Fictitious time integration method of fundamental solutions with Chebyshev polynomials for solving Poisson-type Nonlinear PDEs

Abstract

The fictitious time integration method (FTIM) previously developedby Liu and Atluri (2008a) is combined with the method of fundamental solutions and the Chebyshev polynomials to solve Poisson-type nonlinear PDEs. Themethod of fundamental solutions with Chebyshev polynomials (MFS-CP) is anexponentially-convergent meshless numerical method which is able to solving nonhomogeneous partial differential equations if the fundamental solution and the analytical particular solutions of the considered operator are known. In this study, theMFS-CP isextended to solve Poisson-type nonlinear PDEs by using the FTIM. Inthe solution procedure, the FTIM is introduced to convert a Poisson-type nonlinear PDE into a sequence of linear nonhomogeneous modified Helmholtz equationswhich are then formally solved by the MFS-CP. Several numerical experimentswere carried out to validate the proposed methods.