Solution of Wiener-Hopf and Fredholm integral equations by fast Hilbert and Fourier transforms

TL;DR

This paper introduces advanced FFT-based numerical methods utilizing Hilbert transforms to efficiently solve Wiener-Hopf and Fredholm integral equations, with improved accuracy and iterative solutions, supported by numerical tests and open-source code.

Contribution

It extends existing FFT-based methods for Wiener-Hopf equations using Hilbert transforms and generalizes them to Fredholm equations through iterative coupling.

Findings

Enhanced FFT-based solution accuracy

Effective iterative approach for Fredholm equations

Open-source implementation available

Abstract

We present numerical methods based on the fast Fourier transform (FFT) to solve convolution integral equations on a semi-infinite interval (Wiener-Hopf equation) or on a finite interval (Fredholm equation). We extend and improve a FFT-based method for the Wiener-Hopf equation due to Henery, expressing it in terms of the Hilbert transform, and computing the latter in a more sophisticated way with sinc functions. We then generalise our method to the Fredholm equation reformulating it as two coupled Wiener-Hopf equations and solving them iteratively. We provide numerical tests and open-source code.