A meshless method for the numerical solution of nonlinear weakly singular integral equations using radial basis functions

In this paper a simple and effective method is given for the numerical solution of nonlinear one-and two-dimensional Fredholm integral equations of the second kind with weakly singular kernels. The general framework of the new scheme is based on the collocation method together with radial basis functions (RBFs) constructed on scattered points in which all integrals are computed via quadrature formulae. In order to approximate the singular integrals appeared in the scheme, we introduce a special quadrature formula, since these integrals cannot be estimated by classical integration rules. The method does not require any cell structures, so it is meshless and consequently is independent of the geometry of the domain. We also present the error analysis of the proposed method and demonstrate that the convergence rate of the approach is arbitrarily high for infinitely smooth RBFs. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.