The Wiener-Hopf Technique
The solution of boundary-value problems using integral transforms is comparatively easy for certain simple regions. There are many important problems, however, where the boundary data is of such a form that although an integral transform may be taken sensibly, it does not lead directly to an explicit solution. A typical problem involves a semi-infinite boundary, and may arise in such fields as electromagnetic theory, hydrodynamics, elasticity, and others. The Wiener-Hopf technique, which gives the solution to many problems of this kind, was first developed systematically by Wiener and Hopf in 1931, although the germ of the idea is contained in earlier work by Carleman.1 Although it is most often used in conjunction with the Fourier transform, it is a significant and natural tool for use with the Laplace and Mellin transforms also. As usual, we develop the method in relation to some illustrative problems.
KeywordsEntire Function Diffract Wave Incident Plane Wave Illustrative Problem Reflected Plane Wave
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- 4.See Duffy  or Noble  for more references. In addition to problems in one complex variable, Kraut has considered mixed boundary-value problems which may be resolved using a Wiener-Hopf type of decomposition in two complex variables. See E.A. Kraut, Proc. Amer. Math. Soc., 23 (1969), 24, and further references given there.MathSciNetMATHCrossRefGoogle Scholar