On the numerical solution of two-dimensional integral equations using a meshless local discrete Galerkin scheme with error analysis

This investigation presents a simple and effective method for numerically solving two-dimensional Fredholm integral equations of the second kind on non-rectangular domains. The general framework of the current scheme is based on the Galerkin method together with moving least squares (MLS) technique constructed on scattered points in which all integrals are approximated by a suitable quadrature formula. The MLS approach estimates an unknown function utilizing a locally weighted least square polynomial fitting. The method does not require any cell structures, so it is meshless and consequently independent of the geometry of the domain. The algorithm of the presented scheme is attractive and easy to implement on computers. Furthermore, the error bound and the convergence rate of the presented method are obtained. Illustrative examples clearly show the reliability and the efficiency of the new method and confirm the theoretical error estimates provided in the current paper.