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Springer Monographs in Mathematics

By: P.L. Sachdev, Ch. Srinivasa Rao

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Paperback | 11 November 2009

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244 Pages

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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 solution. 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

here 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, and would

considerably help understand diverse natural phenomena.

Industry Reviews

From the reviews:

"The book is mainly addressed to applied mathematicians, and it may be of interest to physicists, biologists and engineers too. It is very rich with examples and explicit calculations which may become starting points for further advances in the general theory of the large-time behavior of solutions to nonlinear PDEs. Moreover, numerical examples are given. ... The book goes through the papers and the results of several researchers, including the authors themselves, who have made great contributions to the subject in the last decades." (Andrea Marson, Mathematical Reviews, Issue 2011 d)

"This monograph provides a state of the art discussion of several constructive approaches to determine the large time behavior of the solutions. ... The audience the authors have in mind are applied mathematicians, physicists, engineers and biologists who wish to understand asymptotic aspects of these diverse natural phenomena." (G. H¶rmann, Monatshefte f¼r Mathematik, Vol. 162 (2), February, 2011)

"The book goes through the papers and results of many researchers, including the authors themselves. The results and analysis in the book have both analytical and numerical character and are addressed mainly to applied mathematicians. There are very many examples and explicit calculations which may become starting point for further analysis." (Andrey E. Shishkov, Zentralblatt MATH, Vol. 1243, 2012)

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Applied Mathematics Science Physics Classical Mathematics Engineering & Technology Technology in General Maths for Engineers

Introduction	p. 1
References	p. 6
Large Time Asymptotics for Solutions of Nonlinear First-Order Partial Differential Equations	p. 9
Introduction	p. 9
First-order nonlinear partial differential equations - Someexamples	p. 10
Decay estimates for solutions of nonlinear first-order partial differential equations	p. 21
Conclusions	p. 31
References	p. 32
Large Time Asymptotic Analysis of Some Nonlinear Parabolic Equations - Some Constructive Approaches	p. 33
Introduction	p. 33
Travelling waves as asymptotics of solutions of initial value problems - The Burgers equation	p. 34
Profile at infinity - Initial boundary value problem for Burgers equation	p. 38
Travelling waves describing flow in a porous medium	p. 47
Evolution of a stable profile describing cross-field diffusion in toroidal multiple plasma	p. 56
Asymptotic solutions describing fast nonlinear diffusion	p. 63
Large time asymptotic behaviour of periodic solutions of some generalised Burgers equations	p. 74
Asymptotic behaviour of some generalised Burgers equations via balancing argument	p. 82
Evolution of travelling waves in generalised Fisher's equations via matched asymptotic expansions	p. 106
Periodic travelling wave solutions in reaction-diffusion systems	p. 115
Conclusions	p. 122
References	p. 124
Self-similar Solutions as Large Time Asymptotics for Some Nonlinear Parabolic Equations	p. 129
Introduction	p. 129
Self-similar solutions as large time asymptotics for linear heat equation	p. 132
Self-similar solutions as intermediate asymptotics for filtration-absorption model	p. 136
Analysis of a class of similarity solutions of the porous media equation	p. 145
Similarity solutions of nonlinear heat conduction problems as large time asymptotics	p. 154
Large time asymptotics for solutions of the porous media equation	p. 161
Large time behaviour of solutions of a dissipative semilinear heat equation	p. 170
Large time asymptotics for the solutions of a very fast diffusion equation	p. 176
Conclusions	p. 184
References	p. 186
Asymptotics in Fluid Mechanics	p. 189
Introduction	p. 189
Strong explosion in a power law density medium - Self-similar solutions of first and second kind	p. 190
Self-similar solutions for collapsing cavities	p. 199
Large time behaviour of solutions of compressible flow equations with damping	p. 203
Large time behaviour of solutions of unsteady boundary layer equations for an incompressible fluid	p. 211
Asymptotic behaviour of velocity profiles in Prandtl boundary layer theory	p. 219
Conclusions	p. 230
References	p. 230
Index	p. 233
Table of Contents provided by Ingram. All Rights Reserved.

Springer Monographs in Mathematics

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Paperback

Industry Reviews

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