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

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 improve an FFT-based method for the Wiener–Hopf equation due to Henery by expressing it in terms of the Hilbert transform and computing the latter in a more sophisticated way with a sinc function expansion. We further enhance the error convergence using a spectral filter. We then generalize our method to the Fredholm equation by reformulating it as two coupled Wiener–Hopf equations and solving them iteratively. We provide numerical tests and open-source code.