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.
Large Time Asymptotics for Solutions of Nonlinear First-Order Partial Differential Equations.- Large Time Asymptotic Analysis of Some Nonlinear Parabolic Equations #x2013; Some Constructive Approaches.- Self-Similar Solutions as Large Time Asymptotics for Some Nonlinear Parabolic Equations.- Asymptotics in Fluid Mechanics.
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.
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. Hörmann, Monatshefte für 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)
The goals of this text are to prove or disprove the solution of reduced nonlinear ODE's by different analytic methods, and to show that these solutions are intermediate asymptotics of a class of initial/boundary conditions arising from physical considerations.
Introduction.- Large Time Asymptotics for Solutions of Nonlinear First-Order Partial Differential Equations.- Large Time Asymptotic Analysis of Some Nonlinear Parabolic Equations Some Constructive Approaches.- Self-Similar Solutions as Large Time Asymptotics for Some Nonlinear Parabolic Equations.- Asymptotics in Fluid Mechanics.- Index.
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. Hörmann, Monatshefte für 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)
Inhaltsverzeichnis
Large Time Asymptotics for Solutions of Nonlinear First-Order Partial Differential Equations.- Large Time Asymptotic Analysis of Some Nonlinear Parabolic Equations #x2013; Some Constructive Approaches.- Self-Similar Solutions as Large Time Asymptotics for Some Nonlinear Parabolic Equations.- Asymptotics in Fluid Mechanics.
Klappentext
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.
Includes supplementary material: sn.pub/extras
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