`Mathematical Problems in EngineeringVolume 2011, Article ID 171620, 20 pageshttp://dx.doi.org/10.1155/2011/171620`
Research Article

## A Fully Discrete Galerkin Method for a Nonlinear Space-Fractional Diffusion Equation

1Department of Mathematics, Huaibei Normal University, Huaibei 235000, China
2Department of Fundamental Courses, Shanghai Customs College, Shanghai 201204, China
3Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 13 May 2011; Revised 27 July 2011; Accepted 27 July 2011

Copyright © 2011 Yunying Zheng and Zhengang Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. S. Chen and F. Liu, “ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation,” Journal of Applied Mathematics and Computing, vol. 26, no. 1, pp. 295–311, 2008.
2. V. J. Ervin, N. Heuer, and J. P. Roop, “Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation,” SIAM Journal on Numerical Analysis, vol. 45, no. 2, pp. 572–591, 2008.
3. V. J. Ervin and J. P. Roop, “Variational formulation for the stationary fractional advection dispersion equation,” Numerical Methods for Partial Differential Equations, vol. 22, no. 3, pp. 558–576, 2006.
4. B. I. Henry, T. A. M. Langlands, and S. L. Wearne, “Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations,” Physical Review E, vol. 74, no. 3, article 031116, 2006.
5. C. P. Li, A. Chen, and J. J. Ye, “Numerical approaches to fractional calculus and fractional ordinary differential equation,” Journal of Computational Physics, vol. 230, no. 9, pp. 3352–3368, 2011.
6. C. P. Li, Z. G. Zhao, and Y. Q. Chen, “Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion,” Computers & Mathematics with Applications, vol. 62, pp. 855–875, 2011.
7. F. Liu, V. Anh, and I. Turner, “Numerical solution of the space fractional Fokker-Planck equation,” Journal of Computational and Applied Mathematics, vol. 166, no. 1, pp. 209–219, 2004.
8. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
9. R. K. Saxena, A. M. Mathai, and H. J. Haubold, “Fractional reactiondiffusion equations,” Astrophysics and Space Science, vol. 305, no. 3, pp. 289–296, 2006.
10. Y. Y. Zheng, C. P. Li, and Z. G. Zhao, “A note on the finite element method for the space-fractional advection diffusion equation,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1718–1726, 2010.
11. B. Baeumer, M. Kovács, and M. M. Meerschaert, “Fractional reproduction-dispersal equations and heavy tail dispersal kernels,” Bulletin of Mathematical Biology, vol. 69, no. 7, pp. 2281–2297, 2007.
12. S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, “Fractional Bloch equation with delay,” Computers & Mathematics with Applications, vol. 61, no. 5, pp. 1355–1365, 2011.
13. A. V. Chechkin, V. Y. Gonchar, R. Gorenflo, N. Korabel, and I. M. Sokolov, “Generalized fractional diffusion equations for accelerating subdiffusion and truncated Levy flights,” Physical Review E, vol. 78, no. 2, article 021111, 2008.
14. P. D. Demontis and G. B. Suffritti, “Fractional diffusion interpretation of simulated single-file systems in microporous materials,” Physical Review E, vol. 74, no. 5, article 051112, 2006.
15. S. A. Elwakil, M. A. Zahran, and E. M. Abulwafa, “Fractional (space-time) diffusion equation on comb-like model,” Chaos, Solitons & Fractals, vol. 20, no. 5, pp. 1113–1120, 2004.
16. A. Kadem, Y. Luchko, and D. Baleanu, “Spectral method for solution of the fractional transport equation,” Reports on Mathematical Physics, vol. 66, no. 1, pp. 103–115, 2010.
17. R. L. Magin, O. Abdullah, D. Baleanu, and X. J. Zhou, “Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation,” Journal of Magnetic Resonance, vol. 190, no. 2, pp. 255–270, 2008.
18. M. M. Meerschaert, D. A. Benson, and B. Baeumer, “Operator Levy motion and multiscaling anomalous diffusion,” Physical Review E, vol. 63, no. 2 I, article 021112, 2001.
19. W. L. Vargas, J. C. Murcia, L. E. Palacio, and D. M. Dominguez, “Fractional diffusion model for force distribution in static granular media,” Physical Review E, vol. 68, no. 2, article 021302, 2003.
20. V. V. Yanovsky, A. V. Chechkin, D. Schertzer, and A. V. Tur, “Levy anomalous diffusion and fractional Fokker-Planck equation,” Physica A: Statistical Mechanics and its Applications, vol. 282, no. 1-2, pp. 13–34, 2000.
21. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15, Springer, Berlin, Germany, 1994.