Skip to main content
SpringerLink
Log in

Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract.

Fractional calculus has been used to model physical and engineering processes that are best described by fractional differential equations. Therefore designing efficient and reliable techniques for the solution of such equations is an important task. In this paper, we propose an efficient and accurate Galerkin method based on the fractional-order Legendre functions (FLFs) for solving the fractional sub-diffusion equation (FSDE) and the time-fractional diffusion-wave equation (FDWE). The time-fractional derivatives for FSDE are described in the Riemann-Liouville sense, while for FDWE are described in the Caputo sense. To this end, we first derive a new operational matrix of fractional integration (OMFI) in the Riemann-Liouville sense for FLFs. Next, we transform the original FSDE into an equivalent problem with fractional derivatives in the Caputo sense. Then the FLFs and their OMFI together with the Galerkin method are used to transform the problems under consideration into the corresponding linear systems of algebraic equations, which can be simply solved to achieve the numerical solutions of the problems. The proposed method is very convenient for solving such kind of problems, since the initial and boundary conditions are taken into account automatically. Furthermore, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)

  2. K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order (Academic Press, New York, 1974)

  3. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science, 1993)

  4. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley and Sons, 1993)

  5. R. Du, W.R. Cao, Z.Z. Sun, Appl. Math. Model. 34, 2998 (2010)

    Article  MathSciNet  Google Scholar 

  6. F. Liu, V. Anh, I. Turner, J. Comput. Appl. Math. 166, 209 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  7. F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Appl. Math. Comp. 191, 12 (2007)

    Article  MathSciNet  Google Scholar 

  8. C. Tadjeran, M.M. Meerschaert, H.-P. Scheffler, J. Comput. Phys. 213, 205 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  9. P. Zhuang, F. Liu, V. Anh, I. Turner, SIAM J. Numer. Anal. 46, 1079 (2008)

    Article  MathSciNet  Google Scholar 

  10. F. Liu, C. Yang, K. Burrage, J. Comput. Appl. Math. 231, 160 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  11. S. Yuste, L. Acedo, SIAM J. Numer. Anal. 42, 1862 (2005)

    Article  MathSciNet  Google Scholar 

  12. S. Yuste, J. Comput. Phys. 216, 264 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  13. C.M. Chen, F. Liu, I. Turner, V. Anh, J. Comput. Phys. 227, 886 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  14. R. Scherer, S. Kalla, L. Boyadjiev, B. Al-Saqabi, Appl. Numer. Math. 58, 1212 (2008)

    Article  MathSciNet  Google Scholar 

  15. T.A.M. Langlands, B. Henry, J. Comput. Phys. 205, 719 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  16. J.K.R. Metzler, Phys. Rep. 339, 1 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  17. M. Cui, J. Comput. Phys. 228, 7792 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  18. G. Gao, Z. Sun, J. Comput. Phys. 230, 586 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  19. J. Renand, Z. Sun, X. Zhao, J. Comput. Phys. 232, 456 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  20. X. Zhao, Z. Sun, J. Comput. Phys. 230, 6061 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  21. C. Chen, F. Liu, K. Burrage, J. Comput. Phys. 198, 754 (2008)

    MathSciNet  Google Scholar 

  22. D. Murio, Comput. Math. Appl. 56, 1138 (2008)

    Article  MathSciNet  Google Scholar 

  23. T. Langlands, B. Henry, J. Comput. Phys. 205, 719 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  24. P. Zhuang, F. Liu, V. Anh, I. Turner, IMA J. Appl. Math. 74, 645 (2009)

    Article  MathSciNet  Google Scholar 

  25. C. Chen, F. Liu, I. Turner, V. Anh, Numer. Algor. 54, 1 (2010)

    Article  MathSciNet  Google Scholar 

  26. C. Chen, F. Liu, I. Turner, V. Anh, SIAM J. Sci. Comput. 32, 1740 (2010)

    Article  MathSciNet  Google Scholar 

  27. Y. Zhang, Z. Sun, H. Wu, SIAM J. Numer. Anal. 49, 2302 (2011)

    Article  MathSciNet  Google Scholar 

  28. Y. Zhang, Z. Sun, J. Comput. Phys. 230, 8713 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  29. M. Cui, J. Comput. Phys. 231, 2621 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  30. X. Zhao, Q. Xu, Appl. Math. Model. 38, 3848 (2014)

    Article  MathSciNet  Google Scholar 

  31. A. Mohebbi, M. Abbaszadeh, M. Dehghan, Eng. Anal. Boundary Elem. 38, 72 (2014)

    Article  MathSciNet  Google Scholar 

  32. J. Chen, F. Liu, Q. Liu, X. Chen, V. Anh, I. Turner, K. Burrage, Appl. Math. Model. 38, 3695 (2014)

    Article  MathSciNet  Google Scholar 

  33. A.H. Bhrawy, E.H. Dohac, D. Baleanud, S.S. Ezz-Eldien, J. Comput. Phys. 293, 142 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  34. J. Chen, F. Liu, V. Anh, S. Shen, Q. Liu, C. Liao, Appl. Math. Comput. 219, 1737 (2012)

    MathSciNet  Google Scholar 

  35. F. Mainardi, Appl. Math. Lett. 9, 23 (1996)

    Article  MathSciNet  Google Scholar 

  36. W. Wess, J. Math. Phys. 27, 2782 (1996)

    Article  ADS  Google Scholar 

  37. W.R. Schneider, W. Wess, J. Math. Phys. 30, 134 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  38. Y.A. Rossikhin, M.V. Shitikova, Appl. Mech. Rev. 50, 15 (1997)

    Article  ADS  Google Scholar 

  39. Z.Z. Sun, X. Wu, Appl. Numer. Math. 56, 193 (2006)

    Article  MathSciNet  Google Scholar 

  40. M. Cui, J. Comput. Phys. 6, 383 (2013)

    ADS  Google Scholar 

  41. X. Hu, L. Zhang, Comput. Phys. Commun. 182, 1645 (2011)

    Article  ADS  Google Scholar 

  42. X. Hu, L. Zhang, Appl. Math. Comput. 218, 5019 (2012)

    MathSciNet  Google Scholar 

  43. C.F.L. Godinho, J. Weberszpil, J.A. Helayel-Neto, Chaos Solitons Fractals 4, 765 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  44. R. Darzi, B. Mohammadzade, S. Mousavi, R. Beheshti, J. Math. Comput. Sci. 6, 79 (2013)

    Google Scholar 

  45. M.H. Heydari, M.R. Hooshmandasl, F.M.M. Ghaini, F. Fereidouni, Eng. Anal. Bound. Elem. 37, 1331 (2013)

    Article  MathSciNet  Google Scholar 

  46. C. Canuto, M. Hussaini, A. Quarteroni, T. Zang, Spectral methods in fluid dynamics (Springer, 1988)

  47. A. Saadatmandi, M. Dehghan, Comput. Math. Appl. 59, 1326 (2010)

    Article  MathSciNet  Google Scholar 

  48. E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Appl. Math. Model. 36, 4931 (2012)

    Article  MathSciNet  Google Scholar 

  49. E.H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Comput. Math. Appl. 62, 2364 (2011)

    Article  MathSciNet  Google Scholar 

  50. D. Baleanu, A.H. Bhrawy, T.M. Taha, Abstr. Appl. Anal. 2013, 546502 (2013)

    MathSciNet  Google Scholar 

  51. M.H. Heydari, M.R. Hooshmandasl, F.M.M. Ghaini, Comput. Math. Appl. 68, 269 (2014)

    Article  MathSciNet  Google Scholar 

  52. A.H. Bhrawy, M.A. Zaky, R.A.V. Gorder, Numer. Algor. 71, 151 (2016)

    Article  Google Scholar 

  53. A.H. Bhrawy, E.H. Doha, S.S. Ezz-Eldien, M.A. Abdelkawy, Calcolo 53, 1 (2016)

    Article  MathSciNet  Google Scholar 

  54. M.H. Heydari, M.R. Hooshmandasl, F. Mohammadi, Appl. Math. Comput. 234, 267 (2014)

    MathSciNet  Google Scholar 

  55. M.H. Heydari, M.R. Hooshmandasl, F. Mohammadi, Adv. Appl. Math. Mech. 6, 247 (2014)

    Article  MathSciNet  Google Scholar 

  56. M.H. Heydari, M.R. Hooshmandasl, F.M.M. Ghaini, C. Cattani, Phys. Lett. A 379, 71 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  57. M.H. Heydari, M.R. Hooshmandasl, F. Mohammadi, C. Cattani, Commun. Nonlinear Sci. Numer. Simulat. 19, 37 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  58. M.H. Heydari, M.R. Hooshmandasl, F.M.M. Ghaini, C. Cattani, Appl. Math. Comput. 286, 139 (2016)

    MathSciNet  Google Scholar 

  59. C. Cattani, Int. J. Fluid Mech. Res. 5, 1 (2003)

    Google Scholar 

  60. C. Cattani, Comp. Math. Appl. 50, 1191 (2005)

    Article  MathSciNet  Google Scholar 

  61. C. Cattani, A. Kudreyko, Appl. Math. Comput. 215, 4164 (2010)

    MathSciNet  Google Scholar 

  62. C. Cattani, Math. Modell. Anal. 11, 117 (2006)

    MathSciNet  Google Scholar 

  63. C. Cattani, Math. Prob. Eng. 2008, 164808 (2008)

    MathSciNet  Google Scholar 

  64. C. Cattani, Math. Prob. Eng. 2010, 408418 (2010)

    MathSciNet  Google Scholar 

  65. C. Cattani, Math. Prob. Eng. 2012, 502812 (2012)

    MathSciNet  Google Scholar 

  66. S. Kazem, S. Abbasbandy, S. Kumar, Appl. Math. Model. 37, 5498 (2013)

    Article  MathSciNet  Google Scholar 

  67. F. Liu, M.M. Meerschaert, R.J. McGough, P. Zhuang, Q. Liu, Fract. Calc. Appl. Anal. 16, 9 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Yazd University, Yazd, Iran

    M. R. Hooshmandasl & M. H. Heydari

  2. The Laboratory of Quantum Information Processing, Yazd University, Yazd, Iran

    M. R. Hooshmandasl & M. H. Heydari

  3. Engineering School (DEIM), University of Tuscia, Viterbo, Italy

    C. Cattani

Authors
  1. M. R. Hooshmandasl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. H. Heydari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Cattani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. R. Hooshmandasl.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hooshmandasl, M.R., Heydari, M.H. & Cattani, C. Numerical solution of fractional sub-diffusion and time-fractional diffusion-wave equations via fractional-order Legendre functions. Eur. Phys. J. Plus 131, 268 (2016). https://doi.org/10.1140/epjp/i2016-16268-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/i2016-16268-2

Search

Navigation