A constructive method for an approximate solution to scalar Wiener–Hopf equations

Abstract

This paper presents a novel method of approximating the scalar Wiener–Hopf equation, and therefore constructing an approximate solution. The advantages of this method over the existing methods are reliability and explicit error bounds. Additionally, the degrees of the polynomials in the rational approximation are considerably smaller than in other approaches. The need for a numerical solution is motivated by difficulties in computation of the exact solution. The approximation developed in this paper is with a view of generalization to matrix Wiener–Hopf problems for which the exact solution, in general, is not known. The first part of the paper develops error bounds in L p for . These indicate how accurately the solution is approximated in terms of how accurately the equation is approximated. The second part of the paper describes the approach of approximately solving the Wiener–Hopf equation that employs the rational Carathéodory–Fejér approximation. The method is adapted by constructing a mapping of the real line to the unit interval. Numerical examples to demonstrate the use of the proposed technique are included (performed on C hebfun ), yielding errors as small as 10 −12 on the whole real line.