Nonlinear Reaction-Diffusion-Convection: Lie and Conditional Symmetry, exact Solutions and Their Applications

✍ Scribed by Roman Cherniha, Mykola Serov, Oleksii Pliukhin

• Publisher: Chapman and Hall/CRC
• Year: 2017
• Tongue: English
• Leaves: 261
• Series: Chapman & Hall/CRC Monographs and Research Notes in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Content: ""Cover""
""Title ""
""Series Editors""
""copyright""
""Contents""
""Preface""
""List of Figures""
""List of Tables""
""Acronyms""
""Chapter 1 Introduction""
""1.1 Nonlinear reaction-diffusion-convection equations in mathematical modeling""
""1.2 Main methods for exact solving nonlinear reaction-diffusion- convection equations""
""1.3 Lie symmetry of differential equations: historical review, definitions and properties""
""Chapter 2 Lie symmetries of reaction-diffusion-convection equations""
""2.1 Symmetry of the linear diffusion equation"" ""2.2 Symmetry of the nonlinear diffusion equation""""2.3 Equivalence transformations and form-preserving transformations""
""2.3.1 The group of equivalence transformations""
""2.3.2 Form-preserving transformations""
""2.4 Determining equations for reaction-diffusion-convection equations""
""2.5 Complete description of Lie symmetries of reaction-diffusion-convection equations""
""2.5.1 Principal algebra of invariance""
""2.5.2 Necessary conditions for nontrivial Lie symmetry""
""2.5.3 Lie symmetry classification via the Lieâ#x80
#x93
Ovsiannikov algorithm"" ""2.5.4 Application of form-preserving transformation""""2.6 Nonlinear equations arising in applications and their Lie symmetry""
""2.6.1 Heat (diffusion) equations with power-law nonlinearity""
""2.6.2 Diffusion equations with a convective term""
""2.6.3 Nonlinear equations describing three types of transport mechanisms""
""Chapter 3 Conditional symmetries of reaction-diffusion-convection equations""
""3.1 Conditional symmetry of differential equations: historical re-view, definitions and properties""
""3.2 Qâ#x80
#x93
conditional symmetry of the nonlinear heat equation"" ""3.3 Determining equations for finding Q-conditional symmetry of reaction-diffusion-convection equations""""3.4 Qâ#x80
#x93
conditional symmetry of reaction-diffusion-convection equations with constant diffusivity""
""3.5 Qâ#x80
#x93
conditional symmetry of reaction-diffusion-convection equa-tions with power-law diffusivity""
""3.5.1 The case of proportional diffusion and convection coef-ficients""
""3.5.2 The case of different diffusion and convection coeffi-cients""
""3.6 Qâ#x80
#x93
conditional symmetry of reaction-diffusion-convection equations with exponential diffusivity"" ""3.6.1 Solving the nonlinear system (3.166)""""3.6.2 Solving the nonlinear system (3.169)""
""3.7 Nonlinear equations arising in applications and their conditional symmetry""
""Chapter 4 Exact solutions of reaction-diffusion-convection equations and their applications""
""4.1 Classification of exact solutions from the symmetry point of view""
""4.2 Examples of exact solutions for some well-known nonlinear equations""
""4.3 Solutions of some reaction-diffusion-convection equations arising in biomedical applications""
""4.3.1 The Fisher and Murray equations"" ""4.3.2 The Fitzhughâ#x80
#x93
Nagumo equation and its generalizations""