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Partial Differential Equations V

Asymptotic Methods for Partial Differential Equations

  • Editors
  • M. V. Fedoryuk

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 34)

Table of contents

About this book

Introduction

In this paper we shall discuss the construction of formal short-wave asymp­ totic solutions of problems of mathematical physics. The topic is very broad. It can somewhat conveniently be divided into three parts: 1. Finding the short-wave asymptotics of a rather narrow class of problems, which admit a solution in an explicit form, via formulas that represent this solution. 2. Finding formal asymptotic solutions of equations that describe wave processes by basing them on some ansatz or other. We explain what 2 means. Giving an ansatz is knowing how to give a formula for the desired asymptotic solution in the form of a series or some expression containing a series, where the analytic nature of the terms of these series is indicated up to functions and coefficients that are undetermined at the first stage of consideration. The second stage is to determine these functions and coefficients using a direct substitution of the ansatz in the equation, the boundary conditions and the initial conditions. Sometimes it is necessary to use different ansiitze in different domains, and in the overlapping parts of these domains the formal asymptotic solutions must be asymptotically equivalent (the method of matched asymptotic expansions). The basis for success in the search for formal asymptotic solutions is a suitable choice of ansiitze. The study of the asymptotics of explicit solutions of special model problems allows us to "surmise" what the correct ansiitze are for the general solution.

Keywords

Asymptotic expansions Boundary value problem Differentialgleichngen Mechanik der inhomogenen Medien hyperbolic equation matching asymptotic expansions mechanixs of inhomogenous media partial differential equation partial differential equations vergleichbare asymptotische Entwicklungen

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-58423-7
  • Copyright Information Springer-Verlag Berlin Heidelberg 1999
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-63586-1
  • Online ISBN 978-3-642-58423-7
  • Series Print ISSN 0938-0396
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