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September  2019, 18(5): 2765-2787. doi: 10.3934/cpaa.2019124

## Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing

 1 School of Mathematical Sciences, Monash University, VIC 3800, Australia 2 School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia 3 Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China

Received  November 2018 Revised  November 2018 Published  April 2019

Fund Project: The research of the first author is supported in part by the Australian Government through the Australian Research Council Discovery Projects funding scheme (project number DP170100605). The research of the third author is supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF-U1530401.

A time-fractional Fokker–Planck initial-boundary value problem is considered, with differential operator $u_t-\nabla\cdot(\partial_t^{1-\alpha}\kappa_\alpha\nabla u -{\bf{F}}\partial_t^{1-\alpha}u)$, where $0<\alpha <1$. The forcing function ${\bf{F}} = {\bf{F}}(t,x)$, which is more difficult to analyse than the case ${\bf{F}} = {\bf{F}}(x)$ investigated previously by other authors. The spatial domain $\Omega \subset\mathbb{R}^d$, where $d\ge 1$, has a smooth boundary. Existence, uniqueness and regularity of a mild solution $u$ is proved under the hypothesis that the initial data $u_0$ lies in $L^2(\Omega)$. For $1/2<\alpha<1$ and $u_0\in H^2(\Omega)\cap H_0^1(\Omega)$, it is shown that $u$ becomes a classical solution of the problem. Estimates of time derivatives of the classical solution are derived—these are known to be needed in numerical analyses of this problem.

Citation: Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124
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