Large Time Asymptotics for Solutions of Nonlinear Partial Differential Equations.

Format:

Book

Language:

English

Subjects (All):

Differential equations, Nonlinear--Asymptotic theory.

Differential equations, Nonlinear.

Physical Description:

viii, 235 pages : illustrations ; 24 cm

Place of Publication:

[Place of publication not identified] : Springer Verlag, 2009.

Summary:

A large number of physical phenomena are modeled by nonlinear partial differential equations, subject to appropriate initial/boundary conditions; these equations, in general, do not admit exact solutions. The present monograph gives constructive mathematical techniques which bring out large time behavior of solutions of these model equations. These approaches, in conjunction with modern computational methods, help solve physical problems in a satisfactory manner.

The asymptotic methods dealt with here include self-similarity, balancing argument, and matched asymptotic expansions. The physical models discussed in some detail have to relate to porous media equation, heat equation with absorption, generalized Fisher's equation, Burgers equation and its generalizations.

A chapter each is devoted to nonlinear diffusion and fluid mechanics. The present book will be found useful by applied mathematicians, physicists, engineers and biologists.

Contents:

1 Introduction 1

References 6

2 Large Time Asymptotics for Solutions of Nonlinear First-Order Partial Differential Equations 9

2.1 Introduction 9

2.2 First-order nonlinear partial differential equations - Someexamples 10

2.3 Decay estimates for solutions of nonlinear first-order partial differential equations 21

2.4 Conclusions 31

References 32

3 Large Time Asymptotic Analysis of Some Nonlinear Parabolic Equations - Some Constructive Approaches 33

3.1 Introduction 33

3.2 Travelling waves as asymptotics of solutions of initial value problems - The Burgers equation 34

3.3 Profile at infinity - Initial boundary value problem for Burgers equation 38

3.4 Travelling waves describing flow in a porous medium 47

3.5 Evolution of a stable profile describing cross-field diffusion in toroidal multiple plasma 56

3.6 Asymptotic solutions describing fast nonlinear diffusion 63

3.7 Large time asymptotic behaviour of periodic solutions of some generalised Burgers equations 74

3.8 Asymptotic behaviour of some generalised Burgers equations via balancing argument 82

3.9 Evolution of travelling waves in generalised Fisher's equations via matched asymptotic expansions 106

3.10 Periodic travelling wave solutions in reaction-diffusion systems 115

3.11 Conclusions 122

References 124

4 Self-similar Solutions as Large Time Asymptotics for Some Nonlinear Parabolic Equations 129

4.1 Introduction 129

4.2 Self-similar solutions as large time asymptotics for linear heat equation 132

4.3 Self-similar solutions as intermediate asymptotics for filtration-absorption model 136

4.4 Analysis of a class of similarity solutions of the porous media equation 145

4.5 Similarity solutions of nonlinear heat conduction problems as large time asymptotics 154

4.6 Large time asymptotics for solutions of the porous media equation 161

4.7 Large time behaviour of solutions of a dissipative semilinear heat equation 170

4.8 Large time asymptotics for the solutions of a very fast diffusion equation 176

4.9 Conclusions 184

References 186

5 Asymptotics in Fluid Mechanics 189

5.1 Introduction 189

5.2 Strong explosion in a power law density medium - Self-similar solutions of first and second kind 190

5.3 Self-similar solutions for collapsing cavities 199

5.4 Large time behaviour of solutions of compressible flow equations with damping 203

5.5 Large time behaviour of solutions of unsteady boundary layer equations for an incompressible fluid 211

5.6 Asymptotic behaviour of velocity profiles in Prandtl boundary layer theory 219

5.7 Conclusions 230

References 230.

ISBN:

9780387878089

0387878084

OCLC:

428028523