The numerical solution of Fredholm-Hammerstein integral equations by combining the collocation method and radial basis functions

Abstract

Hammerstein integral equations have been applied as a mathematical model in various branches of applied science and engineering. This article investigates an approximate scheme to solve Fredholm-Hammerstein integral equations  of the second kind. The new method utilizes the discrete collocation method together with radial basis functions (RBFs) constructed on scattered points as basis. The discrete collocation method results from the numerical integration of all integrals appeared in the method. We employ the composite Gauss-Legendre integration rule  to estimate the integrals appeared in the  method. Since the scheme   does not need  any  background meshes, it can be identified as a  meshless   method. The algorithm of the presented scheme is attractive and easy to implement on computers. We also provide the error bound and the convergence rate of the presented method.
The results of numerical experiments are compared with analytical solution   to confirm the accuracy and efficiency of the new scheme presented in this paper.