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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 Consequently, the traditional method has been replaced by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.
Singularities in Fluids, Plasmas and Optics, which contains the
proceedings of a NATO Workshop held in Heraklion, Greece, in July
1992, provides a survey of the state of the art in the analysis and
computation of singularities in physical problems drawn from fluid
mechanics, plasma physics and nonlinear optics. The singularities
include curvature singularities on fluid interfaces, the onset of
turbulence in 3-D inviscid flows, focusing singularities for laser
beams, and magnetic reconnection. The highlights of the book
include the nonlinear Schr dinger equation for describing laser
beam focusing, the method of complex variables for the analysis and
computation of singularities on fluid interfaces, and studies of
singularities for the 3-D Euler equations. The book is suitable for
graduate students and researchers in these areas.
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.
Singularities in Fluids, Plasmas and Optics, which contains the
proceedings of a NATO Workshop held in Heraklion, Greece, in July
1992, provides a survey of the state of the art in the analysis and
computation of singularities in physical problems drawn from fluid
mechanics, plasma physics and nonlinear optics. The singularities
include curvature singularities on fluid interfaces, the onset of
turbulence in 3-D inviscid flows, focusing singularities for laser
beams, and magnetic reconnection. The highlights of the book
include the nonlinear Schrodinger equation for describing laser
beam focusing, the method of complex variables for the analysis and
computation of singularities on fluid interfaces, and studies of
singularities for the 3-D Euler equations. The book is suitable for
graduate students and researchers in these areas."
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