Simultaneous Wiener–Hopf equations

Abstract

In Noble's book The Wiener Hopf Technique, Pergamon, 1958, he considers the coupled system of Wiener–Hopf equations (§4.4, pp. 153–154)[Formula: see text]He shows that provided the functions L(α), M(α), Q(α), and R(α) have only simple pole singularities the solution can be reduced to two sets of infinite simultaneous linear algebraic equations. In this article a different approach is used which gives the solution in the form of a Fredholm integral equation of the second kind. This Fredholm integral equation can be reduced to infinite sets of simultaneous linear algebraic equations under the less restrictive conditions that either (i) L(α)/M(α) has no branch points in the lower α-half plane: Im(α) < τ+; or (ii) Q(α)/R(α) has no branch points in the upper α-half plane: Im(α) > τ−. In the special case considered by Noble if L(α)/M(α) (or Q(α)/R(α)) only have simple poles in the lower (upper) half plane then the Fredholm integral equation reduces to one infinite set of simultaneous equations. This extends the Wiener–Hopf technique to yet a larger class of boundary value problems, and simplifies the numerical computations.