A discrete collocation scheme to solve Fredholm integral equations of the second kind in high dimensions using radial kernels

Abstract

In this paper, we propose a kernel-based method to solve multidimensional linear Fredholm integral equations of the second kind over general domains. The discrete collocation method in combination with radial kernels interpolation method is utilized to convert these types of equations to a linear system of equations that can be solved numerically by a suitable numerical method. Integrals appeared in the scheme are approximately computed by the Gauss–Legendre and Monte Carlo quadrature rules. The proposed scheme does not require a structured grid, and thus can be used to solve complex geometry problems based on a set of scattered points that can be arbitrarily chosen. Thus, for the multidimensional linear Fredholm integral equation, an irregular region can be considered. The convergence analysis of the approach is studied for the presented method. The accuracy and efficiency of the new technique are illustrated by several numerical examples.