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 ut − ∇ • (∂ 1−α t κα∇u − F∂ 1−α t 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 u 0 ∈ H 2 (Ω) ∩ H 1 0 (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. 2000 Mathematics Subject Classification. 35R11.

Regularity theory for time-fractional advection–diffusion–reaction equations

“…Mu, Ahmad and Huang [23] obtain analogous estimates using weighted Hölder norms. Recently, Le et al [24] studied (1) for the case G = 0 and a = b = 0, with F = F(x,t). One of their regularity results [24,Theorem 7.3] gives the bound ∂ m t u ≤ Ct −m+1/2 u 0 H 2 (Ω) when g = 0, subject to the restriction 1/2 < α < 1.…”

“…Recently, Le et al [24] studied (1) for the case G = 0 and a = b = 0, with F = F(x,t). One of their regularity results [24,Theorem 7.3] gives the bound ∂ m t u ≤ Ct −m+1/2 u 0 H 2 (Ω) when g = 0, subject to the restriction 1/2 < α < 1. The next section gathers together some technical preliminaries needed for our analysis, which uses delicate energy arguments, a fractional Gronwall inequality and several properties of fractional integrals to prove a priori estimates for the weak solution u of (1)- (3).…”

Implementation of High-Order, Discontinuous Galerkin Time Stepping for Fractional Diffusion Problems

“…The initial-boundary value problem (1.1) is known to be well posed [6,8,11]. Let u, v = Ω uv denote the usual inner product in L 2 (Ω), and let a(u, v) denote the bilinear form associated with A via the first Green identity.…”

Implementation of high-order, discontinuous Galerkin time stepping for fractional diffusion problems

“…The initial-boundary value problem ( 1) is known to be well-posed [5,8,10]. Let u, v = Ω uv denote the usual inner product in L 2 (Ω), and let a(u, v) denote the bilinear form associated with A via the first Green identity.…”