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