Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions

Yang, Qianqian, Turner, Ian, Liu, Fawang and Ilić, Milos (2011) Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions. SIAM Journal on Scientific Computing, 33 3: 1159-1180. doi:10.1137/100800634

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Author Yang, Qianqian
Turner, Ian
Liu, Fawang
Ilić, Milos
Title Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions
Journal name SIAM Journal on Scientific Computing   Check publisher's open access policy
ISSN 1064-8275
Publication date 2011-05-01
Sub-type Article (original research)
DOI 10.1137/100800634
Open Access Status File (Publisher version)
Volume 33
Issue 3
Start page 1159
End page 1180
Total pages 22
Place of publication Philadelphia, PA, United States
Publisher Society for Industrial and Applied Mathematics
Language eng
Formatted abstract
In this paper, a time-space fractional diffusion equation in two dimensions (TSFDE-2D) with homogeneous Dirichlet boundary conditions is considered. The TSFDE-2D is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo fractional derivative tDγ, γ ∈ (0,1), and the second-order space derivatives with the fractional Laplacian −(−Δ)α/2, α ∈ (1,2]. Using the matrix transfer technique proposed by Ilić et al. [M. Ilić, F. Liu, I. Turner, and V. Anh, Fract. Calc. Appl. Anal., 9 (2006), pp. 333-349], the TSFDE-2D is transformed into a time fractional differential system as tDγu = −KαAα/2u, where A is the approximate matrix representation of (−Δ). Traditional approximation of Aα/2 requires diagonalization of A, which is very time-consuming for large sparse matrices. The novelty of our proposed numerical schemes is that, using either the finite difference method or the Laplace transform to handle the Caputo time fractional derivative, the solution of the TSFDE-2D is written in terms of a matrix function vector product f(A)b at each time step, where b is a suitably defined vector. Depending on the method used to generate the matrix A, the product f(A)b can be approximated using either the preconditioned Lanczos method when A is symmetric or the M-Lanzcos method when A is nonsymmetric, which are powerful techniques for solving large linear systems. We give error bounds for the new methods and illustrate their roles in solving the TSFDE-2D. We also derive the analytical solution of the TSFDE-2D in terms of the Mittag-Leffler function. Finally, numerical results are presented to verify the proposed numerical solution strategies.
Keyword Fractional calculus
Caputo fractional derivative
Fractional Laplacian
Finite difference method
Finite element method
Matrix transfer technique
Lanczos method
M-Lanczos method
Matrix function
Q-Index Code E1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Mathematics and Physics
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Created: Wed, 16 Jul 2014, 22:24:46 EST by Jon Swabey on behalf of School of Mathematics & Physics