Mapped Hermite Functions and their applications to two-dimensional weakly singular Fredholm-Hammerstein integral equations

TL;DR

This paper introduces mapped Hermite functions for efficiently solving two-dimensional weakly singular Fredholm-Hammerstein integral equations, achieving exponential convergence and addressing computational challenges of dense matrices.

Contribution

It presents a novel class of mapped Hermite functions and two spectral methods, pioneering direct spectral solutions for multi-point singularity problems.

Findings

Methods achieve exponential convergence rates.

Effective handling of weak singularities at endpoints.

Numerical results confirm high accuracy and efficiency.

Abstract

The Fredholm-Hammerstein integral equations (FHIEs) with weakly singular kernels exhibit multi-point singularity at the endpoints or boundaries. The dense discretized matrices result in high computational complexity when employing numerical methods. To address this, we propose a novel class of mapped Hermite functions, which are constructed by applying a mapping to Hermite polynomials.We establish fundamental approximation theory for the orthogonal functions. We propose MHFs-spectral collocation method and MHFs-smoothing transformation method to solve the two-point weakly singular FHIEs, respectively. Error analysis and numerical results demonstrate that our methods, based on the new orthogonal functions, are particularly effective for handling problems with weak singularities at two endpoints, yielding exponential convergence rate. We position this work as the first to directly study the mapped spectral method for multi-point singularity problems, to the best of our knowledge.