A Constructive Method for Approximate Solution to Scalar Wiener-Hopf Equations

TL;DR

This paper introduces a reliable and explicit error-bounded rational approximation method for scalar Wiener-Hopf equations, enabling accurate solutions with smaller polynomial degrees and potential extension to matrix cases.

Contribution

It develops a new approximation technique using Rational Caratheodory-Fejer approximation with explicit error bounds, improving over existing methods in reliability and polynomial degree.

Findings

Achieved error as small as 10^{-12} on the real line.

Provided Lp error bounds relating solution accuracy to equation approximation.

Demonstrated the method's effectiveness with numerical examples.

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 generalisation to matrix Wiener-Hopf for which the exact solution, in general, is not known. The first part of the paper develops error bounds in Lp for 1<p<\infty. These indicate how accurately the solution is approximated in terms of how accurate the equation is approximated. The second part of the paper describes the approach of approximately solving the Wiener-Hopf equation that employs the Rational Caratheodory-Fejer 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 Chebfun), yielding error as small as 10^{-12} on the whole real line.