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

Kim Ngan Le, William McLean, Martin Stynes

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22 Citations (Scopus)


A time-fractional Fokker–Planck initial-boundary value problem is considered, with differential operator ut − ∇· (∂ t 1α κα∇u− F∂ t 1α 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.

Original languageEnglish
Pages (from-to)2765-2787
Number of pages23
JournalCommunications on Pure and Applied Analysis
Issue number5
Publication statusPublished - 1 Sept 2019


  • Fokker–Planck equation
  • Regularity of solution
  • Riemann–Liouville derivative
  • Time-fractional

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