Meshless Collocation Method Using Hybrid Polynomial Basis Functions And Radial Basis Functions

In this paper,we propose a new approach based on Kansa method with fictitious centres,which is an improvement on the ghost point method.The ghost point method uses radial basis functions(RBFs)to construct interpolation functions to approximate the solution of the equation.In our new method,the radial basis function approximation is augmented by polynomial basis functions.In other words,we use two kinds of functions,radial basis function and polynomial function,as the basis function to discrete the equation.The proposed approach considerably improves not only the accuracy but also the stability of the recently proposed ghost point method using radial basis functions.The difficulty of selecting a good RBF shape parameter is no longer an issue.At the same time,with the addition of polynomial function basis,the choice of radius of the circle/sphere where virtual center points are located is more free.A large number of numerical examples including two dimensional problems,three dimensional problems and Cauchy-Navier equation are presented to show the effectiveness and the improvement of the proposed method over previous methods.