Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014, Article ID 982413, 9 pages
http://dx.doi.org/10.1155/2014/982413
Research Article

Leapfrog/Finite Element Method for Fractional Diffusion Equation

1Department of Fundamental Courses, Shanghai Customs College, Shanghai 201204, China
2School of Mathematical Sciences, Huaibei Normal University, Huaibei 235000, China

Received 18 January 2014; Accepted 17 February 2014; Published 3 April 2014

Academic Editors: C. Li, A. Sikorskii, and S. B. Yuste

Copyright © 2014 Zhengang Zhao and Yunying Zheng. 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. F. Liu, V. Anh, I. Turner, and P. Zhuang, “Time fractional advection-dispersion equation,” Journal of Applied Mathematics and Computing, vol. 13, no. 1-2, pp. 233–245, 2003. View at Google Scholar · View at Scopus
  3. 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 Scopus
  4. S. B. Yuste and L. Acedo, “An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations,” SIAM Journal on Numerical Analysis, vol. 42, no. 5, pp. 1862–1874, 2005. View at Publisher · View at Google Scholar · View at Scopus
  5. G. J. Fix and J. P. Roof, “Least squares finite-element solution of a fractional order two-point boundary value problem,” Computers and Mathematics with Applications, vol. 48, no. 7-8, pp. 1017–1033, 2004. View at Publisher · View at Google Scholar · View at Scopus
  6. 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 · View at Scopus
  7. 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 · View at Scopus
  8. F. H. Zeng, C. P. Li, F. W. Liu, and I. W. Turner, “The use of finite difference/element approaches for solving the time-fractional subdiffusion equation,” SIAM Journal on Scientific Computing, vol. 35, pp. A2976–A3000, 2013. View at Google Scholar
  9. Y. Zheng, C. Li, and Z. Zhao, “A note on the finite element method for the space-fractional advection diffusion equation,” Computers and Mathematics with Applications, vol. 59, no. 5, pp. 1718–1726, 2010. View at Publisher · View at Google Scholar · View at Scopus
  10. Y. Zheng, C. Li, and Z. Zhao, “A fully discrete discontinuous galerkin method for nonlinear fractional fokker-planck equation,” Mathematical Problems in Engineering, vol. 2010, Article ID 279038, 26 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. C. Li, Z. Zhao, and Y. Chen, “Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 855–875, 2011. View at Publisher · View at Google Scholar · View at Scopus
  12. Z. G. Zhao and C. P. Li, “Fractional difference/finite element approximations for the time-space fractional telegraph equation,” Applied Mathematics and Computation, vol. 219, pp. 2975–2988, 2012. View at Google Scholar
  13. Z. G. Zhao and C. P. Li, “A numerical approach to the generalized nonlinear fractional Fokker-Planck equation,” Computers and Mathematics with Applications, vol. 64, pp. 3075–3089, 2012. View at Google Scholar
  14. C. P. Li and F. H. Zeng, “Finite element methods for fractional differential equation,” in Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis, C. P. Li, Y. J. Wu, and R. S. Ye, Eds., pp. 49–68, World Scientific, Singapore, 2013. View at Google Scholar
  15. W. Wyss, “The fractional diffusion equation,” Journal of Mathematical Physics, vol. 27, no. 11, pp. 2782–2785, 1986. View at Google Scholar · View at Scopus
  16. M. F. Shlesinger, B. J. West, and J. Klafter, “Lévy dynamics of enhanced diffusion: application to turbulence,” Physical Review Letters, vol. 58, no. 11, pp. 1100–1103, 1987. View at Publisher · View at Google Scholar · View at Scopus
  17. T. H. Solomon, E. R. Weeks, and H. L. Swinney, “Observation of anomalous diffusion and Lévy flights in a 2-dimensional rotating flow,” Physical Review Letters, vol. 71, no. 24, pp. 3975–3978, 1993. View at Publisher · View at Google Scholar · View at Scopus
  18. 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
  19. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
  20. S. C. Samko, A. A. Kilbas, and O. I. Maxitchev, Integrals and Derivatives of the Fractional Order and Some of Their Applications, Nauka i Tekhnika, Minsk, Belarus, 1987, (Russian).
  21. N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives,” Rheologica Acta, vol. 45, no. 5, pp. 765–771, 2006. View at Publisher · View at Google Scholar · View at Scopus
  22. C. P. Li and Z. G. Zhao, “Introduction to fractional integrability and differentiability,” European Physical Journal, vol. 193, pp. 5–26, 2011. View at Google Scholar
  23. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, Berlin, Germany, 1994.
  24. J. G. Heywood and R. Rannacher, “Finite-element approximation of the nonstationary navier-stokes problem—IV. Error analysis for second-order time discretization,” SIAM Journal on Numerical Analysis, vol. 27, no. 2, pp. 353–384, 1990. View at Google Scholar · View at Scopus