An efficient algorithm based on Gegenbauer wavelets for the solutions of variable-order fractional differential equations

  • Muhammad Usman
  • Muhammad Hamid
  • Rizwan Ul HaqEmail author
  • Wei Wang
Regular Article


The article is devoted to a new computational algorithm based on the Gegenbauer wavelets (GWs) to solve the linear and nonlinear variable-order fractional differential equations. The novel operational matrices for derivatives of positive integer and variable order are derived. New piecewise functions are introduced to obtain the said operational matrices. In the proposed method, the given problem via Gegenbauer wavelets is transformed to a system of algebraic equations. The obtained solutions are endorsing the accuracy and efficiency of the suggested method and are in excellent agreement with the existing literature. The convergence and error bound analysis are presented in our study to show the credibility of the computational method and support the mathematical formulation of the algorithm. The discussed problems reconfirm the appropriateness of said algorithm and perceived that the proposed algorithm is an efficient tool to tackle the nonlinear fractional order problems of complex nature.


  1. 1.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)Google Scholar
  2. 2.
    E.C. De Oliveria, J.A.T. Machado, Math. Probl. Eng. 2014, 238459 (2014)Google Scholar
  3. 3.
    K.B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York, 1974)Google Scholar
  4. 4.
    K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley and Sons, New York, 1993)Google Scholar
  5. 5.
    E. Hesameddini, A. Rahimi, E. Asadollahifard, Commun. Nonlinear Sci. Numer. Simul. 34, 154 (2016)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Y. Li, Y. Yang, Discr. Dyn. Nat. Soc. 63, 1 (2014)Google Scholar
  7. 7.
    M. Chen, L.B. Jia, X.P. Chen, X.Z. Yin, J. Sound Vib. 333, 7183 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    H.T. Pedro, M.H. Kobayashi, J.M. Pereira, C.F. Coimbra, J. Vib. Control 14, 1659 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    L.E. Ramirez, C.F. Coimbra, Int. J. Differ. Equ. 2010, 846107 (2010)Google Scholar
  10. 10.
    J.J. Shyu, S.C. Pei, C.H. Chan, Signal Process 89, 320 (2009)CrossRefGoogle Scholar
  11. 11.
    H.G. Sun, W. Chen, H. Wei, Y.Q. Chen, Eur. Phys. J. ST 193, 185 (2011)CrossRefGoogle Scholar
  12. 12.
    Q.M. Ul Hassan, S.T. Mohyud-Din, Int. J. Biomath. 9, 1650026 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    J. Ahmad, S.T. Mohyud-Din, Int. J. Phys. Sci. 8, 1994 (2013)Google Scholar
  14. 14.
    J. Ahmad, S.T. Mohyud-Din, H.M. Srivastava, X.J. Yang, Waves, Wavelets and Fractals (2015)
  15. 15.
    M. Hamid, M. Usman, T. Zubair, S.T. Mohyud-Din, Ain Shams Eng. J. (2017)
  16. 16.
    X.J. Yang, J.T. Machado, C. Cattani, F. Gao, Commun. Nonlinear Sci. Numer. Simul. 47, 200 (2017)ADSCrossRefGoogle Scholar
  17. 17.
    Y.J. Hao, H.M. Srivastava, H. Jafari, X.J. Yang, Adv. Math. Phys. 2013, 754248 (2013)CrossRefGoogle Scholar
  18. 18.
    M.H. Heydari, M.R. Hooshmandasl, F.M. Ghaini, C. Cattani, J. Comput. Phys. 270, 402 (2014)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    K. Moaddy, A. Freihat, M. Al-Smadi, E. Abuteen, I. Hashim, Soft Comput. 22, 773 (2018)CrossRefGoogle Scholar
  20. 20.
    M. Hamid, M. Usman, T. Zubair, R.U. Haq, W. Wang, Int. J. Heat Mass Transfer 124, 706 (2018)CrossRefGoogle Scholar
  21. 21.
    U. Saeed, M.U. Rehman, M.A. Iqbal, Sci. Res. Essays 9, 571 (2014)CrossRefGoogle Scholar
  22. 22.
    M.H. Heydari, M.R. Hooshmandasl, F. Mohammadi, Appl. Math. Comput. 234, 267 (2014)MathSciNetGoogle Scholar
  23. 23.
    U. Saeed, M.U. Rehman, M.A. Iqbal, Appl. Math. Comput. 264, 431 (2015)MathSciNetGoogle Scholar
  24. 24.
    S.T. Mohyud-Din, M.A. Iqbal, S.M. Hassan, Entropy 17, 6925 (2015)ADSCrossRefGoogle Scholar
  25. 25.
    M. Usman, T. Zubair, M. Hamid, R.U. Haq, W. Wang, Phys. Fluids 30, 023104 (2018)ADSCrossRefGoogle Scholar
  26. 26.
    M.H. Heydari, Z. Avazzadeh, Comput. Appl. Math. (2018)
  27. 27.
    M.M. Izadkhah, J. Saberi-Nadjafi, Math. Methods Appl. Sci. 38, 3183 (2015)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    M.A. Iqbal, M. Shakeel, S.T. Mohyud-Din, M. Rafiq, Adv. Mech. Eng. 9, 1687814017696223 (2017)Google Scholar
  29. 29.
    M. Usman, M. Hamid, S.T. Mohyud-Din, A. Waheed, W. Wang, Int. J. Biomath. 11, 1850048 (2018)MathSciNetCrossRefGoogle Scholar
  30. 30.
    H. Singh, Appl. Math. Comput. 317, 85 (2018)MathSciNetGoogle Scholar
  31. 31.
    M.U. Rehman, U. Saeed, J. Korean Math. Soc. 52, 1069 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    M.M. Izadkhah, J. Saberi-Nadjafi, Math. Methods Appl. Sci. 38, 3183 (2015)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    C. Canuto, M. Hussaini, A. Quarteroni, T. Zang, in Spectral Methods in Fluid Dynamics (Springer, Berlin, 1988)Google Scholar
  34. 34.
    M.H. Heydari, M.R. Hooshmandasl, C. Cattani, G. Hariharan, Fundam. Inform. 151, 255 (2017)CrossRefGoogle Scholar
  35. 35.
    Y.M. Chen, Y.Q. Wei, D.Y. Liu, H. Yu, Appl. Math. Lett. 46, 83 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Muhammad Usman
    • 1
  • Muhammad Hamid
    • 1
  • Rizwan Ul Haq
    • 2
    Email author
  • Wei Wang
    • 1
  1. 1.School of Mathematical SciencesPeking UniversityBeijingChina
  2. 2.Department of Electrical EngineeringBahria UniversityIslamabadPakistan

Personalised recommendations