Numerical simulation for the three-dimension fractional sub-diffusion equation

Applied Mathematical Modelling

Volumn 38, Issue 15-16, 2014, Pages 3695-3705

  Chen, J ^a   Liu, F ^b   Liu, Q ^c   Chen, X ^a   Anh, V ^b   Turner, I ^b   Burrage, K ^b,d  

^a JIMEI UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^c XIAMEN UNIVERSITY   (China)

^d UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Convergence; Fractional alternating direction implicit scheme; Stability; Three dimensional fractional sub diffusion equation

Indexed keywords

ALGORITHMS; COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; NUMERICAL MODELS;

ALTERNATING DIRECTION IMPLICIT SCHEMES; CONVERGENCE; ENERGY METHOD; FRACTIONAL DERIVATIVES; MEMORY EFFECTS; ONE-DIMENSIONAL PROBLEM; THREE-DIMENSION; UNCONDITIONALLY STABLE;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 84904793753     PISSN: 0307904X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apm.2014.03.031     Document Type: Article

References (28)

1. Metzler R., Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 2000, 339:1-77.
2. Kilbas A., Srivastava H., Trujillo J. Theory and Applications of Fractional Differential Equations 2006, Elesvier Science and Technology, Boston.
3. Schneider W., Wyss W. Fractional diffusion and wave equations. J. Math. Phys. 1989, 30:134-144.
4. Mainardi F. The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 1996, 9:23-28.
5. Gorenflo R., Iskenderov A., Luchko Y. Mapping between solutions of fractional diffusion-wave equations. Fract. Calc. Appl. Math. 2000, 3:75-86.
6. Anh V., Leonenko N. Spectral analysis of fractional kinetic equations with random data. J. Stat. Phys. 2001, 104:1349-1387.
7. Agrawal O. Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dyn. 2002, 29:145-155.
8. Huang F., Liu F. The time fractional diffusion and advection-dispersion equation. ANZIAM J. 2005, 46:317-330.
9. Yuste S., Acedo L. An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 2005, 42:1862-1874.
10. Yuste S. Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 2006, 216:264-274.
11. Sun Z., Wu X. A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 2006, 56:193-209.
12. Chen C., Liu F., Turner I., Anh V. A Fourier analysis method for the fractional diffusion equation describing sub-diffusion. J. Comput. Phys. 2007, 227:886-897.
13. Deng W. Numerical algorithm for the time fractional Fokker-Planck equation. J. Comput. Phys. 2007, 227:1510-1522.
14. Deng W. Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal. 2008, 47:204-226.
15. Li X., Xu C. A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 2009, 47:2108-2131.
16. Zhuang P., Liu F., Anh V., Turner I. New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 2008, 46:1079-1095.
17. Zhuang P., Liu F., Anh V., Turner I. Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process. IMA J. Appl. Math. 2009, 74:6445-6467.
18. Liu F., Yang C., Burrage K. Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comput. Appl. Math. 2009, 231:160-176.
19. Zhuang P., Liu F., Anh V., Turner I. Numerical method for the variable-order fractional advection - diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 2009, 47:1760-1781.
20. Chen C., Liu F., Anh V., Turner I. Numerical schemes with high spatial accuracy for a variable-order anomalous sub-diffusion equation. SIAM J. Sci. Comput. 2010, 32:1740-1760.
21. Gao G., Sun Z. A compact difference scheme for the fractional sub-diffusion equations. J. Comput. Phys. 2011, 230:586-595.
22. Zhuang P., Liu F. Implicit difference approximation for the two-dimensional space-time fractional diffusion equation. J. Appl. Math. Comput. 2007, 25:269-282.
23. Zhuang P., Liu F. Finite difference approximation for two-dimensional time fractional diffusion equation. J. Algorithms Comput. Technol. 2007, 1(1):1-15.
24. Liu Q., Gu Y., Zhuang P., Liu F., Nie Y. An implicit RBF meshless approach for time fractional diffusion equations. Comput. Mech. 2001, 48(1):1-12.
25. Chen C., Liu F., Turner I., Anh V. Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation. Numer. Algorithms 2010, 54:1-21.
26. Zhang Y., Sun Z. Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation. J. Comput. Phys. 2011, 230:8713-8728.
27. Zhao X., Sun Z. A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions. J. Comput. Phys. 2011, 230:6061-6074.
28. Sun Z. Numerical Methods of Partial Differential Equations 2005, Science Press, Beijing, (in Chinese).