Nonlinear Reaction-Diffusion-Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their Applications (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

"It is well known that symmetry-based methods are very powerful tools for investigating nonlinear partial differential equations (PDEs), notably for their reduction to those of lower dimensionality (e.g. to ODEs) and constructing exact solutions. This book is devoted to (1) search Lie and conditional (non-classical) symmetries of nonlinear RDC equations, (2) constructing exact solutions using the symmetries obtained, and (3) their applications for solving some biologically and physically motivated problems. The book summarises the results derived by the authors during the last 10 years and those obtained by some other authors."--Provided by publisher Cover 1 Title 2 Series Editors 3 copyright 5 Contents 6 Preface 10 List of Figures 16 List of Tables 18 Acronyms 20 Chapter 1 Introduction 22 1.1 Nonlinear reaction-diffusion-convection equations in mathematical modeling 22 1.2 Main methods for exact solving nonlinear reaction-diffusion- convection equations 24 1.3 Lie symmetry of differential equations: historical review, definitions and properties 29 Chapter 2 Lie symmetries of reaction-diffusion-convection equations 40 2.1 Symmetry of the linear diffusion equation 40 2.2 Symmetry of the nonlinear diffusion equation 42 2.3 Equivalence transformations and form-preserving transformations 50 2.3.1 The group of equivalence transformations 51 2.3.2 Form-preserving transformations 53 2.4 Determining equations for reaction-diffusion-convection equations 57 2.5 Complete description of Lie symmetries of reaction-diffusion-convection equations 60 2.5.1 Principal algebra of invariance 60 2.5.2 Necessary conditions for nontrivial Lie symmetry 60 2.5.3 Lie symmetry classification via the Lie–Ovsiannikov algorithm 70 2.5.4 Application of form-preserving transformation 81 2.6 Nonlinear equations arising in applications and their Lie symmetry 90 2.6.1 Heat (diffusion) equations with power-law nonlinearity 90 2.6.2 Diffusion equations with a convective term 92 2.6.3 Nonlinear equations describing three types of transport mechanisms 94 Chapter 3 Conditional symmetries of reaction-diffusion-convection equations 98 3.1 Conditional symmetry of differential equations: historical re-view, definitions and properties 98 3.2 Q–conditional symmetry of the nonlinear heat equation 104 3.3 Determining equations for finding Q-conditional symmetry of reaction-diffusion-convection equations 108 3.4 Q–conditional symmetry of reaction-diffusion-convection equations with constant diffusivity 113 3.5 Q–conditional symmetry of reaction-diffusion-convection equa-tions with power-law diffusivity 121 3.5.1 The case of proportional diffusion and convection coef-ficients 122 3.5.2 The case of different diffusion and convection coeffi-cients 128 3.6 Q–conditional symmetry of reaction-diffusion-convection equations with exponential diffusivity 132 3.6.1 Solving the nonlinear system (3.166) 139 3.6.2 Solving the nonlinear system (3.169) 146 3.7 Nonlinear equations arising in applications and their conditional symmetry 150 Chapter 4 Exact solutions of reaction-diffusion-convection equations and their applications 156 4.1 Classification of exact solutions from the symmetry point of view 156 4.2 Examples of exact solutions for some well-known nonlinear equations 159 4.3 Solutions of some reaction-diffusion-convection equations arising in biomedical applications 164 4.3.1 The Fisher and Murray equations 164 4.3.2 The Fitzhugh–Nagumo equation and its generalizations 167 4.4 Solutions of reaction-diffusion-convection equations with power-law diffusivity 177 4.4.1 Lie’s solutions of an equation with power-law diffusion and convection 177 4.4.2 Non-Lie solutions of some equations with power-law diffusion and convection 180 4.5 Solutions of reaction-diffusion-convection equations with exponential diffusivity 197 4.5.1 Lie’s solutions of an equation with exponential diffusion and convection 197 4.5.2 Non-Lie solutions of an equation with exponential diffusion and convection 201 4.5.3 Application of the solutions obtained for population dynamics 209 Chapter 5 The method of additional generating conditions for constructing exact solutions 212 5.1 Description of the method and the general scheme of imple-mentation 212 5.2 Application of the method for solving nonlinear reaction-diffusion- convection equations 216 5.2.1 Reduction of the nonlinear equations (5.10) and (5.11) to ODE systems 217 5.2.2 Exact solutions of the nonlinear equations (5.10) and (5.11) 222 5.2.3 Application of the solutions obtained for solving boundary-value problems 232 5.3 Analysis of the solutions obtained and comparison with the known results 237 References 240 Index 260