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
P. Zhuang, F. Liu, V. Anh, I. Turner; Stability and convergence of an implicit numerical method for the non-linear fractional reaction–subdiffusion process. IMA J Appl Math 2009; 74 (5): 645-667. doi: 10.1093/imamat/hxp015
In this paper, we consider the following non-linear fractional reaction–subdiffusion process (NFR-SubDP):
where f(u, x, t) is a linear function of u, the function g(u, x, t) satisfies the Lipschitz condition and 0Dt1–γ is the Riemann–Liouville time fractional partial derivative of order 1 – γ. 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.