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Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation. (English) Zbl 1144.65057
This paper deals with a fractional reaction-subdiffusion equation (FR-subDE) in a bounded domain in which both the motion and the reaction terms are affected by the subdiffusive character of the process. An implicit finite difference method (IFDM) and an explicit finite difference method (EFDM) for the FR-subDE are proposed. The stability and the convergence of the IFDM and EFDM are discussed using a Fourier analysis and the solvability of the IFDM is proved. A comparison between the exact solution and the two numerical solutions shows the agreement of the theoretical analysis with the numerical results.

##### MSC:
 65M06 Finite difference methods (IVP of PDE) 65M12 Stability and convergence of numerical methods (IVP of PDE) 35K57 Reaction-diffusion equations
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##### References:
 [1] Gorenflo, R.; Mainardi, F.: Random walk models for space-fractional diffusion processes. Fract. cal. Appl. anal. 1, 167-191 (1998) · Zbl 0946.60039 [2] Giona, M.; Roman, H. E.: Fractional diffusion equation for transport phenomena in random media. Physica A 185, 87-97 (1992) [3] Liu, F.; Anh, V.; Turner, I.: Numerical solution of the space fractional Fokker -- Planck equation. J. comp. Appl. math. 166, 209-219 (2004) · Zbl 1036.82019 [4] Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. rep. 339, 1-77 (2000) · Zbl 0984.82032 [5] Seki, K.; Wojcik, M.; Tachiya, M.: Fractional reaction -- diffusion equation. J. chem. Phys. 119, 2165-2174 (2003) [6] Henry, B. I.; Wearne, S. L.: Fractional reaction -- diffusion. Physica A 276, 448-455 (2000) [7] Cao, X.; Burrage, K.; Abdullah, F.: A variable stepsize implementation for fractional differential equations. Report (2006) [8] Yuste, S. B.; Acedo, L.; Lindenberg, K.: Reaction front in an A+B$\to C$ reaction -- subdiffusion process. Phys. rev. E 69, 036126 (2004) [9] Yuste, S. B.; Lindenberg, K.: Subdiffusion-limited A+A reactions. Phys. rev. Lett. 87, No. 11, 118301 (2001) [10] Liu, F.; Anh, V.; Turner, I.; Zhuang, P.: Numerical simulation for solute transport in fractal porous media. Anziam j. 45, No. E, 461-473 (2004) · Zbl 1123.76363 [11] Fix, G. J.; Roop, J. P.: Least squares finite element solution of a fractional order two-point boundary value problem. Comput. math. Appl. 48, 1017-1033 (2004) · Zbl 1069.65094 [12] Roop, J. P.: Computational aspects of FEM approximation of fractional advection dispersion equation on bounded domains in R2. J. comp. Appl. math. 193, No. 1, 243-268 (2006) · Zbl 1092.65122 [13] Meerschaert, M.; Tadjeran, C.: Finite difference approximations for fractional advection -- dispersion flow equations. J. comp. Appl. math. 172, 65-77 (2004) · Zbl 1126.76346 [14] Shen, S.; Liu, F.: Error analysis of an explicit finite difference approximation for the space fractional diffusion. Anziam j. 46, No. E, 871-887 (2005) [15] Liu, Q.; Liu, F.; Turner, I.; Anh, V.: Approximation of the Lévy -- Feller advection -- dispersion process by random walk and finite difference method. J. comp. Phys. 222, 57-70 (2007) · Zbl 1112.65006 [16] Liu, F.; Shen, S.; Anh, V.; Turner, I.: Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation. Anziam j. 46, No. E, 488-504 (2005) [17] Zhuang, P.; Liu, F.: Implicit difference approximation for the time fractional diffusion equation. J. appl. Math. comput. 22, No. 3, 87-99 (2006) · Zbl 1140.65094 [18] Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K.: Stability and convergence of the difference methods for the space -- time fractional advection -- diffusion equation. Appl. math. Comp. 191, No. 1, 12-20 (2007) · Zbl 1193.76093 [19] S. Shen, F. Liu, V. Anh, I. Turner, The functional solution and numerical solution of the Riesz fractional advection -- dispersion equation, IMA J. Appl. Math. (2007), in press. · Zbl 1179.37073 [20] Lin, R.; Liu, F.: Fractional high order methods for the nonlinear fractional ordinary differential equation. Nonlinear anal. 66, No. 4, 856-869 (2007) · Zbl 1118.65079 [21] Q. Yu, F. Liu, V. Anh, I. Turner, Solving linear and nonlinear space -- time fractional reaction -- diffusion equations by Adomian decomposition method, Int. J. Numer. Meth. Eng. (2007), in press. · Zbl 1159.76367 [22] Yuste, S. B.; Acedo, L.: An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. anal. 42, No. 5, 1862-1874 (2005) · Zbl 1119.65379 [23] Langlands, T. A. M.; Henry, B. I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. comp. Phys. 205, 719-736 (2005) · Zbl 1072.65123 [24] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008 [25] Lubich, Ch.: Discretized fractional calculus. SIAM J. Math. anal. 17, 704-719 (1986) · Zbl 0624.65015 [26] Chang-Ming Chen, F. Liu, I. Turner, V. Anh, Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comp. Phys. (2007), in press, doi:10.1016/j.jcp.2007.05.012. · Zbl 1165.65053 [27] P. Zhuang, F. Liu, V. Anh, I. Turner, New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation, SIAM J. Numer. Anal. (2007), in press.