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Surveys in Applied Mathematics (Hardcover, 1995 ed.)
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Surveys in Applied Mathematics (Hardcover, 1995 ed.)
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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 Consequently, the traditional method has been replaced by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.
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