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

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.