Projects per year
Abstract
A time-fractional initial-boundary value problem of Fokker–Planck type is considered on the space-time domain Ω × [0 , T] , where Ω is an open bounded domain in Rd for some d≥ 1 , and the order α(x) of the Riemann-Liouville fractional derivative may vary in space with 1 / 2 < α(x) < 1 for all x. Such problems appear naturally in the formulation of certain continuous-time random walk models. Uniqueness of any solution u of the problem is proved under reasonable hypotheses. A semidiscrete numerical method, using finite elements in space to yield a solution uh(t) , is constructed. Error estimates for ‖(u-uh)(t)‖L2(Ω) and ∫0t|∂t1-α(u-uh)(s)|12ds are proved for each t∈ [0 , T] under the assumptions that the following quantities are finite: ‖u(·,0)‖H2(Ω),|u(·,t)|H1(Ω) for each t, and ∫0t[‖u(·,t)‖H2(Ω)2+|∂t1-αu|H2(Ω)2], where u(x, t) is the unknown solution. Furthermore, these error estimates are α-robust: they do not fail when α→ 1 , the classical Fokker–Planck problem. Sharper results are obtained for the special case where the drift term of the problem is not present (which is of interest in certain applications).
Original language | English |
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Article number | 22 |
Number of pages | 16 |
Journal | Journal of Scientific Computing |
Volume | 86 |
Issue number | 2 |
DOIs | |
Publication status | Published - Feb 2021 |
Keywords
- Fokker–Planck equation
- Variable-order fractional derivative
- α-robust
Projects
- 1 Finished
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Discrete functional analysis: bridging pure and numerical mathematics
Droniou, J., Eymard, R. & Manzini, G.
Australian Research Council (ARC), Monash University, Université Paris-Est Créteil Val de Marne (Paris-East Créteil University Val de Marne), University of California System
1/01/17 → 31/12/20
Project: Research