Higher order accurate computational methods for solving one and two dimensional time fractional Fokker-Planck equations

Purpose This study aims to construct higher order accurate computational methods for efficiently solving one-and two-dimensional time fractional Fokker-Planck equations. Design/methodology/approach For one-dimensional time fractional Fokker-Planck equations, L1-2 and L1-2-3 time discretizations are combined with a fourth-order Taylor series based compact difference method (HOC) in space. The second set of schemes employs a fourth-order accurate compact exponential scheme (CES) in space. For two-dimensional problems, an alternating direction implicit (ADI) scheme is developed using the CES method in space, paired with L1 and L1-2 time discretizations. Numerical experiments validate the effectiveness and accuracy of the schemes, and convergence orders are computed for both spatial and temporal discretizations. Findings The proposed methods achieve high accuracy, with spatial convergence of order four and temporal convergence of O(τ3-α) and O(τ4-α), where 0 α 1. Numerical experiments reveal that the designed new methods achieve better accuracy than the existing methods for both one- and two-dimensional problems. The findings highlight the computational feasibility of the methods with high precision for solving time fractional Fokker-Planck equations with variable coefficients. Originality/value This study introduces four novel schemes for solving the one-dimensional time fractional Fokker-Planck equations: L12-HOC, L123-HOC, L12-CES and L123-CES, as well as two advanced ADI schemes for the two-dimensional case: L1-CES and L12-CES. These methods offer a valuable contribution by providing reliable and efficient computational tools for the numerical treatment of time fractional Fokker-Planck equations with variable coefficients.