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

Abstract

A time-fractional Fokker-Planck initial-boundary value problem is considered, with differential operator $u_t-\nabla \cdot ({\partial }_t^{1-\alpha }{\kappa }_{\alpha }\nabla u-\text{F}{\partial }_t^{1-\alpha }u)$, where $0<\alpha <1$. The forcing function $\text{F}=\text{F}(t,x)$, which is more difficult to analyse than the case $\text{F}=\text{F}(x)$ 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.

20 pages, no figures