Searching for an optimal shape parameter for solving a partial differential equation with the radial basis functions method

Abstract

This article presents a procedure for searching for an optimal shape parameter for the solution of partial differential equations with the corresponding initial and boundary conditions, where the solution of the problem is unknown. In recent years, radial basis function methods have emerged as alternative computing methods in the scientific computing community.The numerical solution of partial differential equations has usually been obtained by using finite difference methods, finite element methods (FEMs), boundary elements methods or finite volume methods. In our case, we use the multiquadric radial basis function, Gershgorin’s theorem and the Newton method for searching an optimal shape parameter for solving diffusion equations. More cases are presented, the results of which are compared with the results obtained by the FEM.