The Wiener-Hopf Technique

Abstract

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 sensibly taken, 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. While 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.

The Wiener-Hopf technique is mentioned in a number of books; for a comprehensive review of the method see NOBLE (1958).