This article describes a numerical method for solving nonlinear Fredholm integral equations of the second kind with weakly singular kernels. The scheme estimates the solution by the discrete Galerkin method based on the use of moving least squares (MLS) as a local approximation. In order to approximate the singular integrals appeared in the method, we introduced an accurate special quadrature formula. The proposed scheme is constructed on a set of scattered data and does not require any background mesh, so it is meshless. The method can be easily implemented and its algorithm is simple and effective to solve weakly singular integral equations. The error analysis of the method is provided. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.
