Recently, the fractional Fokker–Planck equations (FFPEs) with multiple internal states are built for the particles undergoing anomalous diffusion with different waiting time distributions for different internal states, which describe the distribution of positions of the particles (Xu and Deng in Math Model Nat Phenom 13:10, 2018). In this paper, we first develop the Sobolev regularity of the FFPEs with two internal states, including the homogeneous problem with smooth and nonsmooth initial values and the inhomogeneous problem with vanishing initial value, and then we design the numerical scheme for the system of fractional partial differential equations based on the finite element method for the space derivatives and convolution quadrature for the time fractional derivatives. The optimal error estimates of the scheme under the above three different conditions are provided for both space semidiscrete and fully discrete schemes. Finally, one- and two-dimensional numerical experiments are performed to confirm our theoretical analysis and the predicted convergence order.

最近，针对经历不同时间，不同内部状态等待时间分布的异常扩散粒子，建立了具有多个内部状态的分数福克-普朗克方程（FFPE），它们描述了粒子位置的分布（数学模型Nat中的Xu和Deng Phenom 13:10，2018）。本文首先研究了具有两个内部状态的FFPE的Sobolev正则性，包括具有光滑初始值和非光滑初始值的齐次问题以及具有消失的初始值的不齐次问题，然后设计了分数部分系统的数值方案。基于有限元方法的微分方程用于空间导数，而卷积正交则用于时间分数导数。对于空间半离散方案和完全离散方案，都提供了在上述三种不同条件下该方案的最佳误差估计。最后，进行一维和二维数值实验以确认我们的理论分析和预测的收敛阶。