Novel numerical methods for timespace fractional reaction diffusion equations in two dimensions
Yang, Qianqian, Moroney, Timothy J., Burrage, Kevin, Turner, Ian, & Liu, Fawang (2011) Novel numerical methods for timespace fractional reaction diffusion equations in two dimensions. Australian and New Zealand Industrial and Applied Mathematics Journal, 52, C395C409.

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Abstract
We consider timespace fractional reaction diffusion equations in two dimensions. This equation is obtained from the standard reaction diffusion equation by replacing the first order time derivative with the Caputo fractional derivative, and the second order space derivatives with the fractional Laplacian. Using the matrix transfer technique proposed by Ilic, Liu, Turner and Anh [Fract. Calc. Appl. Anal., 9:333349, 2006] and the numerical solution strategy used by Yang, Turner, Liu, and Ilic [SIAM J. Scientific Computing, 33:11591180, 2011], the solution of the timespace fractional reaction diffusion equations in two dimensions can be written in terms of a matrix function vector product $f(A)b$ at each time step, where $A$ is an approximate matrix representation of the standard Laplacian. We use the finite volume method over unstructured triangular meshes to generate the matrix $A$, which is therefore nonsymmetric. However, the standard Lanczos method for approximating $f(A)b$ requires that $A$ is symmetric. We propose a simple and novel transformation in which the standard Lanczos method is still applicable to find $f(A)b$, despite the loss of symmetry. Numerical results are presented to verify the accuracy and efficiency of our newly proposed numerical solution strategy.
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ID Code:  46269 

Item Type:  Journal Article 
Refereed:  Yes 
Additional Information:  This paper first given at a conference: The 15th Biennial Computational Techniques and Applications Conference CTAC2010, UNSW, Sydney, 28 Nov  1 Dec 2010 
Keywords:  timespace fractional reaction diffusion equation, finite volume method, matrix transfer technique, Lanczos method 
ISSN:  14468735 
Subjects:  Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Biological Mathematics (010202) Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > NUMERICAL AND COMPUTATIONAL MATHEMATICS (010300) > Numerical Solution of Differential and Integral Equations (010302) 
Divisions:  Past > QUT Faculties & Divisions > Faculty of Science and Technology Past > Schools > Mathematical Sciences 
Copyright Owner:  Copyright 2011 Australian Mathematical Society 
Deposited On:  29 Sep 2011 23:35 
Last Modified:  29 Sep 2011 23:36 
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