The fractional Fokker-Planck system with multiple internal states is derived in [Xu and Deng, Math. Model. Nat. Phenom., $\mathbf{13}$, 10 (2018)], where the space derivative is Laplace operator. If the jump length distribution of the particles is power law instead of Gaussian, the space derivative should be replaced with fractional Laplacian. This paper focuses on solving the two state Fokker-Planck system with fractional Laplacian. We first provide a priori estimate for this system under different regularity assumptions on the initial data. Then we use $L_1$ scheme to discretize the time fractional derivatives and finite element method to approximate the fractional Laplacian operators. Furthermore, we give the error estimates for the space semidiscrete and fully discrete schemes without any assumption on regularity of solutions. Finally, the effectiveness of the designed scheme is verified by numerical experiments.

中文翻译：

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具有两个内部状态的时空分数 Fokker-Planck 系统的数值算法

具有多个内部状态的分数福克-普朗克系统在 [Xu and Deng, Math. 模型。纳特。Phenom., $\mathbf{13}$, 10 (2018)]，其中空间导数是拉普拉斯算子。如果粒子的跳跃长度分布是幂律而不是高斯分布，则空间导数应该用分数拉普拉斯算子代替。本文重点研究用分数拉普拉斯算子求解二态 Fokker-Planck 系统。我们首先在初始数据的不同规律性假设下为该系统提供先验估计。然后我们使用$L_1$方案来离散时间分数阶导数和有限元方法来逼近分数阶拉普拉斯算子。此外，我们给出了空间半离散和完全离散方案的误差估计，没有对解的规律性做任何假设。最后，