Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation

被引:142
作者
Chen, Chang-ming [2 ]
Liu, F. [1 ,3 ]
Burrage, K. [4 ]
机构
[1] Queensland Univ Technol, Sch Math Sci, Brisbane, Qld 4001, Australia
[2] Xiamen Univ, Sch Math Sci, Xiamen 361006, Peoples R China
[3] S China Univ Technol, Sch Math Sci, Guangzhou 510640, Peoples R China
[4] Univ Queensland, Dept Math, Brisbane, Qld 4072, Australia
基金
中国国家自然科学基金; 澳大利亚研究理事会;
关键词
space-time fractional derivatives; fractional advection-diffusion equation; implicit difference method; explicit difference method; stability; convergence; Fourier analysis;
D O I
10.1016/j.amc.2007.09.020
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Various fields of science and engineering deal with dynamical systems that can be described by fractional partial differential equations (FPDE), for example, systems biology, chemistry and biochemistry applications due to anomalous diffusion effects in constrained environments. However, effective numerical methods and numerical analysis for FPDE are still in their infancy. In this paper, we consider a fractional reaction-subdiffusion equation (FR-subDE) in which both the motion and the reaction terms are affected by the subdiffusive character of the process. Using the relationship between the Riemann-Liouville and Grunwald-Letnikov definitions of fractional derivatives, an implicit and an explicit difference methods for the FR-subDE are presented. The stability and the convergence of the two numerical methods are investigated by a Fourier analysis. The solvability of the implicit finite difference method is also proved. The high-accuracy algorithm is structured using Richardson extrapolation. Finally, a comparison between the exact solution and the two numerical solutions is given. The numerical results are in excellent agreement with our theoretical analysis. (C) 2007 Elsevier Inc. All rights reserved.
引用
收藏
页码:754 / 769
页数:16
相关论文
共 27 条
[1]  
CAO X, 2006, VARIABLE STEPSIZE IM
[2]   A Fourier method for the fractional diffusion equation describing sub-diffusion [J].
Chen, Chang-Ming ;
Liu, F. ;
Turner, I. ;
Anh, V. .
JOURNAL OF COMPUTATIONAL PHYSICS, 2007, 227 (02) :886-897
[3]   Least squares finite-element solution of a fractional order two-point boundary value problem [J].
Fix, GJ ;
Roop, JP .
COMPUTERS & MATHEMATICS WITH APPLICATIONS, 2004, 48 (7-8) :1017-1033
[4]   FRACTIONAL DIFFUSION EQUATION FOR TRANSPORT PHENOMENA IN RANDOM-MEDIA [J].
GIONA, M ;
ROMAN, HE .
PHYSICA A, 1992, 185 (1-4) :87-97
[5]  
Gorenflo R., 1998, Fract. Calc. Appl. Anal., V1, P167
[6]   Fractional reaction-diffusion [J].
Henry, BI ;
Wearne, SL .
PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS, 2000, 276 (3-4) :448-455
[7]   The time fractional diffusion equation and the advection-dispersion equation [J].
Huang, F ;
Liu, F .
ANZIAM JOURNAL, 2005, 46 :317-330
[8]   The accuracy and stability of an implicit solution method for the fractional diffusion equation [J].
Langlands, TAM ;
Henry, BI .
JOURNAL OF COMPUTATIONAL PHYSICS, 2005, 205 (02) :719-736
[9]   Fractional high order methods for the nonlinear fractional ordinary differential equation [J].
Lin, R. ;
Liu, F. .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2007, 66 (04) :856-869
[10]   Call admission policies based on calculated power control setpoints in SIR-based power-controlled DS-CDMA cellular networks [J].
Liu, DR ;
Zhang, Y ;
Hu, SQ .
WIRELESS NETWORKS, 2004, 10 (04) :473-483