Encyclopaedia of Mathematical Sciences

Partial Differential Equations V

Asymptotic Methods for Partial Differential Equations

Editors: Fedoryuk, M.V. (Ed.)

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About this book

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.

Table of contents (6 chapters)

  • Equations with Rapidly Oscillating Solutions

    Fedoryuk, M. V.

    Pages 1-51

  • Asymptotic Expansion as t → ∞ of the Solutions of Exterior Boundary Value Problems for Hyperbolic Equations and Quasiclassical Approximations

    Vainberg, B. R.

    Pages 53-89

  • The Higher-Dimensional WKB Method or Ray Method. Its Analogues and Generalizations

    Babich, V. M.

    Pages 91-131

  • Semiclassical Asymptotics of Eigenfunctions

    Lazutkin, V. F.

    Pages 133-171

  • The Boundary Layer

    Il’in, A. M.

    Pages 173-210

Buy this book

eBook $149.00
price for USA (gross)
  • ISBN 978-3-642-58423-7
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $199.00
price for USA
  • ISBN 978-3-540-53371-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $199.00
price for USA
  • ISBN 978-3-642-63586-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Partial Differential Equations V
Book Subtitle
Asymptotic Methods for Partial Differential Equations
Editors
  • M.V. Fedoryuk
Translated by
Wolf, S.A., Joel, J.S.
Series Title
Encyclopaedia of Mathematical Sciences
Series Volume
34
Copyright
1999
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-58423-7
DOI
10.1007/978-3-642-58423-7
Hardcover ISBN
978-3-540-53371-9
Softcover ISBN
978-3-642-63586-1
Series ISSN
0938-0396
Edition Number
1
Number of Pages
VII, 247
Additional Information
Original Russian edition published by VINITI, Moscow, 1988
Topics