Advertisement

The European Physical Journal Special Topics

, Volume 222, Issue 8, pp 1901–1914 | Cite as

Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions

  • H. Ye
  • F. LiuEmail author
  • I. Turner
  • V. Anh
  • K. Burrage
Regular Article

Abstract

Fractional partial differential equations with more than one fractional derivative in time describe some important physical phenomena, such as the telegraph equation, the power law wave equation, or the Szabo wave equation. In this paper, we consider two- and three-dimensional multi-term time and space fractional partial differential equations. The multi-term time-fractional derivative is defined in the Caputo sense, whose order belongs to the interval (1,2],(2,3],(3,4] or (0,m], and the space-fractional derivative is referred to as the fractional Laplacian form. We derive series expansion solutions based on a spectral representation of the Laplacian operator on a bounded region. Some applications are given for the two- and three-dimensional telegraph equation, power law wave equation and Szabo wave equation.

Keywords

European Physical Journal Special Topic Fractional Derivative Homogeneous Dirichlet Boundary Condition Telegraph Equation Caputo Sense 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D.A. Benson, S.W. Wheatcraft, M.M. Meerschaert, Water Resour. Res. 36, 1403 (2000)ADSCrossRefGoogle Scholar
  2. 2.
    D.A. Benson, S.W. Wheatcraft, M.M. Meerschaert, Water Resour. Res. 36, 1413 (2000)ADSCrossRefGoogle Scholar
  3. 3.
    J. Chen, F. Liu, V. Anh, J. Math. Anal. Appl. 338, 1364 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    I.H. Dimovski, Convolutional Calculus (Bulgarian Academy of Sciences, Sofia, 1982)Google Scholar
  5. 5.
    E.C. Eckstein, J.A. Goldsten, M. Leggas, Electron. J. Differential Equation 3, 39(1999)Google Scholar
  6. 6.
    R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, River Edge, New Jersey, 2000)Google Scholar
  7. 7.
    M. Ilic, F. Liu, I. Turner, V. Anh, Fractional Calculus Appl. Anal. 9, 333 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    H. Jiang, F. Liu, I. Turner, K. Burrage, J. Math. Anal. Appl. 389, 1117 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    H. Jiang, F. Liu, I. Turner, K. Burrage, Comput. Math. Appl. (2012), doi:  10.1016/j.camwa.2012.02.042
  10. 10.
    H. Jiang, F. Liu, M.M. Meerschaert, et al., Electr. J. Math. Anal. Appl. 1, 1 (2013)Google Scholar
  11. 11.
    J.K. Kelly, R.J. McGough, M.M. Meerschaert, J. Acoust. Soc. Am. 124, 2861 (2008)ADSCrossRefGoogle Scholar
  12. 12.
    C.P. Li, D.L. Qian, Y.Q. Chen, Discr. Dyn. Nature Soc. ID 562494, 15 pages (2011)Google Scholar
  13. 13.
    C.P. Li, Z.G. Zhao, Eur. Phys. J. Special Topics 193, 5 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    F. Liu, V. Anh, I. Turner, J. Comput. Appl. Math. 166, 209 (2004)MathSciNetADSCrossRefzbMATHGoogle Scholar
  15. 15.
    F. Liu, M M. Meerschaert, R.J. McGugh, et al., Fract. Calculus Appl. Anal. 16, 9 (2013)CrossRefGoogle Scholar
  16. 16.
    Y. Luchko, R. Gorenflo, ACTA Math. Vietnamica 24, 207 (1999)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Y. Luchko, J. Math. Anal. Appl. 374, 538 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)MathSciNetADSCrossRefzbMATHGoogle Scholar
  19. 19.
    S. Momani, Z. Odibat, Numer. Meth. Partial Differ. Eq. 24, 1416 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)Google Scholar
  21. 21.
    M. Stojanovic, J. Comp. Appl. Math. 235, 3121 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    X. Wang, F. Liu, J. Fuzhou University 35, 520 (2007)zbMATHGoogle Scholar
  23. 23.
    Q. Yang, F. Liu, I. Turner, Appl. Math. Modell. 34, 200 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    G.M. Zaslavsky, Phys. Rep. 371, 461 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  25. 25.
    Y. Zhang, D.A. Benson, D.M. Reeves, Adv. Water Res. 32, 561 (2009)CrossRefGoogle Scholar
  26. 26.
    P. Zhuang, F. Liu, V. Anh, I. Turner, SIAM J. Numer. Anal. 47, 1760 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Department of Applied MathematicsDonghua UniversityShanghaiPR China
  2. 2.Mathematical SciencesQueensland University of TechnologyBrisbaneAustralia
  3. 3.Department of Computing Science and OCISBOxford UniversityOxfordUK

Personalised recommendations