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A Fourier method for the fractional diffusion equation describing sub-diffusion. (English) Zbl 1165.65053
The paper focuses on the initial-boundary value problem of the fractional diffusion equation describing sub-diffusion. A brief review of relevant literature is included, thus setting the paper in context. A paper by T. A. M. Langlands and B. I. Henry [J. Comput. Phys. 205, No. 2, 719–736 (2005; Zbl 1072.65123)] in which they proposed an implicit numerical scheme (\(L_1\) approximation) and discussed its accuracy and stability, is referred to. The motivation behind this paper is to build on this work, by deriving the global accuracy of the presented implicit scheme and establishing unconditional stability for all \(\lambda\) in the range \(0<\lambda \leq 1\). A Fourier method is used. In Section 2 an implicit difference approximation scheme (IDAS) is presented and its unconditional stability and \(L_2\)-convergence are investigated in Sections 3 and 4. In Section 5 the implicit difference scheme is written in matrix form and is proved to be uniquely solvable. The paper concludes with two numerical examples, each describing sub-diffusion (one wit a non-homogeneous term and the second with a homogeneous term), to confirm their theoretical results. The examples demonstrate that the IDAS is unconditionally stable and convergent ant that it can be applied to simulate fractional dynamical systems. The authors state that the Fourier method technique used to analyse stability and convergence can be extended to other fractional partial differential equations.
Reviewer: Pat Lumb (Chester)

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35B35 Stability in context of PDEs
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[1] 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
[2] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (), 223-276
[3] Mainardi, F., Fractional relaxation – oscillation and fractional diffusion-wave phenomena, Chaos soliton fract., 7, 9, 1461-1477, (1996) · Zbl 1080.26505
[4] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[5] Gorenflo, R.; Mainardi, F.; Moretti, D.; Paradisi, P., Time fractional diffusion: a discrete random walk approach, Nonlinear dyn., 29, 129-143, (2002) · Zbl 1009.82016
[6] Gorenflo, R.; Mainardi, F., Random walk models for space fractional diffusion processes, Fractional calc. appl. anal., 1, 167-191, (1998) · Zbl 0946.60039
[7] Mainardi, F.; Luchko, Yu.; Pagnini, G., The fundamental solution of the space – time fractional diffusion equation, Fractional calc. appl. anal., 4, 2, 153-192, (2001) · Zbl 1054.35156
[8] Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., Application of a fractional advection – dispersion equation, Water resour. res., 36, 6, 1403-1412, (2000)
[9] Benson, D.A.; Wheatcraft, S.W.; Meerschaert, M.M., The fractional-order governing equation of levy motion, Water resour. res., 36, 6, 1413-1423, (2000)
[10] Wyss, W., The fractional diffusion equation, J. math. phys., 27, 2782-2785, (1986) · Zbl 0632.35031
[11] Schneider, W.R.; Wyss, W., Fractional diffusion and wave equations, J. math. phys., 30, 134-144, (1989) · Zbl 0692.45004
[12] Huang, F.; Liu, F., The time fractional diffusion and advection – dispersion equation, Anziam j., 46, 317-330, (2005) · Zbl 1072.35218
[13] 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
[14] Yuste, S.B.; Acedo, L., An explicit finite difference method and a new von neumman-type stability analysis for fractional diffusion equations, SIAM J. numer. anal., 42, 5, 1862-1874, (2005) · Zbl 1119.65379
[15] 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
[16] Liu, F.; Anh, V.; Turner, I.; Zhuang, P., Numerical simulation for solute transport in fractal porous media, Anziam j., 45, E, 461-473, (2004) · Zbl 1123.76363
[17] Meerschaert, M.; Tadjeran, C., Finite difference approximations for fractional advection – dispersion flow equations, J. comp. appl. math., 172, 65-77, (2004) · Zbl 1126.76346
[18] Shen, S.; Liu, F., Error analysis of an explicit finite difference approximation for the space fractional diffusion, Anziam j., 46, E, 871-887, (2005)
[19] 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, E, 488-504, (2005) · Zbl 1082.60511
[20] Roop, J.P., Computational aspects of FEM approximation of fractional advection dispersion equations on boundary domains in R2, J. comput. appl. math., 193, 1, 243-268, (2006) · Zbl 1092.65122
[21] Liu, Q.; Liu, F.; Turner, I.; Anh, V., Approximation of the levy – feller advection – dispersion process by random walk and finite difference method, J. phys. comp., 222, 57-70, (2007) · Zbl 1112.65006
[22] Zhuang, P.; Liu, F., Implicit difference approximation for the time fractional diffusion equation, J. appl. math. comput., 22, 3, 87-99, (2006) · Zbl 1140.65094
[23] F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage , Stability and convergence of the difference methods for the space – time fractional advection – diffusion equation, Appl. Math. Comput., (2007), in press. · Zbl 1193.76093
[24] Zhang, H.; Liu, F.; Anh, V., Numerical approximation of levy – feller diffusion equation and its probability interpretation, J. comput. appl. math., 206, 1098-1115, (2007) · Zbl 1125.26014
[25] So, F.; Liu, K.L., A study of the subdiffusive fractional fokker – plank equation of bistable systems, Physica A, 331, 378-390, (2004)
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