Germano, Guido;
Phelan, Carolyn E;
Marazzina, Daniele;
Fusai, Gianluca;
(2025)
Solution of Wiener–Hopf and Fredholm integral equations by fast Hilbert and Fourier transforms.
IMA Journal of Applied Mathematics
, Article hxaf021. 10.1093/imamat/hxaf021.
(In press).
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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.
| Type: | Article |
|---|---|
| Title: | Solution of Wiener–Hopf and Fredholm integral equations by fast Hilbert and Fourier transforms |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1093/imamat/hxaf021 |
| Publisher version: | https://doi.org/10.1093/imamat/hxaf021 |
| Language: | English |
| Additional information: | Copyright © The Author(s) 2025. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. |
| Keywords: | Wiener–Hopf, Fredholm, integral equation, fast Fourier transform, fast Hilbert transform |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10217744 |
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