A new fractional finite volume method for solving the fractional diffusion equation

Applied Mathematical Modelling

Volumn 38, Issue 15-16, 2014, Pages 3871-3878

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

^a QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^b XIAMEN UNIVERSITY   (China)

^c UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Finite volume method; Fractional diffusion equation; Nonlinear source term; Space time dependent variable coefficient; Two sided space fractional derivative

Indexed keywords

BOUNDARY CONDITIONS; NONLINEAR EQUATIONS; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; SOLUTE TRANSPORT;

DEPENDENT VARIABLES; DIRICHLET BOUNDARY CONDITION; FRACTIONAL DERIVATIVES; FRACTIONAL DIFFERENTIAL EQUATIONS; FRACTIONAL DIFFUSION EQUATION; NONLINEAR SOURCE TERM; RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVES; SPACE-FRACTIONAL DIFFUSIONS;

FINITE VOLUME METHOD;

EID: 84904742021     PISSN: 0307904X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apm.2013.10.007     Document Type: Article

References (28)

1. Benson D.A., Wheatcraft S.W., Meerschaert M.M. Application of a fractional advection-dispersion equation. Water Resour. Res. 2000, 36(6):1403-1412.
2. Benson D.A., Wheatcraft S.W., Meerschaert M.M. The fractional-order governing equation of levy motion. Water Resour. Res. 2000, 36(6):1413-1423.
3. Liu F., Anh V., Turner I. Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 2004, 16(6):209-219.
4. Metzler R., Klafter J. The fractional Fokker-Planck equation: dispersive transport in an external force field. J. Mol. Liq. 2000, 86:219-228.
5. Meerschaert M.M., Sikorskii A. Stochastic models for fractional calculus. De Gruyter Studies in Mathematics 2012, 43. De Gruyter, Berlin.
6. Leonenko N.N., Meerschaert M.M., Sikorskii A. Fractional Pearson diffusions. J. Math. Anal. Appl. 2013, 403(2):532-546.
7. Kochubei A.N. Fractional-parabolic systems. Potential Anal. 2012, 37(1):1-30.
8. Zaslavsky G.M. Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 2002, 371:461-580.
9. Zhang Y., Benson D.A., Meerschaert M.M., Labolle E.M. Space-fractional advection-dispersion equations with variable parameters: diverse formulas, numerical solutions, and application to the MADE-site data. Water Resour. Res. 2007, 43:W05439.
10. Zhang Y., Benson D.A., Reeves D.M. Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications. Adv. Water Resour. 2009, 32:561-581.
11. Baeumer B., Kovály M., Meerschaert M.M. Numerical solutions for fractional reaction-diffusion equations. Comput. Math. Appl. 2008, 55:2212-2226.
12. Meerschaert M.M., Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 2004, 172:65-77.
13. Meerschaert M.M., Tadjeran C. Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 2006, 56(1):80-90.
14. Liu F., Zhuang P., Anh V., Turner I., Burrage K. Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 2007, 191:12-21.
15. Shen S., Liu F., Anh V., Turner I. The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation. IMA J. Appl. Math. 2008, 73:850-872.
16. Shen S., Liu F., Anh V. Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation. Numer. Algorithms 2011, 56:383-403.
17. S. Shen, F. Liu, V. Anh, I. Turner, A second-order accuracy numerical approximation for the Riesz space fractional advection-dispersion equation, in: Proceedings of the Fifth Symposium on Fractional Differentiation and Its Applications, May 14-17, 2012, Hohai University, Nanjing, China.
18. 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(2):1079-1095.
19. 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(1):160-176.
20. Chen W., Ye L., Sun H. Fractional diffusion equations by the Kansa method. Comput. Math. Appl. 2010, 59:1614-1620.
21. Fu Z., Chen W., Yang H. Boundary particle method for Laplace transformed time fractional diffusion equations. J. Comput. Phys. 2013, 235:52-66.
22. 2. J. Comput. Appl. Math. 2006, 193(1):243-268.
23. Zheng Y., Li C., Zhao Z. A note on the finite element method for the space-fractional advection diffusion equation. Comput. Math. Appl. 2010, 59:1718-1726.
24. Li C.P., Zhao Z.G., Chen Y.Q. Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 2011, 62:855-875.
25. A. Bueno-Orovio, D. Kay, K. Burrage, Fourier spectral methods for fractional-inspace reaction-diffusion equations, J. Comp. Phys., 2013, in press.
26. Zhang X., Crawford J.W., Deeks L.K., Shutler M.I., Bengough A.G., Young I.M. A mass balance based numerical method for the fractional advection-dispersion equation: theory and application. Water Resour. Res. 2005, 41:1-10.
27. H. Hejazi, T. Moroney, F. Liu, A finite volume method for solving the time-space fractional advection-dispersion equation, in: Proceedings of the Fifth Symposium on Fractional Differentiation and Its Applications, May 14-17, Hohai University, Nanjing, China (MS11, Paper ID 038).
28. Baeumer B., Kovály M., Meerschaert M.M. Fractional reproduction-dispersal equations and heavy tail dispersal kernels. Bull. Math. Biol. 2007, 69:2281-2297.