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 Grunwald–Letnikov definitions of fractional derivatives, an implicit and an explicit difference methods for the FR-subDE 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.

[1]  Fawang Liu,et al.  Numerical simulation for solute transport in fractal porous media , 2004 .

[2]  Fawang Liu,et al.  Fractional high order methods for the nonlinear fractional ordinary differential equation , 2007 .

[3]  S. Wearne,et al.  Fractional Reaction-Diffusion , 2000 .

[4]  Fawang Liu,et al.  New Solution and Analytical Techniques of the Implicit Numerical Method for the Anomalous Subdiffusion Equation , 2008, SIAM J. Numer. Anal..

[5]  B. Henry,et al.  The accuracy and stability of an implicit solution method for the fractional diffusion equation , 2005 .

[6]  Fawang Liu,et al.  A Fourier method for the fractional diffusion equation describing sub-diffusion , 2007, J. Comput. Phys..

[7]  Fawang Liu,et al.  Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation , 2007, Appl. Math. Comput..

[8]  J. P. Roop.  Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2 , 2006 .

[9]  Massimiliano Giona,et al.  Fractional diffusion equation for transport phenomena in random media , 1992 .

[10]  I. Turner,et al.  Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation , 2005 .

[11]  Fawang Liu,et al.  Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends , 2005 .

[12]  Katja Lindenberg,et al.  Reaction front in an A+B-->C reaction-subdiffusion process. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  C. Lubich.  Discretized fractional calculus , 1986 .

[14]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[15]  Fawang Liu,et al.  Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method , 2007, J. Comput. Phys..

[16]  G. Fix,et al.  Least squares finite-element solution of a fractional order two-point boundary value problem , 2004 .

[17]  Kazuhiko Seki,et al.  Fractional reaction-diffusion equation , 2003 .

[18]  Fawang Liu,et al.  Numerical solution of the space fractional Fokker-Planck equation , 2004 .

[19]  Santos B. Yuste,et al.  An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations , 2004, SIAM J. Numer. Anal..

[20]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[21]  Santos B. Yuste,et al.  Subdiffusion-limited A+A reactions. , 2001 .

[22]  Fawang Liu,et al.  Implicit difference approximation for the time fractional diffusion equation , 2006 .

[23]  Fawang Liu,et al.  Solving linear and non‐linear space–time fractional reaction–diffusion equations by the Adomian decomposition method , 2008 .

[24]  Fawang Liu,et al.  The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation , 2008 .