In this paper, the numerical solution of fractional convection-diffusion equation in the Caputo sense in time has been investigated. Based on the discretization of the Caputo derivative and the direct Taylor expansion method, a mesoscopic finite difference scheme (MesoFD) is established, and a sufficient condition for the stability of the scheme is provided. The effectiveness of the algorithm and the theoretical analysis results are verified through two-dimensional and three-dimensional numerical examples. In addition we have applied this method to solve fractional Cahn-Hilliard equation and the numerical simulation results are in good agreement with those reported in the existing literature.
 

mesoscopic finite difference scheme, fractional convection-diffusion equation, Caputo derivative