A meshless local Galerkin method for solving Volterra integral equations deduced from nonlinear fractional differential equations using the moving least squares technique

Abstract

The current investigation studies a numerical method to solve Volterra integral equations of the second kind arising in the single term fractional differential equations with initial conditions. The proposed method estimates the solution of the mentioned Volterra integral equations using the discrete Galerkin method based on the moving least squares (MLS) approach constructed on scattered points. The MLS methodology is an effective technique for the approximation of an unknown function that involves a locally weighted least squares polynomial fitting. We compute fractional integrals appeared in the method by a suitable integration rule based on the non-uniform composite Gauss-Legendre quadrature formula. Since the scheme does not need any background meshes, it can be identified as a meshless method. The scheme is simple and effective to solve fractional differential equations and its algorithm can be easily implemented on computers. The error bound and the convergence rate of the presented method are obtained. Finally, numerical examples are included to show the validity and efficiency of the new technique.