About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 596184, 25 pages
http://dx.doi.org/10.1155/2012/596184
Research Article

Finite Element Solutions for the Space Fractional Diffusion Equation with a Nonlinear Source Term

Department of Mathematics Education, Seoul National University, Seoul 151-748, Republic of Korea

Received 26 April 2012; Revised 6 July 2012; Accepted 17 July 2012

Academic Editor: Bashir Ahmad

Copyright © 2012 Y. J. Choi and S. K. Chung. 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.

Linked References

  1. K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002. View at Publisher · View at Google Scholar
  2. A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. I,” Applicable Analysis, vol. 78, no. 1-2, pp. 153–192, 2001. View at Publisher · View at Google Scholar
  3. A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. II,” Applicable Analysis, vol. 81, no. 2, pp. 435–493, 2002. View at Publisher · View at Google Scholar
  4. R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000. View at Publisher · View at Google Scholar
  5. R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A, vol. 37, no. 31, pp. R161–R208, 2004. View at Publisher · View at Google Scholar
  6. K. B. Oldham and J. Spanier, The Fractional Calculus, Dover Publications, New York, NY, USA, 2002.
  7. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  8. B. Baeumer, M. Kovács, and M. M. Meerschaert, “Numerical solutions for fractional reaction-diffusion equations,” Computers & Mathematics with Applications, vol. 55, no. 10, pp. 2212–2226, 2008. View at Publisher · View at Google Scholar
  9. W. Deng, “Finite element method for the space and time fractional Fokker-Planck equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 204–226, 2008. View at Publisher · View at Google Scholar
  10. Z. Q. Deng, V. P. Singh, and L. Bengtsson, “Numerical solution of fractional advection-dispersion equation,” Journal of Hydraulic Engineering, vol. 130, no. 5, pp. 422–431, 2004. View at Publisher · View at Google Scholar · View at Scopus
  11. Y. Lin and C. Xu, “Finite difference/spectral approximations for the time-fractional diffusion equation,” Journal of Computational Physics, vol. 225, no. 2, pp. 1533–1552, 2007. View at Publisher · View at Google Scholar
  12. F. Liu, A. Anh, and I. Turner, “Numerical solution of the space fractional Fokker-Planck equation,” Journal of Computational and Applied Mathematics, vol. 166, pp. 209–219, 2004.
  13. 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. View at Publisher · View at Google Scholar
  14. M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for fractional advection-dispersion flow equations,” Journal of Computational and Applied Mathematics, vol. 172, no. 1, pp. 65–77, 2004. View at Publisher · View at Google Scholar
  15. M. M. Meerschaert, H.-P. Scheffler, and C. Tadjeran, “Finite difference methods for two-dimensional fractional dispersion equation,” Journal of Computational Physics, vol. 211, no. 1, pp. 249–261, 2006. View at Publisher · View at Google Scholar
  16. M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for two-sided space-fractional partial differential equations,” Applied Numerical Mathematics, vol. 56, no. 1, pp. 80–90, 2006. View at Publisher · View at Google Scholar
  17. V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias, and H. R. Hicks, “Numerical methods for the solution of partial differential equations of fractional order,” Journal of Computational Physics, vol. 192, no. 2, pp. 406–421, 2003. View at Publisher · View at Google Scholar
  18. H. W. Choi, S. K. Chung, and Y. J. Lee, “Numerical solutions for space fractional dispersion equations with nonlinear source terms,” Bulletin of the Korean Mathematical Society, vol. 47, no. 6, pp. 1225–1234, 2010. View at Publisher · View at Google Scholar
  19. W. Deng, “Numerical algorithm for the time fractional Fokker-Planck equation,” Journal of Computational Physics, vol. 227, no. 2, pp. 1510–1522, 2007. View at Publisher · View at Google Scholar
  20. W. Deng and C. Li, “Finite difference methods and their physical constraints for the fractional Klein-Kramers equation,” Numerical Methods for Partial Differential Equations, vol. 27, no. 6, pp. 1561–1583, 2011. View at Publisher · View at Google Scholar
  21. T. A. M. Langlands and B. I. Henry, “The accuracy and stability of an implicit solution method for the fractional diffusion equation,” Journal of Computational Physics, vol. 205, no. 2, pp. 719–736, 2005. View at Publisher · View at Google Scholar
  22. 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,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 12–20, 2007. View at Publisher · View at Google Scholar
  23. P. Zhuang, F. Liu, V. Anh, and I. Turner, “New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation,” SIAM Journal on Numerical Analysis, vol. 46, no. 2, pp. 1079–1095, 2008. View at Publisher · View at Google Scholar
  24. 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. View at Publisher · View at Google Scholar
  25. J. P. Roop, “Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2,” Journal of Computational and Applied Mathematics, vol. 193, no. 1, pp. 243–268, 2006. View at Publisher · View at Google Scholar
  26. 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, 2007. View at Publisher · View at Google Scholar
  27. D. Braess, Finite Elements, Cambridge University Press, Cambridge, UK, 2nd edition, 2001.
  28. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1994.
  29. Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005. View at Publisher · View at Google Scholar
  30. S. Tang and R. O. Weber, “Numerical study of Fisher's equation by a Petrov-Galerkin finite element method,” Australian Mathematical Society B, vol. 33, no. 1, pp. 27–38, 1991. View at Publisher · View at Google Scholar