About this Journal Submit a Manuscript Table of Contents
Advances in Mathematical Physics
Volume 2013 (2013), Article ID 129404, 5 pages
http://dx.doi.org/10.1155/2013/129404
Research Article

A Weighted Average Finite Difference Method for the Fractional Convection-Diffusion Equation

School of Mathematical Sciences, Anhui University, Hefei, Anhui 230601, China

Received 18 January 2013; Revised 8 April 2013; Accepted 3 June 2013

Academic Editor: R. de la Llave

Copyright © 2013 Lijuan Su and Pei Cheng. 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. M. de la Sen, “Positivity and stability of the solutions of Caputo fractional linear time-invariant systems of any order with internal point delays,” Abstract and Applied Analysis, vol. 2011, Article ID 161246, 25 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. H. Yang, “Existence of mild solutions for a class of fractional evolution equations with compact analytic semigroup,” Abstract and Applied Analysis, vol. 2012, Article ID 903518, 15 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232–236, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. Raberto, E. Scalas, and F. Mainardi, “Waiting-times and returns in high-frequency financial data: an empirical study,” Physica A, vol. 314, no. 1–4, pp. 749–755, 2002. View at Publisher · View at Google Scholar · View at Scopus
  5. L. Sabatelli, S. Keating, J. Dudley, and P. Richmond, “Waiting time distributions in financial markets,” The European Physical Journal B, vol. 27, no. 2, pp. 273–275, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  6. B. Baeumer, D. A. Benson, M. M. Meerschaert, and S. W. Wheatcraft, “Subordinated advection-dispersion equation for contaminant transport,” Water Resources Research, vol. 37, no. 6, pp. 1543–1550, 2001. View at Publisher · View at Google Scholar · View at Scopus
  7. R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, New York, NY, USA, 2006.
  8. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at MathSciNet
  9. F. Mainardi, “Fractional relaxation-oscillation and fractional diffusion-wave phenomena,” Chaos, Solitons and Fractals, vol. 7, no. 9, pp. 1461–1477, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. B. I. Henry and S. L. Wearne, “Fractional reaction-diffusion,” Physica A, vol. 276, no. 3-4, pp. 448–455, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  11. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “The fractional-order governing equation of Levy motion,” Water Resources Research, vol. 36, no. 6, pp. 1413–1423, 2000. View at Publisher · View at Google Scholar · View at Scopus
  12. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  13. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  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 · View at Zentralblatt MATH · View at MathSciNet
  15. Ch. Lubich, “Discretized fractional calculus,” SIAM Journal on Mathematical Analysis, vol. 17, no. 3, pp. 704–719, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. 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 · View at Zentralblatt MATH · View at MathSciNet
  17. S. B. Yuste, “Weighted average finite difference methods for fractional diffusion equations,” Journal of Computational Physics, vol. 216, no. 1, pp. 264–274, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. 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 · View at Zentralblatt MATH · View at MathSciNet
  19. 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 · View at Zentralblatt MATH · View at MathSciNet
  20. K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1994. View at MathSciNet
  21. L. Su, W. Wang, and Z. Yang, “Finite difference approximations for the fractional advection-diffusion equation,” Physics Letters A, vol. 373, pp. 4405–4408, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH