Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2017, Article ID 8716752, 8 pages
https://doi.org/10.1155/2017/8716752
Research Article

A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

Correspondence should be addressed to Muzhou Hou; moc.anis@uohzumuoh

Received 4 June 2017; Revised 16 August 2017; Accepted 22 August 2017; Published 24 September 2017

Academic Editor: Mariano Torrisi

Copyright © 2017 Taohua Liu and Muzhou Hou. 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. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
  2. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Frcational Integrals and Derivatives: Theory and Applications, Gordon Breach, London, Uk, 1993.
  3. I. M. Sokolov, J. Klafter, and A. Blumen, “Fractional kinetics,” Physics Today, vol. 55, no. 11, pp. 48–54, 2002. View at Google Scholar · View at Scopus
  4. R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Danbury, Conn, USA, 2006.
  5. M. Raberto, E. Scalas, and F. Mainardi, “Waiting-times and returns in high-frequency financial data: an empirical study,” Physica A: Statistical Mechanics and Its Applications, vol. 314, no. 1–4, pp. 749–755, 2002. View at Publisher · View at Google Scholar · View at Scopus
  6. X. Liu and Z. Liu, “Existence results for fractional semilinear differential inclusions in Banach spaces,” Journal of Applied Mathematics and Computing, vol. 42, no. 1-2, pp. 171–182, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. X. Liu and Z. Liu, “On the 'bang-bang' principle for a class of fractional semilinear evolution inclusions,” Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol. 144, no. 2, pp. 333–349, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  8. X. Liu, Z. Liu, and X. Fu, “Relaxation in nonconvex optimal control problems described by fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 409, no. 1, pp. 446–458, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. X. Yin, L. Li, and S. Fang, “Second-order accurate numerical approximations for the fractional percolation equations,” The Journal of Nonlinear Sciences and Applications, vol. 10, no. 08, pp. 4122–4136, 2017. View at Publisher · View at Google Scholar
  10. H. Chen, S. Gan, and D. Xu, “A fractional trapezoidal rule type difference scheme for fractional order integro-differential equation,” Journal of Fractional Calculus and Applications, vol. 7, no. 1, pp. 133–146, 2016. View at Google Scholar · View at MathSciNet
  11. H. Chen, S. Gan, D. Xu, and Q. Liu, “A second-order BDF compact difference scheme for fractional-order Volterra equation,” International Journal of Computer Mathematics, vol. 93, no. 7, pp. 1140–1154, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  12. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  13. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advection-dispersion equation,” Water Resources Research, vol. 36, no. 6, pp. 1403–1412, 2000. View at Publisher · View at Google Scholar · 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 MathSciNet · View at Scopus
  15. J. Jia and H. Wang, “Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions,” Journal of Computational Physics, vol. 293, pp. 359–369, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. B. Guo, Q. Xu, and Z. Yin, “Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions,” Applied Mathematics and Mechanics. English Edition, vol. 37, no. 3, pp. 403–416, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. L. Feng, P. Zhuang, F. Liu, I. Turner, and Q. Yang, “Second-order approximation for the space fractional diffusion equation with variable cofficient,” Progress in Fractional Differentiation and Applications, vol. 1, pp. 23–35, 2015. View at Google Scholar
  18. A. Böttcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Springer, New York, NY, USA, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  19. R. M. Gray, “Toeplitz and circulant matrices: a review,” Foundations and Trends in Communications and Information Theory, vol. 2, no. 3, pp. 155–239, 2006. View at Publisher · View at Google Scholar · View at Scopus
  20. P. J. Davis, Circulant Matrices, John Wiley & Sons, New York, NY, USA, 1979. View at MathSciNet