The Numerical Solution of Fredholm-Hammerstein Integral Equations by Combining the Collocation Method and Radial Basis Functions

Hammerstein integral equations have been arisen from mathematical models in various branches of applied sciences and engineering. This article investigates an approximate scheme to solve Fredholm-Hammerstein integral equations of the second kind. The new method uses the discrete collocation method together with radial basis functions (RBFs) constructed on scattered points as a basis. The discrete collocation method results from the numerical integration of all integrals appeared in the approach. 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 interesting 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 confirm the accuracy and efficiency of the new scheme presented in this paper and are compared with the Legendre wavelet technique.