On the non-commutative factorization of Wiener-Hopf kernels of Khrapkov type

Abstract
There are a great many problems in mathematical physics and engineering which, when modelled mathematically, are reduced to Wiener–Hopf functional equations defined in some region of a complex plane. In simple models this equation is scalar, but in more complicated situations inherent coupling, etc., can often give rise to coupled systems of equations, and so the resulting Wiener–Hopf kernels are of matrix form. The crucial step in the solution of any Wiener–Hopf equation is to decompose the kernel, which can be a quite general function of the complex transform variable, into a product of two factors which are regular in overlapping half–planes. This can be accomplished explicitly for scalar kernels, and may be generalized to a particular class of matrix kernels, namely those which permit a commutative decomposition. Although this class is wide, and therefore contains some valuable physical problems, it is found that many such kernels yield factorization elements with exponentially large behaviour at infinity in their half–planes of analyticity. This prevents a later step in the Wiener–Hopf procedure being carried through, and therefore introduces a serious limitation to the usefulness of Khrapko's approach. Previous researchers have offered schemes to eliminate the exponential behaviour of the Khrapkov factors, but, for reasons discussed herein, these have problems in their successful implementation. This article offers a new procedure for quenching the exponential growth, which is based on a method proposed recently by the author for fully coupled kernels which do not have commutative factors. The approach is two–step; first a commutative factorization is obtained which is algebraic at infinity as required, but which has spurious branch–cuts in the regions of intended analyticity. Second, an approximate procedure is used to eliminate the offending branch–points by introduction of Padé approximants for a certain scalar function, and by inclusion of a regularizing meromorphic matrix. For one particular example kernel, in which an exact factorization happens to be known, the explicit approximate kernel factors thus derived are shown to yield very close agreement to the exact factors. The application of this work to problems in the area of fluid–structural interaction is discussed.

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