New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation

SIAM Journal on Numerical Analysis

Volumn 46, Issue 2, 2008, Pages 1079-1095

  Zhuang, P ^a   Liu, F ^b,c   Anh, V ^a   Turner, I ^a  

^a XIAMEN UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^c SOUTH CHINA UNIVERSITY OF TECHNOLOGY   (China)

Author keywords

Anomalous subdiffusion equation; Convergence; Fractional integro differential equation; Implicit numerical method; Stability

Indexed keywords

DIFFERENTIAL EQUATIONS; DIGITAL ARITHMETIC; ELECTRIC NETWORK ANALYSIS; FLOW PATTERNS; INTEGRAL EQUATIONS; INTEGRODIFFERENTIAL EQUATIONS; NUMBER THEORY; NUMERICAL METHODS; SEMICONDUCTOR DOPING;

ANALYTICAL TECHNIQUES; ANOMALOUS DIFFUSIONS; ANOMALOUS SUBDIFFUSION EQUATION; CONVERGENCE; DIMENSIONAL PROBLEMS; ENERGY METHODS; FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION; FRACTIONAL ORDERS; GENERALIZED DIFFUSION EQUATIONS; IMPLICIT NUMERICAL METHOD; MATHEMATICAL APPROACHES; NEW SOLUTIONS; NUMERICAL EXAMPLES; NUMERICAL RESULTS; ORDER OF CONVERGENCES; SOLUTION TECHNIQUES; STABILITY AND CONVERGENCES; SUBDIFFUSION; THEORETICAL ANALYSES;

CONVERGENCE OF NUMERICAL METHODS;

EID: 55549107511     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/060673114     Document Type: Article

References (27)

1. R. METZLER AND J. KLAFTER, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000) pp. 1-77.
2. R. KUTNER, A. PEKALSKI, AND K. SZNAJD-WERON, Anomalous Diffusion, Basics to Application, Springer, Berlin, 1999.
3. S.B. YUSTE AND L. ACEDO, Some exact results for the trapping of subdiffusive particles in one dimension, Phys. A, 336 (2004), pp. 334-346.
4. S.B. YUSTE AND L. ACEDO, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42 (2005), pp. 1862-1874.
5. T.A.M. LANGLANDS AND B.I. HENRY, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205 (2005), pp. 719-736.
6. W.R. SCHNEIDER AND W. WYSS, Fractional diffusion and wave equations, J. Math. Phys., 30 (1989), pp. 134-144.
7. F. HUANG AND F. LIU, The time fractional diffusion and advection-dispersion equation, ANZIAM J., 46 (2005), pp. 317-330.
8. F. MAINARDI, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9 (1996), pp. 23-28
9. V.V. ANH AND N.N. LEONENKO, Spectral analysis of fractional kinetic equations with random data, J. Stat. Phys., 104 (2001), pp. 1349-1387.
10. R. GORENFLO, A. ISKENDEROV, AND YU. LUCHKO, Mapping between solutions of fractional diffusion-wave equations, Fract. Calc. Appl. Math., 3 (2000), pp. 75-86.
11. A.A. KILBAS, M. SAIGO, AND R.K. SAXENA, Solutions of Volterra integro-differential equations with generalised Mittag-Lefler function in the kernels, J. Integral Equations, 14 (2002), pp. 377-396.
12. F. LIU, V. ANH, AND I. TURNER, Numerical Solution of the Space Fractional Fokker-Planck Equation, J. Comput. Appl. Math., 166 (2004), pp. 209-219.
13. F. LIU, V. ANH, I. TURNER, AND P. ZHUANG, Numerical simulation for solute transport in fractal porous media, ANZIAM J., 45 (2004), pp. 461-473.
14. W. MCLEAN, I. H. SLOAN, AND V. THOMÉE, TIME DISCRETIZATION VIA LAPLACE TRANSFORMATION OF AN INTEGRO-DIFFERENTIAL EQUATION OF PARABOLIC TYPE, NUMER. MATH., 102 (2006), PP. 497-522.
15. W. MCLEAN, K. MUSTRAPHA, AND V. THOMEE, Numerical methods for some fractional-order partial differential equations, in Proceedings of the International Symposium on Fractional Calculus, Dunedin, University of Otago, 2006, pp. 9-13.
16. 2, J. Comput. Appl. Math., 193 (2006), pp. 243-268.
17. M. MEERSCHAERT AND C. TADJERAN, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), pp. 65-77.
18. S. SHEN AND F. LIU, Error analysis of an explicit finite difference approximation for the space fractional diffusion, ANZIAM J., 46 (2005), pp. 871-887.
19. Q. LIU, F. LIU, I. TURNER, AND V. ANH, Approximation of the Levy-Feller advection-dispersion process by random walk and finite difference method, J. Phys. Comp., 222 (2007), pp. 57-70.
20. H. ZHANG, F. LIU, AND V. ANH, Numerical approximation of Levy-Feller diffusion equation and its probability interpretation, J. Comput. Appl. Math., 206 (2007), pp. 1098-1115.
21. F. LIU, S. SHEN, V. ANH, AND I. TURNER, Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, ANZIAM J., 46 (2005), pp. 488-504.
22. P. ZHUANG AND F. LIU, Implicit difference approximation for the time fractional diffusion equation, J. Appl. Math. Comput., 22 (2006), pp. 87-99.
23. F. LIU, P. ZHUANG, V. ANH, I. TURNER, AND K. BURRAGE, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput., 191 (2007), pp. 12-20.
24. R. LIN AND F. LIU, Fractional high order methods for the nonlinear fractional ordinary differential equation, Nonlinear Anal., 66 (2007), pp. 856-869.
25. X. CAO, K. BURRAGE, AND F. ABDULLAH, A variable stepsize implementation for fractional differential equations, BIT, submitted.
26. K.B. OLDHAM AND J. SPANIER, The Fractional Calculus, Academic Press, New York and London, 1974.
27. I. SOKOLOV, J. KLAFTER, AND A. BLUMEN, Fractional Kinetics, Physics Today, November, (2002), pp. 48-54; also available online from http://www.physicstoday.org.