Higher-order accurate numerical techniques on non-uniform mesh for solving time-fractional Fokker-Planck equations exhibiting weak initial singularity

This paper develops adaptive numerical schemes to approximate solutions of one- and two-dimensional time fractional Fokker-Planck equations, addressing the weak singularity at the initial time. The proposed methodology employs graded, tanh, and harmonic L1-2 meshes to discretize the fractional time derivative, achieving a time convergence order of 3−α, where 0<α<1. Fourth-order accurate Taylor series and exponential compact schemes are used for spatial discretization. Furthermore, the study extends to solving the 2D time-fractional Fokker-Planck equations with weak initial singularity using graded and tanh L1-2 meshes with a fourth-order ADI scheme. Numerical experiments are conducted to demonstrate the improved accuracy and effectiveness of the proposed schemes.