 =

3

𝜋2

(2𝛾+1)𝑏02

𝜋𝛾

, 𝐴 = 𝑎1 = (2𝛾+1) ;

𝛾𝐸

𝛾𝑆

[−2𝑎 − 4𝑎𝛼 − 𝛾𝐸 ]

𝑄(𝑅) =

𝐶(𝑅) =

4𝛼

1

4𝛼

1

+ 4𝛼] 𝛾𝑆 [(𝑟1 ⁄𝑟0 )2 −1] 𝑙𝑛(𝑅) + 𝛾𝑆 ] 𝑙𝑛(𝑟1 ⁄𝑟0 )

𝛾𝐸 [(𝑟1 ⁄𝑟0 )2 −1] 𝑙𝑛(𝑅) − 𝛾𝐸 𝑙𝑛(𝑟1 ⁄𝑟0 )

[−𝛾𝑆 𝑅 +

2

[𝛾𝐸 𝑅 2 −

Limiting case result: Saturated (zero order) catalytic kinetics: In this case, 𝐶 > 𝐾𝑀 and 𝛼𝐶 >> 1,

𝜓 = 2𝜋

The normalized current is

𝛾𝑆

𝐶(𝑅) = 1 − 3𝑎 − 4𝑎𝛼 − 2.667𝑎2 𝛼 − 𝛾𝐸 − 0.667𝑎𝛾𝐸 − 𝑎𝑅 2 − 1.333𝑎2 𝛼𝑅 3 − 0.333𝛾𝐸 𝑎𝑅 3 + (4𝑎 + 4𝑎𝛼 + 4𝑎2 𝛼 + 𝛾𝐸 + 𝛾𝐸 𝛼)𝑅 𝛾 𝑄(𝑅) = 𝐸 (1 − 𝐶(𝑅))

𝐷=

𝑙0 4𝑏

/𝑏)2 /16𝜋)

4𝑏

𝑙0

, 𝑏0 =

= 𝐴 + 𝐶 = 𝑏0

√𝜋𝛾 ,𝑎 2𝛾+1 1

𝜋𝛾

, 𝑎0 =

∗𝑏 4𝑛𝐹𝐷𝐴 𝑐𝐴

𝐵 = 𝑎0 √𝜋 = (2𝛾+1) ; 𝐶 = 𝑏0 − 𝑎1 =

𝛾=

where

𝜑𝑠𝑠 (𝐾) =

Dimensionless current for disc electrode (𝛾 = 0.5) 𝜑𝑠𝑠 (𝐾) = 0.7854 + 0.7854𝐾 1/2 + 0.2146 𝑒𝑥𝑝( − 0.6934𝐾 −1/2) Dimensionless current for ring electrode (𝛾 > 0.5) when K=0:

𝜑𝑠𝑠 (𝐾) =

𝜑𝑠𝑠 (𝐾) =

𝜑𝑠𝑠 (𝐾) =

The steady state current for EC reaction is given as follows:

Expressions for concentration and current or flux

Table 4.2. (Continued)

S. Reference No

Experimental techniques and enzymatic scheme

Nonlinear differential equations with initial and boundary conditions

Analytical techniques

= 𝐷𝑄

(1 − [ 𝐾0 (𝜒𝑟0 )𝐼0 (𝜒𝑟1 )−𝐾0 (𝜒𝑟1 )𝐼0 (𝜒𝑟0 )

])

𝑎=

,

𝐶𝐶∗

𝐶𝐶

,𝑄 =

𝐶𝐶∗

𝐶𝑄

𝑅=

𝑟 𝑟0

,𝛼 =

𝐶𝐶∗ 𝐾𝑀

, 𝛾𝐸 =

𝐷𝐶 𝐾𝑀

𝑘𝑐𝑎𝑡 𝐶𝐸 𝑟02

, 𝛾𝑆 =

(−𝛾𝐸 (𝑟1⁄𝑟0)2 − 3 𝑟1⁄𝑟0 − 𝛾𝐸 (𝑟1⁄𝑟0) + 9 + 12𝛼 + 2𝛾𝐸 − √𝐴) 2(4𝛼(𝑟1⁄𝑟0) − 8𝛼 + 4𝛼(𝑟1⁄𝑟0)2)𝐴 = 81 + 216𝛼 + 36𝛾𝐸 − 12(𝑟1⁄𝑟0)2 𝛾𝐸 − 30𝛾𝐸 (𝑟1⁄𝑟0) − 54(𝑟1⁄𝑟0) + 24(𝑟1⁄𝑟0)2 𝛾𝐸 𝛼 − 48𝛾𝐸 𝛼 + 4𝛾𝐸 2 − 3𝛾𝐸 2 (𝑟1⁄𝑟0)2 − 4𝛾𝐸 2(𝑟1 ⁄𝑟0) + 24𝛾𝐸 𝛼(𝑟1⁄𝑟0) − 72𝛼(𝑟1⁄𝑟0) + 6 𝛾𝐸 (𝑟1⁄𝑟0)3 + 9(𝑟1⁄𝑟0)2 + (𝑟1⁄𝑟0)4𝛾𝐸 2 + 2𝛾𝐸 2 (𝑟1⁄𝑟0)3 + 144𝛼 2

𝐷𝑄 𝐾𝑀

𝑘𝑐𝑎𝑡 𝐶𝐸 𝑟02

where𝜒 2 = 𝑘𝑐𝑎𝑡 𝑐𝐸 ⁄𝐷𝐶 𝐾𝑀 ,𝐶 =

(𝜒𝑟0)3⁄2 {√𝜒𝑟0 𝑒𝑥𝑝( 𝜒𝑟1 − 𝜒𝑟0) − √𝜒𝑟1} − (𝜒𝑟0)√𝜒𝑟1{√𝜒𝑟0 − √𝜒𝑟1 𝑒𝑥𝑝( 𝜒𝑟0 − 𝜒𝑟1)} ≈ 2𝜋 [ ] (𝜒𝑟0) 𝑒𝑥𝑝( 𝜒𝑟1 − 𝜒𝑟0) − (𝜒𝑟1) 𝑒𝑥𝑝( 𝜒𝑟0 − 𝜒𝑟1) 𝑤ℎ𝑒𝑛𝜒𝑟0𝑎𝑛𝑑𝜒𝑟1 → ∞

The sensor response is 𝐼 𝑗= 𝑛𝐹𝐿𝐷𝐶 𝑐𝐶∗ 2𝜋𝜒𝑟0{𝐾1(𝜒𝑟0)[𝐼0(𝜒𝑟1) − 𝐼0(𝜒𝑟0)] − 𝐼1(𝜒𝑟0)[𝐾0 (𝜒𝑟0) − 𝐾0 (𝜒𝑟1)]} = [𝐾0(𝜒𝑟0)𝐼0(𝜒𝑟1) − 𝐾0 (𝜒𝑟1)𝐼0(𝜒𝑟0 )] ≈ 𝜋(𝜒𝑟0)2𝑤ℎ𝑒𝑛𝜒𝑟0𝑎𝑛𝑑𝜒𝑟1 → 0

𝑐𝐶∗

]

𝐼0 (𝜒𝑟) [𝐾0 (𝜒𝑟0 )−𝐾0 (𝜒𝑟1 )]+𝐾0 (𝜒𝑟) [𝐼0 (𝜒𝑟1 )−𝐼0 (𝜒𝑟0 )]

𝐾0 (𝜒𝑟0 )𝐼0 (𝜒𝑟1 )−𝐾0 (𝜒𝑟1 )𝐼0 (𝜒𝑟0 )

𝐷𝐶

𝐼 (𝜒𝑟) [𝐾0 (𝜒𝑟0 )−𝐾0 (𝜒𝑟1 )]+𝐾0 (𝜒𝑟) [𝐼0 (𝜒𝑟1 )−𝐼0 (𝜒𝑟0 )]

𝑐𝐶∗ 𝑐𝑄 (𝑟)

=[0

𝑐𝐶 (𝑟)

Unsaturated (first order) catalytic kinetics: In this case 𝑐𝐶 < 𝐾𝑀 .

2𝛼 𝑙𝑛(𝑟1 ⁄𝑟0 )

The normalized current is 𝜋𝛾𝑆 [(𝑟1⁄𝑟0)2 − 1 − 2 𝑙𝑛(𝑟1⁄𝑟0)] 𝜓=

Expressions for concentration and current or flux

Reference

(Eswari and Rajendran 2010)

S. No

6

𝐸2

𝑘2

𝐾𝑐𝑎𝑡

𝐸1 ↔ [𝐸1 𝑆] → 𝑃 +

𝑘1

Experimental techniques and enzymatic scheme Amperometric 𝑆+

Nonlinear differential equations with initial and boundary conditions 𝐷𝑆 𝑑 𝑑𝑐𝑆 𝑘𝑐𝑎𝑡 𝑐𝐸 𝑐𝑆 (𝑟 )− =0 𝑟 𝑑𝑟 𝑑𝑟 𝑐𝑆 + 𝐾𝑀 𝐷𝐻 𝑑 𝑑𝑐𝐻 𝑘𝑐𝑎𝑡 𝑐𝐸 𝑐𝑆 (𝑟 )+ − 𝑣(𝑟) 𝑟 𝑑𝑟 𝑑𝑟 𝑐𝑆 + 𝐾𝑀 =0 Boundary conditions are: 𝑑𝑐𝑆 𝑟 = 𝑟0: = 0, 𝑐𝐻 = 0 𝑑𝑟 𝑟 = 𝑟1: 𝑐𝑆 = 𝑐𝑆∗, 𝑐𝐻 = 0 HPM

Analytical techniques

𝐾1 (𝜒𝑟0 )𝐼0 (𝜒𝑟)+𝐼1 (𝜒𝑟0 )𝐾0 (𝜒𝑟)

]

= 𝑔𝐵 (𝜒𝑟0, 𝑟1⁄𝑟0) ≡

𝐼

[1 −

] 𝐾1 (𝜒𝑟0 )𝐼0 (𝜒𝑟1 )+𝐼1 (𝜒𝑟0 )𝐾0 (𝜒𝑟1 )

𝐾1 (𝜒𝑟0 )𝐼0 (𝜒𝑟0 )+𝐼1 (𝜒𝑟0 )𝐾0 (𝜒𝑟0 )

= 𝑔𝐵 (𝜒𝑟0, 𝑟1⁄𝑟0) ≈

≡

2𝜋 𝑙𝑛(𝑟1 ⁄𝑟0 )

when 𝜒𝑟0 and 𝜒𝑟1 → ∞

𝐾1 (𝜒𝑟0 )𝐼0 (𝜒𝑟1 )+𝐼1 (𝜒𝑟0 )𝐾0 (𝜒𝑟1 ) 𝑝 [𝑒𝑥𝑝(𝑝 𝜒𝑟1 )+𝑒𝑥𝑝(𝑝 𝜒𝑟0 )]

2𝜋(𝜒𝑟0 )2 𝐾1 (𝜒𝑟0 )𝐼0 (𝜒𝑟0 )+𝐼1 (𝜒𝑟0 )𝐾0 (𝜒𝑟0 ) [𝑒𝑥𝑝(𝑝 𝜒𝑟1 )−𝑒𝑥𝑝(𝑝 𝜒𝑟0 )]

𝐶𝑆 = 4(𝑐𝑆 − 𝑐𝑆∗)⁄(𝜒𝑟0)2 𝐾𝑀 =

𝑟02

𝑟12

−

𝑟02

𝑟2

𝑟

𝑟

− 2 𝑙𝑛 ( 1 )

Zero order catalytic kinetics: in this case 𝑐𝑆 > K𝑀 Electrode reaction at conducting material in the film when𝑣(𝑟) = 𝑘 𝑐𝐻

=

Electrode reaction at conducting material in the film when 𝑣(𝑟) = 𝑘 𝑐𝐻 𝐼 = 𝑔𝐵 (𝜒𝑟0, 𝜒𝑟1, 𝑝) 𝑛𝐹𝐿𝐷𝑆 𝑐𝑆∗

𝑛𝐹𝐿𝐷𝑆 𝑐𝑆∗

2𝜋 𝑙𝑛(𝑟1 ⁄𝑟0 )

2𝜋 [1 𝑙𝑛( 𝑟1⁄𝑟0) 2 − (𝜒𝑟0)2 𝑙𝑛( 𝜒𝑟0) −{ }] when 𝜒𝑟0 and 𝜒𝑟1 → 0 2 − (𝜒𝑟0)2 𝑙𝑛( 𝜒𝑟1)

𝐼 = 𝑔𝐵 (𝜒𝑟0, 𝑟1⁄𝑟0) 𝑛𝐹𝐿𝐷𝑆 𝑐𝑆∗

𝑛𝐹𝐿𝐷𝑆 𝑐𝑆∗

𝐼

The current is 𝐼 𝐾1(𝜒𝑟0)𝐼0(𝜒𝑟1) + 𝐼1(𝜒𝑟0)𝐾0(𝜒𝑟1) = 𝑔𝐴 (𝑟1⁄𝑟0 , 𝜒𝑟0) ≡ 2𝜋𝜒𝑟1 [ ] 𝑛𝐹𝐷𝑆 𝑐𝑆∗𝐿 𝐾1(𝜒𝑟0)𝐼0(𝜒𝑟1) + 𝐼1(𝜒𝑟0)𝐾0(𝜒𝑟1) ≈ 2𝜋𝜒𝑟1 𝑡𝑎𝑛ℎ[ 𝜒(𝑟1 − 𝑟0)] when 𝜒𝑟0 → ∞ ≈ 𝜒 2𝜋(𝑟12 − 𝑟02) when 𝜒𝑟0 → 0 Electrode reaction at electrode only when 𝑣(𝑟) = 0

𝐾1 (𝜒𝑟0 )𝐼0 (𝜒𝑟1 )+𝐼1 (𝜒𝑟0 )𝐾0 (𝜒𝑟1 )

𝑐𝑆 = 𝑐𝑆∗ [

First order catalytic kinetics: 𝑐𝑆 < K𝑀 Electrode reaction at conducting material in the film when 𝑣(𝑟) = 0

Expressions for concentration and current or flux

Table 4.2. (Continued)

S. No

Reference

Experimental techniques and enzymatic scheme

Nonlinear differential equations with initial and boundary conditions

Analytical techniques

=

1+𝑒𝑥𝑝[√𝑘⁄𝐷𝐻 (𝜒𝑟0 +𝜒𝑟1 −2𝜒𝑟)]

1+𝑒𝑥𝑝[−√𝑘⁄𝐷𝐻 (𝜒𝑟0 −𝜒𝑟1 )]

2𝜋√𝑘⁄𝐷𝐻 𝜒𝑟0 [𝑒𝑥𝑝[−√𝑘⁄𝐷𝐻 (𝜒𝑟0 −𝜒𝑟1 )−1]

𝑒𝑥𝑝[√𝑘⁄𝐷𝐻 (𝜒𝑟0 −𝜒𝑟)]+𝑒𝑥𝑝[√𝑘⁄𝐷𝐻 (𝜒𝑟1 −𝜒𝑟)]

𝑐𝐻 𝐷𝐻 (𝜒𝑟0 )2 𝐷𝑆 𝐾𝑀

𝑟12 ⁄𝑟02 𝑙𝑛(𝑟0 ⁄𝑟) 𝑙𝑛(𝑟⁄𝑟1 ) + ] 𝑙𝑛(𝑟1 ⁄𝑟0 ) 𝑙𝑛(𝑟1 ⁄𝑟0 ) 2⁄ 2 𝑟1 𝑟0 + 𝑟0⁄𝑟1] 𝑙𝑛( 𝑟1⁄𝑟0)

= 0.25 [𝑟 2⁄𝑟02 +

]

(𝑐𝑆 −𝑐𝑆∗ ) 𝑟 𝑟0

𝑟2

+ 0.5

𝑟12

−

𝑟1

𝐷𝐻

𝑘

, 𝜒 2 = 𝑘𝑐𝑎𝑡 𝑐𝐸 ⁄𝐷𝑆 𝐾𝑀 .

1+𝑒𝑥𝑝[−𝑝(𝜒𝑟0 −𝜒𝑟1 )]

2𝜋 𝑝𝜒𝑟0 [𝑒𝑥𝑝[−𝑝(𝜒𝑟0 −𝜒𝑟1 )−1]

𝑒𝑥𝑝[𝑝 (𝜒𝑟0 −𝜒𝑟)+𝑒𝑥𝑝[𝑝 (𝜒𝑟1 −𝜒𝑟)

𝑟02 𝑟02 𝑟0 1+𝑒𝑥𝑝[𝑝 (𝜒𝑟0 +𝜒𝑟1 −2𝜒𝑟)]

− 0.5

]

1 𝑟1 𝑟 2 𝑟02

= [

𝑐𝐻 𝐷𝐻 (𝜒𝑟0 )2 𝐷𝑆 𝑐𝑆∗

−

𝑟02

𝑟2

𝐼 𝑟1 = 𝜋 (1 − ) (𝜒𝑟0)2𝑛𝐹𝐿𝐷𝑆 𝑐𝑆∗ 𝑟0

𝐶𝐻𝑃 =

−

𝑟0

𝑟1

𝑟

𝑟0

+ ]

Electrode reaction at electrode only when 𝑣(𝑟) = 0

Where 𝑝 = √

= =1−[

=

𝜒2 𝐷𝑆 𝑐𝑆∗

𝑐𝑆∗ (𝜒𝑟0 )2 𝑐𝐻 𝑘

𝑛𝐹𝐿𝜒2 𝐷𝑆 𝑐𝑆∗ 𝐷𝐻

𝐼𝑘

𝐶𝐻𝑃 =

𝐶𝑆 𝑃 =

𝐼 𝜋 = [2 − (𝜒𝑟0)2𝑛𝐹𝐿𝐷𝑆 𝐾𝑀 2 For all values of 𝐾𝑀 Electrode reaction at conducting material in the film when 𝑣(𝑟) = 𝑘 𝑐𝐻

𝐶𝐻 =

Electrode reaction at electrode only when 𝑣(𝑟) = 0

𝑛𝐹𝐿𝜒2 𝐷𝑆 𝐾𝑀 𝐷𝐻

𝐼𝑘

𝐶𝐻 = 𝑐𝐻 𝑘⁄𝜒 2 𝐷𝑆 𝐾𝑀 = 1 − [

Expressions for concentration and current or flux

120

L. Rajendran, R. Swaminathan and M. Chitra Devi

REFERENCES Aoki, Koichi, Katsuya Honda, Koichi Tokuda, and Hiroaki Matsuda. 1985. “Voltammetry at Microcylinder Electrodes.” Journal of Electroanalytical Chemistry and Interfacial Electrochemistry 186 (1–2): 79–86. https://doi.org/10.1016/0368-1874(85)85756-7. Carslaw, Horatio Scott, and John Conrad Jaeger. 1959. “Conduction of Heat in Solids.” Oxford: Clarendon Press, 1959, 2nd Ed. Chen, Lifu, Chuhong Lin, and Richard G. Compton. 2018. “Single Entity Electrocatalysis: Oxygen Reduction Mediated via Methyl Viologen Doped Nafion Nanoparticles.” Physical Chemistry Chemical Physics 20 (23): 15795–806. https://doi.org/10.1039/ C8CP02311E. Devi, M Chitra, P Pirabaharan, L Rajendran, and Marwan Abukhaled. 2020. “An Efficient Method for Finding Analytical Expressions of Substrate Concentrations for Different Particles in an Immobilized Enzyme System.” Reaction Kinetics, Mechanisms and Catalysis, March, 1–19. https://doi.org/10.1007/s11144-020-01757-0. Eswari, A, and L Rajendran. 2010. “Analytical Solution of Steady-State Current an Enzyme-Modified Microcylinder Electrodes.” Journal of Electroanalytical Chemistry 648 (1): 36–46. https://doi.org/10.1016 /j.jelechem.2010.07.002. Eswari, A, and L Rajendran. 2011. “Analytical Expressions Pertaining to the Concentration of Catechol, o-Quinone and Current at PPOModified Microcylinder Biosensor for Diffusion-Kinetic Model.” Journal of Electroanalytical Chemistry 660 (1): 200–208. https://doi.org/10.1016/j.jelechem.2011.06.033. Harfield, John C., Christopher Batchelor-McAuley, and Richard G. Compton. 2012. “Electrochemical Determination of Glutathione: A Review.” The Analyst 137 (10): 2285. https://doi.org/10.1039/ c2an35090d.

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Krailas, Satida, and Q. Tuan Pham. 2002. “Macrokinetic Determination and Water Movement in a Downward Flow Biofilter for Methanol Removal.” Biochemical Engineering Journal 10 (2): 103–13. https://doi.org/10.1016/S1369-703X(01)00165-6. Le, Haonan, Enno Kätelhön, and Richard G. Compton. 2020. “Reversible Voltammetry at Cylindrical Electrodes: Validity of a One-Dimensional Model.” Journal of Electroanalytical Chemistry 859. https://doi.org/10.1016/j.jelechem.2020.113865. Li, Xiuting, Christopher Batchelor-McAuley, and Richard G. Compton. 2019. “Silver Nanoparticle Detection in Real-World Environments via Particle Impact Electrochemistry.” ACS Sensors 4 (2): 464–70. https://doi.org/10.1021/acssensors.8b01482. Margret PonRani, Vincent Michael Raj, Lakshmanan Rajendran, and Raju Eswaran. 2011. “Analytical Expression of the Substrate Concentration in Different Part of Particles with Immobilized Enzyme and Substrate Inhibition Kinetics.” Analytical and Bioanalytical Electrochemistry 3 (5): 507–20. Molina, A., J. González, E. Laborda, and R. G. Compton. 2015. “Analytical Solutions for Fast and Straightforward Study of the Effect of the Electrode Geometry in Transient and Steady State Voltammetries: Single- and Multi-Electron Transfers, Coupled Chemical Reactions and Electrode Kinetics.” Journal of Electroanalytical Chemistry 756: 1–21. https://doi.org/10.1016/j. jelechem.2015.07.030. Privett, Benjamin J., Jae Ho Shin, and Mark H. Schoenfisch. 2010. “Electrochemical Sensors.” Analytical Chemistry 82 (12): 4723–41. https://doi.org/10.1021/ac101075n. Rahamathunissa, G., C. A. Basha, and L. Rajendran. 2007. “The Theory of Reaction-Diffusion Processes at Cylindrical Ultramicroelectrodes.” Journal of Theoretical and Computational Chemistry 6 (2): 301–7. https://doi.org/10.1142/S0219633607 003076.

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Rajendran, L. 2006. “Microring Electrode: Transient and Steady-State Chronoamperometric Current for First-Order EC Reactions.” Electrochimica Acta 51 (21): 4439–46. https://doi.org/10.1016/j. electacta.2005.12.033. Varadharajan, Govindhan, and Lakshmanan Rajendran. 2011. “Analytical Expression for Methanol Concentration within a Biofilm Phase of a Biofilter Bed under Steady State Condition.” International Journal of Mathematical Archive 2 (11): 2389–2402. Venugopal, Kodhandapani, Alagu Eswari, and Lakshmanan Rajendran. 2011. “Mathematical Model for Steady State Current at PpoModified Micro-Cylinder Biosensors.” Journal of Biomedical Science and Engineering 04 (09): 631–41. https://doi.org/10. 4236/jbise.2011.49079.

Chapter 5

STEADY AND NON-STEADY NONLINEAR REACTION-DIFFUSION IN A SPHERE 5.1. INTRODUCTION If we limit ourselves to cases where the diffusion is radial, the nonlinear diffusion equation for a constant coefficient for the density/concentration/ heat of a substance, a population in spherical coordinate system takes the form 𝜕𝑎 𝜕𝑡 𝜕𝑏 𝜕𝑡

𝜕2 𝑎

2 𝜕𝑎

𝜕2 𝑏

2 𝜕𝑏

= 𝐷 𝜕𝑥 2 + 𝑥 𝜕𝑥 − 𝑓(𝑎, 𝑏, 𝑡) = 𝐷 𝜕𝑥 2 + 𝑥 𝜕𝑥 + 𝑔(𝑎, 𝑏, 𝑡)

(5.1)

(5.2)

For the case of steady state conditions the above equations becomes 𝑑2𝑎

2 𝑑𝑎

𝜕2 𝑏

2 𝑑𝑏

𝐷𝑎 𝑑𝑥 2 + 𝑥 𝑑𝑥 − 𝑓(𝑎, 𝑏, 𝑡) = 0 𝐷𝑏 𝑑𝑥 2 + 𝑥 𝑑𝑥 + 𝑔(𝑎, 𝑏, 𝑡) = 0

(5.3)

(5.4)

124

L. Rajendran, R. Swaminathan and M. Chitra Devi Table 5.1. Simple differential equation in spherical coordinate system

Type of equation steady state 𝑓(𝑎, 𝑏) = 0

Equation 𝑑2 𝑎(𝑥)

steady state 𝑓(𝑎, 𝑏) = 𝑘

𝑑𝑥 2 𝑑2 𝑎(𝑥)

steady state 𝑓(𝑎, 𝑏) = 𝑘𝑎(𝑥)

𝑑𝑥 2 𝑑2 𝑎(𝑥) 𝑑𝑥 2 𝑑2 𝑎(𝑥)

steady state 𝑓(𝑎, 𝑏) = 𝑘𝑎2 (𝑥) Non steady state 𝑓(𝑎, 𝑏) = 0

𝑑𝑥 2 𝜕𝐶𝑃 𝜕𝑡

+ + + +

Solution

2 𝑑𝑎(𝑥) 𝑥 𝑑𝑥 2 𝑑𝑎(𝑥) 𝑥 𝑑𝑥 2 𝑑𝑎(𝑥) 𝑥 𝑑𝑥 2 𝑑𝑎(𝑥) 𝑥 𝑑𝑥

= 𝐷𝑝

𝜕 2 𝐶𝑃 𝜕𝑟 2

=0

𝑎(𝑥) =

𝑐1

=𝑘

𝑎(𝑥) =

𝑐1

= 𝑘𝑎(𝑥) 2

= 𝑘𝑎 (𝑥)

+

2𝐷𝑃 𝜕𝐶𝑃

𝑡 = 0 ; 𝐶𝑃 = 𝐶𝑃𝑏 , 𝑟 = 𝑎 ; 𝐶𝑃 = 𝐶𝑃𝑃 , 𝑟 → ∞, 𝐶𝑃 → 𝐶𝑃𝑏

𝑟

𝜕𝑟

𝑎(𝑥) =

𝑥 𝑥

+ 𝑐2 + 𝑐2 +

𝑐1 𝑒 −𝑥√𝑘 𝑥

+

𝑘𝑥 2 6 𝑐2 𝑒 𝑥√𝑘 𝑥√𝑘

Second order nonlinear. No solution 𝑎

𝐶𝑃 (𝑟, 𝑡) = 𝐶𝑃𝑃 [1 − 𝑒𝑟𝑓𝑐(𝑚)] 𝑟

where 𝑚 =

𝑟−𝑎 √4𝐷𝑃 𝑇

The various limiting case of the above equation and corresponding solutions are given the Table 5.1.

5.2. CONCENTRATION OF SPHERICAL CATALYTIC PARTICLE IN IMMOBILIZED ENZYME SYSTEM The mass balance equation for substrate and product and corresponding boundary conditions in spherical co-ordinates are (Carrasco et al. 2008), 𝐷𝑆 𝑑

(𝑟 2

𝑑𝐶𝑆

(𝑟 2

𝑑𝐶𝑃

𝑟=0⇒

𝑑𝐶𝑆

𝑟 2 𝑑𝑟 𝐷𝑃 𝑑 𝑟 2 𝑑𝑟

𝑑𝑟

) = 𝑉𝑆

𝑑𝑟

𝑑𝑟

(5.5)

) = −𝑉𝑆 = 0;

𝑑𝐶𝑃 𝑑𝑟

(5.6) =0

𝑟 = 𝑅 ⇒ 𝐶𝑆 = 𝐶𝑆𝑅 ; 𝐶𝑃 = 𝐶𝑃𝑅

(5.7) (5.8)

Steady and Non-Steady Non-Linear Reaction-Diffusion …

125

In dimensionless form, the differential Eqns. (5.5) and (5.6) becomes (Femila Mercy Rani, Sevukaperumal, and Rajendran 2015): 1 𝑑 𝜌2 𝑑𝜌

𝑑𝑆

𝑆

(𝜌2 𝑑𝜌) − 𝜑 𝛼+𝑆 = 0

(5.9)

where 𝑆 is the normalized substrate concentration and 𝜌 is the dimensionless radial coordinate and 𝜑 and 𝛼 are the dimensionless parameters. The boundary conditions are represented as follows (Femila Mercy Rani, Sevukaperumal, and Rajendran 2015): 𝜌 = 0;

𝑑𝑆 𝑑𝜌

=0

(5.10)

𝜌 = 1; 𝑆 = 1

(5.11)

Our goal is to apply the MADM (George Adomian 1994; G. E. Adomian and Adomian 1984; G Adomian 1976) to the model, under steady-state conditions Eqn. (5.9) can be takes the following form for spherical particle, we first write the Eqn. (5.9) in operator form 𝐿(𝑆) = 𝜑𝑁(𝑆)

(5.12)

𝑑2

𝑆

where 𝐿 = 𝜌−1 𝑑𝜌2 𝜌 and 𝑁(𝑆) = 𝛼+𝑆. Where the inverse operator is given by 𝜌

𝜌

𝐿−1 (𝑆) = 𝜌−1 ∫0 ∫0 𝜌. (𝑆)𝑑𝜌𝑑𝜌

(5.13)

Applying the above inverse operator on both sides of the Eqn. (5.12) yields 𝑆(𝜌) = 𝑎𝜌 + 𝑏 + 𝜑𝐿−1 (𝑁(𝑆))

(5.14)

126

L. Rajendran, R. Swaminathan and M. Chitra Devi

where a, b are an integrating constants, which can be obtained from the boundary conditions(5.10) and (5.11) 𝑆(𝜌) = ∑∞ 𝑛=0 𝑆𝑛 (𝜌)

(5.15)

In terms of Adomian polynomials 𝐴𝑛 , the nonlinear term 𝑁(𝑆) can be written as an infinite series. 𝑁(𝑆) = ∑∞ 𝑛=0 𝐴𝑛 (𝜌)

(5.16)

where the Adomian polynomials 𝐴𝑛 is 1 𝑑𝑛

𝑛 𝐴𝑛 = 𝑛! 𝑑𝜆𝑛 𝑁(∑∞ 𝑛=0(𝜆 𝑆𝑛 ))|

𝜆=0

(5.17)

where 𝜆 ∈ [0,1] is a hypothetical parameter. Substituting Eqns.(5.15) and (5.16) in (5.14) gives −1 ∑∞ ∑∞ 𝑛=0 𝑆𝑛 (𝜌) = 𝑎𝜌 + 𝑏 + 𝜑𝐿 𝑛=0 𝐴𝑛

(5.18)

By equating the terms in the linear system of Eqn. (5.18), we obtain the following recurrence formula: 𝑆0 (𝜌) = 𝑎𝜌 + 𝑏

(5.19)

𝑆𝑛+1 (𝜌) = 𝜑𝐿−1 (𝐴𝑛 ) ; n ≥ 0

(5.20)

where 𝐴𝑛 are the Adomian polynomials of𝑆0 , 𝑆1 , 𝑆2 , 𝑆3 , … , 𝑆𝑛 . We can find the first few𝐴𝑛 as follows: 𝑆

0 𝐴0 = 𝑁(𝑆0 ) = 𝛼+𝑆

0

(5.21)

Steady and Non-Steady Non-Linear Reaction-Diffusion … 𝑑

𝛼𝑆

𝐴1 = 𝑑𝜆 [𝑁(𝑆0 + 𝜆𝑆1 )] = (𝛼+𝑆1 )2

127 (5.22)

0

We identify the zeroth component as 𝑆0 (𝜌) = 𝑎𝜌 + 𝑏

(5.23)

The initial approximations (Boundary conditions Eqns. (5.10) and (5.11) are as follows: 𝑑𝑆0

|

= 0, 𝑆0 (1) = 1

(5.24)

|

= 0, 𝑆𝑖 (1) = 0; 𝑖 = 1,2,3, …

(5.25)

𝑑𝜌 𝜌=0

and 𝑑𝑆𝑖

𝑑𝜌 𝜌=0

Solving the Eqn. (5.23) using the boundary conditions (Eqns.(5.24) and (5.25)), we get 𝑆0 = 1

(5.26)

To obtain 𝑆1 , 𝑆2 we substitute Eqn. (5.21)and (5.22) in (5.20) then, we get 1

𝑆1 (𝜌) = 𝜑𝐿−1 (𝐴0 ) = 𝜑 (𝛼+1) [𝜌2 − 1] 𝑆2 (𝜌) = 𝜑𝐿−1 (𝐴1 ) = 𝜑 2

𝛼 (𝛼+1)3

𝜌4

(5.27) 𝜌2

7

[120 − 36 + 360]

The remaining polynomials can be generated easily.

(5.28)

128

L. Rajendran, R. Swaminathan and M. Chitra Devi

Figure 5.1. Effect of Thiele module 𝜑 on the concentration of substrate 𝑆 using Eqn. (5.30) when 𝛼 = 1. Solid line represents the simulation results and (∗ symbol) represent analytical results.

Figure 5.2. Effect of the parameter 𝛼 on the concentration of substrate 𝑆 using Eqn. (5.30) when 𝛼 = 1. Solid line represents the simulation results and (∗ symbols) represent analytical results.

𝑆(𝜌) = 𝑆0 (𝜌) + 𝑆1 (𝜌) + 𝑆2 (𝜌) + ⋯

(5.29)

Steady and Non-Steady Non-Linear Reaction-Diffusion … 1

𝛼

𝜌4

𝜌2

7

𝑆(𝜌) = 1 + 𝜑 (𝛼+1) [𝜌2 − 1] + 𝜑 2 (𝛼+1)3 [120 − 36 + 360]

129 (5.30)

The Eqn. (5.30) is the new closed form of analytical expressions of a concentration of the substrate 𝑆(𝜌) for all values of the parameter. The obtained analytical solution is compared with numerical simulation result (Matlab coding) for various values of parameter 𝜑 and 𝛼 in Figures 5.1-5.2. An acceptable agreement is noted.

5.3. ANALYTICAL SOLUTION OF VARIOUS NONLINEAR BOUNDARY VALUE PROBLEMS IN SPHERICAL COORDINATES Nonlinear partial differential equations (PDEs) in spherical coordinate play an essential role in describing many physical, industrial, and biological processes. Their solutions could be considerably facilitated by using appropriate coordinate transformations. Nonlinear reaction-diffusion in spherical coordinates occurs in Lithium-ion battery (Zeng 2015), Ultramicroelectrodes (Molina et al. 2010; Athimoolam, Lakshmanan, and Alwarappan 2011; A. Eswari and Rajendran 2011), Glucose enzyme reaction (Saranya et al. 2018) and hydrology (Mortensen et al. 2012). Particularly in electrochemistry, the nonlinear term depends upon reversible, irreversible, Michalis-Menten, and non-Michael-Menten kinetics, etc. Various methods for solving these nonlinear equations are discussed with examples in the chapter-2. The latest contribution of steady and non-steady state reaction-diffusion processes in spherical coordinates in physical chemistry with distinct reaction mechanisms and experimental techniques with the corresponding solutions is given in Table-5.2 and 5.3.

(Renugadevi, Porous solid reaction Sevukaperumal, systems and Rajendran 2016)

3

GA + H2𝑂2

G- Glucose GA- Gluconic acid G- Glucose oxidase

G+O2 →

(Saranya et al. 2018)

𝐺𝑜

𝑟 𝑑𝑟

𝑖

𝐷𝑒 (𝐾𝑚 +𝑐𝑠 +𝐾𝑠 )

𝑑𝑟

𝑑𝑐𝑠

=0,

𝑑𝑟 2

𝑑2 𝐶ℎ

𝑑2 𝐶𝑎 𝑑𝑟 2

+

+

𝑑𝑟 2

)+

𝑟 𝑑𝑟

2 𝑑𝐶ℎ

)+

𝑟 𝑑𝑟

𝐶𝑂𝑋 (𝐾𝑔 +𝐶𝑔 )+𝐶𝑔 𝐾𝑂𝑋

𝑣𝑂𝑋 𝑐𝑔 𝑐𝑂𝑋 𝑉𝑚𝑎𝑥

𝑣ℎ 𝑐𝑔 𝑐𝑂𝑋 𝑉𝑚𝑎𝑥

𝐶𝑂𝑋 (𝐾𝑔 +𝐶𝑔 )+𝐶𝑔 𝐾𝑂𝑋

𝑣𝑎 𝑐𝑔 𝑐𝑂𝑋 𝑉𝑚𝑎𝑥

)+

𝐶𝑂𝑋 (𝐾𝑔 +𝐶𝑔 )+𝐶𝑔 𝐾𝑂𝑋

𝑣𝑔 𝑐𝑔𝑐𝑂𝑋 𝑉𝑚𝑎𝑥

𝐶𝑂𝑋 (𝐾𝑔 +𝐶𝑔 )+𝐶𝑔 𝐾𝑂𝑋

)+

𝑟 𝑑𝑟

2 𝑑𝐶𝑂𝑋

2 𝑑𝐶𝑎

+

𝑟 𝑑𝑟

2 𝑑𝐶𝑔

+

𝑟

𝑅

𝑐 𝑐0

1−𝑦

𝑑𝑦

𝑑𝑥 𝑥=0

𝑦(1) = 1, [ ]

𝑇0 𝑚𝑎𝑥

𝛥𝑇

=[ ] 𝑅𝑇0

𝑄

𝐾𝑇0

𝑐0 𝐻𝐷

(𝑐0 − 𝑐), 𝛾 =

=0

]

1+𝛽(1−𝑦)

The boundary conditions are

𝐾

𝐻𝐷

𝐷

𝑘

= 𝜑0 2 𝑦 𝑒𝑥𝑝 [𝛾𝛽

, 𝑥 = , 𝜑0 = 𝑅√ 0 , 𝛽 =

𝑥 𝑑𝑥

2 𝑑𝑦

𝛥𝑇 = 𝑇 − 𝑇0 = −

𝑦=

where

𝑑𝑥 2

𝑑2 𝑦

The boundary conditions are 𝑑𝐶𝑔 𝑑𝐶𝑂𝑋 𝑑𝐶𝑎 𝑑𝐶ℎ 𝑟 = 0, = 0, = 0, = 0, =0 𝑑𝑟 𝑑𝑟 𝑑𝑟 𝑑𝑟 ∗ 𝑟 = 𝑆, 𝐶𝑔 = 𝐶𝑔∗, 𝐶𝑂𝑋 = 𝐶𝑂𝑋 , 𝐶𝑎 = 𝐶𝑎∗, 𝐶ℎ = 𝐶ℎ∗

𝐷ℎ (

𝐷𝑎 (

+

𝑑2 𝐶𝑂𝑋

𝐷𝑂𝑋 (

𝐷𝑔 (

𝑑2 𝐶𝑔 𝑑𝑟 2

𝑟 = 𝑅𝑝 , 𝑐𝑆 = 𝑐𝑆0

𝑟 = 0,

The boundary conditions are

𝑑𝑟

MADM

HPM

Experimental techniques Nonlinear differential equations with initial Analytical and enzymatic scheme and boundary conditions techniques 𝑉𝑚 𝑐𝑠 Enzyme flow calorimetry 𝑑2 𝑐𝑠 + 2 𝑑𝑐𝑠 − MHFM = 0 2 2 𝑐

2

S. Reference No 1 (Devi et al. 2020)

𝐾𝑚

𝑐𝑆0

𝑣𝑔 𝛾𝑔

𝑣𝑔 𝛾𝑔 𝑣ℎ 𝛾ℎ

𝑐2

(𝑢(𝑅) − 1)

𝑎𝑛𝑑 𝑐 =

(𝑢(𝑅) − 1)

(𝑢(𝑅) − 1)

𝑣𝑔 𝛾𝑔 𝑣𝑎 𝛾𝑎

𝛷2 1+𝑐0 + 𝛽0

𝑐𝑠 𝐾𝑚

,𝑥 =

𝑟 𝑅𝑝

, 𝛷 = 𝑅𝑝 √

,v 

6

𝜑02

+

𝜂=

|

3 dy

𝜑02 dx 𝑥=1

C a*

Ca

7 360

= 1−

Ch

, 

𝑆

−𝐷.𝐶ℎ∗

15

𝜑02 (1−𝛾𝛽)

𝑉𝑚

,𝛽 =

(√𝑘 𝑐𝑜𝑡 ℎ √𝑘 − 1)

𝐾𝑚 𝐷𝑒

C g*

, 

6

1

36

] 𝑥2 +

120

𝜑02 (1−𝛾𝛽)

; k  v g  g /(  1   ) 𝜑02 (1−𝛾𝛽)

* COx

K OX

𝑥4

V S2 V S2 r , R  ,  g  max * ,  OX  max * , s Dg C g DOX COX Kg

C h*

=

𝜑02 (1 − 𝛾𝛽) + 𝜑02 [ −

Dh C h*

)

𝑆 𝜕𝑅 𝑅=1

𝐶ℎ∗ 𝜕𝐻

,H  Vmax S 2

,w  ,, h 

* COx

= −𝐷 (

The utilization factor

𝑦(𝑥) = 1 −

Da C a*

)

𝜕𝑟 𝑟=𝑆

𝜕𝐶ℎ

COX Vmax S 2

C g*

Cg

a 

u

where

(𝐽ℎ ) = −𝐷 (

The flux of hydrogen peroxide from the surface is

𝐻(𝑅) ≈ 1 +

𝑤(𝑅) ≈ 1 +

𝑣(𝑅) ≈ 1 +

𝑅 𝑠𝑖𝑛 ℎ(√𝑘) 𝑣𝑂𝑋 𝛾𝑂𝑋

𝑠𝑖𝑛 ℎ(√𝑘𝑅)

, 𝑐0 =

𝑢(𝑅) ≈

𝑘𝑚

𝐾𝑖

𝑏2 + 2𝑏 𝑡𝑎𝑛ℎ(𝑏) =

𝑐(𝑥) = 𝑐0

𝑐𝑜𝑠ℎ(𝑏𝑥) 𝑐𝑜𝑠ℎ(𝑏) where the parameter b is obtained from the following relation

Expressions for concentration and current

Table 5.2. Recent contribution of steady-state reaction-diffusion equations in the sphere

5

Experimental techniques and enzymatic scheme Oxidation of ethanol to CO2 1 𝐶2𝐻6𝑂 + 𝑂2 2 → 𝐶2𝐻4 𝑂 + 𝐻2 𝑂 5 𝐶2𝐻4𝑂 + 𝑂2 2 → 2𝐶𝑂2 + 2𝐻2 𝑂 𝐶4𝐻8𝑂2 + 5𝑂2 → 4𝐶𝑂2 + 4𝐻2 𝑂 + 𝑧 𝑑𝑧

2 𝑑𝐶𝐸𝐴

=

𝑟3 𝐷𝑒𝑓,𝐸𝐴

𝑑𝑧

= 0 (𝑗 ≡ 𝐸𝑡, 𝐴𝑐𝑜𝑟𝐸𝐴)

) , ℎ = (𝑉𝑚 /2𝐷𝑆0 )2 𝐿, 𝐴 = 𝑆/𝑆0 ,

𝛼1 = 𝐾𝑚 /𝑆0

𝐷

𝑘𝑚 𝐿

𝑆ℎ = (

1

At 𝑧 = 𝑅, 𝐶𝑗 = 𝐶𝑗 𝑏 (𝑗 ≡ 𝐸𝑡, 𝐴𝑐𝑜𝑟𝐸𝐴)

At 𝑧 = 0,

𝑑𝐶𝑗

The boundary conditions are

1+𝐾𝑐,𝐸𝑡 𝐶𝐸𝑡 +𝐾𝑐,𝐴𝑐 𝐶𝐴𝑐 +𝐾𝑐,𝐸𝐴 𝐶𝐸𝐴

𝑘𝑟𝑒𝑓3 𝑒𝑥𝑝[−(𝐸3 ⁄𝑅𝑔 )(1⁄𝑇−1⁄𝑇𝑟𝑒𝑓 )]𝐶𝐸𝐴

1 + 𝐾𝑐,𝐸𝑡 𝐶𝐸𝑡 + 𝐾𝑐,𝐴𝑐 𝐶𝐴𝑐 + 𝐾𝑐,𝐸𝐴 𝐶𝐸𝐴

𝑘𝑟𝑒𝑓2 𝑒𝑥𝑝 [−(𝐸2 ⁄𝑅𝑔 ) (1⁄𝑇 − 1⁄𝑇 𝑟𝑒𝑓 )] 𝐶𝐴𝑐

1 + 𝐾𝑐,𝐸𝑡 𝐶𝐸𝑡 + 𝐾𝑐,𝐴𝑐 𝐶𝐴𝑐 + 𝐾𝑐,𝐸𝐴 𝐶𝐸𝐴

𝑘𝑟𝑒𝑓1 𝑒𝑥𝑝 [−(𝐸1 ⁄𝑅𝑔 )(1⁄𝑇 − 1⁄𝑇𝑟𝑒𝑓 )] 𝐶𝐸𝑡

𝑟3 =

=

𝑟2

𝑟1 =

Where

𝑑𝑧 2

𝑑2 𝐶𝐸𝐴

𝑑 2𝐶𝐴𝐶 2 𝑑𝐶𝐴𝐶 𝑟2 − 𝑟1 + = 𝑑𝑧 2 𝑧 𝑑𝑧 𝐷𝑒𝑓,𝐴𝐶

ADM

Nonlinear differential equations with initial Analytical and boundary conditions techniques MADM 𝑑 2𝐶𝐸𝑡 2 𝑑𝐶𝐸𝑡 𝑟1 + = 𝑑𝑧 2 𝑧 𝑑𝑧 𝐷𝑒𝑓,𝐸𝑡

(Renuga Devi, Michaelis-Menten enzyme 𝑑2 𝐴 + 2 𝑑𝐴 − 2ℎ2 𝐴 = 0 𝑑𝑥 2 𝑥 𝑑𝑥 𝛼1 +𝐴 Sevukaperumal, Kinetics Boundary conditions: 𝑘𝑜𝑓𝑓 𝑘𝑐𝑎𝑡 and Rajendran 𝐸 + 𝑆 ↔ 𝐸𝑆 → 𝐸 𝑑𝐴 𝑘𝑜𝑥 2015) 𝑥 = 0, =0 𝑑𝑥 +𝑃 𝑑𝐴 𝑥 = 1, = 𝑆ℎ (1 − 𝐴) 𝑑𝑥 𝑤ℎ𝑒𝑟𝑒

S. Reference No 4 (V. Meena, Praveen, and Rajendran 2016) 6

]+

6

] + [𝜑3 2 𝛾3 2𝑀 2 − 𝜑3 𝛾3 𝑀1]𝑀(𝑦)

,𝑊 = 𝑏 𝐶𝐸𝐴 1

𝐶𝐸𝐴

−

,

ℎ2 𝛼1 +1

𝑙(ℎ) = − [

where

3

1

36

+

7

] 360

+

𝛼1 +1

ℎ2

𝑆ℎ (𝛼1 +1)

𝑚(ℎ)𝛼1

] , 𝑚(ℎ) = 𝑆ℎ (𝛼1 +1)

2ℎ2

𝐴(𝑥) = 1 + [𝑙(ℎ) + 𝑚(ℎ)] − 2

120

𝑏 𝐶𝐸𝑡

𝐶𝐸𝑡

,𝑉 =

[

+

9

+

18

𝑆ℎ 𝑙(ℎ)

+ 60

18 𝑆ℎ 𝑚(ℎ)𝑥 4

+ +

𝑚(ℎ) 15 𝑆ℎ 𝑙(ℎ)𝑥 2

𝑙(ℎ)

60

𝑆ℎ 𝑚(ℎ)

, 𝑀1 = (𝜑1𝛾1 (𝛼1 − 𝛼2 ) + 𝛼2 𝜑2 𝛾2 + 𝛼3𝜑3 𝛾3 )𝑀3 ,

1 +𝛼2 +𝛼3 ) 𝑦4 𝑦2

𝑀(𝑦) = [

𝑀 = (1+𝛼

𝑏 𝐶𝐴𝑐

𝐶𝐴𝑐

𝑅

𝑏 𝑏 𝑏 𝛼1 = 𝐾𝑐,𝐸𝑡 𝐶𝐸𝑡 , 𝛼2 = 𝐾𝑐,𝐴𝑐 𝐶𝐴𝑐 , 𝛼3 = 𝐾𝑐,𝐸𝐴𝐶𝐸𝐴 ,𝑦 = ,𝑈 =

𝑧

Where the dimensionless parameters are 𝑅 2𝐾𝑟𝑒𝑓𝑖 𝜑𝑖 = , 𝛾 = 𝑒𝑥𝑝 [−(𝐸𝑖 ⁄𝑅𝑔 ) (1⁄𝑇 − 1⁄𝑇 𝑟𝑒𝑓 )] (𝑖 = 1,2,3), 𝐷𝑒𝑓,𝐸𝑡 𝑖

𝑊(𝑦) = 1 + 𝑀𝜑3 𝛾3 [

𝑦 2 −1

(𝜑 𝛾 (𝜑 𝛾 − 𝜑1𝛾1 ) − 𝜑1 2𝛾1 2 )𝑀2 [ 2 2 2 2 ] 𝑀(𝑦) −(𝜑2 𝛾2 − 𝜑1𝛾1 )𝑀1

6

𝑦 2 −1

] + [𝜑1 2𝛾1 2 𝑀2 − 𝜑1 𝛾1 𝑀]𝑀(𝑦)

𝑉(𝑦) = 1 + 𝑀(𝜑2 𝛾2 − 𝜑1 𝛾1 ) [

𝑈(𝑦) = 1 + 𝑀𝜑1 𝛾1 [

𝑦 2 −1

Expressions for concentration and current

]

Enzyme flow calorimetry

(Margret PonRani, Rajendran, and Eswaran 2011)

(Praveen and Rajendran 2015)

6

7

+ 𝑟 𝑑𝑟

2 𝑑𝑐𝑠

−

𝑉𝑚 𝑐𝑠 𝑐2

𝑟 = 0, 𝑑𝑟

𝑑𝑐𝑠

=0

= 0;𝑟 = 𝑅𝑝 , 𝑐𝑆 = 𝑐𝑆0

𝑖

𝐷𝑒 (𝐾𝑚 +𝑐𝑠 +𝐾𝑠 )

The boundary conditions

𝑑𝑟 2

𝑑2 𝑐𝑠

Nonlinear differential equations with initial and boundary conditions

𝐻

𝐻

𝑑𝑟 2

𝑟

𝑑𝑟

= 0;

𝐻

=

𝑆

𝑌𝑋∗

𝑠 𝐶𝑔𝑟𝑎𝑛𝑢𝑙𝑒

𝜇𝑋 𝛼 ∗ ∗𝑐𝑒𝑙𝑙 𝑌𝑋/𝑆

.

+ 𝛽𝑋𝑐𝑒𝑙𝑙 ,

+ 𝑚𝑋𝑐𝑒𝑙𝑙 ,

𝑠 𝐾𝑠 +𝐶𝑔𝑟𝑎𝑛𝑢𝑙𝑒

=

𝜇 = 𝜇𝑚𝑎𝑥

𝐻2 𝜑𝑔𝑟𝑎𝑛𝑢𝑙𝑒

𝑠 𝜑𝑔𝑟𝑎𝑛𝑢𝑙𝑒 =

𝜇𝑋𝑐𝑒𝑙𝑙

when 𝑟 = 𝑅

Where

𝐻 𝐶𝑔 2

𝑠 2 𝐶𝑔𝑟𝑎𝑛𝑢𝑙𝑒 = 𝐶𝑙𝑠 ; 𝐶𝑔𝑟𝑎𝑛𝑢𝑙𝑒 =

𝐻

𝑑𝑟

2 𝑑𝐶𝑔𝑟𝑎𝑛𝑢𝑙𝑒

0 when 𝑟 = 0

𝑑𝑟

𝑠 𝑑𝐶𝑔𝑟𝑎𝑛𝑢𝑙𝑒

Boundary conditions:

𝐻2 𝜑𝑔𝑟𝑎𝑛𝑢𝑙𝑒

+

=

2 𝑑2 𝐶𝑔𝑟𝑎𝑛𝑢𝑙𝑒

2 2 2𝐷𝑔𝑟𝑎𝑛𝑢𝑙𝑒 𝑑𝐶𝑔𝑟𝑎𝑛𝑢𝑙𝑒

𝐻

2 𝐷𝑔𝑟𝑎𝑛𝑢𝑙𝑒

𝐻

𝑠 𝑑2 𝐶𝑔𝑟𝑎𝑛𝑢𝑙𝑒 Two-phase flow transport in an 𝑠 𝐷𝑔𝑟𝑎𝑛𝑢𝑙𝑒 + 𝑑𝑟 2 immobilized-cell 𝑠 𝑠 2𝐷𝑔𝑟𝑎𝑛𝑢𝑙𝑒 𝑑𝐶𝑔𝑟𝑎𝑛𝑢𝑙𝑒 𝑠 = 𝜑𝑔𝑟𝑎𝑛𝑢𝑙𝑒 photobioreactor 𝑟 𝑑𝑟

Experimental techniques and enzymatic scheme

S. Reference No

MADM

HPM

Analytical techniques

𝑐𝑠 𝐾𝑚

𝑟 𝑅𝑝

𝐻

𝐶𝑔 2

𝐻

𝑅 2 𝑚𝑋𝑐𝑒𝑙𝑙 𝐶𝑙𝑠

, 𝛼1 =

𝐾𝑠

𝐶𝑙𝑠

𝑠 𝐷𝑔𝑟𝑎𝑛𝑢𝑙𝑒 𝐾𝑠 𝐶𝑔 2 2 𝐶𝑔𝑟𝑎𝑛𝑢𝑙𝑒

𝐻

𝛾3 =

, 𝛹1 = (

.

𝑚𝑙 𝑅

3(1+𝛼1 )

, 𝛾4 =

2

𝑠 𝛼𝐷𝑔𝑟𝑎𝑛𝑢𝑙𝑒 𝐶𝑙𝑠

, 𝛷1 −

(2+𝜔) 6(1+𝛼1 )

𝐻

𝐻

2 𝐷𝑔𝑟𝑎𝑛𝑢𝑙𝑒 𝐶𝑔 2

𝑅 2 𝑚𝑋𝑐𝑒𝑙𝑙

∗

.

𝑚𝑔 𝑅 𝐻

𝑟

𝑅

,𝜁 = ,𝑢 =

2

𝐶𝑔𝑠

,𝑣 =

𝑅 2𝑚𝑋𝑐𝑒𝑙𝑙 ,𝛾 𝑠 𝐷𝑔𝑟𝑎𝑛𝑢𝑙𝑒 𝐾𝑠 2

𝑠 𝐶𝑔𝑟𝑎𝑛𝑢𝑙𝑒

, 𝛾1 =

)

6(1+𝛼1 )

(𝛾3 +𝛾4 )(2+𝜔)

𝐾𝑚

𝑐𝑆0

2 𝛼𝐷𝑔𝑟𝑎𝑛𝑢𝑙𝑒 𝐶𝑔 2

𝐻

𝛷2 +

, 𝑐0 =

) 𝑎𝑛𝑑𝛹2 = (

, 𝛹2 =

) (𝜁 2 − 1)

𝐾𝑖 𝑘𝑚

𝑋𝑐𝑒𝑙𝑙 𝜇𝑚𝑎𝑥 𝑅 𝛼 𝑋𝑐𝑒𝑙𝑙 𝜇𝑚𝑎𝑥 𝑅 𝐶𝑙𝑆 ,𝛷 = ∗ 𝑠 ∗ 𝑠 𝐻 𝑌𝑋/𝑠 𝐷𝑔𝑟𝑎𝑛𝑢𝑙𝑒 𝐾𝑠 2 𝑌𝑋/𝑠 𝐷𝑔𝑟𝑎𝑛𝑢𝑙𝑒 𝐾𝑠 𝐶𝑔 2 𝑅 2𝑚𝑋𝑐𝑒𝑙𝑙 = 𝑠 𝐷𝑔𝑟𝑎𝑛𝑢𝑙𝑒 𝐶𝑙𝑆

𝑀 𝐻2

𝑀𝐶𝑂2

𝛷1 =

𝜔=

−1 3(1+𝛼1 )

Where

𝛹1 =

6

1+𝛼1 (𝛾1 +𝛾2 )

6 1+𝛼1 1 𝛷2 +𝛾3 +𝛾4

𝑣(𝜁) = 1 + (

,𝛽 =

) (𝜁 2 − 1)

𝐾𝑚 𝐷𝑒

𝑉𝑚

(𝑥 2 − 1)

, 𝛷 = 𝑅𝑝 √

1 𝛷1 +𝛾1 +𝛾2

,𝑥 =

𝛷 2 𝛽𝑐0 3(𝛽+𝛽𝑐0 +𝑐02 )

𝑢(𝜁) = 1 + (

𝑐=

where

𝑐(𝑥) = 𝑐0 +

Expressions for concentration and current

Table 5.2. (Continued)

Reversible Michaelies-Menten kinetics

(Joy et al. 2011)

(Usha, Anitha, and Rajendran 2012)

8

9

Amperometric enzyme electrode

Experimental techniques and enzymatic scheme

S. Reference No

(𝑟 𝑑𝑟

) = −𝑉𝑠

𝐾𝑚 +𝐶𝑠 +(𝐾𝑚 ⁄𝐾𝑝 )𝐶𝑝

(𝑟 𝑑𝑟

2 𝑑𝑆

)= 𝑌𝑋/𝑝 (𝐾𝑜 +𝑆)

𝜌 2𝑑𝜌 𝑈+𝛼 𝜇𝑓𝑚𝑎𝑥

The boundary conditions 𝑑𝑆 = 0 𝑎𝑡 𝑟 = 𝑟𝑝 𝑑𝑟 𝑆 = 𝑆𝑏 𝑎𝑡 𝑟 = 𝑟𝑏

𝑟 2 𝑑𝑟

𝐷𝑓 𝑑

𝜂 = 3(𝛼 + 1) ∫0

1 𝑈

The effectiveness factor is

𝑉𝑠 =

𝑉𝑚 (𝐶𝑠 −(𝐶𝑝 ⁄𝐾𝑒𝑞 ))

The boundary conditions are 𝑑𝐶𝑝 𝑑𝐶𝑠 𝑟=0⇒ = 0; = 0; 𝑑𝑟 𝑑𝑟 𝑟 = 𝑅 ⇒ 𝐶𝑠 = 𝐶𝑠𝑅 ; 𝐶𝑝 = 𝐶𝑝𝑅

𝑟 2 𝑑𝑟

) = 𝑉𝑠

MADM

HAM

𝑑𝐶𝑠

𝑑𝑟 2 𝑑𝐶𝑝

(𝑟 2

𝐷𝑠 𝑑 𝑟 2 𝑑𝑟 𝐷𝑝 𝑑

Analytical techniques

Nonlinear differential equations with initial and boundary conditions 6𝛼

ℎ𝜙

𝜙ℎ𝐴

(𝛼+1)

)]

𝑅

𝐶𝑆𝑅 −𝐶𝑆𝐸

𝜑2 6(1+𝛼)

𝑆𝑏

𝑆

[1 −

Where 𝑢 =

+

ℎ𝜙

𝜌 4 − ((1 + ℎ)𝛼 + ℎ + 6

ℎ𝜙

) 𝜌2 +

(1+

1 𝐷𝑆 ) 𝐾𝑒𝑞 𝐷𝑃 𝐷 𝐾 𝑆𝑅 −𝐶𝑆𝐸 )𝐷𝑆 (1+ 𝑀 𝑆 ) 𝐾𝑃 𝐷𝑃

𝑅 2 𝑉𝑚

,𝛼 =

𝐾

𝑃

𝐾𝑀 𝐷𝑆 ) 𝐾𝑃 𝐷𝑃

𝐾𝑀 + 𝐾𝑀 𝐶𝑃𝐸 +𝐶𝑆𝐸 (𝐶𝑆𝑅 −𝐶𝑆𝐸 )(1+

;𝜒 =

𝑟𝑏

𝑟

6(1+𝛼)2

;𝜆 =

𝜑4 𝜆

𝑟𝑏

;𝛼 =

𝐾𝑜

𝜑4

𝜒4 𝐷𝑓 𝑌𝑥/𝑝 𝐾𝑜

𝜇𝑓𝑚𝑎𝑥

120(1+𝛼)3

; 𝜑 = 𝑟𝑏 √

𝜒3 + 𝑆𝑏

36(1+𝛼)3 𝑟𝑝

] 𝜒2 −

𝜑 2 (1−2𝜆)

𝜑2(1 − 2𝜆) 𝜑4 (18𝜆3 − 40𝜆2 + 30𝜆 − 7) − 6(1 + 𝛼) 360(1 + 𝛼)3 𝜑2 𝜆 𝜑2 (9𝜆2 − 20𝜆 + 10) − [1 − ]𝜒 3(1 + 𝛼) 60(1 + 𝛼)2

𝑟

, 𝜌 = , 𝜑 = (𝐶

𝐶𝑆 −𝐶𝑆𝐸

𝑢(𝜒) = 1 −

𝑈=

[

[𝜙ℎ𝐴 + 18𝛼 2(𝐴 − 𝜙ℎ𝐵) − 108𝛼 3 𝐵(𝛼 + 1)]

60

ℎ𝜙 6𝛼2 20

Where, 𝑙 = 𝜙 2 − 6𝜙(𝛼 + 1) ,

𝜂(𝛼, 𝜙) =

7ℎ𝜙

(𝜌 2 − 1) +

((1 + ℎ)𝛼 + ℎ +

𝑈(𝜌) = 1 −

Expressions for concentration and current

Experimental techniques and enzymatic scheme Bioelectrolytic reaction

S. Reference No

10 (Praveen, Valencia, and Rajendran 2014)

𝑉𝑚 𝑆 𝐾𝑆 +𝑆

𝑆(1+𝑆/𝐾𝐼 )+𝐾𝑆

𝑉𝑚 𝑆

𝑉𝑚 𝑆 𝐾𝑆 (1+𝑃/𝐾𝐼 )+𝑆

(1+𝑃/𝐾𝐼 )(𝐾𝑆 +𝑆)

𝑉𝑚 𝑆

′ (𝑆−𝑆 ) 𝑉𝑚 𝑒

𝐾𝑆 +(𝑆−𝑆𝑒 )

Boundary conditions: 𝑑𝑆 𝑑𝑃 = = 0 when 𝑟 = 0 𝑑𝑟 𝑑𝑟 𝑆 = 𝑆0 , 𝑃 = 𝑃0 𝑤ℎ𝑒𝑛 𝑟 = 𝑅

𝑉=

Reversible Michaelis-Menten reaction:

𝑉=

Total non-competitive product inhibition

𝑉=

Total competitive product inhibition:

𝑉=

Uncompetitive substrate inhibition:

𝑉=

𝑑 2 𝑃 2 𝑑𝑃 𝐷𝑃 ( 2 + )+𝑉 = 0 𝑑𝑟 𝑟 𝑑𝑟 Simple Michaelis-Menten:

Nonlinear differential equations with initial and boundary conditions 𝑑 2𝑆 2 𝑑𝑆 𝐷𝑆 ( 2 + )−𝑉 = 0 𝑑𝑟 𝑟 𝑑𝑟 MADM

Analytical techniques

2

40

63 2

3

𝛽0

,

𝛽0

,

𝛽0

𝛾(𝜍) = [𝛾0 +

3

𝛷2 𝐴 2 𝑃 1

2 3

+

63

40

27

(𝛽0 −𝛽𝑒 )

𝛽0 (1+𝛽0 +𝛾0 )

, 𝐵1 =

+

3

40

63

27

9

(1+𝛽0 +𝛾0 )3 27

(

𝑉𝑚 3 𝐷𝑆 𝐾𝑆

𝑅

(1+𝛾0 )

𝑅

=

𝑉𝑚

𝐾𝑆

2

𝛷𝑆2 𝛽0

)

1 2

𝐾𝐼

𝑃

4

𝑅

𝑟

𝐾𝐼

𝐾𝑆

40

𝛷𝑆2

𝐾𝐼

𝑃0

, 𝛷𝑆 =

(1+𝛽0 +𝛽0 𝛼+𝛽02 𝛼)3

, 𝛾0 =

) , 𝐵1 = −

,𝜍 = ,𝛼 =

5 (1+𝛽0 +𝛾0 )2

1

,𝛾 =

−

3 𝐷𝑃 𝐾𝐼

) , 𝛷𝑃 = (

1 2

𝑆

(1+𝛽0 +𝛾0 )3 𝑆 , 𝛽0 𝐾

𝑆0

5 (1+𝛽0 +𝛾0 )2

3

2 𝛷𝑆2 (1+𝛾0 )−𝛷𝑃 𝛽0

where 𝛽 =

𝐵2 =

𝜂′ = (1 −

+

40

𝛷2 𝛽 𝐵 ] 𝜍 2 4 𝑃 0 2

9

𝛷𝑆4 𝐵] + [ 𝛷𝑆2 𝐴 − 𝛷𝑆4 𝐵] 𝜍 2 + [ 𝛷𝑆4 𝐵] 𝜍 4

3

−

4

3 63 3 9 𝛾(𝜍) = [𝛾0 + 𝛷𝑃2 𝐴 + 𝛷𝑃2 𝛽0 𝐵2 ] + [− 𝛷𝑃2 𝐴 + 𝛷𝑃2 𝛽0 𝐵2 ] 𝜍 2 2 40 2 4 27 + [ 𝛷𝑃2 𝛽0 𝐵2 ] 𝜍 4 40

2

3

𝛽(𝜍) = [𝛽0 − 𝛷𝑆2 𝐴 +

9

2 𝛷𝑆2 (1+𝛾0 )+𝛷𝑃 𝛽0

[− 𝛷𝑃2 𝐴1 2

2

Total non-competitive product inhibition

5

3

𝛷𝑆2 𝛽0 𝐵1 ] + [ 𝛷𝑆2 𝐴1 − 𝛷𝑆2 𝐵1 ] 𝜍 2 + [ 𝛷𝑆2 𝛽0 𝐵1 ] 𝜍 4 𝛷2 𝛽 𝐵 ] 40 𝑃 0 2

40 63

𝜂 ′ = (1 − 𝛷𝑆2 𝑄) , 𝐴1 =

3

[ 𝛷𝑃2 𝛽0 𝐵2 ] 𝜍 4 40

,

(1+𝛽0 )3 (1+𝛽0 +𝛽02 𝛼)3 (1+𝛽0 +𝛽0 𝛼+𝛽02 𝛼)3 (1+𝛽0 −𝛽𝑒 )3

𝛽(𝜍) = [𝛽0 − 𝛷𝑆2 𝐴1 +

27

4

9

𝛷𝑆4 𝐵] + [ 𝛷𝑆2 𝐴 − 𝛷𝑆4 𝐵] 𝜍 2 + [ 𝛷𝑆4 𝐵] 𝜍 4

𝛽0 𝛽0 𝛽0 (𝛽0 − 𝛽𝑒 ) , , , & (1 + 𝛽0 ) (1 + 𝛽0 + 𝛽02 𝛼) (1 + 𝛽0 + 𝛽0 𝛼 + 𝛽02 𝛼) (1 + 𝛽0 − 𝛽𝑒 )

𝛷2 𝑄) 5 𝑆

3

Total competitive product inhibition

𝐵=

𝐴=

𝜂 = (1 −

′

𝛽(𝜍) = [𝛽0 − 𝛷𝑆2 𝐴 +

3

Concentration for Simple Michaelis-Menten kinetics, Uncompetitive substrate reaction and Reversible Michaelis-Menten reaction is

Expressions for concentration and current

Table 5.2. (Continued)

Flow calorimetry, glucoamylase kinetics

11 (Sevukaperumal, Eswari, and Rajendran 2013)

12 (A. Meena and Amperometric Rajendran 2010b)

Experimental techniques and enzymatic scheme

S. Reference No +

𝜕𝑟 2 𝑉𝑚 𝑐𝑠𝑝

𝜕2 𝑐𝑠𝑝

)−

𝐿

=

𝑘𝑐𝛴 𝑎2 𝐷𝑠

𝑑𝜉 𝑚

= 𝑓𝑠

𝑓𝑅

𝐷𝐸

=

𝑘𝜅𝑠 ∞ 𝑎2

;

𝑓𝐸

𝑓𝑅

𝜎2

𝜆

=

= 𝜌𝜎2

𝜆

; 𝛾𝑠 =

𝑠 𝑏 𝑠∞ 𝑟 ;𝑣 = ; 𝛼 = = ; 𝑘𝑠 ∞ 𝑐𝛴 𝐾𝑀 ; 𝜉 𝐿

𝛾𝐸 =

𝑢=

where

𝑑𝜉 𝑚

𝜌( ) }

𝑑𝑣

𝜓∑ = 𝜓 + 𝜓𝐸 = 𝜎 {( ) −

𝑑𝑢

The net normalized current density is

𝑑𝜉

𝑑𝜉

𝐿

= 0; 1 − 𝑢 = 𝜎 ( )

𝑑𝑢

𝑎+𝐿

𝑑𝑣

= 𝑚; 𝑣 = 1; 𝑢 = 0𝜉 =

1 + 𝑚;

𝜉=

𝑎

𝑑 2 𝑢 2 𝑑𝑢 𝛾𝑠 𝑢𝑣 + − =0 𝑑𝜉 2 𝜉 𝑑𝜉 1 + 𝛼𝑢 2 𝑑 𝑣 2 𝑑𝑣 𝛾𝐸 𝑢𝑣 + − =0 𝑑𝜉 2 𝜉 𝑑𝜉 1 + 𝛼𝑢

The initial and Boundary conditions: 𝑑𝑐𝑆𝑃 = 0 𝑎𝑡 𝑟 = 0 𝑠𝑟 𝑐𝑆𝑃 = 𝑐𝑆 𝑎𝑡 𝑟 = 𝑅𝑃

=0

𝑟 𝜕𝑟

2 𝜕𝑐𝑠𝑝

2 /𝐾 ) 𝐾𝑚 +𝑐𝑠𝑝 +(𝑐𝑠𝑝 1

𝐷𝑒 (

Nonlinear differential equations with initial and boundary conditions

HPM

HPM

HAM

ADM

Analytical techniques

𝑅𝑝

60

;𝑈 =

(

7𝛾𝐸2 (1−𝛽) 360(1+𝛼+𝛽)3

𝐵=

𝑚2 3

𝛾𝑠 𝑚(1+𝑚)4

(

−

𝑚

1

𝐾𝑚

𝑐𝑆2

−

𝜉

1

;𝛽 =

+

3(1+𝑚+𝑚𝜎)

𝛾𝑠 𝑚2 (1+𝑚)2

𝐾𝑖 𝐾𝑚

𝑐𝑆2

6

𝛾𝐸2 𝑥 2

1

𝜉

1

𝑚

)

120

𝛾𝐸2 𝑥 4

(ℎ + 2 + ℎ(1 +

6

),𝐶 =

6(1+𝑚+𝑚𝜎)

𝛾𝐸 (1+𝑚)4 (𝑚−2)

6𝑚(1+𝑚+𝑚𝜎)

6

𝑥2

)( ) +

(1+𝑚)2 (𝛾𝑆 (1+𝑚+2𝜎+3𝑚𝜎)−6(1+𝑚+𝑚𝜎)

{(1+𝑚)(2𝑚−1)+𝜎(𝑚−2)}

𝑚2 (1+𝑚+𝑚𝜎)

𝜌𝛾𝐸 (1+𝑚)2 (3𝑚+2)

(1+𝑚+𝑚𝜎)2

where

𝜓∑ = 𝜎 (

6

ℎ𝛾𝐸

)−(

[(𝜉 − 2𝑚)(𝜉 − 𝑚)] + 𝐶 ( − )

1+𝑚+𝑚𝜎

7𝛾𝐸2 (1−𝛽) 36(1+𝛼+𝛽)3

− 1 + 𝛼 + 𝛽) ;𝛼 =

6

𝛾𝐸

60

7𝑔𝛾𝐸

−

+ 𝐵) ( − ) +

𝐷𝑒 𝐾𝑚

𝑅𝑝2 𝑉𝑚

𝑚(1+𝑚)2

; 𝛾𝐸 =

+(

𝑐𝑆

𝑐𝑆𝑃

The net normalized current density is

6(1+𝑚+𝑚𝜎)

𝛾𝐸 (1+𝑚)2

6(1+𝑚+𝑚𝜎)

120

ℎ2 𝛾𝐸2 𝑥 4

𝛾𝐸

6(1+𝛼+𝛽)

+(

− 1 + 𝛼 + 𝛽) − (

𝛾𝑆 (1+𝑚)2 (𝜉 2 −3𝑚𝜉)

𝑣 =1+

𝑢=

Where 𝑥 =

2

)) 𝑥 +

6 𝑟

+

(ℎ + 2 + ℎ(1 + 𝛼 + 𝛽) +

𝛾𝐸 7𝛾𝐸

36

𝑈(𝑥) = 1 +

𝛼 + 𝛽) +

ℎ2 𝛾𝐸2

6

ℎ𝛾𝐸

20

𝑥4

𝑈(𝑥) = 1 +

𝛾𝐸 6(1+𝛼+𝛽)

( ) 3

6(1+𝛼+𝛽)

𝛾𝐸2 (1−𝛽)

𝑈(𝑥) = 1 −

Expressions for concentration and current

13 (A Eswari and Rajendran 2010)

S. Reference No

𝑘2

𝑘𝑐𝑎𝑡

𝑆 + 𝐸1 ⟺ [𝐸1 𝑆] → 𝑃 + 𝐸2

𝑘1

Experimental techniques and enzymatic scheme

𝑑𝑟

(𝑟

𝑑𝑐𝑆

𝑑𝑟

)+

)−

𝑑𝑟 2 𝑑𝑐𝐻

(𝑟 2 𝑐𝑆 +𝐾𝑀

𝑐𝑆 +𝐾𝑀 𝑘𝑐𝑎𝑡 𝑐𝐸 𝑐𝑆

𝑘𝑐𝑎𝑡 𝑐𝐸 𝑐𝑆

𝑛𝐹𝐴

1

= 𝐷𝐻 (𝑑𝑐𝐻 /𝑑𝑟)𝑟=𝑟0

𝑑𝑐𝑆 𝑟 = 𝑟0; = 0, 𝑐𝐻 = 0 𝑑𝑟 𝑟 = 𝑟1; 𝑐𝑆 = 𝑐𝑆′ , 𝑐𝐻 = 0

𝑟2

𝑟 2 𝑑𝑟 𝐷𝐻 𝑑

𝐷𝑆 𝑑

=0

=0

Nonlinear differential equations with initial and boundary conditions Reduction of Order and HPM

Analytical techniques

(𝜒𝑟0 −1)

(𝜒𝑟0 +1) 𝑒𝑥𝑝(2𝜒𝑟0 )

=

𝑟0 (𝛼𝑒 𝜒𝑟0 +𝑒 −𝜒𝑟0 ) 𝑟0 −𝑟1

𝑟0 2+2𝜒𝑟0

=

2𝑟 𝜒𝑟 1+2𝜒𝑟0 − 0𝑟 0 1

𝑟1

for 𝜒𝑟0 and 𝜒𝑟1 → ∞ for all values of 𝜒𝑟0

𝑟1 −𝑟0

(𝑟1 −𝑟0 )

𝑟0

=

𝑟0 𝑐𝑆 −𝑐𝑆∗ 𝑐𝑆∗ (𝜒𝑟0 )2

=

𝑟 2

(𝜒𝑟0 )2 𝑛𝐹𝐴𝐷𝑆 𝑐𝑆∗

𝐶𝐻𝑃 =

𝑟 2

=

𝑟0

𝑟0

𝑟1

𝑟 𝑟0

2𝑟0

𝑟 2

𝑟0

−( ) −

𝑟0

𝑟 2

𝑟

2

𝑟

+

𝑟0

𝑟1

𝑟0

𝑟

+ 1)

𝑟0

− 0.5 ( ) + 0.5 ( 1 ) −

𝑟0

𝑟0

𝑟

𝑟0 𝑟1

−2 𝑟1 2

𝑟1

𝑟0

−( ) + +( ) −2

𝑟1 2

𝑟0

𝑐𝐻 𝐷𝐻 1 𝑟 𝑟 = ( 12 𝐷𝑆 𝑐𝑆∗ (𝜒𝑟0 )2 2 𝑟0 𝐼𝑟0 𝑟1 −𝑟0

𝐶𝑆𝑃 =

For all case of 𝐾𝑀

(𝜒𝑟0 )2 𝑛𝐹𝐴𝐷𝑆 𝑐𝑆∗

𝑟1

𝑐𝐻 𝐷𝐻 𝐷𝑆 𝑐𝑆∗ (𝜒𝑟0 )2 𝐼𝑟0

𝐶𝐻𝑃 =

2

=( ) −

𝑟 𝐾𝑀 0

= ( 1) − ( ) + 2

6(𝑐 −𝑐𝑆∗ )

𝐶𝑆 = (𝜒𝑟𝑆)2

𝑟1 2

𝑟0

𝑟1

𝑟𝑟0

−1

))}

−

𝛼𝑒 𝜒𝑟 +𝑒 −𝜒𝑟

𝛼𝑒 𝜒𝑟 +𝑒 −𝜒𝑟

𝑟0 (𝛼𝑒 𝜒𝑟1 +𝑒 −𝜒𝑟1 )

𝑟

Zero order catalytic kinetics: In this case𝑐𝑆 > K 𝑀

𝑛𝐹𝐴𝐷𝑆 𝐶𝑆

𝑟0 𝑟

𝑛𝐹𝐴𝐷𝑆 𝐶𝑆 𝐼𝑟0

= 𝑔 (𝜒𝑟0, 1 ) =

𝑟

+

= 𝑔 (𝜒𝑟0, 1 ) = 2 for 𝜒𝑟0 and 𝜒𝑟1 → 0

𝐼𝑟0 𝑛𝐹𝐴𝐷𝑆 𝐶𝑆 𝐼𝑟0

𝑟

)−( 1(

𝑟(𝑟1 −𝑟0 )

𝑟1 (𝑟1 −𝑟)

𝑟1

𝐷𝑆 𝐾𝑀

𝑘𝑐𝑎𝑡 𝑐𝐸

)+(

𝑟0 (𝛼𝑒 𝜒𝑟1 +𝑒−𝜒𝑟1 )

[

,𝜒 = √

𝜒𝑟0(𝛼𝑒 𝜒𝑟0 + 𝑒 −𝜒𝑟0 )]

𝑛𝐹𝐴𝐷𝑆 𝐶𝑆

𝐼𝑟0

The current is

𝛼=

𝑟(𝑟1 −𝑟0 ) 𝛼𝑒 𝜒𝑟1 +𝑒 −𝜒𝑟1

(

𝑟1 (𝑟1 −𝑟) 𝛼𝑒 𝜒𝑟0 +𝑒 −𝜒𝑟0

𝑟[𝛼 𝑒𝑥𝑝(𝜒𝑟)+𝑒𝑥𝑝(−𝜒𝑟1 )]

𝑟1 [𝛼 𝑒𝑥𝑝(𝜒𝑟)+𝑒𝑥𝑝(−𝜒𝑟)]

={

Where

𝐷𝑆 𝐶𝑆∗

𝐷𝐻 𝑐𝐻

𝑐𝑆 /𝑐𝑆′ =

Expressions for concentration and current

Table 5.2. (Continued)

Fluidized bed biofilm bioreactors

14 (Alagu Eswari and Rajendran 2012)

15 (Athimoolam, Chronoamperometry 𝑘1 Lakshmanan, and 𝑃⟺𝑄 𝑘−1 Alwarappan 2011)

Experimental techniques and enzymatic scheme

S. Reference No

𝑑𝑟 2

𝑑2 𝑆 2𝐷𝑠𝑓 𝑑𝑆

𝑑𝑟

=

𝜕𝑟 2

𝜕2 𝐶𝑃

+

𝜕𝑟 2

𝜕2 𝐶𝑄

+ 𝑟

𝜕𝑟

2𝐷𝑄 𝜕𝐶𝑄

𝜕𝑟

− 𝑘−1𝐶𝑄 +

− 𝑘1 𝐶𝑃 +

𝐶𝑄 →

−𝐷

|

|

𝜕𝑟 𝑟=𝑎

𝜕𝑟 𝑟=𝑎

𝜕𝐶𝑄

𝐽𝑃 = −𝐷

𝜕𝐶𝑃

𝐶𝑄𝑏

and 𝐽𝑄 =

The flux is defined as:

𝑟 → ∞ ; 𝐶𝑃 →

𝐶𝑃𝑏 ,

𝑟 = 𝑎 ; 𝐶𝑃 = 𝐶𝑃𝑠 , 𝐶𝑄 = 𝐶𝑄𝑠

𝑘1𝐶𝑃 = 0 The initial and boundary conditions are 𝑡 = 0 ; 𝐶𝑃 = 𝐶𝑃𝑏 , 𝐶𝑄 = 𝐶𝑄𝑏

𝐷𝑄

𝑟

2𝐷𝑃 𝜕𝐶𝑃

𝑘−1𝐶𝑄 = 0

𝐷𝑝

𝑑𝑟

The boundary conditions are 𝑑𝑆 𝑑𝐶 = = 0𝑎𝑡𝑟 = 𝑟𝑝 𝑑𝑟 𝑑𝑟 𝑑𝑆 𝐷𝑠𝑓 = 𝐾𝑆 (𝑆𝑏 − 𝑆)𝑎𝑡𝑟 = 𝑟𝑝 + 𝛿 𝑑𝑟 𝑑𝑆 𝐷0𝑓 = 𝐾0(𝐶𝑏 − 𝐶)𝑎𝑡𝑟 = 𝑟𝑝 + 𝛿

𝐶 𝐾0 +𝐶

𝑟 𝑑𝑟 𝜇𝑚𝑎𝑥

+ 2

2𝐷𝑜𝑓 𝑑𝑆

𝑌𝑋/𝑂 𝑆+𝐾𝑆 +𝑆 2 /𝐾𝑖

𝜌𝑣

𝐷𝑜𝑓

𝑑2 𝑆

=

𝐶 𝐾0 +𝐶

𝑟 𝑑𝑟 𝜇𝑚𝑎𝑥

+

𝑌𝑋/𝑆 𝑆+𝐾𝑆 +𝑆 2 /𝐾𝑖

𝜌𝑣

𝐷𝑠𝑓

Nonlinear differential equations with initial and boundary conditions

Laplace Transformation, Duhamel’s Theorem

HAM

Analytical techniques

∗ +(𝐾∗ /𝐾∗ )+𝐾∗ +𝐾∗ +(1/𝐾∗ )+1)(𝑥+𝑟 /𝛿)2 6𝐵𝑖𝑆 (𝐾𝑆∗ 𝐾𝑂 𝑝 𝑂 𝑖 𝑂 𝑆 𝑖

ℎ𝜑𝑠 (4𝐵𝑖𝑆 (𝑟𝑝 /𝛿)+𝐵𝑖𝑆 +3𝐵𝑖𝑆 (𝑟𝑝 /𝛿)2 +6(𝑟𝑝 /𝛿)2 +2+6(𝑟𝑝 /𝛿))

∗ /𝐾∗ )+𝐾∗ +𝐾∗ +(1/𝐾∗ )+1)(𝑥+𝑟 /𝛿)2 6𝐵𝑖𝑂 (𝐾𝑆∗ 𝐾𝑂∗ +(𝐾𝑂 𝑝 𝑖 𝑂 𝑆 𝑖

𝑎

𝑟

𝜅

(1 − ) +

𝐽𝑄 =

𝐷

𝐷

𝑎

𝜅𝑎

𝜅

𝐷

𝜅

𝜅

𝜅

𝐷

𝜅𝑟

𝜅𝑟

𝑒𝑥𝑝 [(𝑎 −

𝑒𝑥𝑝 [(𝑎 −

(𝑘1 𝐶𝑃𝑠 −𝑘−1 𝐶𝑄𝑠 )𝑎

(𝑘1 𝐶𝑃𝑠 −𝑘−1 𝐶𝑄𝑠 )𝑎

𝐷

(𝐶𝑃𝑠 + 𝐶𝑄𝑠 ) −

(𝐶𝑃𝑠 + 𝐶𝑄𝑠 ) +

+ ( + 𝑘−1√ ) 𝐶𝑄𝑠 − 𝑘1√ 𝐶𝑃𝑠

𝑎

𝐷

𝜅𝑟

𝑘1 𝑎

𝜅𝑟

𝑘−1 𝑎

+ ( + 𝑘1√ ) 𝐶𝑃𝑠 − 𝑘−1√ 𝐶𝑄𝑠 −𝐷𝑘1 𝐶 𝑏

𝜅𝑎

−𝐷𝑘−1 𝐶 𝑏

The flux is given by 𝐽𝑃 =

𝑟

𝑎

(1 − ) +

𝑘1 𝐶 𝑏

𝑟)√𝜅⁄𝐷 ]

𝐶𝑄 (𝑟) =

𝑟)√𝜅⁄𝐷 ]

𝜅

𝑘−1 𝐶 𝑏

K S * Sb K K  K  , K i  , K O*  O , BiS  S , BiO  O Sb Ki Cb Dsf Dof

K S*  𝐶𝑃 (𝑟) =

r  r S C   max  2   max  2 , C*  , x  , s  ,O  , Cb Cb  Dsf YX / S Sb DOf YX / S Cb

S* 

Where

∗ +(𝐾∗ /𝐾∗ )+𝐾∗ +𝐾∗ +(1/𝐾∗ )+1)(𝑥+𝑟 /𝛿) 6(𝐾𝑆∗ 𝐾𝑂 𝑝 𝑂 𝑖 𝑂 𝑆 𝑖

−

−

ℎ𝜑𝑂 (4𝐵𝑖𝑂 (𝑟𝑝 /𝛿)+𝐵𝑖𝑂 +3𝐵𝑖𝑂 (𝑟𝑝 /𝛿)2 +6(𝑟𝑝 /𝛿)2 +2+6(𝑟𝑝 /𝛿))

ℎ𝜑0 (3𝑥 2 (𝑟𝑝 /𝛿)+𝑥 3 )

𝐶 ∗(𝑥) = 1 +

∗ +(𝐾∗ /𝐾∗ )+𝐾∗ +𝐾∗ +(1/𝐾∗ )+1)(𝑥+𝑟 /𝛿) 6(𝐾𝑆∗ 𝐾𝑂 𝑝 𝑂 𝑖 𝑂 𝑆 𝑖

ℎ𝜑𝑠 (3𝑥 2 (𝑟𝑝 /𝛿)+𝑥 3 )

𝑆 ∗(𝑥) = 1 +

Expressions for concentration and current

Experimental techniques and enzymatic scheme

17 (Senthamarai and Amperometric Rajendran 2010)

16 (A. Meena and Amperometric Rajendran 2010a)

S. Reference No

2 𝑑𝑣

𝐿

𝑎+𝐿

𝐿

−

𝛾𝐸 𝑢𝑣 1+𝛼𝑢

𝑑𝜉

𝑑𝑣

= 𝑓𝑠

𝑓𝑅

𝐷𝐸

=

𝑘𝜅𝑠 ∞ 𝑎2 𝑓𝐸

𝑓𝑅

𝜎2

𝜆

=

=

𝜆 𝜌𝜎2

; 𝛾𝑠 =

𝑑𝑟 2

+ 𝑟 𝑑𝑟

2𝐷𝑠 𝑑𝑏

− 𝑘𝑠𝑏 = 0

𝑖 𝑑𝑠

𝑛𝐹𝐴

𝐷𝐸

𝑠

=

𝑘𝑠 ∞

𝑘𝑠 ∞ 𝑎2

𝑢=

Where

𝜌𝜎2

𝜆

𝑏 𝑐𝛴

𝐷𝑠

𝑘𝑐𝛴 𝑎2

𝑎

𝑟

=

=

𝜎2

𝜆

; 𝜒 = ; 𝛾𝐸 =

; 𝛾𝑠 =

;𝑣 =

𝑑𝑟 𝑟=0

𝑑𝑏

= −𝐷𝐸 ( )

𝑑𝑟 𝑟=𝑎

𝐷𝑠 ( )

𝐼=

The current value is

𝑑𝑟

The boundary conditions are 𝑑𝑠 𝑟 = 0𝑏 = 𝑐𝛴 ; =0 𝑑𝑟 𝑑𝑏 𝑟 = 𝑎 = 0; 𝑠 = 𝑘𝑠 ∞

𝐷𝑠

𝑑2 𝑠

𝑑 2 𝑏 2𝐷𝐸 𝑑𝑏 𝐷𝐸 2 + − 𝑘𝑠𝑏 = 0 𝑑𝑟 𝑟 𝑑𝑟

𝐷𝑠

𝑘𝑐𝛴

𝑎2

𝛾𝐸 =

𝑑𝑢

= 0; 1 − 𝑢 = 𝜎 (𝑑𝜉 )

=0

𝑠 𝑏 𝑟 𝑢 = ∞ ; 𝑣 = ; 𝛼 = 𝑠 ∞ /𝐾𝑀 ; 𝜉 = ; 𝑘𝑠 𝑐𝛴 𝐿

= 1 + 𝑚;

𝑑𝜉

𝑑𝑢

=0

= 𝑚; 𝑣 = 1;

𝜉 𝑑𝜉 𝑎

where

𝜉=

𝜉=

𝑑𝜉 2

+

VIM

HPM

𝑑 2 𝑢 2 𝑑𝑢 𝛾𝑠 𝑢𝑣 + − =0 𝑑𝜉 2 𝜉 𝑑𝜉 1 + 𝛼𝑢 𝑑2 𝑣

Analytical techniques

Nonlinear differential equations with initial and boundary conditions

2

+ 𝑚) +

𝜉

1

𝑚

+ 𝑚)3 ( − )| + |

1

4

𝜒2

𝜒=1

𝑙𝑚𝛾𝐸 +

8(𝜆⁄𝜎 )

= |𝑙 − 1| + 2 (1 − 𝑚)𝛾𝑠

3

𝜒3

𝑙 = (360+4𝛾𝑠

(360−𝛾 𝛾𝐸 −30𝛾𝑠 +180𝛾𝐸 ) ,𝑚 𝑠 𝛾𝐸 +30𝛾𝑠 +180𝛾𝐸 )

Where

4

𝑑𝜉 𝜉=𝑚

𝑑𝑣

= −𝜌𝜎 ( )

8

𝜒4

=

(1 − 𝑙 + 𝑙𝑚)𝛾𝐸 +

=

𝛾𝐸 +2

𝛾𝐸

𝜎2 𝜌)+2

(𝜆⁄𝜎2 𝜌)

= (𝜆⁄

2(𝜆⁄𝜎 2 ) (1 + (23⁄𝜌))(𝜆⁄𝜎 2 )2 − when (𝜆⁄𝜎 2 ) → 0 3 72 (16𝜌2 −45𝜌+450) 5 2 𝜓= +𝜌− when (𝜆⁄𝜎 ) → ∞ 2 𝜓=

𝑑𝑢

1

(𝑙𝛾𝐸 − 2𝑚) −

Normalized current 𝜓 = (𝑑𝜒)

1 1+𝑚

+

𝜎

]+

3(1+𝛼)

𝜎𝛾𝑠

(

(1+𝑚)2

(1+𝑚)3 −𝑚3

− 𝑚2 )|

(1+𝑚)2

5

𝜒5

(𝑙 − 1)𝛾𝐸 𝑚 +

𝜒2 𝜒3 𝜒4 𝜒5 (2𝑙 − 2 + 𝑙𝛾𝑠 ) − 𝑙𝑚𝛾𝑠 + (1 − 𝑙 + 𝑙𝑚)𝛾𝑠 + (𝑙 − 1)𝛾𝑠 𝑚 4 3 8 5 6 𝜒 + 𝑚𝛾𝑠 (1 − 𝑙) 12

𝑚𝛾𝐸 (1 − 𝑙) 12

𝜒6

𝑣(𝜒) = 1 +

𝑢(𝜒) = 𝑙 +

𝑑𝜉 𝜉=1+𝑚

𝑑𝑢

𝜓 = 1 − 𝑢𝜉=1+𝑚 = 𝜎 ( )

−

1 𝛾 (𝜉 2 6(1+𝛼) 𝐸

1 1 𝛾 𝑚3 [ 3(1+𝛼) 𝑠 𝜉

The normalized current density is then given by

1 𝛾 (1 3(1+𝛼) 𝐸

− (1 + 𝑚) )

2

1 𝛾 𝜎(1 3(1+𝛼) 𝑠

𝑣(𝜉) = 1 + |

𝛾 (𝜉 6(1+𝛼) 𝑠

1

𝑢(𝜉) = 1 −

Expressions for concentration and current

Table 5.2. (Continued)

)

1

(Athimoolam, Lakshmanan, and Alwarappan 2011)

S. Reference No

−𝑛𝑒 −

𝑛𝑒

𝑘−1

𝑃 ↔ 𝑄 ↔− 𝑍

𝑘1

CE mechanism scheme 𝑃 + 𝑛𝑒 − ↔ 𝑄

𝑘−1

𝑄 + 𝑍 ↔ 𝑃 + Products

𝑘1

𝑟

𝜕𝑟

𝐶𝑃𝑠 , 𝐶𝑄

𝐽𝑄 = −𝐷

𝐽𝑃 = −𝐷

| and

𝜕𝑟 𝑟=𝑎

|

𝜕𝑟 𝑟=𝑎 𝜕𝐶𝑄

𝜕𝐶𝑃

The flux is defined as:

→ 𝐶𝑄𝑏

𝑟 → ∞ ; 𝐶𝑃 → 𝐶𝑃𝑏 , 𝐶𝑄

= 𝐶𝑄𝑠

𝑟 = 𝑎 ; 𝐶𝑃 =

=

𝐶𝑄𝑏

The initial and boundary conditions are 𝑡 = 0 ; 𝐶𝑃 = 𝐶𝑃𝑏 , 𝐶𝑄

Experimental techniques Nonlinear differential and enzymatic scheme equations with initial and boundary conditions Chronoamperometry, first- 𝜕𝐶𝑃 = 𝐷 𝜕2 𝐶𝑃 + 𝑝 𝜕𝑟 2 𝜕𝑡 order reaction mechanism 2𝐷 𝑃 𝜕𝐶𝑃 − 𝑘1 𝐶𝑃 + 𝑘−1𝐶𝑄 𝑘1 𝑟 𝜕𝑟 𝑃 ↔ 𝑄 𝜕𝐶𝑄 𝜕2 𝐶𝑄 𝑘−1 = 𝐷𝑄 2 + 𝜕𝑡 𝜕𝑟 EC mechanism scheme 2𝐷𝑄 𝜕𝐶𝑄 − 𝑘−1𝐶𝑄 + 𝑘1𝐶𝑃 𝑃 + 𝑛𝑒 − ↔ 𝑄 Duhamel’s Theorem, Reduction of Order, Laplace Transformation

Analytical technique

𝜅𝑟

𝑟

+

2𝜅𝑟

(𝑘1 𝐶𝑃𝑠 −𝑘−1 𝐶𝑄𝑠 )𝑎

))

𝑟−𝑎

))

) − 𝐹(𝑟, 𝜅, 𝑡)

2√𝐷𝑡

𝑒𝑥𝑝 [(𝑎 − 𝑟)√𝜅⁄𝐷 ]

2√𝐷𝑡

𝑟−𝑎

𝑟−𝑎

2√𝐷𝑡

) + 𝐹(𝑟, 𝜅, 𝑡)

erfc (

2√𝐷𝑡

𝑟−𝑎

𝑟−𝑎

𝜅

𝑎

( +

−𝐷𝑘−1 𝐶 𝑏 1

𝜅

𝑎

( +

−𝐷𝑘1 𝐶 𝑏 1

𝐻(𝜅, 𝑡)

𝐽𝑄 =

1

1

𝐷

+ √𝜅𝑡)

𝐷

𝑎

𝐷

𝜅

𝐷

𝜅

𝑎

𝜅

𝜅

𝑘−1 𝐷

𝑘1 𝐷

(𝐶𝑃𝑠 +

(𝐶𝑃𝑠 + 𝐶𝑄𝑠 ) −

𝜅√𝜋𝐷𝑡

𝜅√𝜋𝐷𝑡

) + ( + 𝑘1√ ) 𝐶𝑃𝑠 − 𝑘−1√ 𝐶𝑄𝑠 +

𝐷

𝑟−𝑎 2√𝐷𝑡

) + ( + 𝑘−1√ ) 𝐶𝑄𝑠 − 𝑘1 √ 𝐶𝑃𝑠 +

√𝜋𝐷𝑡

√𝜋𝐷𝑡

𝐶𝑄𝑠 ) + 𝐻(𝜅, 𝑡) and

𝐽𝑃 =

𝐷

− √𝜅𝑡) ; 𝑚+ = erfc (

The fluxes are as follows:

2√𝐷𝑡

𝑚− = erfc (

(𝑘1𝐶𝑃𝑠 − 𝑘−1𝐶𝑄𝑠 )𝑎 𝑒𝑥𝑝 [(𝑎 − 𝑟)√𝜅⁄𝐷 ] 𝜅𝑟 and 𝐹(𝑟, 𝜅, 0) = 0. 𝐹(𝑟, 𝜅, ∞) =

{𝑚 + 𝑒𝑥𝑝 [−2(𝑎 − 𝑟)√𝜅⁄𝐷 ] m+ }

−

𝑟

𝑎

erfc (

(1 −

𝐶𝑄𝑠 )

𝜅

𝑘1 𝐶 𝑏

𝐹(𝑟, 𝜅, 𝑡) =

where

+

(𝐶𝑃𝑠 𝜅𝑟

𝑘1 𝑎

𝑎

𝜅

(1 − erfc (

𝑘−1 𝐶 𝑏

(𝐶𝑃𝑠 + 𝐶𝑄𝑠 ) erfc (

𝐶𝑄 (𝑟, 𝑡) =

+

𝑘−1 𝑎

𝐶𝑃 (𝑟, 𝑡) =

Expressions for concentration and current

Table 5.3. A recent contribution of non-steady-state reaction-diffusion equations in the sphere

S. Reference No

Experimental techniques Nonlinear differential and enzymatic scheme equations with initial and boundary conditions

Analytical technique

]

1

(

𝐶𝑃𝑆

+

𝐶𝑏

𝐶𝑄𝑆

√𝜋𝐷𝑡

𝐾+1 𝐶 𝑏

1

[

𝜅

𝐷

√𝜋𝐷𝑡

𝜅

)] (1 +

𝐷

√𝜋𝐷𝑡

𝑎

)+

(

𝐶𝑃𝑆 𝐾+1 𝐶 𝑏

1

−

− √ erfc (√𝜅𝑡)] The current is

+ √ erf (√𝜅𝑡))]

𝐾

=[ −

𝑒𝑥𝑝[−𝜅𝑡]

) [1 + 𝑎 (

𝑑𝑟 𝑟=𝑎

𝑑𝐶𝑄

=[

𝜅

𝐷(𝑘1 𝐶𝑃𝑠 −𝑘−1 𝐶𝑄𝑠 ) 𝑒𝑥𝑝[−𝜅𝑡]

−1

1

(

𝐶𝑃𝑆

𝛿𝑑

𝑎

𝐶𝑄𝑆 𝐶𝑏

1

𝑎

1

−2

1

−1

(

𝐶𝑃𝑆

𝐶𝑏

𝐶𝑄𝑆

𝛿𝑟

𝑎

𝜅

𝐷

1

𝑎

−1

𝑖(𝑡)𝑎 𝑛𝐹𝐴𝐷

= {1 + 𝑎 [

√𝜋𝐷𝑡

𝑒𝑥𝑝[−𝜅𝑡]

𝜅

+ √ erf (√𝜅𝑡)]} ( 𝐷

− 𝐹(𝑟, 𝜅, 𝑡)

1+𝐾

𝐶𝑏

− 𝐶𝑃𝑆 )

} +𝑚+ 𝑒𝑥𝑝 (−(𝑎 − 𝑟)√𝜅⁄𝐷 )

𝜅

𝑘1 𝐶 𝑏

𝑚− 𝑒𝑥𝑝 ((𝑎 − 𝑟)√𝜅⁄𝐷 )

the current 𝑖(𝑡) = 𝑛𝐹𝐴𝑗𝑄 is as follows:

2𝜅𝑟

{

+ 𝐹(𝑟, 𝜅, 𝑡) ; 𝐶𝑄 (𝑟, 𝑡) =

(𝑘1 𝐶𝑃𝑠 −𝑘−1 (𝐶 𝑏 −𝐶𝑃𝑠 ))𝑎

𝜅

𝑘−1 𝐶 𝑏

𝐹(𝑟, 𝜅, 𝑡) =

Where

𝐶𝑃 (𝑟, 𝑡) =

𝑎

−

−

𝛿𝑑

𝑎

𝛿𝑑

− 1) (

) [1 + ( − 1) erf ((

−𝐾

and 𝛿𝑟 = ( + √ )

𝛿𝑟

𝑎

𝐾+1 𝐶 𝑏

𝜋) + (

𝛿𝑑

)] ( ) + − 1)

+

√𝜋)] where 𝛿𝑑 = (𝑎 + √𝜋𝐷𝑡)

− 1) (

2

𝐾+1 𝐶 𝑏

The concentration of species 𝑃 and 𝑄 for EC’ mechanism as

1)

𝑎 𝛿𝑟

1) 𝑒𝑥𝑝 (− (

1

𝐾

=[ −

𝑖(𝑡)𝑎 𝑛𝐹𝐴𝐷𝐶 𝑏

The current in terms of diffusion layer thickness 𝛿𝑑 and reaction layer thickness 𝛿𝑟 is

𝐾

𝐶𝑄𝑆 𝐶𝑏

𝑛𝐹𝐴𝐷𝐶 𝑏

𝑖(𝑡)𝑎

given by

where 𝐻(𝜅, 𝑡) =

Expressions for concentration and current

Table 5.3 (Continued)

S. Reference No

Experimental techniques Nonlinear differential and enzymatic scheme equations with initial and boundary conditions

Analytical technique

1−𝐾𝑒 𝜂

= ((1+𝐾)(1+𝑒 𝜂 )) {1 + 𝑎 [ √𝜋𝐷𝑡

𝑒𝑥𝑝[−𝜅𝑡]

𝜅

𝐷

+ √ erf (√𝜅𝑡)]}

+

𝑘1√𝜋𝐷𝑡

𝑘−1𝑎

+ 𝑎𝑧(𝜅, 𝑡))

𝑎2 𝑧(𝜅, 𝑡) 𝑎2 √𝜅 +( + ) (𝑐 𝑏 − 𝐶𝑄𝑠 ) 𝐷√𝜋𝑡 √𝜋𝐷𝑡 𝑎2 𝑘−1 𝑎2 𝑘−1𝑧(𝜅, 𝑡) 𝑘−1 −𝐶𝑄𝑠 ( + ) (1 + ) 𝑘1 ( 𝐷√𝜋𝑡𝜅 𝜅√𝜋𝑡𝐷

)

𝜅 𝜅 (𝑎√ + 𝐷 𝑘1

𝑛𝐹𝐴𝐷𝐶 𝑏

𝑖𝐿 (𝑡)

=

𝜅

1 𝜅 1 1 ) 1+( +√ +𝑧(𝜅,𝑡))⁄𝐾( + 𝑎 𝐷 𝑎 √𝜋𝐷𝑡

{ +√ +𝑧(𝜅,𝑡)}⁄𝐾 𝑎 𝐷

1

limiting current density becomes:

When the electrode is polarized to an extreme value 𝐶𝑄𝑠 = 0 the approximate

𝑖(𝑡)𝑎 = 𝑛𝐹𝐴𝐷

𝜅 𝑎 𝜅 (𝑎√ + 1 + + 𝑎𝑧(𝜅, 𝑡)) (𝑐 𝑏 − 𝐶𝑄𝑠 ) 𝐷 𝑘1 √𝜋𝐷𝑡

−𝑛𝐹𝐴𝐽𝑄 becomes

𝜅

𝐷

√𝜋𝐷𝑡

− √ erfc (√𝜅𝑡). Now the current density 𝑖 =

𝑒𝑥𝑝[−𝜅𝑡]

√𝐷𝜅+𝑘1 𝑎√𝜅+√𝐷𝑘1 𝑎𝑧(𝜅,𝑡)+

𝑘−1 𝑎 √𝜋𝑡

𝑘−1 𝑎 1 )𝐶 𝑏 +(√𝜅− +√𝐷𝑧(𝜅,𝑡))𝑘−1 𝑎𝐶𝑄𝑆 √𝜋𝑡 √𝜋𝑡

(√𝐷𝑘−1 +

where 𝑧(𝜅, 𝑡) =

𝐶𝑃𝑆 =

The concentration of species 𝑃 and 𝑄 for CE Mechanism we can find 𝐶𝑃𝑆 (the surface concentration of species P) as follows:

𝑛𝐹𝐴𝐷𝐶 𝑏

𝑖𝐿 (𝑡)𝑎

The limiting current in terms of potential is

Expressions for concentration and current

2

(A Eswari and Rajendran 2011)

S. Reference No

=𝐷 𝜕𝑟 2

𝜕2 𝐶𝑍

𝜕𝑟

+ 𝑟 𝜕𝑟

2𝐷 𝜕𝐶𝑍

𝑟 𝜕𝑟

+

𝑘𝐶𝑅 𝐶𝑍 The initial and boundary conditions are 𝑡 = 0; 𝐶0 = 𝐶0∗, 𝐶𝑅 = 0, 𝐶𝑍 = 𝐶𝑍∗ 𝜕𝐶𝑅 𝑟 = 𝑟0; 𝐶0 = 0, −𝐷 𝜕𝑟 𝜕𝐶0 𝜕𝐶𝑍 =𝐷 , =0 𝜕𝑟 𝜕𝑟 𝑟 → ∞; 𝐶0 → 𝐶0∗, 𝐶𝑅 → 0, 𝐶𝑍 → 𝐶𝑍∗

𝜕𝑡

𝜕𝐶𝑍

𝑘𝐶𝑅 𝐶𝑍

𝜕𝑡

Experimental techniques Nonlinear differential and enzymatic scheme equations with initial and boundary conditions 𝜕𝐶0 𝜕2 𝐶 2𝐷 𝜕𝐶0 Chronoamperometric = 𝐷 20 + + 𝜕𝑡 𝜕𝑟 𝑟 𝜕𝑟 𝑂 + 𝑛𝑒 − ↔ 𝑅 𝑘𝐶𝑍 𝐶𝑅 𝑘 𝑅 + 𝑍 → 𝑂 + Products 𝜕𝐶𝑅 𝜕2 𝐶 2𝐷 𝜕𝐶𝑅 = 𝐷 2𝑅 + + HPM, Laplace Transform Method, Reduction of Order Method

Analytical technique

2√𝜏

)]

𝑢= 𝐶0∗

𝐶0

Where ,𝜌 =

𝜏

𝜋

2

𝜌

𝜏

4𝜏

𝑟 𝑟0

,𝑣 =

𝐶0∗

𝐶𝑅

,𝜏 =

𝑟02

𝐷𝑡

, 𝛾𝐸 =

𝐷

𝑘𝐶𝑍∗ 𝑟02

, 𝛾𝑆 =

)]

𝐷

𝑘𝐶0∗ 𝑟02

2√𝜏

𝜌−1

)−

𝐶𝑍∗

𝐶𝑍

)] +

,𝑤 =

2√𝜏

𝜌−1

4𝜏

(𝜌−1)2

) − (𝜌 − 1)𝑒𝑟𝑓𝑐 (

𝜋

( − 1) [2√ 𝑒𝑥𝑝 (−

(𝜌−1)2

2

𝛾𝐸 1

[2√ 𝑒𝑥𝑝 (−

𝛾𝑠

)−

) − 𝑒𝑥𝑝(𝜌 − 1 + 𝜏) 𝑒𝑟𝑓𝑐 (√𝜏 +

2√𝜏

2𝜌

[𝑒𝑟𝑓𝑐 (

𝜌−1

𝛾𝑆

𝑤(𝜌, 𝜏) = 1 −

𝑣(𝜌, 𝜏) = 1 − 𝑢(𝜌, 𝜏)

(𝜌 − 1)𝑒𝑟𝑓𝑐 (

𝜌−1

2√𝜏

𝜌−1

𝑢(𝜌, 𝜏) = 1 − 𝜌𝑒𝑟𝑓𝑐 (

Expressions for concentration and current

Table 5.3. (Continued)

4

3

(Swaminathan 2019)

𝜕𝑟 2

+ 𝑟 𝜕𝑟

2𝐷𝑆 𝜕𝑠

− 𝑘𝑠𝑏 = 𝜕𝑡

𝜕𝑠

𝑘𝑠 ∞

𝑠

2 𝜕𝐶𝑆

𝜕𝑡

]−

𝑎2

𝐷𝑡

𝑅 𝜕𝑅 𝜕𝐶𝑆

+ =

;

;𝜏 =

𝐷𝐸

;𝜌 =

∞

𝜕𝑅

𝜕𝐶𝑆

=0

,𝜏 = 𝐷𝑆 𝐶𝑠∗

2

𝑟1 2

𝐷𝑆 𝑡

𝐾𝑐𝑎𝑡 𝐶𝐸 𝑟𝑚 𝑟1

𝑟1

𝑅

𝐶𝑠 𝐶𝑠∗ 𝐶𝑠∗

𝐾𝑚

,𝑟 =

,α =

Where 𝑢 =

𝑅 = 𝑟1; 𝐶𝑆 = 𝐶𝑠∗

0;

,𝑘 =

The initial and boundary conditions are 𝑡 = 0; 𝑐𝑠 = 0, 𝑅 =

𝐾𝑚 + 𝐶𝑆

𝐾𝑐𝑎𝑡 𝐶𝐸 𝐶𝑆

Michaelis-Menten kinetics 𝐷 [ 𝑆

𝐷𝑆 𝜕2 𝐶𝑆 𝜕𝑅 2

𝑘𝑐∑ 𝑎2

𝑘𝜅𝑠 ∞ 𝑎2

𝑐∑

𝑏

= 0; 𝑠 = 𝑘𝑠

;𝑣 =

𝜕𝑟

; 𝛾𝐸 =

𝛾𝑠 =

𝑎

𝑟

𝑢=

where

𝑟 = 𝑎;

𝜕𝑏

𝑡 = 0; 𝑏 = 𝑐∑ ; 𝑠 = 0, 𝑟 𝜕𝑠 = 0; 𝑏 = 𝑐∑ ; = 0, 𝜕𝑟

The initial and boundary conditions are

𝐷𝑆

𝜕2 𝑠

𝜕𝑏 = 𝜕𝑡

Experimental techniques Nonlinear differential and enzymatic scheme equations with initial and boundary conditions 𝜕 2 𝑏 2𝐷𝐸 𝜕𝑏 (Shanmugarajan et al. Amperometric 𝐷𝐸 2 + − 𝑘𝑠𝑏 𝜕𝑟 𝑟 𝜕𝑟 2010)

S. Reference No

Laplace Transform, Reduction of Order Method

Laplace Transform, Reduction of Order Method

Analytical technique

𝜌

𝛾𝐸

𝑠𝑖𝑛ℎ(√𝛾𝑆 𝜌)

+ 𝜌

2𝜋

𝐸

∑∞ 𝑛=1

𝑛 𝑠𝑖𝑛(𝑛𝜋𝜌) (−1)𝑛 (𝑛2 𝜋 2 +𝛾𝑆 )

𝑑𝑢

𝑑𝑣 (𝑛−1)4

𝑟 𝑠𝑖𝑛ℎ(√𝑚) 2 2

𝑠𝑖𝑛ℎ(√𝑚𝑟)

+

𝑟

𝑢=

𝐶𝑠 𝐶𝑠∗

,𝑟 =

𝑅 𝑟1

,𝜏 =

𝑟1 2

𝐷𝑆 𝑡

√𝛾𝐸 (1−√𝛾𝐸 ) 𝑒𝑥𝑝(√𝛾𝐸 ) − 2(√𝛾𝐸 𝑠𝑖𝑛ℎ(√𝛾𝐸 )−𝑐𝑜𝑠ℎ(√𝛾𝐸 ))

,α =

𝐶𝑠∗

𝐾𝑚

,𝑘 =

𝑛 𝑠𝑖𝑛(𝑛𝜋𝑟) (𝑛2 𝜋 2 +𝑚)

𝐷𝑆 𝐶𝑠∗

𝐾𝑐𝑎𝑡 𝐶𝐸 𝑟𝑚 𝑟1 2

−𝑛+1 ∑∞ 𝑛=1(−1)

𝑛2 (𝑛2 𝜋 2 +𝛾𝑆 )

𝑒𝑥𝑝( − ((𝑛 − 1)2𝜋 2 + 𝛾𝐸 )𝜏)

2𝜋

((𝑛−1)2 𝜋 2 +𝛾𝐸 )

−

× 𝑒𝑥𝑝( − (𝑛 𝜋 + 𝑚)𝜏) 𝑤ℎ𝑒𝑟𝑒 𝑚 = 𝑘/(1 + 𝛼)

𝑢(𝑟, 𝜏) =

𝛾𝐸 𝜋 4 ∑∞ 𝑛=1

𝛾𝐸

2

= −√

= √𝛾𝑆 𝑐𝑜𝑡ℎ( √𝛾𝑆 ) − 1 + 2𝜋 2 ∑∞ 𝑛=1

𝑑𝜌 𝜌=0

𝜓 = −( )

𝛾𝑆 )𝜏]

𝑑𝜌 𝜌=1

𝜓=( )

+ 𝛾𝐸 )𝜏) 𝑒𝑥𝑝[−(𝑛2 𝜋 2 +

× 𝑒𝑥𝑝( − (𝑛2 𝜋 2 + 𝛾𝑆 )𝜏)

𝑒𝑥𝑝(√𝛾𝐸 𝜌) (1−𝛾𝐸 ) 𝑒𝑥𝑝(√𝛾𝐸 ) 𝑐𝑜𝑠ℎ(√𝛾𝐸 𝜌) + 𝜌√𝛾𝐸 𝜌√𝛾𝐸 (√𝛾𝐸 𝑠𝑖𝑛ℎ(√𝛾𝐸 )−𝑐𝑜𝑠ℎ(√𝛾𝐸 )) 2 (𝑛−1) 𝑐𝑜𝑠((𝑛−1)𝜋𝜌) ∑∞ 𝑒𝑥𝑝( − ((𝑛 − 1)2𝜋 2 𝑛=1 ((𝑛−1)2 𝜋2 +𝛾 )

𝜌 𝑠𝑖𝑛ℎ(√𝛾𝑆 )

The non-steady-state current is given by

−2

𝜋2

𝑣(𝜌, 𝜏) =

𝑢(𝜌, 𝜏) =

Expressions for concentration and current

5 2

𝜕2 𝑐𝐴

−

𝑘2 𝑐𝐻 𝑐𝐴 The initial and Boundary conditions 𝑡 = 0; 𝑐𝐻 = 𝑐𝐻∞ , 𝑐𝐻𝐴 ∞ = 𝑐𝐻𝐴 , 𝑐𝐴 = 𝑐𝐴∞ 𝑟 = 𝑟𝑠 ; 𝑐𝐻 = 0, 𝑑𝑐𝐻𝐴 /𝑑𝑟 = 0, 𝑑𝑐𝐴 /𝑑𝑟 = 0 𝑟 = ∞; 𝑐𝐻 = 𝑐𝐻∞ , 𝑐𝐻𝐴 = ∞ 𝑐𝐻𝐴 , 𝑐𝐴 = 𝑐𝐴∞ The dimensionless current at the microdisc electrode 𝐼𝑆 = −𝑛𝐹𝐴𝐷𝐻 /𝑟𝑆 (𝑑𝑣/ 𝑑𝜌)𝜌=1

= 𝐷𝐴 2 + 𝜕𝑡 𝜕𝑟 2𝐷𝐴 𝜕𝑐𝐴 + 𝑘1𝑐𝐻𝐴 𝑟 𝜕𝑟

𝜕𝑐𝐴

𝑘2 𝑐𝐻 𝑐𝐴

𝜕𝑐𝐻

𝜕2 𝑐 = 𝐷𝐻 2𝐻 + 𝜕𝑡 𝜕𝑟 2𝐷𝐻 𝜕𝑐𝐻 + 𝑘1 𝑐𝐻𝐴 − 𝑟 𝜕𝑟

𝑘2 𝑐𝐻 𝑐𝐴

𝜕𝑐𝐻𝐴

𝜕2 𝑐 = 𝐷𝐻𝐴 𝐻𝐴 + 𝜕𝑡 𝜕𝑟 2 2𝐷𝐻𝐴 𝜕𝑐𝐻𝐴 − 𝑘1𝑐𝐻𝐴 + 𝑟 𝜕𝑟

Experimental techniques Nonlinear differential and enzymatic scheme equations with initial and boundary conditions

𝑘1 (Alagu Eswari, Usha, 𝐻𝐴 ↔ 𝐻 + + 𝐴− 𝐻 + + 𝑘2 and Rajendran 2011) 1 𝑒 − ↔ 𝐻2

S. Reference No

HPM, Reduction of Order, Laplace Transformation

Analytical technique

1

(𝜀1 −1)𝜌

𝛾𝐸 𝜀1

𝛾𝑆 =

𝜀1 =

∞ 𝐷𝐻𝐴 𝑐𝐻𝐴

∞ ∞ 2 𝑘2 𝑐𝐻 𝑐𝐴 𝑟𝑆

𝐷𝐻𝐴

𝑐𝐻

, 𝛾𝐸1 =

𝐷𝐻𝐴

∞ 𝑐𝐴

𝑐𝐴

𝐷𝐻𝐴

𝑘1 𝑟𝑆2

,

4𝜏

∞ 𝐷𝐻𝐴 𝑐𝐻

𝑟𝑠

2√𝜀2 𝜏

,

∞ 𝐷𝐻𝐴 𝑐𝐴

∞ 2 𝑘1 𝑐𝐻𝐴 𝑟𝑆

𝑟𝑠2

√𝜀1 𝜏 𝐷𝐻𝐴 𝑡

𝑒𝑥𝑝 (−

4𝜏𝜀1

4𝜏

, 𝛾𝑆1 =

𝐷𝐻𝐴

∞ 2 𝑘1 𝑐𝐴 𝑟𝑆

, 𝛾𝑆2 =

∞ 2 𝑘1 𝑐𝐴 𝑟𝑆

𝐷𝐻𝐴

)]

)] +

4𝜏𝜀1

(𝜌−1)2

(𝜌−1)2

)]

𝑒𝑥𝑝 (−

)] (𝜀2 √𝜀1 − 𝜀1√𝜀2 )

√𝜋𝜏

1

1

√𝜋𝜏 (𝜌−1)2

𝑒𝑥𝑝 (−

)−

) − 𝜀1 𝜌−1

0.28217𝛾𝐸1

,𝜏 =

+

, 𝛾𝐸2 =

,𝜌 =

√𝜀1 𝜏 𝑟

1 √𝜋𝜏 (𝜌−1)2

( − 1) [

2√𝜀1 𝜌

0.56419

∞ 2 𝑘1 𝑐𝐻𝐴 𝑟𝑆

, 𝛾𝐸 =

,𝑤 = 𝐷𝐻

∞ 𝑐𝐻

=

,𝑣 =

𝐷𝐻 ′𝜀2

∞ 𝑐𝐻𝐴

𝑐𝐻𝐴

𝜓 = 𝐼𝑆 𝑟𝑆 /𝑛𝐹𝐴𝐷𝐻 𝐶𝐻∞ = 1 + 𝑢=

1

)+

1

[√𝜀1 𝜀2 𝑒𝑥𝑝 (− √𝜋𝜏

2√𝜀1 √𝜏

𝛾𝐸1

[𝑒𝑥𝑝( 𝜌 − 1) 𝑒𝑥𝑝( 𝜀2 𝜏)𝑒𝑟𝑓𝑐 (√𝜀2 𝜏 +

(𝜀1 −𝜀2 )𝜌

𝛾𝐸2

The dimensionless current

(𝜀1 −𝜀2 )𝜌

𝛾𝐸 𝜀1

𝑤(𝜌, 𝜏) = 1 −

𝜌

(𝜌−1)2

2√𝜏

𝜌−1

𝑒𝑥𝑝( 𝜌 − 1) 𝑒𝑥𝑝( 𝜏)𝑒𝑟𝑐 (√𝜏 +

[𝑒𝑥𝑝( 𝜌 − 1) 𝑒𝑥𝑝( 𝜏)𝑒𝑟𝑓𝑐 (√𝜏 +

𝛾𝐸 √𝜀1 1 (𝜌−1)2 [ 𝑒𝑥𝑝 (− )− (𝜀1 −1)𝜌 √𝜋𝜏 4𝜏

𝑣(𝜌, 𝜏) = 1 − 𝑒𝑟𝑓𝑐 (

2√𝜏

)] +

𝜌−1

𝑢(𝜌, 𝜏) = 1 +

Expressions for concentration and current

Table 5.3. (Continued)

Steady and Non-Steady Non-Linear Reaction-Diffusion …

145

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Journal of Electroanalytical Chemistry 651 (2): 173–84. https://doi.org/10.1016/j.jelechem.2010.11.027. Eswari, A, and L Rajendran. 2010. “Analytical Solution of Steady State Current at a Microdisk Biosensor.” Journal of Electroanalytical Chemistry 641 (1–2): 35–44. https://doi.org/10.1016/j.jelechem. 2010.01.015. Eswari, A, and L Rajendran. 2011. “Analytical Expressions of Concentration and Current in Homogeneous Catalytic Reactions at Spherical Microelectrodes: Homotopy Perturbation Approach.” Journal of Electroanalytical Chemistry 651 (2): 173–84. https://doi. org/10.1016/j.jelechem.2010.11.027. Eswari, Alagu, and Lakshmanan Rajendran. 2012. “Approximate Analytical Solution of the Concentration of Phenol and Oxygen and Rate of Phenol Degradation in Fluidized Bed Bioreactor.” Biochemical Engineering Journal 68: 42–53. https://doi.org/ 10.1016/j.bej.2012.07.005. Eswari, Alagu, Seetharaman Usha, and Lakeshmanan Rajendran. 2011. “Approximate Solution of Non-Linear Reaction Diffusion Equations in Homogeneous Processes Coupled to Electrode Reactions for CE Mechanism at a Spherical Electrode.” American Journal of Analytical Chemistry 02 (02): 93–103. https://doi.org/10. 4236/ajac.2011.22010. Femila Mercy Rani, J, S Sevukaperumal, and L Rajendran. 2015. “Analytical Expression Pertaining to Concentration of Substrate and Effectiveness Factor for Immobilized Enzymes with Reversible Michaelis Menten Kinetics.” Asian Journal of Science and Applied Technology 4 (1): 10–16. Joy, Rathinasamy Angel, Athimoolam Meena, Shunmugham Loghambal, and Lakshmanan Rajendran. 2011. “A Two-Parameter Mathematical Model for Immobilizedenzymes and Homotopy Analysis Method.” Natural Science 03 (07): 556–65. https://doi. org/10.4236/ns.2011.37078.

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Margret PonRani, Vincent Michael Raj, Lakshmanan Rajendran, and Raju Eswaran. 2011. “Analytical Expression of the Substrate Concentration in Different Part of Particles with Immobilized Enzyme and Substrate Inhibition Kinetics.” Analytical and Bioanalytical Electrochemistry 3 (5): 507–20. Meena, A., and L. Rajendran. 2010a. “Analytical Solution of System of Coupled Non-Linear Reaction Diffusion Equations. Part I: Mediated Electron Transfer at Conducting Polymer Ultramicroelectrodes.” Journal of Electroanalytical Chemistry 647 (2): 103–16. https://doi.org/10.1016/j.jelechem.2010.06.013. Meena, A, and L Rajendran. 2010b. “Analytical Solution of System of Coupled Non-Linear Reaction Diffusion Equations. Part II: Direct Reaction of Substrate at Underlying Microdisc Surface.” Journal of Electroanalytical Chemistry 650 (1): 143–51. https://doi.org/10. 1016/j.jelechem.2010.08.009. Meena, V, T Praveen, and L Rajendran. 2016. “Mathematical Modeling and Analysis of the Molar Concentrations of Ethanol, Acetaldehyde and Ethyl Acetate inside the Catalyst Particle.” Kinetics and Catalysis 57 (1): 125–34. https://doi.org/10.1134/S0023158 416010092. Molina, Ángela, Francisco Martínez-Ortiz, Eduardo Laborda, and Richard G. Compton. 2010. “Characterization of Slow Charge Transfer Processes in Differential Pulse Voltammetry at Spherical Electrodes and Microelectrodes.” Electrochimica Acta 55 (18): 5163–72. https://doi.org/10.1016/j.electacta.2010.04.024. Mortensen, Jeff, Sayaka Olsen, Jean Yves Parlange, and Aleksey S. Telyakovskiy. 2012. “Approximate Similarity Solution to a Nonlinear Diffusion Equation with Spherical Symmetry.” International Journal of Numerical Analysis and Modeling 9 (1): 105–14. Praveen, T, and L Rajendran. 2015. “Theoretical Analysis through Mathematical Modeling of Two-Phase Flow Transport in an

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Immobilized-Cell Photobioreactor.” Chemical Physics Letters 625: 193–201. https://doi.org/10.1016/j.cplett.2015.01.007. Praveen, T, Pedro Valencia, and L Rajendran. 2014. “Theoretical Analysis of Intrinsic Reaction Kinetics and the Behavior of Immobilized Enzymes System for Steady-State Conditions.” Biochemical Engineering Journal 91: 129–39. https://doi.org/10. 1016/j.bej.2014.08.001. Renuga Devi, M, S Sevukaperumal, and L Rajendran. 2015. “NonLinear Reaction Diffusion Equation with Michaelis-Menten Kinetics and Adomian Decomposition Method.” Applied Mathematics 5 (1): 21–32. https://doi.org/10.5923/j.am. 20150501.04. Renugadevi, Mayathevar, Saminathan Sevukaperumal, and Lakshmanan Rajendran. 2016. “The Approximate Analytical Solution of Non-Linear Equation for Simultaneous Internal Mass and Heat Diffusion Effects.” Natural Science 08 (06): 284–94. https://doi.org/10.4236/ns.2016.86033. Saranya, K., V. Mohan, R. Kizek, C. Fernandez, and L. Rajendran. 2018. “Unprecedented Homotopy Perturbation Method for Solving Nonlinear Equations in the Enzymatic Reaction of Glucose in a Spherical Matrix.” Bioprocess and Biosystems Engineering 41 (2): 281–94. https://doi.org/10.1007/s00449-017-1865-0. Senthamarai, R., and L. Rajendran. 2010. “System of Coupled NonLinear Reaction Diffusion Processes at Conducting PolymerModified Ultramicroelectrodes.” Electrochimica Acta 55 (9): 3223– 35. https://doi.org/10.1016/j.electacta.2010.01.013. Sevukaperumal, Swaminathan, Alagu Eswari, and Lakshmanan Rajendran. 2013. “Solution of Non-Linear Boundary Value Problems in Immobilized Glucoamylase Kinetics.” Natural Science 05 (04): 478–94. https://doi.org/10.4236/ns.2013.54061. Shanmugarajan, Anitha, Subbiah Alwarappan, Rajendran Lakshmanan, and Ashok Kumar. 2010. “Solutions of the Coupled Reaction

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and Diffusion Equations within Polymer-Modified Ultramicroelectrodes.” Journal of Physical Chemistry A 114 (26):7030–37. https://doi.org/10.1021/jp1025224. Swaminathan, R. 2019. “Reaction/Diffusion Equation with MichaelisMenten Kinetics in Microdisk Biosensor: Homotopy Perturbation Method Approach.” International Journal of Electrochemical Science, April, 3777–91. https://doi.org/10.20964/2019.04.13. Usha, Seetharaman, Shanmugarajan Anitha, and Lakshmanan Rajendran. 2012. “Approximate Analytical Solution of Non-Linear Reaction Diffusion Equation in Fluidized Bed Biofilm Reactor.” Natural Science 04 (12): 983–91. https://doi.org/10.4236/ns.2012. 412127. Zeng, Yi. 2015. “Mathematical Modeling of Lithium-Ion Intercalation Particles and Their Electrochemical Dynamics.”

Chapter 6

NONLINEAR CONVECTION-DIFFUSION PROBLEMS 6.1. INTRODUCTION The convection-diffusion equation provides a beneficial and vital mathematical model in natural sciences and engineering (Mickens 1999). These applications include air transport, soil pollutant adsorption, neutron dispersion, food manufacturing, a stream of drilling fluids, and charged species reactants etc. Convection diffusion process also occurs in the modelling of biological, chemical systems and semiconductors. This equation is called the drift – diffusion equation in semiconductor physics. The formulation of the convection-diffusion-reaction is divided into three stages (Makungu, Haario, and Mahera 2012). (i). The first process is called convection, which is due to material movement from one place to another. (ii). The second phase is called diffusion, which is due to transferring materials from the high concentration region to a low concentration region. (iii). The final process is called reaction and is induced by decay, adsorption, and the reaction of substances with other

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components (Makungu, Haario, and Mahera 2012). This equation takes the general form 𝜕𝑐 𝜕𝑡

= 𝐷𝛻. (𝛻𝑐) − 𝛻. (𝑉𝐶 ) + 𝑅

(6.1)

In the above equation c is the concentration of species for mass transfer or temperature for heat transfer. D denotes the coefficient of diffusions or thermal diffusivity. V is the velocity, and it is a function of space and time. R describes sources or sinks of the quantity. It is a nonlinear term. ∇2 is the Laplacian opreator and ∇c represents the gradient of the concentration. For steady-state conditions the above equation becomes 𝐷𝛻. (𝛻𝐶) + 𝛻. (𝑉𝐶 ) − 𝑅(𝐶) = 0

(6.2)

when the reaction term 𝑅 = 0, and velocity is 𝑉 = 𝑎𝑥 2 , the equation becomes 𝑑2 𝐶 𝑑𝑥 2

𝑑𝐶

+ 𝑎𝑥 2 𝑑𝑥 = 0

(6.3)

The solution of the above equation is

𝐶(𝑥) =

1 𝑎𝑥3 ) 3 3 2/3 3 3

𝑐1 𝑥𝛤( , 3

√𝑎𝑥

+ 𝑐2

(6.4)

The constant c1 and c2 can be obtained from the given boundary conditions.

Non-Linear Convection-Diffusion Problems

153

6.2. ELECTROCHEMICAL CONVECTION DIFFUSION PHENOMENA AT THE ROTATING DISK ELECTRODE The system of convection diffusion equation in EC’ mechanism 𝑘

(𝐴 ± 𝑒 − ⇔ 𝐵,𝐵 + 𝑆 → 𝐴 + 𝑌can be described as follows (Okuonghae 2006): 𝑑2 𝐴

𝑑𝐴

𝑑2 𝐵

𝑑𝐵

−𝐷𝐴 𝑑𝑥 2 − 𝐶𝑥 2 𝑑𝑥 − 𝑘𝑠 𝐵 = 0

(6.5)

−𝐷𝐵 𝑑𝑥 2 − 𝐶𝑥 2 𝑑𝑥 + 𝑘𝑠 𝐵 = 0

(6.6)

The boundary conditions are 𝐴(𝑥 = 0) = 0, 𝐴(𝑥 = 𝐿) = 𝐴∞

(6.7)

𝐵(𝑥 = 0) = 𝐴∞ , 𝐵(𝑥 = 𝐿) = 0

(6.8)

Here it is also assumed that 𝐷𝐴 is not equal to 𝐷𝐵 . Using the following the dimensionless variables (R. Saravanakumar, Pirabaharan, and Rajendran 2019), 𝐴

𝐵

𝐶

1 3

𝐶

1 3

𝑘𝑠

𝐶

−2 3

𝐷

𝑎 = 𝐴 , 𝑏 = 𝐴 , 𝜉 = (𝐷 ) 𝑥, 𝑙 = (𝐷 ) 𝐿, 𝑘 = 𝐷 (𝐷 ) , 𝜇 = 𝐷𝐴 ∞

∞

𝐴

𝐴

𝐴

𝐴

𝐵

(6.9) the Eqns. (6.5) and (6.6) reduces to the following dimensionless form (R. Saravanakumar, Pirabaharan, and Rajendran 2019): 𝑑2 𝑎 𝑑𝜉 2

𝑑𝑎

+ 𝜉 2 𝑑𝜉 + 𝑘𝑏 = 0

(6.10)

154

L. Rajendran, R. Swaminathan and M. Chitra Devi 𝑑2 𝑏 𝑑𝜉 2

𝑑𝑏

+ 𝜇𝜉 2 𝑑𝜉 − 𝑘𝜇𝑏 = 0

(6.11)

The dimensionless boundary conditions are, 𝑎(𝜉 = 0) = 0, 𝑎(𝜉 = 𝑙) = 1

(6.12)

𝑏(𝜉 = 0) = 1, 𝑏(𝜉 = 𝑙) = 0 The dimensionless current in this case is

(6.13)

𝐼𝐸𝐶′ = 𝐽 = −

𝑑𝑏

|

(6.14)

𝑑𝜉 𝜉=0

Concentration of species A and B can be calculated by solving the Eqns. (6.10) and (6.11) using Taylor series method. Consider Taylor’s series for the dimensionless concentration of 𝑎(𝜉) and 𝑏(𝜉). 𝑞

𝑑𝑝 𝑎

𝑎(𝜉) = ∑𝑝=0 (𝑑𝜉𝑝 |

𝑞

𝜉𝑝

𝜉=0

) 𝑝!

𝜉=0

) 𝑝!

𝑑𝑝 𝑏

𝑏(𝜉) = ∑𝑝=0 (𝑑𝜉𝑝 |

(6.15)

𝜉𝑝

(6.16)

From the boundary conditions (Eqn. (6.12)), we get𝑎(0) = 0. Let us consider, 𝑎′(0) = 𝑚 where 𝑚 is constant and let 𝑑𝑝 𝑏

|

𝑑𝜉 𝑝 𝜉=0

𝑑𝑝 𝑎

|

𝑑𝜉 𝑝 𝜉=0

= 𝐴𝑝 ,

= 𝐵𝑝 we get 𝑞

𝜉𝑝

𝑎(𝜉) = ∑𝑝=0 𝐴𝑝 𝑝!

(6.17)

Non-Linear Convection-Diffusion Problems 𝑞

155

𝜉𝑝

𝑏(𝜉) = ∑𝑝=0 𝐵𝑝 𝑝!

(6.18)

where, 𝐴0 = 0, 𝐴1 = 𝑚, 𝐴2 = −𝑘, 𝐴3 = −𝑘𝐵1 , 𝐴4 = −2𝑚 − 𝑘𝐵2 , 𝐴5 = (6 − 𝐵3 )𝑘, 𝐴6 = (12𝐵1 − 𝑘𝐵4 )𝑘, 𝐴7 = 40𝑚 + (20𝐵2 − 𝐵5 )𝑘

(6.19)

𝐵0 = 1, 𝐵1 = 𝑛, 𝐵2 = 𝑚𝑢𝑘, 𝐵3 = 𝜇𝑘𝑛, 𝐵4 = −2𝜇𝑛 + 𝜇 2 𝑘 2 , 𝐵5 = −6𝜇 2 𝑘 + 𝜇 2 𝑘 2 𝑛 𝐵6 = −14𝜇 2 𝑘𝑛 + 𝜇 3 𝑘 3 , 𝐵7 = 40𝜇2𝑛 − 26𝜇 3 𝑘 2 + 𝜇 3 𝑘 3 𝑛 (6.20) Substituting we get the concentration of species A and B as follows: 𝑎(𝜉) = 𝑚𝜉 −

𝑘𝜉 2 2

−

𝑘𝑛𝜉 3 6

−

(2𝑚+𝑘 2 𝜇)𝜉 4 24

+

(6−𝑘𝜇𝑛)𝑘𝜉 5 120

+

(12𝑛+(2𝑛−𝜇2 𝑘 2 ))𝑘𝜉 6

(6.20)

720

𝑏(𝜉) = 1 + 𝑛𝜉 + (14𝜇2 𝑛−𝜇3 𝑘 2 )𝑘𝜉 6 720

𝜇𝑘𝜉 2 2

+

𝜇𝑘𝑛𝜉 3 6

−

(2𝑛−𝜇2 𝑘 2 )𝜇𝜉 4 24

−

(6𝜇2 −𝜇2 𝑘𝑛)𝑘𝜉 5 120

−

(6.21)

where 30𝑙4 𝑘 2 𝜇 + 360𝑘𝑙2 + 120𝑘𝑛𝑙3 −1 𝑚(𝑙, 𝑘, 𝜇) = 60𝑙(𝑙3−12) (−1216 𝑘𝑛 − 2𝑙6 𝜇𝑘𝑛 − 36𝑙5 𝑘 + 6𝑙5 𝑘 2 𝜇𝑛) +𝑙6 𝜇2 𝑘 3 + 720 (6.22)

156

L. Rajendran, R. Swaminathan and M. Chitra Devi

Figure 6.1. Effect of the parameter 𝑘 on concentration profile 𝑎(𝜉) using Eqn. (6.17).

Figure 6.2. Effect of the parameter 𝜇 on concentration profile 𝑎(𝜉) using Eqn. (6.17).

Non-Linear Convection-Diffusion Problems 1 720+𝜇𝑘𝑙(30𝜇𝑘𝑙 3 +360𝑙+𝜇2 𝑘 2 𝑙 5 −36𝜇𝑙 4 )

𝑛(𝑙, 𝑘, 𝜇) = − 2 (60𝜇𝑘𝑙2−30𝑙3𝜇+3𝑙4𝜇2𝑘 2−7𝑙5𝜇2𝑘)𝑙+360𝑙

157 (6.23)

The unknown constant “𝑚 and 𝑛” are obtained using the boundary condition 𝑎(𝜉 = 𝑙) = 1 and 𝑏(𝜉 = 𝑙) = 0. Our analytical expression for the concentration of species A is compared with simulation results (Matlab) in Figures 6.1 and 6.2 for various values of parameters. A satisfactory agreement is noted. The dimensionless current becomes 𝐼𝐸𝐶′ = 𝐽 = −

𝑑𝑏

|

𝑑𝜉 𝜉=0

= −𝑛(𝑙, 𝑘, 𝜇)

(6.24)

6.3. NONLINEAR CONVECTION DIFFUSION EQUATIONS AND CORRESPONDING ANALYTICAL SOLUTIONS IN VARIOUS FIELDS OF CHEMICAL SCIENCES Nonlinear convection – diffusion equations with nonlinear sources are used to describe several physical, mechanical and biological processes and phenomena (Kudryashov and Sinelshchikov 2017). The equation convection-diffusion is a reasonably simple equation which describes flows or describes a system which changes stochastically. Therefore, in certain ways, the same or identical equation occurs similar to space flows. This equation is specifically connected to the Black – Scholes equation, Navier – Stokes equations, and Fokker-Planck equation. Tables 6.1 and 6.2 provide a recent contribution to mathematical modeling of non-steady state and steady state convectiondiffusion equations in the chemical and physical sciences respectively.

(Jansi Rani et al. 2017)

(R Saravanakumar et al. 2018)

1

2

S. Reference No

Hydrodynamic voltammetry

Experimental techniques and enzymatic scheme Heterogeneous electron transfer at an RDE 𝑂+𝑒 ↔ 𝑅 + 𝑣𝑧 𝜕𝑧

𝜕𝑐𝑖

= 𝐷𝑖

𝑣 −1𝛺2 𝑧 3+. ..

𝜕𝑧

)

+ 4

𝜏𝜁

1

{

1 4

1

+ 𝜏−

𝑐𝑖

=𝐷 𝜕𝑧 2

𝜕2 𝑐 𝜕𝑧

𝑧=0 𝜕𝑐

− 𝑎𝑧 2

)

𝜕𝑧 𝑧=0

𝑗(𝜏) = 𝐷 ( )

𝜕𝑐

The current is = 𝑐0(𝑎𝐷 2 )1/3 ( 𝜕𝜁

𝜕𝐶𝑁

𝜁=0

)

where 𝑎 = 0.51023𝜈 −1/2𝛺3/2 The initial and boundary conditions are 𝑐(𝑧, 0) = 𝑐0, 𝑐(∞, 𝑡) = 𝑐0, 𝑐(0, 𝑡) = 0

𝜕𝑡

𝜕𝑐

= −𝐷𝑂 (

)+

𝜁

2𝜏

𝜁3 24

√𝜏 𝜁 2 ( 4

𝜁𝜏

4

+ ]−

4√𝜋

𝜁 2 √𝜏

−𝜁2

𝑗(𝜏) 𝑐0 (𝑎𝐷2 )1/3

=

1 √𝜋𝜏

𝜏

4

+ + 0.0166666𝜏 5/2

𝐶𝑁 =

1

𝑎

𝐷

1 3

, 𝜏 = (𝐷𝑎2 )3 , 𝜁 = ( ) 𝑧

𝜏1/2 +1.3763𝜏+1.8942𝜏3/2

0.5642+0.7765𝜏1/2 +1.0687𝜏+1.4709𝜏3/2

𝑐0

𝑐

Where

𝐽(𝜏) =

The current for all time as a rational approximation:

𝐽(𝜏) =

−

4

𝑘𝜁 3

𝑒 4𝜏 + 𝐶𝑁2(𝜁, 𝜏)

The dimensionless current density for small values of time

2𝜏

𝐶𝑁 (𝜁, 𝜏) = 𝑒𝑟𝑓 ( ) at𝑎 = 0

2𝜏 𝜁

𝜁

𝑘 = 0.8175 × 𝑆𝑐 −1/3 = 0.8175 × ( )

𝐶𝑁 (𝜁, 𝜏) = 𝑒𝑟𝑓 ( ) + 𝑒𝑟𝑓𝑐 ( ) [

6

0.09375

√𝜋

2 /4𝜏)

𝑘𝜏 3 +

𝑒 (−𝜁

−

2

𝑘𝜏𝜁

)

  (𝑖 ≡ O, R), 𝑎 = 0.51023𝜈 −1/2𝛺3/2 ,

𝜕𝑧

𝑐𝑏

−

𝐹𝐴

𝐷

𝜏 = (𝐷𝑎2 )1/2𝑡, 𝜁 = 𝑧 ( )

, 𝜃𝑖 =

0.016666 5/2 𝜏 √𝜋

+[

𝑎 1/3

2√𝜋

𝑘𝜏3/2

0.0107142𝑘 2𝜏 7/2]}

1+𝑒 𝜂 √𝜋𝜏

Where

√𝜋

1

=

𝑣 −1/3

HPM and Padé Approximation

𝜁3 24

𝜃𝑅 (𝜁, 𝜏) = 1 − 𝜃𝑂 (𝜁, 𝜏) Transient current-potential response at an RDE is 𝜕𝜃𝑂 (𝜁, 𝜏) 𝜓(𝜏) = − ( ) 𝜕𝜁 𝜁=0

)(

𝐷

𝜕𝑧 𝑧=0

𝜕𝑐𝑅

𝜁

2√𝜏

) + 𝑒𝑟𝑓𝑐 (

𝜕𝑐𝑂

0′

= −𝐷𝑅 (

𝜁

2√𝜏

𝜃𝑂 (𝜁, 𝜏) = 𝑒𝑟𝑓 (

Expressions for concentration and current

𝑖(𝑡)

𝑅𝑇

𝑧=0

HPM

Analytical techniques

The current response (𝑖(𝑡)) is obtained by

𝜂=

𝐹

)

(𝐸 − 𝐸 )

𝜕𝑐𝑂

where

𝐷𝑂 (

The initial and boundary conditions are: 𝑐𝑂 (𝑧, 0) = 𝑐𝑏 ,  𝑐𝑅 (𝑧, 0) = 0 𝑐𝑂 (∞, 𝑡) = 𝑐𝑏 ,  𝑐𝑅 (∞, 𝑡) = 0 𝑐𝑂 (0, 𝑡) = 𝑒 𝜂 𝑐𝑅 (0, 𝑡)

3

1

𝛺3/2𝑧 2 +

(𝑖 ≡ O, R)

𝜕𝑧 2 −1/2

𝜕2 𝑐𝑖

𝑣𝑧 = −0.51023𝑣

𝜕𝑡

𝜕𝑐𝑖

Nonlinear differential equations with initial and boundary conditions.

Table 6.1. Recent contributions to mathematical modelling of non-steady state convection-diffusion equations

Experimental Techniques and Enzymatic Scheme Heterogeneous electron transfer at an RDE 𝑂+𝑒 ↔ 𝑅

2 (R SaravanaHydrodynamic kumar et al. 2018) Voltammetry

1 (Jansi Rani et al. 2017)

S. Reference No

𝜕𝑧 2

𝜕2 𝑐𝑖

− 𝑣𝑧 𝜕𝑧

𝜕𝑐𝑖

𝑣 −1 𝛺2 𝑧 3

𝜕𝑧

𝑑𝑧 2

𝑧=0

− 𝑎𝑧 2 𝑑𝑧

𝑑𝑐

=0

0′

= −𝐷𝑅 ( 𝜕𝑧

𝜕𝑐𝑅

where 𝑎 = 0.51023𝜈 −1/2 𝛺3/2 The initial and boundary conditions are 𝑐(∞) = 𝑐0 , 𝑐(0) = 0

𝐷

𝑅𝑇 𝑑2 𝑐

𝜂=

𝐹

)

(𝐸 − 𝐸 )

𝜕𝑐𝑂

where

𝐷𝑂 (

) 𝑧=0

The boundary conditions are: 𝑐𝑂 (∞) = 𝑐𝑏 ,  𝑐𝑅 (∞) = 0, 𝑐𝑂 (0) = 𝑒 𝜂 𝑐𝑅 (0)

3

1

HPM

=0 (𝑖 ≡ O, R) HPM

𝑣𝑧 = −0.51023𝑣 −1/2 𝛺3/2 𝑧 2 +

𝐷𝑖

Nonlinear Differential Analytical Equations with Initial and Techniques Boundary Conditions

𝜁3

3 𝜁3

2

3 2

3

𝜁3

3 𝜁3

3

1

3

1

3

𝜁3

3

3 /3)

(0.0242656𝜁 6 − 0.00970645𝜁 3 +

𝛼 32/3

𝜁=0

=

𝑗(𝜏) 𝑐0 (𝑎𝐷2 )1/3

3

= 0.7765

3

1 1+𝑒 𝜂

𝛤( , )+𝛽

1 𝜁3

)

∞

(0.77645 − 0.283037𝑘 −

incomplete gamma function.

Where 𝛼 = 0.7765 and 𝛽 = 1, 𝛤(𝑥, 𝑦) = ∫𝑦 𝑡 𝑥−1 𝑒 −𝑡 𝑑𝑡 is the

𝐽(𝜏) =

The flux is

𝐶𝑁 (𝜁) =

2)

𝜕𝜁

𝜕𝜃𝑂 (𝜁,𝜏)

0.0805585𝑘

𝜓𝑆𝑆 = − (

0.070759𝜁 2 − 0.0291193) 𝜃𝑅 (𝜁, 𝜏) = 1 − 𝜃𝑂 (𝜁, 𝜏) The stationary limiting current for the steady- state voltammetry is:

3

0.098123𝛤 ( + )] + 𝑒 (−𝜁

0.269198𝛤 ( + )] + 𝑘 2 [−0.0387285𝛤 ( + ) +

3

1

𝜃𝑂 (𝜁) = 1 − 0.3733 𝛤 ( + ) − 𝑘 [0.13607𝛤 ( + ) +

Expressions for Concentration and Current or Flux

Table 6.2. Recent contributions to mathematical modelling of steady-state convection-diffusion equations

Nonlinear Differential Analytical Equations with Initial and Techniques Boundary Conditions

Padé 2𝐹(𝜉) + 𝐻 ′ (𝜉) = 0 𝐹 ″ (𝜉) − 𝐻𝐹 ′ (𝜉) − 𝐹 2 (𝜉) + Approximation 𝐺 2 (𝜉) = 0 𝐺 ″ (𝜉) − 𝐻(𝜉)𝐺 ′ (𝜉) − 2𝐹(𝜉)𝐺(𝜉) = 0 𝐻 ″ (𝜉) − 𝐻(𝜉)𝐻′ (𝜉) − 𝑃 ′ (𝜉) =0 The boundary conditions are: 𝐻 = 𝐹 = 𝑃 = 0, 𝐺 = 1 𝑎𝑡 𝜉 = 0, and 𝐹 = 𝐺 = 0, 𝐻 = −𝛼, 𝑃 = 1 𝑎𝑡 𝜉 = ∞ 4 (Rajendran 2006) Mass transfer; Disc 𝑃 𝑧 𝜕𝑐 = 𝛻 2 𝑐 Two-Point Padé 𝑆 𝜕𝑥 electrode; Channel Approximation The boundary conditions are flow 𝜕𝑐 = 0 on 𝑧 = 0, 𝜌 𝜕𝑧 = (𝑥 2 + 𝑦 2 )1/2 > 1 𝑐 = 1 on 𝑧 = 0, 𝜌 < 1 𝑐 = 0 𝑎𝑠 𝑟 = (𝑥 2 + 𝑦 2 + 𝑧 2 )1/2 → ∞

Experimental Techniques and Enzymatic Scheme 3 (Chitra Devi et al. Fluid viscosity, heat 2017) transfer of powerlaw fluid over rotating disk

S. Reference No

1+𝑞1 𝜉+𝑞2 𝜉 2 +𝑞3 𝜉3 +𝑞4 𝜉 4 +𝑞5 𝜉 5

𝑝0 +𝑝1 𝜉+𝑝2 𝜉 2 +𝑝3 𝜉 3 +𝑝4 𝜉4 +𝑝5 𝜉5

1/3

+2.157𝑃𝑆

1/6

2/3

+1.1631623𝑃𝑆 1/3

1/2

+0.53925𝑃𝑆

+0.811246𝑃𝑆 1+0.376096𝑃𝑆

it becomes exact as 𝑃𝑆 → 0𝑎𝑛𝑑𝑃𝑆 → ∞

𝐼=(

4+1.504386𝑃𝑆

1/6

For the disc electrode the current is ) 𝑛𝐹[𝑐]𝐷𝑎

𝐹(𝜉), 𝐺(𝜉) and 𝐻(𝜉)denoted radial, axial, velocity. The Padé approximation coefficients𝑝0 to 𝑝5 and𝑞0 to 𝑞5 are given in Ref.(Chitra Devi et al. 2017) for 𝐹(𝜉), 𝐺(𝜉) 𝑎𝑛𝑑 𝐻(𝜉) respectively.

𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =

Expressions for Concentration and Current or Flux

Table 6.2. (Continued)

Experimental Techniques and Enzymatic Scheme 5 (Visuvasam et al. First-order redox 2018) reaction

S. Reference No

𝑑𝑧

=

𝑑𝑧

𝑑𝑐

= 𝐷𝑓 𝑑𝑧

𝐷𝑝 𝑑𝑐̂

,𝑧 = 0

𝑣) , 𝛾 = 2𝑘𝛺/𝑣. The boundary conditions are, 𝑐 = 1, 𝑧 → ∞ 𝑐̂ = 0, 𝑧 → −∞ 𝑐 = 𝑐̂ , 𝑧 = 0

3 2

where 𝛼 = 0.51, 𝛽 = 2𝑘ℎ(𝛺/

(−𝛾𝑍 − 𝛽)

HPM

= 𝛺

𝑘𝑟

3

9𝜈

=

4𝑎2

+

2𝑎

𝑚

, 𝑙0 =

.

]

ℎ

[

𝑎

𝑣 1/2 𝜆 𝑘1 +𝑙0 𝛺

=( )

𝐼 𝐼𝑀

−

𝑙1 𝑎2

+

2𝑎4

𝑏

𝐷

𝐾

𝜆

𝑣 −1/2 ℎ 𝛺

], 𝐼 = 𝐼𝑅 ( )

2.6789𝐷𝑎+3

3𝛾

𝑎

𝑘1 +𝑙0

=[

=

−

𝐷

𝑙1

,

𝑎2

𝐺𝑘1 𝑎

𝛼𝜆

,𝑏 =−

,𝐺 =

, 𝑎 = √ , 𝑙2 =

3+2.6789𝐷𝑎

3

𝐷𝑝 𝐷𝑓

𝐷

𝐼√ =

𝐾

𝐺𝐷 𝐺 𝐵 +0.8930𝐵−0.75𝐾+0.5 √𝐾𝐷 √𝐾𝐷 2

(1+0.8930√𝐾𝐷)

1+0.8930√𝐾𝐷+0.4514𝐵√𝐾𝐷−0.8930

The above equation can be re-written in terms of the dimensionless parameters B, G, D and K:

2𝑎4

𝑏

𝐼𝑅 =

, 𝑘1 =

1 6

) ,𝐷 =

3

2

−3𝑙2 −2.6789𝐷𝑙1

3

𝐷𝑘1 𝑎

9𝛺9

𝜈5 𝛼2 𝐷𝑓4

= 2𝑘ℎ (

𝑦2 ]

The expressions for the current:

𝐷

𝑏

2.6789

𝑙0 +𝑙2

1 3

3𝛽 𝛼𝜆2

4𝑎

𝑏

) , 𝑚1 = −

𝑚 = − 𝑘1 𝑎, 𝑙1 = −

𝐵

1

) ,𝐵=

𝛼 2 𝐷𝑓

(

−𝐷(𝑙1 +𝑙𝑜 𝑎)

𝛼𝜆𝛺

3𝑘𝑟

),

1 3

𝛼𝜈

3𝐷𝑓 3

1.3541𝐵𝑚1 ,

𝑘2 =

𝐾=

9

𝛼 2 𝜈 2 𝐷𝑓

2𝑘𝛺 (

𝜆

y = ,𝜆 = (

𝑧

where

𝑐̂ (𝑦) = 𝑒 𝑎𝑦 [𝑘1 + 𝑙0 + 𝑙1 𝑦 +

3

𝑐(𝑦) = 1 + (𝑚1 + 𝑘2 )𝛤 ( , 𝑦 3 ) − 𝐵𝑚1 𝛤 ( , 𝑦 3 )

1

𝑣 𝑑𝑧 2 𝐷𝑝 𝑑2 𝑐̂ 𝑘 − 𝑟 𝑐̂ 𝑣 𝑑𝑧 2 𝛺

=

𝐷𝑓 𝑑 2 𝑐

𝑑𝑐

𝑑𝑧 𝑑𝑐̂

(−𝛼𝑍 2 − 𝛽)

Expressions for Concentration and Current or Flux

Nonlinear Differential Analytical Equations with Initial and Techniques Boundary Conditions

+

6 (Rajendran Saravanakumar et al. 2020)

S. Reference No

𝜂 𝑑𝑥

𝜇

The boundary conditions 𝜇 𝑧𝑃 = 𝑎𝑡 𝑥 = 0 𝜇 𝑏 𝑧𝑅 𝜇 = 1 𝑎𝑡 𝑥 → ∞ 𝑏

𝐷𝑖

Experimental Nonlinear Differential Analytical Techniques Equations with Initial and Techniques and Enzymatic Boundary Conditions Scheme 𝑑2 𝜇 𝑧𝑖 Steady-state HPM + 𝑖 [𝑐𝑖 (𝜇 − 𝑑𝑥 2 2𝐹𝐷𝑖 voltammetry 𝑧𝑗 𝑧𝑘 𝑏 1 𝑑𝜇 1 𝑑𝜇 𝑅 − (𝑧𝑃 − 𝑧𝑅 )𝑒 − → 𝑧𝑅 𝑧𝐶 𝜇 ) 𝜇2 𝑑𝑥 + 𝜇 𝑑𝑥 ] + 𝑃 𝐾 𝜔3 𝑑 𝑑𝜇 =0 √ 𝐾

1⁄ 3

√𝜂 )

𝑑

,𝜌 = 𝜇

= 𝑏

𝜇

,𝜆 = )𝑐 𝑏 𝑧𝑅 (𝑧𝑅 −𝑧𝐶 𝑅

𝜇

2𝐹𝜇𝑏 √𝜔

1

𝑑𝜌

|

𝑧𝐶 −𝑧𝑅 𝑑𝜒 𝜒=0

=

1.1198(𝑧𝑅−𝑧𝑃 ) 𝑧𝑅 (𝑧𝐶 −𝑧𝑅 )+0.5661𝑧𝑃 (𝑧𝑅 −𝑧𝑃 )

𝑖𝑙𝑒𝑣𝑖𝑐ℎ

𝑖

=[

1 0.5661𝑧𝑃 (𝑧𝑅 −𝑧𝑃 ) 𝑧𝑅 (𝑧𝐶 −𝑧𝑅 )

1+

]

The ratio of two steady state currents is

𝜆=

Normalized current density .

𝑖

3

𝑖

𝜂

1⁄ 3

√ ) 𝐾𝐷2 𝑑

(

𝑧𝑝 1 ( − 1) [𝑢1 (0)𝑢1 (𝜒) (𝑢1 (0))2 𝑧𝑅 + 𝜆𝑧𝑝 {𝑢1 (0)𝑢2 (𝜒) − 𝑢2 (0)𝑢1 (𝜒)}],

1  u1 (  )    ,  3 , 𝑢2 (𝜒) = 𝛤 (23 , 𝜒 3), 3  1 u1 (0)    , 𝑢2 (0) = 𝛤 (23).  3

3𝐷𝑖

𝜒 = 𝑥(

where

𝜌(𝜒) = 1 +

Expressions for Concentration and Current or Flux

Table 6.2. (Continued)

,

7 (R. Saravanakumar, Pirabaharan, and Rajendran 2019)

S. Reference No

Experimental Techniques and Enzymatic Scheme E reaction, 𝐴 + 𝑒− → 𝐵 𝑑𝑥 2

𝑑2 𝐴

+ 𝑣𝑥 𝑑𝑥

𝑑𝐴

= 0, 0
0.

, 𝑠𝑒 (𝑙𝑒 , 𝑡) = 𝑠𝑚 (𝑙𝑒 , 𝑡), 𝑡 > 0, 𝜕𝑥 𝑥=𝑙𝑒

𝜕𝑝𝑚

𝜕𝑥 𝑥=𝑙𝑒

𝜕𝑠𝑚

= 𝑑𝑝𝑚

= 𝑑𝑠𝑚 𝜕𝑥 𝑥=𝑙𝑒

𝜕𝑝𝑒

𝜕𝑥 𝑥=𝑙𝑒

𝜕𝑠𝑒

𝑑𝑝𝑒

𝑑𝑠𝑒

𝑠𝑚 (𝑙𝑚 + 𝑙𝑒 , 𝑡) = 𝑠0 , 𝑝𝑚 (𝑙𝑚 + 𝑙𝑒 , 𝑡) = 0, 𝑡 > 0,

𝑑𝑠𝑒

Initial condition: when 𝑡 = 0 𝑠𝑒 (𝑥, 0) = 0, 𝑝𝑒 (𝑥, 0) = 0, 𝑥 ∈ [0, 𝑙𝑒 ], 𝑠𝑚 (𝑥 = 0) = 0, 𝑝𝑚 (𝑥 = 0) = 0, 𝑥 ∈ [𝑙𝑚 + 𝑙𝑒 ] 𝑠𝑚 (𝑥 = 𝑙𝑚 + 𝑙𝑒 ) = 𝑠0 , 𝑝𝑚 (𝑥 = 𝑙𝑚 + 𝑙𝑒 ) = 0. Boundary condition

𝜕𝑡

𝜕𝑡 𝜕𝑝𝑚

𝜕𝑠𝑚

Experimental Numerical techniques Differential equations and boundary conditions techniques and enzymatic scheme Amperometric biosensor Non-uniform Discrete 𝜕𝑠𝑒 = 𝑑 𝜕2 𝑠𝑒 − 𝑣𝑚𝑎𝑥𝑠𝑒 𝑠𝑒 𝜕𝑥 2 𝜕𝑡 𝑘𝑀 +𝑠𝑒 𝑘1 Grid 𝑆 + 𝐸 ↔ 𝐸𝑆 → 𝑃 + 𝐸 𝜕𝑝𝑒 𝜕2 𝑝 𝑣 𝑠 𝑘−1 = 𝑑𝑝𝑒 2𝑒 − 𝑚𝑎𝑥 𝑒 , 𝑥 ∈ (0, 𝑙𝑒 )

Table 7.1. Recent numerical method contribution to nonlinear reaction-diffusion equations.

Reference

(Romas Baronas, Kulys, and Ivanauskas 2004)

(Lesław K. Bieniasz 2004)

(Romas Baronas et al. 2013)

S. no

2

3

4

⇔ 𝐸𝑆𝑖 → 𝐸 + 𝑃𝑖 where 𝑖 = 1, . . . , 𝑘

𝑘2 𝑖

Amperometric 𝐸 + 𝑆𝑖 𝑘1𝑖

Chronoamperometric Finite Difference Method

Five-Point Method

𝜕𝑡

𝜕𝑥

𝐾𝑀 +𝑃

|

𝜕𝑥 𝑥=0

𝜕𝑃

= 𝐷𝑃𝑖

= 𝐷𝑠𝑖

,

𝜕𝑥 2

𝑉𝑖𝑚𝑎𝑥

𝜕𝑡

𝑉𝑖𝑚𝑎𝑥 𝐾𝑀 𝑖 (1+∑𝑘 𝑗=1𝑆𝑗 ⁄𝐾𝑀 𝑖 )

,

𝜕𝑦

𝜕𝑐(𝑦,𝑡) 𝜕𝑐(𝑦,𝑡)

𝐾𝑀 𝑖 (1+∑𝑘 𝑗=1𝑆𝑗 ⁄𝐾𝑀 𝑖 )

−

−

𝜕 2 𝑃𝑖

𝜕𝑥 2

𝜕2 𝑆𝑖

𝜕𝑥 2

𝜕2 𝑐(𝑦,𝑡)

|

|

𝜕𝑥 𝑥=0

𝜕𝑆𝑖

= 0, 𝑖 = 1, … , 𝑘, 𝑆𝑖 (𝑑, 𝑡) = 𝑆0 𝑖 , 𝑃𝑖 (𝑑, 𝑡) = 0, 𝑖 = 1, … , 𝑘.

𝐷𝑠𝑖

Boundary condition: when 𝑡 > 0 𝑃𝑖 (0, 𝑡) = 0, 𝑖 = 1, … , 𝑘,

Initial condition: when 𝑡 = 0 𝑆𝑖 (𝑥, 0) = 0, 𝑃𝑖 (𝑥, 0) = 0, 𝑥 ∈ [0, 𝑑), 𝑆𝑖 (𝑑, 0) = 𝑆0 𝑖 , 𝑃𝑖 (𝑑, 0) = 0, 𝑖 = 1, … , 𝑘

𝜕𝑡

𝜕𝑃𝑖

𝜕𝑡

𝜕𝑆𝑖

𝐷(𝑡)

𝜕𝑆 𝜕𝑥 𝑥=0

= 𝐹 (𝑦, 𝑡, 𝑐(𝑦, 𝑡),

= −𝐷𝑠

𝑃(𝑑, 𝑡) = 0.

𝐷𝑃

Initial condition: when 𝑡 = 0 𝑆(𝑥, 0) = 0, 𝑆(𝑑, 0) = 𝑆0 , 𝑥 ∈ [0, 𝑑), 𝑃(𝑥, 0) = 0, 𝑥 ∈ [0, 𝑑] Boundary condition: when 𝑡 > 0 𝑆(𝑥, 𝑡) = 0, 𝑆(𝑑, 𝑡) = 𝑆0 , 𝑥 ∈ [0, 𝑑),

); 𝑐(𝑥, 0) = 𝑐 0

Experimental Numerical techniques Differential equations and boundary conditions techniques and enzymatic scheme 𝜕𝑆 𝜕2 𝑆 𝑉 Amperometric biosensor Finite Difference = 𝐷𝑠 2 + 𝑚𝑎𝑥 𝜕𝑡 𝜕𝑥 𝐾𝑀 +𝑃 𝐸 Method 𝑆→𝑃→𝑆 𝜕𝑃 𝜕2 𝑃 𝑉 = 𝐷𝑃 2 − 𝑚𝑎𝑥

Reference

(D Britz et al. 2009)

(L K Bieniasz 2005)

(Tamaki, Ito, and Yamaguchi 2009)

S. no

5

6

7 𝑘2

𝑘−1

𝑘3

𝐸red + 2MOx → 2MRed + 𝐸OX

+𝑃

𝐸Ox + 𝑆 ↔ 𝐸𝑆 → 𝐸Red

𝑘1

Ping-pong mechanism

Chronoamperometric

Finite Difference Method

Singularity Correction Method.

𝜕𝑠 𝜕𝑋

= 0, 𝑝 = 0,

0

= 𝐷[ 𝜕𝑦 2

𝜕2 (𝑦,𝑡)

+ 𝑣(𝑦)

𝜕𝑦

] + 𝐹(𝑦, 𝑡, 𝑐(𝑦, 𝑡))

𝜕𝑐(𝑦,𝑡)

= 𝐷𝑀

𝜕𝑥 2

𝜕2 𝐶𝑀𝑅(𝑥,𝑡)

−2

Kcat 𝐶𝐸𝑛𝑧 1+𝐾𝑆 ⁄𝐶𝑆 +𝐾𝑀 ⁄𝐶𝑀

𝑥 = 𝑙, 𝐶𝑆 = 𝐶𝑆𝑏 ;

𝜕𝑥

𝜕𝐶𝑀𝑅

= 0.

The initial and boundary conditions are: 𝑡 = 0, 𝐶𝑆 = 0; 𝐶𝑀𝑅 = 𝐶𝑀𝑇 − 𝑀0∗ ,

𝜕𝑡

𝜕𝐶𝑀𝑅 (𝑥,𝑡)

𝑐(𝑥, 0) = 𝑐 , 𝑐(∞, 𝑡) = 𝑐 𝑏 (𝑡), 𝑐 0 𝑓𝑜𝑟 𝑡 = 0 𝑐(0, 𝑡) = { 1 . 𝑐 𝑓𝑜𝑟 𝑡 = 0 2 𝜕𝐶𝑆 (𝑥, 𝑡) 𝜕 𝐶𝑆 (𝑥, 𝑡) K cat 𝐶𝐸𝑛𝑧 = 𝐷𝑆 − 𝜕𝑡 𝜕𝑥 2 1 + 𝐾𝑆 ⁄𝐶𝑆 + 𝐾𝑀 ⁄𝐶𝑀

𝜕𝑥 2

𝜕2 𝑐(𝑦,𝑡)

𝑡 ≥ 0, 𝑥 = 𝑑: 𝑠 = 𝑠0 , 𝑝 = 0.

𝑡 > 0, 𝑥 = 0:

Experimental Numerical techniques Differential equations and boundary conditions techniques and enzymatic scheme 𝜕𝑠 𝜕2 𝑠 𝑠 Amperometric Finite Difference = 𝐷𝑠 2 − 𝑉𝑚 𝜕𝑡 𝜕𝑥 𝐾𝑚 +𝑠 Method 𝐸 + 𝑆 ↔ 𝐸𝑆 → 𝐸 + 𝑃 𝜕𝑝 𝜕2𝑝 𝑠 = 𝐷𝑠 2 + 𝑉𝑚 𝜕𝑡 𝜕𝑥 𝐾𝑚 + 𝑠 The initial and boundary conditions are: 𝑡 = 0,0 ≤ 𝑥 < 𝑑: 𝑠 = 0, 𝑝 = 0,

Table 7.1. (Continued)

Reference

(Ašeris, Baronas, and Kulys 2012)

(Gooding and Hall 1996)

S. no

8

9

2 1 𝐸2

𝑃2

𝑘2

𝑘3

𝑘−1

𝐸red + 𝑀𝑒𝑑ox → 𝐸OX + 𝑀𝑒𝑑red

𝑘4

𝐸red + 𝑂2 → 𝐸OX + 𝐻2 𝑜2

+𝑃

𝐸Ox + 𝑆 ↔ 𝐸𝑆 → 𝐸Red

𝑘1

𝑆1 + 𝑆2 →

1

Shooting Method

𝜕𝑥 2

𝜕𝑥 2 𝜕2 𝑃2,𝑒

+

+

=0

𝑘21 𝑆1,𝑒+𝑘22 𝑆2,𝑒

2 𝑘21 𝑘22 𝑒2 𝑆1,𝑒𝑆2,𝑒

|

𝜕𝑥 𝑥=0

𝜕𝑐𝑒

= 0, 𝑃1,𝑒 (𝑥 = 0) = 0, 𝑐 = 𝑆1 , 𝑆2 , 𝑃2 ,

=0

𝑖𝑠 𝑑 𝑛𝑒 𝐹𝐷𝑃1 ,𝑒𝐴𝑆10

=

𝑑𝑦

𝑑𝑃1𝑁

𝑦=0

|

dy2

dy2

=

𝑑𝑦

𝑑[𝑆]

=

𝑑𝑦

𝑑[𝑀𝑒𝑑𝑟𝑒𝑑 ]

=0 [MedOX ] = [MedOX]𝑏 = 𝐾𝑂 [MedOX ]∞ , [𝑆] = [𝑆]𝑏 = 𝐾𝑆 [𝑆]∞ , [Medred ] = 0.

At the electrode, y = d

𝑑𝑦

𝑑[𝑀𝑒𝑑𝑂𝑋 ]

At the far wall, y = 0

𝛽𝑆 [𝑆]

= − 𝑘4[𝐸red ][MedOX ] = − 𝑘2 [𝐸𝑇 ] ( 𝛽𝑆 = (𝑘−1 + 𝑘2 )/𝑘1 and 𝛽𝑂 = 𝑘2 /𝑘4.

where

𝐷𝑀

𝑑2 [Medred ]

[MedOX ]

−1

𝑘−1 𝑘1

+ 1)

−1

) = 𝑘2 [𝐸𝑇 ] (

[MedOX ]

𝛽𝑂

𝑘−1 +𝑘2

+

= 𝑘1[𝐸OX ][𝑆] − 𝑘−1[ES] = [𝐸OX ][𝑆] (𝑘1 −

+ 1)

𝑑2 [𝑆]

[𝑆]

𝛽𝑆

+

−1 𝑑 2 [MedOX ] 𝑘2 𝑘1 𝛽𝑆 𝛽𝑂 = 𝑘4 [𝐸red ][MedOX ] = [𝐸OX ][𝑆] = 𝑘2 [𝐸𝑇 ] ( + + 1) dy2 𝑘−1 + 𝑘2 [𝑆] [MedOX ]

𝛽𝑂

𝐷𝑆

𝐷𝑀

𝜓=

𝑆𝑖,𝑒 (𝑥 = 𝑑𝑒 + 𝑑𝑑 ) = 𝑆𝑖0 , 𝑃𝑖,𝑒 (𝑥 = 𝑑𝑒 + 𝑑𝑑 ) = 0, 𝑖 = 1,2. Current density

𝐷𝐶𝑒

Boundary conditions:

𝐷𝑃2,𝑒

𝐷𝑃1,𝑒

𝑘21 𝑆1,𝑒 +𝑘22 𝑆2,𝑒

𝑘1 𝑒1 𝑆1,𝑒

𝜕𝑥

𝜕2 𝑃1,𝑒

Experimental Numerical techniques Differential equations and boundary conditions techniques and enzymatic scheme 𝜕2 𝑆 𝑘 𝑘 𝑒 𝑆 𝑆 Amperometric parallel Finite Difference 𝐷𝑆1,𝑒 1,𝑒 − 𝑘1 𝑒1 𝑆1,𝑒 − 21 22 2 1,𝑒 2,𝑒 = 0 𝜕𝑥 2 𝑘21 𝑆1,𝑒 +𝑘22 𝑆2,𝑒 substrates conversion Method 𝜕2 𝑆2,𝑒 𝑘21 𝑘22 𝑒2 𝑆1,𝑒𝑆2,𝑒 𝐸1 1 𝐷 − =0 𝑆2,𝑒 2 𝑆 → 𝑃

Reference

(R Baronas, Ivanauskas, and Kulys 2002)

S. no

10 𝜕𝑡

𝜕𝑥

𝐾𝑀 +𝑆

Initial condition: when 𝑡 = 0 𝑆(𝑥, 0) = 0, 𝑆(𝑑, 0) = 𝑆0 , 𝑥 ∈ [0, 𝑑), 𝑃(𝑥, 0) = 0, 𝑥 ∈ [0, 𝑑] Boundary condition: when 𝑡 > 0 0, 𝑡 > 𝑇𝐹 𝜕𝑆 𝑆(𝑑, 𝑡) = { , | = 0, 𝑆0 , 𝑡 ≤ 𝑇𝐹 𝜕𝑥 𝑥=0 𝑃(0, 𝑡) = 0, 𝑃(𝑑, 𝑡) = 0

Experimental Numerical techniques Differential equations and boundary conditions techniques and enzymatic scheme 𝜕𝑆 𝜕2 𝑆 𝑉 Amperometric Finite Difference = 𝐷𝑠 2 − 𝑚𝑎𝑥 𝜕𝑡 𝜕𝑥 𝐾𝑀 +𝑆 𝐸 Method 𝑆→𝑃 𝜕𝑃 𝜕2 𝑃 𝑉 = 𝐷𝑃 2 + 𝑚𝑎𝑥

Table 7.1. (Continued)

Numerical Methods

177

efficiently, and quickly. We can learn a wide variety of techniques to solve nonlinear PDEs by simplifying and transforming the equations and solutions, arbitrary functions, and parameters. Relationships are also provided between the different types of solutions, various methods, and approaches. The results obtained in Maple, Matlab, and Mathematica allow for a deeper understanding of the subject.

7.5. ANALYTICAL AND NUMERICAL (MATLAB) SOLUTIONS OF THE COUPLED REACTION AND DIFFUSION EQUATIONS WITHIN POLYMER-MODIFIED ULTRAMICROELECTRODES Consider the coupled reaction/diffusion equations in polymermodified ultramicroelectrodes (Rebouillat, Lyons, and Flynn 1999) 𝜕𝑢 𝜕𝑡 𝜕𝑣 𝜕𝑡

𝜕2 𝑢

2 𝜕𝑢

𝜕2 𝑣

2 𝜕𝑣

= 𝜕𝜌2 + 𝜌 𝜕𝜌 − 𝛾𝐸 𝑢𝑣 = 𝜕𝜌2 + 𝜌 𝜕𝜌 − 𝛾𝑆 𝑢𝑣

(7.1)

(7.2)

The initial and boundary conditions are 𝑡 = 0; 𝑢 = 0; 𝑣 = 0

(7.3)

𝜌 = 0;

𝑑𝑢 𝑑𝜌

= 0; 𝑣 = 1

(7.4)

𝜌 = 1;

𝑑𝑣 𝑑𝜌

= 0; 𝑢 = 1

(7.5)

where 𝑢 is the concentrations of oxidized mediator and 𝑣 is the substrate. We aim to solve the above equation for 𝛾𝐸 = 𝛾𝑆 = 𝛾 using

178

L. Rajendran, R. Swaminathan and M. Chitra Devi

new homotopy perturbation method (NHPM). The nonlinear reactiondiffusion equations in the planar plane are 𝜕𝑢 𝜕𝑡 𝜕𝑣 𝜕𝑡

=

=

𝜕2 𝑢 𝜕𝜌2 𝜕2 𝑣 𝜕𝜌2

− 𝛾𝑢𝑣

(7.6)

− 𝛾𝑆 𝑢𝑣

(7.7)

The boundary condition is the same as defined in Eqns.(7.3)-(7.5). To find the solution of Eqns.(7.6)and(7.7) , the homotopy is constructed as follows: (1 − 𝑝) [

𝜕2 𝑢 𝜕𝜌2

− 𝛾𝑢𝑣𝜌=0 − 𝜕𝑡 ] + 𝑝 [𝜕𝜌2 − 𝛾𝑢𝑣 − 𝜕𝑡 ] = 0

𝜕2 𝑢

(1 − 𝑝) [

𝜕2 𝑣 𝜕𝜌2

− 𝛾𝑢𝜌=1 𝑣 −

𝜕𝑢

𝜕𝑣 ] 𝜕𝑡

+ 𝑝[

𝜕2 𝑣 𝜕𝜌2

𝜕𝑢

− 𝛾𝑢𝑣 −

𝜕𝑣 ] 𝜕𝑡

=0

(7.8)

(7.9)

on simplification we get (1 − 𝑝) [

𝜕2 𝑢 𝜕𝜌2

− 𝛾𝑢 −

𝜕𝑢 ]+ 𝜕𝑡

(1 − 𝑝) [

𝜕2 𝑣 𝜕𝜌2

− 𝛾𝑣 −

𝜕𝑣 ] 𝜕𝑡

𝜕2 𝑢 𝜕𝜌2

− 𝛾𝑢𝑣 −

𝜕𝑢 ] 𝜕𝑡

=0

(7.10)

𝜕2 𝑣 𝜕𝜌2

− 𝛾𝑢𝑣 −

𝜕𝑣 ] 𝜕𝑡

=0

(7.11)

𝑝[

+𝑝[

The approximate solution of Eqns. (7.6) and (7.7) are 𝑢 = 𝑢0 + 𝑝 𝑢1 + 𝑝2 𝑢2 +. . ..

(7.12)

𝑣 = 𝑣0 + 𝑝 𝑣1 + 𝑝2 𝑢2 +. . ..

(7.13)

Substituting Eqn. (7.12) in Eqn. (7.10) and Eqn.(7.13) in (7.11), then comparing the coefficients of like powers of p yields:

Numerical Methods 𝜕 2 𝑢0

− 𝛾𝑢0 −

𝜕𝑢0

− 𝛾𝑢1 −

𝜕𝑢1

𝑝0 : 𝜕𝜌20 − 𝛾𝑣0 −

𝜕𝑣0

𝑝0 : 𝑝1 :

𝜕𝜌2 𝜕 2 𝑢1 𝜕𝜌2 𝜕2 𝑣

𝜕2 𝑣

𝑝1 : 𝜕𝜌21 − 𝛾𝑣1 −

𝜕𝑡

𝜕𝑡

𝜕𝑡

𝜕𝑣1 𝜕𝑡

179

=0

(7.14)

+ 𝛾𝑢0 (1 − 𝑣0 ) = 0

(7.15)

=0

(7.16)

+ 𝛾𝑣0 (1 − 𝑢0 ) = 0

(7.17)

we are subjecting Eqns. (7.14)-(7.17) to Laplace transformation to t we have, 𝑝0 : 𝑝1 :

𝜕 2 𝑢0 𝜕𝜌2 𝜕 2 𝑢1 𝜕𝜌2

− 𝛾𝑢0 − 𝑠𝑢0 = 0

(7.18)

− 𝛾𝑢1 − 𝑠𝑢1 + 𝛾𝑢0 (1 − 𝑣0 ) = 0

(7.19)

𝜕2 𝑣

𝑝0 : 𝜕𝜌20 − 𝛾𝑣0 − 𝑠𝑣0 = 0 𝜕2 𝑣

𝑝1 : 𝜕𝜌21 − (𝛾 + 𝑠)𝑣1 + 𝛾𝑣0 (1 − 𝑢0 ) = 0

(7.20)

(7.21)

The transformation of boundary conditions is 𝜌 = 0;

𝑑𝑢0

𝜌 = 1;

𝑑𝑣0

𝑑𝜌

𝑑𝜌

1 𝑑𝑢

= 0; 𝑣0 = 𝑠 ; 𝑑𝜌𝑖 = 0; 𝑣𝑖 = 0; 𝑖 = 1,2,3 … 1 𝑑𝑣

= 0; 𝑢0 = 𝑠 ; 𝑑𝜌𝑖 = 0; 𝑢𝑖 = 0; 𝑖 = 1,2,3 …

(7.22)

(7.23)

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L. Rajendran, R. Swaminathan and M. Chitra Devi

where 𝑠 is the Laplace variable and an over bar indicates a Laplacetransformed quantity. Solving the Eqns. (7.18)-(7.21), and using the boundary conditions (7.22) and (7.23) we can find the following results. 𝑐𝑜𝑠ℎ( 𝛾+𝑠𝜌)

𝑢0 = 𝑠 𝑐𝑜𝑠ℎ(√

(7.24)

√𝛾+𝑠)

𝑣0 =

𝑐𝑜𝑠ℎ(√𝛾+𝑠𝜌)

𝑠𝑖𝑛ℎ( 𝛾+𝑠𝜌)

+ 𝑠 𝑐𝑜𝑡ℎ(√

𝑠

(7.25)

√𝛾+𝑠)

According to the HPM, we can conclude that 𝑢 (𝜌,s) = lim 𝑢 (𝜌,s) = 𝑢0 + 𝑢1 + 𝑢2 …

(7.26)

𝑣 (𝜌,s) = lim 𝑣 (𝜌,s) = 𝑣0 + 𝑣1 + 𝑣2 …

(7.27)

𝑝→1

𝑝→1

Considering only the first iteration result and putting Eqns. (7.24) into Eqns. (7.26) and Eqns. (7.25) into Eqns. (7.27). 𝑐𝑜𝑠ℎ( 𝛾+𝑠𝜌)

𝑢 (𝜌,s) = 𝑠 𝑐𝑜𝑠ℎ(√

(7.28)

√𝛾+𝑠)

𝑣 (𝜌,s) =

𝑐𝑜𝑠ℎ(√𝛾+𝑠𝜌) 𝑠

𝑠𝑖𝑛ℎ( 𝛾+𝑠𝜌)

+ 𝑠 𝑐𝑜𝑡ℎ(√

(7.29)

√𝛾+𝑠)

Using residues theorem, we can take the inverse Laplace transform we get 𝑢(𝑥, 𝑡) = 𝑢𝑠𝑠 − 𝑢 𝑇𝑅 =

𝑐𝑜𝑠ℎ(√𝛾𝜌) − 𝑐𝑜𝑠ℎ(√𝛾)

(𝑛+0.5) 𝑐𝑜𝑠ℎ((𝑛+0.5)𝜋𝜌) ∞ 𝑛 (𝑛+0.5)2 𝜋2 −√𝛾 2𝜋 ∑𝑛=0(−1) { 2 2

}

× 𝑒𝑥𝑝((−(𝑛 + 0.5) 𝜋 − √𝛾)𝑡)

Similarly using the same procedure we can find 𝑣(𝑥, 𝑡).

(7.30)

Numerical Methods

181

7.6. NUMERICAL SIMULATION Numerical simulations are required to study the behavior of systems and to check the accuracy of the obtained approximate analytical solution (Eqn. (7.30)). Thus the differential equations (Eqns. (7.6) and (7.7)) are run on a computer following a program known as bvp4c (Program-2), which is a finite-difference code that implements the three-stage Lobatto IIIA formula included in the influential MATLAB software package. Figures 7.1-7.3 show a comparison between the approximate analytical results and numerical data for the concentration oxidized mediator for various values of parameters. A satisfactory agreement is noted. Programme 1: Matlab program to find the sum of the series of Eqn. (7.30) function u=u(x) t=1; x=linspace(0,1); % x denotes rho ust0=0; for n=0:1:1000; gamma=1; ust0=ust0+((1)^(n)*(n+1/2)*cos((n+1/2)*pi*x))/(((n+1/2)^2*pi^2+sqrt(gamma)))*e xp((-(n+1/2)^2*pi^2-sqrt(gamma))*t); end uss=cosh(sqrt(gamma)*x)./(cosh(sqrt(gamma))); ust=((2*pi)*ust0); u=uss-ust; plot(x,u,’b’) ----------------------------------------------------------------------------------

182

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Programme 2: Matlab program to find the numerical simulation of Eqns.(7.6)-(7.7) function pdex4 m = 0; x = [0:0.1:1] t = linspace(0,1); sol = pdepe(m,@pdex4pde,@pdex4ic,@pdex4bc,x, t); u1 = sol(:, :, 1); u2 = sol(:, :, 2); figure plot(x,u1(end,:),’:bo’) ylabel(‘Normalized concentration u(x)’); xlabel(‘Normalized distance x’); figure plot(x,u2(end,:),’:bo’) ylabel(‘Normalized concentration v(x)’); xlabel(‘Normalized distance x’); function [c,f,s] = pdex4pde(x,t,u,DuDx) c = [1;1]; f = DuDx; gamma = 1; G=-gamma*u(1)*u(2); H=-gamma*u(1)*u(2); s = [G;H]; function u0 = pdex4ic(x) u0 = [0; 0]; function [pl,ql,pr,qr] = pdex4bc(xl,ul,xr,ur,t) pl = [0; ul(2)-1]; ql = [1;0];

Numerical Methods

183

pr = [ur(1)-1;0]; qr = [0;1];

Figure 7.1. Normalized concentration oxidized mediator 𝑢(𝜌) calculated using Eqn. (7.30). Profiles are obtainable for fixed value of the parameters 𝑡 = 1 and various values of the normalized time parameter 𝛾.

Figure 7.2. Normalized concentration oxidized mediator u(ρ) calculated using Eqn. (7.30). Profiles are obtainable for fixed value of the parameters t = 2 and various values of the normalized time parameter γ.

184

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Figure 7.3. Normalized concentration oxidized mediator u(𝜌) calculated using Eqn. (7.30). Profiles are obtainable for fixed value of the parameters𝛾 and various values of the normalized time parameter 𝑡.

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Numerical Methods

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Britz, D, R Baronas, E. Gaidamauskaite, and F. Ivanauskas. 2009. “Further Comparisons of Finite Difference Schemes for Computational Modelling of Biosensors.” Nonlinear Analysis: Modelling and Control 14 (4): 419–33. https://doi.org/10.15388/ na.2009.14.4.14467. Britz, Dieter, and Jörg Strutwolf. 2016. Digital Simulation in Electrochemistry. Digital Simulation in Electrochemistry. Monographs in Electrochemistry. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-30292-8. Gooding, J J, and E. A. H. Hall. 1996. “Practical and Theoretical Evaluation of an Alternative Geometry Enzyme Electrode.” Journal of Electroanalytical Chemistry 417 (1–2): 25–33. https://doi.org/10. 1016/S0022-0728(96)04752-3. Li, Dongfang, Chengjian Zhang, and Jinming Wen. 2015. “A Note on Compact Finite Difference Method for Reaction–Diffusion Equations with Delay.” Applied Mathematical Modelling 39 (5–6): 1749–54. https://doi.org/10.1016/j.apm.2014.09.028. Liu, Yuan, Yingda Cheng, Shanqin Chen, and Yong-Tao Zhang. 2019. “Krylov Implicit Integration Factor Discontinuous Galerkin Methods on Sparse Grids for High Dimensional Reaction-Diffusion Equations.” Journal of Computational Physics 388: 90–102. https://doi.org/https://doi.org/10.1016/j.jcp.2019.03.021. Michael, E. P. E., J. Dorning, and Rizwan-Uddin. 2001. “Studies on Nodal Integral Methods for the Convection-Diffusion Equation.” Nuclear Science and Engineering 137 (3): 380–99. https://doi.org/ 10.13182/NSE137-380. Mickens, Ronald E. 1999. “Nonstandard Finite Difference Schemes for Reaction-Diffusion Equations.” Numerical Methods for Partial Differential Equations 15 (2): 201–14. https://doi.org/10.1002/ (SICI)1098-2426(199903)15:23.0.CO;2-H. Mittal, R. C., and R. K. Jain. 2012. “Redefined Cubic B-Splines Collocation Method for Solving Convection–Diffusion Equations.”

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ABOUT THE AUTHORS

Dr. L. Rajendran, PhD Professor and Head of Department, Department of Mathematics, Academy of Maritime Education and Training (AMET), (Deemed to be University), Chennai, Tamil Nadu, India Email: [email protected]

Dr. Lakshmanan Rajendran is a Professor and Head of the Department of Mathematics, Academy of Maritime Education and Training (Deemed to be University), Chennai, India. He has published 120 papers in international/SCI journals and 100 papers in national

190

About the Authors

journals. He has also written four books. He serves as a reviewer in many International Journals. He completed 6 research projects from various funding agencies in India. More than 35 students completed their PhD under his guidance. He visited Germany and Poland, under INSA fellowships.

Dr. R. Swaminathan, PhD Principal and Head of the Department, Department of Mathematics, Vidhyaa Giri College of Arts and Science, Puduvayal, Karaikudi, Tamil Nau, India

Dr. R. Swaminathan, a renowned academic specialising in the field of Mathematics, has put his life and soul into Education to produce topquality students. Currently serves as the Principal and Head of the Department of Mathematics, Vidhyaa Giri College of Arts and Science in Puduvayal, Karaikudi. His vast experience has led him to Author Mathematics textbooks for State Universities and also for School Education. Well over 34years of his life have been spent as an educationist. Recipient of Dr. S. Radhakrishnan State award for Best Teacher for the year 2002-2003. In addition, he was also awarded the BOLT-BROAD-OUTLOOK-LEARNER, TEACHER award in 2003. TATA Consultancy Services honoured him by recognizing him with the “Education India’s Best Teacher” award in 2005.

About the Authors

191

M. Chitra Devi Research Assistant, Department of Mathematics, Anna University, University College of Engineering, Dindigul, India Ms. M. Chitra Devi, was born on 13th April 1992 at Madurai, Tamil Nadu, India. She received her Bachelor's degree in Mathematics at E.M.G. Yadava Women's College, Madurai in the year 2010-2013, Master's degree in Mathematics at Fatima College, Madurai, during 2013-2015 & Master of Philosophy in Mathematics at Saraswathi Narayanan College, Madurai in the year 2015-2016. She has worked as a Project Assistant in DST-SERB funded project entitled “Modeling of Non-Linear Convection-Diffusion Processes in Hydrodynamic Electrodes (EMR/2015/002279)” during 2017-2018. Currently, she is doing her PhD on “Analysis of Nonlinear Differential Equations in Rotating Disc Electrode” at Anna University, University college of Engineering, Dindigul, Tamilnadu, India under the supervision of Dr. P. Pirabaharan, Assistant Professor, Department of Mathematics, Anna University, University College of Engineering – Dindigul, Tamil Nadu, India of and Co-supervision of Dr. L. Rajendran, Professor and Head, Department of Mathematics, AMET Deemed to be University, Kanathur, Chennai, Tamil Nadu, India. She has published 4 papers in international/SCI journals and 2 papers in national journals. She has also written one book chapter.

INDEX A Adomian decomposition method (ADM), 7, 12, 16, 26, 27, 29, 60, 91, 92, 93, 97, 131, 135, 148 amperometric, 6, 10, 11, 19, 33, 59, 60, 73, 78, 79, 80, 83, 84, 85, 86, 87, 88, 91, 92, 94, 96, 97, 98, 115, 116, 118, 133, 135, 138, 143, 172, 173, 174, 175, 176, 185 Amperometric biosensor, 6, 172, 173

B biofuel cell, 4, 6, 11, 97, 98, 187 biosensor, 6, 96, 97, 98, 120, 146, 149, 172, 173, 184, 185 Burgers equation, 3, 6, 55

C Cauchy, 2, 3 channel flow, 166 chronoamperometric, 77, 112, 113, 114, 116, 122, 142, 167, 173, 174, 185

convection-diffusion-reaction, 151, 187 cyclic-voltammetry, 74 cylindrical coordinate, 111

D Danckwerts’ method, 78, 79, 112, 113 diffusive wave in fluid dynamics, 6 Dirichlet, 2, 3 disc electrode, 55, 113, 116, 160, 166, 191 Disc electrode, 160 Duffing equation, 50, 52, 58 dynamics, 3, 5, 6, 10, 12, 15, 60, 149, 185

E electroactive polymers, 46 electrochemical immobilization, 33 exp-function method, 16, 35, 36, 39, 55, 56, 57, 58, 60, 61

F F-expansion, 16

194

Index

finite difference method, 60, 173, 174, 175, 176, 186 first-order reaction, 108, 139 first-order redox reaction, 161 Fisher equation, 3 Fisher–Kolmogorov equation, 6, 185 fluid viscosity, 160

mediated bioelectrocatalysis, 98 Michaelis-Menten kinetics, 59, 91, 96, 97, 134, 143, 148, 149 Mixed, 2, 3, 96 modified Lindstedt-Poincare method, 51

G

Neumann, 2, 3 nonlinear equations, ix, 5, 6, 7, 17, 26, 32, 40, 65, 72, 129, 148 non-uniform discrete grid, 172

glucose oxidase enzyme, 40

H heterogeneous catalysis, 37 homotopy analysis method (HAM), 7, 10, 16, 21, 22, 23, 24, 25, 57, 59, 80, 82, 133, 135, 137, 146 homotopy perturbation method (HPM), 12, 16, 17, 18, 20, 23, 60, 73, 74, 76, 77, 78, 80, 83, 84, 85, 86, 87, 88, 89, 90, 91, 94, 96, 98, 103, 106, 115, 118, 130, 132, 135, 136, 138, 142, 144, 148, 149, 158, 159, 161, 162, 163, 164, 165, 166, 178, 180 hydrodynamic voltammetry, 159 hyperbolic function method, 16, 38, 39, 42

J Jacobi-elliptic function expansion, 16

L Laplace transform technique, 43, 81, 93

M Maclaurin series, 45, 46, 47, 48, 66, 67

N

O oxidation of glucose, 93

P parameter-expanding methods, 16, 50 parameterized perturbation method, 16, 52 pattern, 6, 12 pattern dynamics, 6 phenyl glycidyl ether, 11, 64, 95, 98 planar, 63, 78, 83, 95, 178 Poisson–Boltzmann equation, 6 polymer-modified ultramicroelectrodes, 4, 19, 60, 148, 149, 177 potentiometric, 10, 74, 85, 96

Q quantum field theory, 5, 15

R reaction-diffusion equation, i, iii, vii, x, 1, 4, 5, 9, 13, 63, 72, 73, 101, 112, 130, 139, 172, 178, 185, 186, 187

Index reduction of order, 136, 139, 142, 143, 144 Robin, 2, 3 rotating disk electrode, 56, 61, 153, 166, 167

S shifting formula, 112 ship dynamics, 5 ships, 6 shooting method, 43, 175 sine-cosine method, 16 singularity correction method., 174 spherical coordinate, 111, 123, 129 steady-state biofilters, 24 substrate inhibition kinetics, 28, 59, 121, 147

T Tanh-sech, 16 Taylor Series and Padé Approximation Method, 16, 45

195

travelling waves, 6 Troesch’s problem, 43, 56, 58, 60, 61 Two-Point Padé Approximation, 160, 166 two-scale transform, 43

U ultramicroelectrodes, 4, 19, 60, 121, 129, 147, 148, 149, 177, 187 unsaturated (first-order) catalytic kinetic, 107

V Van der Pol’s equation, 6 variable separable method, 79 variational fractal theory, 16, 42 variational iteration method (VIM), x, 9, 11, 16, 32, 34, 57, 58, 78, 80, 91, 94, 97, 116, 138 voltammetry, 6, 8, 10, 74, 76, 78, 81, 91, 93, 95, 120, 121, 147, 158, 159, 162, 167