The Conical Radial Basis Function for Partial Differential Equations

Abstract

The performance of the parameter-free conical radial basis functions accompanied with the Chebyshev node generation is investigated for the solution of boundary value problems. In contrast to the traditional conical radial basis function method, where the collocation points are placed uniformly or quasi-uniformly in the physical domain of the boundary value problems in question, we consider three different Chebyshev-type schemes to generate the collocation points. This simple scheme improves accuracy of the method with no additional computational cost. Several numerical experiments are given to show the validity of the newly proposed method.

The Quasi-Optimal Radial Basis Function Collocation Method: A Technical Note

“…As is known to all, the selection of shape parameter in the traditional MQ-RBF is very important in dealing with partial differential equations, and its small variation may cause severe differences to solutions [2]. e selection of optimal shape parameter in the RBF methods has long been a challenging task [3][4][5][6][7]. Several attempts and considerable progress have been made in literatures [8][9][10][11], but this question still remains a bottleneck for the MQ method application to practical problems.…”