Surveys in Applied Mathematics

1. Front Matter

   Pages i-xii

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2. Asymptotic Methods for Partial Differential Equations: The Reduced Wave Equation and Maxwell’s Equations

   • Joseph B. Keller, Robert M. Lewis

   Pages 1-82

3. Whiskered Tori for Integrable Pde’s: Chaotic Behavior in Near Integrable Pde’s

   • David W. McLaughlin, Edward A. Overman II

   Pages 83-203

4. Diffusion in Random Media

   • George C. Papanicolaou

   Pages 205-253

5. Back Matter

   Pages 255-264

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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 Consequently, the traditional method has been replaced by methods for direct solution of problems either numerically or asymptotically.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.

• Applied Mathematics
• Maxwell's equations
• equation
• mathematics
• wave equation
• partial differential equations

• Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, USA

  Joseph B. Keller

• Courant Institute of Mathematical Sciences, New York University, New York, USA

  David W. McLaughlin

• Department of Mathematics, Stanford University, Stanford, USA

  George C. Papanicolaou

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