On the Solution of Wiener-Hopf Problems Involving Noncommutative Matrix Kernel Decompositions

Many problems in physics and engineering with semi-infinite boundaries or interfaces are exactly solvable by the Wiener–Hopf technique. It has been used successfully in a multitude of different disciplines when the Wiener–Hopf functional equation contains a single scalar kernel. For complex boundary value problems, however, the procedure often leads to coupled equations which therefore have a kernel of matrix form. The key step in the technique is to decompose the kernel into a product of two functions, one analytic in an upper region of a complex (transform) plane and the other analytic in an overlapping lower half-plane. This is straightforward for scalar kernels but no method has yet been proposed for general matrices. In this article a new procedure is introduced whereby Padé approximants are employed to obtain an approximate but explicit noncommutative factorization of a matrix kernel. As well as being simple to apply, the use of approximants allows the accuracy of the factorization to be increased almost indefinitely. The method is demonstrated by way of example. Scattering of acoustic waves by a semi-infinite screen at the interface between two compressible media with different physical properties is examined in detail. Numerical evaluation of the approximate factorizations together with results on an upper bound of the absolute error reveal that convergence with increasing Padé number is extremely rapid. The procedure is computationally efficient and of simple analytic form and offers high accuracy (error < 10−7% typically). The method is applicable to a wide range of initial or boundary value problems.