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# Surveys in Applied Mathematics

### Surveys in Applied Mathematics

Hardcover Published: 31st August 1995
ISBN: 9780306449932
Number Of Pages: 264

### Hardcover

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Partial differential equations play a central role in many branches of science and engineering. Therefore it is important to solve problems involving them. One aspect of solving a partial differential equation problem is to show that it is well-posed, i. e. , that it has one and only one solution, and that the solution depends continuously on the data of the problem. Another aspect is to obtain detailed quantitative information about the solution. The traditional method for doing this was to find a representation of the solution as a series or integral of known special functions, and then to evaluate the series or integral by numerical or by asymptotic methods. The shortcoming of this method is that there are relatively few problems for which such representations can be found. Consequently, the traditional method has been replaced by methods for direct solution of problems either numerically or asymptotically. This article is devoted to a particular method, called the "ray method," for the asymptotic solution of problems for linear partial differential equations governing wave propagation. These equations involve a parameter, such as the wavelength. . , which is small compared to all other lengths in the problem. The ray method is used to construct an asymptotic expansion of the solution which is valid near . . = 0, or equivalently for k = 21r I A near infinity.

 Asymptotic Methods for Partial Differential Equations: The Reduced Wave Equation and Maxwell's Equations p. 1 Asymptotic Methods for the Reduced Wave Equation p. 3 Asymptotic Methods for Maxwell's Equations p. 60 Whiskered Tori for Integrable Pde's: Chaotic Behavior in Near Integrable Pde's p. 83 Numerical Experiments p. 88 Elementary Analysis p. 128 Integrable Structure of NLS p. 134 Whiskered Tori for Focusing NLS p. 151 Numerical Measurements - Spectral Transform p. 168 Analysis of the Perturbed NLS System p. 182 Conclusion p. 198 Diffusion in Random Media p. 205 One-Dimensional Conductors p. 206 Multidimensional Diffusion p. 211 General Theory of Effective Conductivity for Random Media p. 236 Index p. 255 Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780306449932
ISBN-10: 0306449935
Series: Surveys in Applied Mathematics : Book 1
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 264
Published: 31st August 1995