Boundary particle method for Laplace transformed time fractional diffusion equations

摘要

This paper develops a novel boundary discretization meshless approach, Laplace transformed boundary particle method (LTBPM), for numerical modeling of time fractional diffusion equations. The present approach implements Laplace transform technique to obtain the corresponding time-independent inhomogeneous equation and then employs a truly boundary-only meshless boundary particle method (BPM) to solve the Laplace-transformed inhomogeneous problem. Unlike the other boundary discretization methods, the BPM does not require any inner nodes, since the recursive composite multiple reciprocity technique is used to reduce an inhomogeneous problem to a series of higher-order homogeneous problems. Finally, the Stehfest numerical Laplace inversion can retrieve the numerical solutions of time fractional diffusion equations from the corresponding BPM solutions. The present method avoids enormous computing costs for the simulation of a long history fractional systems and remedies the low accuracy at the initial instants of time encountered in the other traditional methodologies. Numerical experiments demonstrate that the LTBPM is highly accurate, computationally efficient, and numerically stable for 2D and 3D time fractional diffusion equations.