Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative

Journal of Computational Physics

Volumn 231, Issue 4, 2012, Pages 1743-1750

  Çelik, Cem ^a   Duman, Melda ^a  

^a DOKUZ EYLUL UNIVERSITY   (Turkey)

Author keywords

Crank Nicolson method; Fractional centered difference; Riesz fractional derivative operator

Indexed keywords

DIFFUSION; PARTIAL DIFFERENTIAL EQUATIONS;

CRANK-NICOLSON METHODS; FINITE DOMAINS; FRACTIONAL CENTERED DIFFERENCE; FRACTIONAL DERIVATIVE OPERATOR; FRACTIONAL DERIVATIVES; FRACTIONAL DIFFUSION EQUATION; NUMERICAL RESULTS; RIESZ FRACTIONAL DERIVATIVE OPERATOR; SECOND-ORDER ACCURACY; UNCONDITIONALLY STABLE;

NUMERICAL METHODS;

EID: 84855207635     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2011.11.008     Document Type: Article

References (23)

1. Schneider W.R., Wyss W. Fractional diffusion and wave equations. Journal of Mathematical Physics 1989, 30(1):134-144.
2. Wyss W. The fractional diffusion equation. Journal of Mathematical Physics 1986, 27(11):2782-2785.
3. Metzler R., Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Physics Reports 2000, 339(1):1-77.
4. Metzler R., Klafter J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics A: Mathematical and General 2004, 37:R161.
5. Sabatier J., Agrawal O.P., Machado J.A.T. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering 2007, Springer, Netherlands.
6. Magin R.L. Fractional Calculus in Bioengineering 2006, Begell House Publisher., Inc., Connecticut.
7. Scalas E., Gorenflo R., Mainardi F. Fractional calculus and continuous-time finance. Physica A: Statistical Mechanics and its Applications 2000, 284(1-4):376-384.
8. Gorenflo R., Mainardi F. Random walk models for space-fractional diffusion processes. Fractional Calculus and Applied Analysis 1998, 1(2):167-191.
9. S.G. Samko, A.A. Kilbas, O.I. Marichev. Fractional integrals and derivatives: theory and applications. Transl. from the Russian. New York, Gordon and Breach, 1993.
10. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations 2006, Elsevier Science Ltd, Amsterdam.
11. Podlubny I. Fractional Differential Equations 1999, Academic press, New York.
12. Tuan V.K., Gorenflo R. Extrapolation to the limit for numerical fractional differentiation. Zeitschrift für angewandte Mathematik und Mechanik 1995, 75(8):646-648.
13. Meerschaert M.M., Tadjeran C. Finite difference approximations for two-sided space-fractional partial differential equations. Applied Numerical Mathematics 2006, 56(1):80-90.
14. Tadjeran C., Meerschaert M.M., Scheffler H.P. A second-order accurate numerical approximation for the fractional diffusion equation. Journal of Computational Physics 2006, 213(1):205-213.
15. Shen S., Liu F., Anh V. Numerical approximations and solution techniques for the space-time Riesz-caputo fractional advection-diffusion equation. Numerical Algorithms 2011, 56:383-403.
16. Shen S., Liu F., Anh V., Turner I. The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation. IMA Journal of Applied Mathematics 2008, 73:850-872.
17. Zhuang P., Liu F., Anh V., Turner I. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM Journal on Numerical Analysis 2009, 47(3):1760-1781.
18. Yang Q., Liu F., Turner I. Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Applied Mathematical Modelling 2010, 34(1):200-218.
19. Yang Q., Turner I., Liu F. Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation. Australian and New Zealand Industrial and Applied Mathematics Journal (ANZIAM) 2009, 50:C800-C814.
20. Zhang H., Liu F. Numerical simulation of the Riesz fractional diffusion equation with a nonlinear source term. Journal of Applied Mathematics and Informatics 2008, 26(1-2):1-14.
21. Zhang H., Liu F., Anh V. Galerkin finite element approximation of symmetric space-fractional partial differential equations. Applied Mathematics and Computation 2010, 217(6):2534-2545.
22. Ortigueira M.D. Riesz potential operators and inverses via fractional centred derivatives. International Journal of Mathematics and Mathematical Sciences 2006, 1-12.
23. Kincaid D., Cheney W. Numerical Analysis 1991, Brooks/Cole Pub., California.