An implicit MLS meshless method for 2-D time dependent fractional diffusion-wave equation

It is well accepted that fractional partial differential equations (FPDE) can be used to model many processes for which the normal partial differential equations (PDE) fail to describe precisely. Numerical approaches seem to be promising alternatives when exact solution of FPDE is difficult to derive. However, numerical solution of FPDE encounters new challenges brought in by the fractional order derivatives. In this paper we consider the 2D time dependent fractional diffusion-wave equation (FDWE) with Caputo derivative in temporal direction. We discretize the fractional order derivative with finite difference method (FDM) and present a moving least squares (MLS) meshless approximation in spatial directions which can be used to handle more complex problem domain. The convergence and stability properties of semi-discretized scheme related to time are theoretically analyzed. Finally, we conduct several numerical experiments to test our method for both regular and irregular node point distribution on rectangular and circular domain. The results indicate that the proposed method is accurate and efficient.