A fictitious points one–step MPS–MFS technique

Abstract

The method of fundamental solutions (MFS) is a simple and efficient numerical technique for solving certain homogenous partial differential equations (PDEs) which can be extended to solving inhomogeneous equations through the method of particular solutions (MPS). In this paper, radial basis functions (RBFs) are considered as the basis functions for the construction of a particular solution of the inhomogeneous equation. A hybrid method coupling these two methods using both fundamental solutions and RBFs as basis functions has been effective for solving a large class of PDEs. In this paper, we propose an improved fictitious points method in which the centres of the RBFs are distributed inside and outside the physical domain of the problem and which considerably improves the performance of the MPS–MFS. We also describe various techniques to deal with the several parameters present in the proposed method, such as the location of the fictitious points, the source location in the MFS, and the estimation of a good value of the RBF shape parameter. Five numerical examples in 2D/3D and for second/fourth–order PDEs are presented and the performance of the proposed method is compared with that of the traditional MPS–MFS.