Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation

Various fields of science and engineering deal with dynamical systems that can be described by fractional partial differential equations (FPDE), for example, systems biology, chemistry and biochemistry applications due to anomalous diffusion effects in constrained environments. However, effective numerical methods and numerical analysis for FPDE are still in their infancy. In this paper, we consider a fractional reaction-subdiffusion equation (FR-subDE) in which both the motion and the reaction terms are affected by the subdiffusive character of the process. Using the relationship between the Riemann-Liouville and Grünwald-Letnikov definitions of fractional derivatives, an implicit and an explicit difference methods for the FRsubDE are presented. The stability and the convergence of the two numerical methods are investigated by a Fourier analysis. The solvability of the implicit finite difference method is also proved. The high-accuracy algorithm is structured using Richardson extrapolation. Finally, a comparison between the exact solution and the two numerical solutions is given. The numerical results are in excellent agreement with our theoretical analysis.