In this work, we investigate a variational formulation for a time-fractional Fokke–Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion. The model involves a fractional-order Caputo derivative in time, and thus inherently nonlocal. The study follows the Wasserstein gradient flow approach pioneered by [R. Jordan, D. Kinderlehrer, and F. Otto, SIAM J. Math. Anal., 29(1):1–17, 1998]. We propose a JKO-type scheme for discretizing the model, using the L1 scheme for the Caputo fractional derivative in time, and establish the convergence of the scheme as the time step size tends to zero. Illustrative numerical results in one- and two-dimensional problems are also presented to show the approach.

中文翻译：

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时间分数Fokker-Planck方程的Wasserstein梯度流公式

在这项工作中，我们研究了时间分数Fokke-Planck方程的变分公式，该方程在研究异常缓慢扩散的复杂物理系统时出现。该模型在时间上涉及分数阶Caputo导数，因此固有地是非局部的。这项研究遵循了[R. Jordan，D。Kinderlehrer和F.Otto，SIAM J. Math。[J. Anal。，29（1）：1-17，1998]。我们为时间离散化Caputo分数导数使用L1方案，提出了一种JKO型方案，用于离散化模型，并随着时间步长趋于零，建立了方案的收敛性。一维和二维问题的说明性数值结果也被提出来展示该方法。