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IMA Journal of Applied Mathematics Advance Access published online on April 6, 2009
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxp015
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© The Author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Stability and convergence of an implicit numerical method for the non-linear fractional reaction–subdiffusion process

P. Zhuang

School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China

F. Liu{dagger}

School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia and School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, People's Republic of China

V. Anh and I. Turner

School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia

{dagger} Email: f.liu{at}qut.edu.au, fwliu{at}xmu.edu.cn

Received on July 22, 2008; Revision received December 21, 2008. Accepted on February 24, 2009

In this paper, we consider the following non-linear fractional reaction–subdiffusion process (NFR-SubDP):

Formula
where f(u, x, t) is a linear function of u, the function g(u, x, t) satisfies the Lipschitz condition and 0Dt1–{gamma} is the Riemann–Liouville time fractional partial derivative of order 1 – {gamma}. We propose a new computationally efficient numerical technique to simulate the process. Firstly, the NFR-SubDP is decoupled, which is equivalent to solving a non-linear fractional reaction–subdiffusion equation (NFR-SubDE). Secondly, we propose an implicit numerical method to approximate the NFR-SubDE. Thirdly, the stability and convergence of the method are discussed using a new energy method. Finally, some numerical examples are presented to show the application of the present technique. This method and supporting theoretical results can also be applied to fractional integrodifferential equations.

Keywords: fractional reaction–subdiffusion equation; implicit numerical method; convergence and stability; energy method.


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