TY - JOUR
T1 - Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing
AU - Le, Kim Ngan
AU - McLean, William
AU - Stynes, Martin
PY - 2019/9/1
Y1 - 2019/9/1
N2 - 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.
AB - 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.
KW - Fokker–Planck equation
KW - Regularity of solution
KW - Riemann–Liouville derivative
KW - Time-fractional
UR - http://www.scopus.com/inward/record.url?scp=85064243451&partnerID=8YFLogxK
U2 - 10.3934/cpaa.2019124
DO - 10.3934/cpaa.2019124
M3 - Article
AN - SCOPUS:85064243451
SN - 1534-0392
VL - 18
SP - 2765
EP - 2787
JO - Communications on Pure and Applied Analysis
JF - Communications on Pure and Applied Analysis
IS - 5
ER -