Jacobian Spectral Collocation Method for Spatio-Temporal Coupled Fokker-Planck Equation with Variable-Order Fractional Derivative

Fractional Fokker–Planck equation plays an important role in describing anomalous dynamics. In this paper, we consider a spectral method for the spatio-temporal coupled fractional Fokker-Planck equation with variable-order fractional derivative. After introducing a suitable intermediate variable, the original equation can be split into two parts and decoupled. Then a spectral collocation method is employed to the resulting equation. Specifically, the points collocated both in temporal and spatial domains are related to fractional Jacobi functions, and the so-called fractional interpolation functions are used as basis functions for temporal semi-discretization. A recurrence relation is derived for fast generation of the fractional differentiation matrix. Regularity analysis for the solution as well as the convergence estimates for the spectral approximation are established for the constant-order case. Several numerical tests demonstrate the efficiency of the method and support the theoretical results.