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

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.