A Fourier method for the fractional diffusion equation describing sub-diffusion

Journal of Computational Physics

Volumn 227, Issue 2, 2007, Pages 886-897

  Chen, Chang Ming ^a   Liu, F ^b,c   Turner, I ^b   Anh, V ^b  

^a XIAMEN UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^c SOUTH CHINA UNIVERSITY OF TECHNOLOGY   (China)

Author keywords

Convergence; Fourier method; Fractional diffusion equation; Implicit difference approximation; Stability; Sub diffusion

Indexed keywords

DYNAMICAL SYSTEMS; FOURIER TRANSFORMS; PARTIAL DIFFERENTIAL EQUATIONS;

APPROXIMATION SCHEME; CONVERGENCE; EXACT SOLUTION; FOURIER METHODS; FRACTIONAL DIFFUSION EQUATION; FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS; IMPLICIT DIFFERENCE APPROXIMATIONS; ORDER OF CONVERGENCE; STABILITY AND CONVERGENCE; SUB-DIFFUSION;

DIFFUSION;

EID: 36149001420     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2007.05.012     Document Type: Article

References (25)

1. Metzler R., and Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000) 1-77
2. Gorenflo R., and Mainardi F. Fractional calculus: integral and differential equations of fractional order. In: Carpinteri A., and Mainardi F. (Eds). Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer Verlag, Wein and New York 223-276
3. Mainardi F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Soliton Fract. 7 9 (1996) 1461-1477
4. Podlubny I. Fractional Differential Equations (1999), Academic Press, New York
5. Gorenflo R., Mainardi F., Moretti D., and Paradisi P. Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29 (2002) 129-143
6. Gorenflo R., and Mainardi F. Random walk models for space fractional diffusion processes. Fractional Calc. Appl. Anal. 1 (1998) 167-191
7. Mainardi F., Luchko Yu., and Pagnini G. The fundamental solution of the space-time fractional diffusion equation. Fractional Calc. Appl. Anal. 4 2 (2001) 153-192
8. Benson D.A., Wheatcraft S.W., and Meerschaert M.M. Application of a fractional advection-dispersion equation. Water Resour. Res. 36 6 (2000) 1403-1412
9. Benson D.A., Wheatcraft S.W., and Meerschaert M.M. The fractional-order governing equation of Levy motion. Water Resour. Res. 36 6 (2000) 1413-1423
10. Wyss W. The fractional diffusion equation. J. Math. Phys. 27 (1986) 2782-2785
11. Schneider W.R., and Wyss W. Fractional diffusion and wave equations. J. Math. Phys. 30 (1989) 134-144
12. Huang F., and Liu F. The time fractional diffusion and advection-dispersion equation. ANZIAM J. 46 (2005) 317-330
13. Langlands T.A.M., and Henry B.I. The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comp. Phys. 205 (2005) 719-736
14. Yuste S.B., and 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 (2005) 1862-1874
15. Liu F., Anh V., and Turner I. Numerical solution of the space fractional Fokker-Planck equation. J. Comp. Appl. Math. 166 (2004) 209-219
16. Liu F., Anh V., Turner I., and Zhuang P. Numerical simulation for solute transport in fractal porous media. ANZIAM J. 45 E (2004) 461-473
17. Meerschaert M., and Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J. Comp. Appl. Math. 172 (2004) 65-77
18. Shen S., and Liu F. Error analysis of an explicit finite difference approximation for the space fractional diffusion. ANZIAM J. 46 E (2005) 871-887
19. Liu F., Shen S., Anh V., and Turner I. Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation. ANZIAM J. 46 E (2005) 488-504
20. 2. J. Comput. Appl. Math. 193 1 (2006) 243-268
21. Liu Q., Liu F., Turner I., and Anh V. Approximation of the Levy-Feller advection-dispersion process by random walk and finite difference method. J. Phys. Comp. 222 (2007) 57-70
22. Zhuang P., and Liu F. Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 22 3 (2006) 87-99
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.
24. Zhang H., Liu F., and Anh V. Numerical approximation of Levy-Feller diffusion equation and its probability interpretation. J. Comput. Appl. Math. 206 (2007) 1098-1115
25. So F., and Liu K.L. A study of the subdiffusive fractional Fokker-Plank equation of bistable systems. Physica A 331 (2004) 378-390