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 − ∇· (∂ _t ¹ − ^α κα∇u− F∂ _t ¹ − ^α u), 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 ⊂ R ^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 u ₀ lies in L ² (Ω). For 1/2 < α < 1 and u0 ∈ H ² (Ω) ∩ H ₀ ¹ (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.