The Implication of Local Thin Plate Splines for Solving Nonlinear Mixed Integro-Differential Equations Based on the Galerkin Scheme

Abstract

In this article, we investigate the construction of a computational method for solving nonlinear  mixed Volterra-Fredholm integro-differential equations of the second kind. The method firstly converts these types of integro-differential equations to a class of nonlinear  integral equations and then utilizes the  locally supported thin plate splines as a basis in the discrete Galerkin method to estimate the solution. The local thin plate splines are known as a type of the free shape parameter radial basis functions constructed on a small set of nodes in the support domain of any node which establish a stable technique to approximate an unknown function. The presented method in comparison with the method based on the globally supported thin plate splines for solving integral equations is well-conditioned and  uses much less computer memory. Moreover, the algorithm of the presented approach is attractive and easy to implement on computers. The numerical method developed in the current paper does not require any cell structures, so it is meshless. Finally, numerical examples are considered to demonstrate the validity and efficiency of the new method.