A finite difference method for an initial-boundary value problem with a Riemann-Liouville-Caputo spatial fractional derivative

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:381

Article

Luis Gracia, Jose^1,2  Stynes, Martin³ 

[1] Univ Zaragoza, IUMA, Zaragoza, Spain

[2] Univ Zaragoza, Dept Appl Math, Zaragoza, Spain

[3] Beijing Computat Sci Res Ctr, Appl & Computat Math Div, Beijing 100193, Peoples R China

关键词: Fractional differential equation;    Time-dependent problem;    Riemann-Liouville-Caputo fractional derivative;    Weak singularity;    Discrete comparison principle;    Steady-state problem;   

DOI  :  10.1016/j.cam.2020.113020

来源: Elsevier

【 摘 要 】

An initial-boundary value problem with a Riemann-Liouville-Caputo space fractional derivative of order alpha is an element of ( 1, 2) is considered, where the boundary conditions are reflecting. A fractional Friedrichs' inequality is derived and is used to prove that the problem approaches a steady-state solution when the source term is zero. The solution of the general problem is approximated using a finite difference scheme defined on a uniform mesh and the error analysis is given in detail for typical solutions which have a weak singularity near the spatial boundary x = 0. It is proved that the scheme converges with first order in the maximum norm. Numerical results are given that corroborate our theoretical results for the order of convergence of the difference scheme, the approach of the solution to steady state, and mass conservation. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

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