Stability and Convergence of an Effective Numerical Method for the Time-Space Fractional Fokker-Planck Equation with a Nonlinear Source Term

Fractional Fokker-Planck equations FFPEs have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term TSFFPENST , which involve the Caputo time fractional derivative CTFD of order α ∈ 0, 1 and the symmetric Riesz space fractional derivative RSFD of order μ ∈ 1, 2 . Approximating the CTFD and RSFD using the L1-algorithm and shifted Grünwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.