Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation

Liu, F., Zhuang, P., Anh, V., Turner, I. and Burrage, K. (2007) Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Applied Mathematics and Computation, 191 1: 12-20. doi:10.1016/j.amc.2006.08.162


Author Liu, F.
Zhuang, P.
Anh, V.
Turner, I.
Burrage, K.
Title Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation
Journal name Applied Mathematics and Computation   Check publisher's open access policy
ISSN 0096-3003
Publication date 2007-08-01
Sub-type Article (original research)
DOI 10.1016/j.amc.2006.08.162
Open Access Status Not Open Access
Volume 191
Issue 1
Start page 12
End page 20
Total pages 9
Place of publication New York, U.S.
Publisher Elsevier
Language eng
Subject 0102 Applied Mathematics
Formatted abstract
In this paper, we consider a space–time fractional advection dispersion equation (STFADE) on a finite domain. The STFADE is obtained from the standard advection dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α  (0, 1], and the first-order and second-order space derivatives by the Riemman–Liouville fractional derivatives of order β  (0, 1] and of order γ  (1, 2], respectively. For the space fractional derivatives  and , we adopted the Grünwald formula and the shift Grünwald formula, respectively. We propose an implicit difference method (IDM) and an explicit difference method (EDM) to solve this equation. Stability and convergence of these methods are discussed. Using mathematical induction, we prove that the IDM is unconditionally stable and convergent, but the EDM is conditionally stable and convergent. Numerical results are in good agreement with theoretical analysis.
Keyword Space–time fractional derivatives
Fractional advection–diffusion equation
Implicit difference method
Explicit difference method
Stability
Convergence
Q-Index Code C1

Document type: Journal Article
Sub-type: Article (original research)
Collections: Excellence in Research Australia (ERA) - Collection
Institute for Molecular Bioscience - Publications
 
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Created: Thu, 03 Sep 2009, 20:26:35 EST by Mr Andrew Martlew on behalf of Institute for Molecular Bioscience