# Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions

Yang, Qianqian, Moroney, Timothy J., Burrage, Kevin, Turner, Ian, & Liu, Fawang (2011) Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions. *Australian and New Zealand Industrial and Applied Mathematics Journal*, *52*, C395-C409.

## Abstract

We consider time-space 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., 9:333--349, 2006] and the numerical solution strategy used by Yang, Turner, Liu, and Ilic [SIAM J. Scientific Computing, 33:1159--1180, 2011], the solution of the time-space 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 non-symmetric. 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 |
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Item Type: | Journal Article |

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: | time-space fractional reaction diffusion equation, finite volume method, matrix transfer technique, Lanczos method |

ISSN: | 1446-8735 |

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: | 30 Sep 2011 09:35 |

Last Modified: | 30 Sep 2011 09:36 |

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