Partial Differential Equations V

1. Front Matter

   Pages i-vii

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2. Equations with Rapidly Oscillating Solutions

   • M. V. Fedoryuk

   Pages 1-51

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

   • B. R. Vainberg

   Pages 53-89

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

   • V. M. Babich

   Pages 91-131

5. Semiclassical Asymptotics of Eigenfunctions

   • V. F. Lazutkin

   Pages 133-171

6. The Boundary Layer

   • A. M. Il’in

   Pages 173-210

7. The Averaging Method for Partial Differential Equations (Homogenization) and Its Applications

   • N. S. Bakhvalov, G. P. Panasenko, A. L. Shtaras

   Pages 211-239

8. Back Matter

   Pages 241-247

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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.

• 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

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