Wasserstein gradient flow formulation of the time-fractional Fokker-Planck equation

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.