A numerical method for solving a system of Volterra–Fredholm integral equations of the second kind based on the meshless method

Abstract

In this paper, an efficient numerical method is proposed for solving a nonlinear system of Volterra–Fredholm integral equations of the second kind, using two-dimensional radial basis functions (RBFs). This method is based on interpolation by radial basis functions including multiquadric (MQ), using Legendre–Gauss–Radau nodes and weights. The proposed method does not require any background mesh or cell structures, so it is meshless and consequently independent of the geometry of domain. Newton method is employed for solving the nonlinear system obtained with MQ RBF collocation method. Also a theorem is proved for convergence analysis. Some numerical examples are presented and the results are compared with the analytical solution to demonstrate the validity and the applicability of the proposed method.