Projects per year
Abstract
A timefractional initialboundary value problem of Fokker–Planck type is considered on the spacetime domain Ω × [0 , T] , where Ω is an open bounded domain in R^{d} for some d≥ 1 , and the order α(x) of the RiemannLiouville fractional derivative may vary in space with 1 / 2 < α(x) < 1 for all x. Such problems appear naturally in the formulation of certain continuoustime 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 u_{h}(t) , is constructed. Error estimates for ‖(uuh)(t)‖L2(Ω) and ∫0t∂t1α(uuh)(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+∂t1αuH2(Ω)2], where u(x, t) is the unknown solution. Furthermore, these error estimates are αrobust: 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 

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
 Variableorder fractional derivative
 αrobust
Projects
 1 Finished

Discrete functional analysis: bridging pure and numerical mathematics
Droniou, J., Eymard, R. & Manzini, G.
Australian Research Council (ARC), Monash University, Université ParisEst Créteil Val de Marne (ParisEast Créteil University Val de Marne), University of California System
1/01/17 → 31/12/20
Project: Research