A new approach based on the genetic algorithm for finding a good shape parameter in solving partial differential equations by Kansa's method

Many radial basis function (RBF) methods contain a free shape parameter that plays an important role for the accuracy of the method. In most papers the authors end up choosing this shape parameter by trial and error or some other ad hoc means. In this paper, we propose applying the genetic algorithm to determine a good shape parameter of radial basis functions for the solution of partial differential equations. We use meshless collocation method based on the radial basis function (Kansa's method) to solve partial differential equations. Due to the severely ill-conditioned matrix arising from using RBF, we also consider the truncated singular value decomposition method (TSVD) for solving system of linear equations which is obtained from Kansa's method. Numerical results show that the proposed algorithm based on the genetic optimization is effective and provides a reasonable shape parameter along with acceptable accuracy of the solution.