Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable Coefficients

This paper presents a practical numerical method, an implicit finite-difference scheme for solving a two-dimensional time-space fractional Fokker-Planck equation with space-time depending on variable coefficients and source term, which represents a model of a Brownian particle in a periodic potential. The Caputo derivative and the Riemann-Liouville derivative are considered in the temporal and spatial directions, respectively. The Riemann-Liouville derivative is approximated by the standard Grunwald approximation and the shifted Grunwald approximation. The stability and convergence of the numerical scheme are discussed. Finally, we provide a numerical example to test the theoretical analysis.