A meshless method for generalized linear or nonlinear Poisson-type problems

This paper presents a meshless method, based on coupling virtual boundary collocation method (VBCM) with the radial basis functions (RBF) and the analog equation method (AEM), to analyze generalized linear or nonlinear Poisson-type problems. In this method, the AEM is used to construct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator, instead of other complicated ones which are needed in conventional BEM, can be employed. While global RBF is used to approximate fictitious body force which appears when the analog equation method is introduced, and VBCM are utilized to solve homogeneous solution based on the superposition principle. As a result, a new meshless method is developed for solving nonlinear Poisson-type problems. Finally, some numerical experiments are implemented to verify the efficiency of the proposed method and numerical results are in good agreement with the analytical ones. It appears that the proposed meshless method is very effective for nonlinear Poisson-type problems.