The combination of meshless method based on radial basis functions with a geometric numerical integration method for solving partial differential equations: Application to the heat equation

Abstract

In this paper a new scheme is investigated for solving partial differential equations via combination of radial basis functions (RBFs) and group preserving scheme (GPS), which takes advantage of two powerful methods. In this method, we use Kansas approach to approximate the spatial derivatives and then we apply GPS method to approximate first-order time derivative. An advantage of the developed method is that it can be applied to problems with non-regular geometrical domains. To show the efficiency of this method, some heat equations are solved in one, two and three dimension spaces. The two-dimensional version of heat equation on different geometries such as the rectangular, triangular and circular domains is solved. The three-dimensional case is solved on the cubical and spherical domains. To show the high accuracy of the method, a comparison study of the present method and method used in the paper of Dehghan [1] is given.