The European Physical Journal Special Topics

, Volume 222, Issue 8, pp 1987–1998 | Cite as

A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations

  • J. Quintana-Murillo
  • S. B. YusteEmail author
Regular Article


An implicit finite difference method with non-uniform timesteps for solving fractional diffusion and diffusion-wave equations in the Caputo form is presented. The non-uniformity of the timesteps allows one to adapt their size to the behaviour of the solution, which leads to large reductions in the computational time required to obtain the numerical solution without loss of accuracy. The stability of the method has been proved recently for the case of diffusion equations; for diffusion-wave equations its stability, although not proven, has been checked through extensive numerical calculations.


European Physical Journal Special Topic Fractional Derivative Fractional Calculus Adaptive Method Absorb Boundary Condition 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    K.B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York, 1974)Google Scholar
  2. 2.
    R. Hilfer (ed.), Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000)Google Scholar
  3. 3.
    I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (Academic Press, San Diego, 1999)Google Scholar
  4. 4.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)Google Scholar
  5. 5.
    R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)MathSciNetADSCrossRefzbMATHGoogle Scholar
  6. 6.
    R. Metzler, J. Klafter, J. Phys. A-Math. Gen. 37, R161 (2004)MathSciNetADSCrossRefzbMATHGoogle Scholar
  7. 7.
    R. Klages, G. Radons, I.M. Sokolov (eds.), Anomalous Transport: Foundations and Applications (Elsevier, Amsterdam, 2008)Google Scholar
  8. 8.
    R.L. Magin, O. Abdullah, D. Baleanu, X.J. Zhou, J. Mag. Reson. 190, 255 (2008)ADSCrossRefGoogle Scholar
  9. 9.
    I.M. Sokolov, J. Klafter, A. Blumen, Phys. Today 55, 48 (2002)CrossRefGoogle Scholar
  10. 10.
    B.I. Henry, T.A.M. Langlands, S. Wearne, Phys. Rev. Lett. 100, 128103 (2008)ADSCrossRefGoogle Scholar
  11. 11.
    S.B. Yuste, E. Abad, K. Lindenberg, Reactions in Subdiffusive Media and Associated Fractional Equations, in Fractional Dynamics. Recent Advances, edited by J. Klafter, S.C. Lim, R. Metzler (World Scientific, Singapore, 2011)Google Scholar
  12. 12.
    S.B. Yuste, E. Abad, K. Lindenberg, Phys. Rev. E 82, 061123 (2010)ADSCrossRefGoogle Scholar
  13. 13.
    E. Barkai, R. Metzler, J. Klafter, Phys. Rev. E 61, 132 (2000)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    S.B. Yuste, L. Acedo, Physica A 336, 334 (2004)ADSCrossRefGoogle Scholar
  15. 15.
    A.M.A. El-Sayed, M. Gaber, Phys. Lett. A 359, 175 (2006)MathSciNetADSCrossRefzbMATHGoogle Scholar
  16. 16.
    H. Jafari, S. Momani, Phys. Lett. A 370, 388 (2007)MathSciNetADSCrossRefzbMATHGoogle Scholar
  17. 17.
    S.S. Ray, Phys. Scripta 75, 53 (2007)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Nonlinear Dynam. 29, 129 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    S.B. Yuste, L. Acedo, SIAM J. Numer. Anal. 42, 1862 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    S.B. Yuste, J. Comput. Phys. 216, 264 (2006)MathSciNetADSCrossRefzbMATHGoogle Scholar
  21. 21.
    V.E. Lynch, B.A. Carreras, D. del-Castillo-Negrete, K.M. Ferreira-Mejias, H.R. Hicks, J. Comput. Phys. 192, 406 (2003)MathSciNetADSCrossRefzbMATHGoogle Scholar
  22. 22.
    M.M. Meerschaert, C. Tadjeran, J. Comput. Appl. Math. 172, 65 (2004)MathSciNetADSCrossRefzbMATHGoogle Scholar
  23. 23.
    C.M. Chen, F. Liu, I. Turner, V. Anh, J. Comput. Phys. 227, 886 (2007)MathSciNetADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Z.Z. Sun, X. Wu, Appl. Numer. Math. 56, 193 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    I. Podlubny, A.V. Chechkin, T. Skovranek, Y. Chen, B.M. Vinagre, J. Comput. Phys. 228, 3137 (2009)MathSciNetADSCrossRefzbMATHGoogle Scholar
  26. 26.
    M. Cui, J. Comput. Phys. 228, 7792 (2009)MathSciNetADSCrossRefzbMATHGoogle Scholar
  27. 27.
    H. Brunner, L. Ling, M. Yamamoto, J. Comput. Phys. 229, 6613 (2010)MathSciNetADSCrossRefzbMATHGoogle Scholar
  28. 28.
    T. Skovranek, V.V. Verbickij, Y. Tarte, I. Podlubny, Discretization of fractional-order operators and fractional differential equations on a non-equidistant mesh, Article no. FDA10-062, edited by I. Podlubny, B.M. Vinagre Jara, YQ. Chen, V. Feliu Batlle, I. Tejado Balsera, Proceedings of FDA10 (The 4th IFAC Workshop Fractional Differentiation and its Applications, Badajoz, 2010), p. 18Google Scholar
  29. 29.
    K. Mustapha, W. McLean, Numer. Algorithms 56, 159 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    K. Mustapha, J. AlMutawa, Numer. Algorithms 61, 1017 (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    S.B. Yuste, J. Quintana-Murillo, Comput. Phys. Comm. 182, 2594 (2012)MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    D.A. Murio, Comput. Math. Appl. 56, 1138 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    F. Liu, P. Zhuang, V. Anh, I. Turner, ANZIAM J. 47, C48 (2006)MathSciNetGoogle Scholar
  34. 34.
    S.B. Yuste, J. Quintana-Murillo, Phys. Scripta T136, 014025 (2009)ADSCrossRefGoogle Scholar
  35. 35.
    J. Quintana-Murillo, S.B. Yuste, J. Comput. Nonlin. Dyn. 6, 021014 (2011)CrossRefGoogle Scholar
  36. 36.
    R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (SIAM, Philadelfia, 2007)Google Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidad de ExtremaduraBadajozSpain

Personalised recommendations