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 uₜ-∇·(∂ₜ1-ακ_α∇ u -F∂ₜ1-αu), where 0<α <1. The forcing function F = F(t,x), which is more difficult to analyse than the case F = F(x) investigated previously by other authors. The spatial domain Ω ᵈ, where d≥ 1, has a smooth boundary. Existence, uniqueness and regularity of a mild solution u is proved under the hypothesis that the initial data u₀ lies in L²(Ω). For 1/2<α<1 and u₀∈ H²(Ω)∩ H₀¹(Ω), 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.