The MAPS with polynomial basis functions for solving axisymmetric time-fractional equations

Abstract

This study presents a parallel meshless approach for time-fractional equations in the axisymmetric geometry. In the present parallel meshless approach, the method of approximate particular solutions (MAPS), in conjunction with Laplace transform, extended precision arithmetic (EPA) and multiple scale technique (MST) is implemented. In this application, the Laplace transform is employed to eliminate the dependence on time. The MAPS is used to obtain the solution of time-independent equation in Laplace space. Furthermore, the solution of time dependent equation is restored by Stehfest’s algorithm. Here the multiple scale technique (MST) and extended precision arithmetic (EPA) are applied to alleviate the effect of an ill-conditioned matrix on a numerical inverse Laplace transform. Then multi-cores parallel computing can improve effectively computing speed of the proposed meshless approach. In the numerical implementation, both the 2D axisymmetric equations and the original 3D equations are considered and compared.