An energy regularization of the MQ-RBF method for solving the Cauchy problems of diffusion-convection-reaction equations

The accuracy of the Kansa type multi-quadric radial basis function (MQ-RBF) method is heavily dependent on the distribution of source points. A proper choice of source points can enhance the stability and accuracy. In this paper we propose an energy regularization technique to choose the source points and the weighting factors preceding the MQ-RBFs in the numerical solution of the Cauchy problem for the steady-state diffusion-convection-reaction equation in an arbitrary plane domain. We derive an inequality, and the energy RBF (ERBF) method can preserve the energy when the inequality is satisfied. It is a criterion to pick up the source points and weighting factors. Through numerical tests under large noises, we find that the performance of the ERBF is good.