A SHARP α-ROBUST L1 SCHEME ON GRADED MESHES FOR TWO-DIMENSIONAL TIME TEMPERED FRACTIONAL FOKKER-PLANCK EQUATION

In this paper, we are concerned with the numerical solution for the two-dimensional time fractional Fokker-Planck equation with the tempered fractional derivative of order alpha.alpha. Although some of its variants are considered in many recent numerical analysis works, there are still some significant differences. Here we first provide the regularity estimates of the solution. Then a modified L1 scheme inspired by the middle rectangle quadrature formula on graded meshes is employed to compensate for the singularity of the solution at t -> 0(+), while the five-point difference scheme is used in space. Stability and convergence are proved in the sense of L-infinity norm, getting a sharp error estimate O(tau(min{2-alpha,r alpha})) on graded meshes. Furthermore, the constant multipliers in the analysis do not blow up as the order of Caputo fractional derivative alpha alpha approaches the classical value of 1. Finally, we perform the numerical experiments to verify the effectiveness and convergence orders of the presented schemes.