Finding a good shape parameter of RBF to solve PDEs based on the particle swarm optimization algorithm

Abstract

The present study aims at integrating the Particle Swarm Optimization (PSO) algorithm with Kansa’s method based on meshless collocation methods in order to determine a good shape parameter of Radial Basis Function (RBF) for solving partial differential equations (PDEs). For this purpose, we use a two-staged experimental design. While in the first stage, PSO algorithm was used to determine an optimal shape parameter for the related RBFs, in the second stage, we employed Kansa’s method to estimate the RMS error for specifying approximate solutions. To study the performance of the proposed algorithm, we offer numerical results for two examples of partial differential equations and show the effectiveness of the proposed method. Numerical results demonstrated the performance superiority of the new algorithm model. The findings also indicated that the evolutionary algorithm model is more effective than the golden section search algorithm in finding a good shape parameter of RBF.