SeMA Journal

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Numerical solution of the Bagley–Torvik equation using Laguerre polynomials

  • Tianfu JiEmail author
  • Jianhua Hou


The Laplace transform and Laguerre polynomials are applied to the solution of the Bagley–Torvik equation. We first convert the fractional differential equation into the algebraic equation by Laplace transform. Then, we find the numerical inversion of Laplace transform by Laguerre polynomials. The fractional derivative is described in the Caputo sense. Numerical examples are provided to demonstrate the accuracy and efficiency of the method.


Bagley–Torvik equation Laguerre polynomial Laplace transform Caputo derivative 

Mathematics Subject Classification

26A33 34A08 34K28 



  1. 1.
    Balaji, S., Hariharan, G.: An efficient operational matrix method for the numerical solutions of the fractional Bagley-Torvik equation using wavelets. J. Math. Chem. 57(8), 1885–1901 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bhrawy, A.H., Alofi, A.S.: The operational matrix of fractional integration for shifted Chebyshev polynomials. Appl. Math. Lett. 26(1), 25–31 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Çenesiz, Y., Keskin, Y., Kurnaz, A.: The solution of the Bagley–Torvik equation with the generalized Taylor collocation method. J. Frankl. Inst. 347(2), 452–466 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cuomo, S., D’Amore, L., Murli, A., Rizzardi, M.: Computation of the inverse Laplace transform based on a collocation method which uses only real values. J. Comput. Appl. Math. 198(1), 98–115 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cuomo, S., D’Amore, L., Rizzardi, M., Murli, A.: A modification of Weeks’ method for numerical inversion of the Laplace transform in the real case based on automatic differentiation. Lect. Notes Comput. Sci. Eng. 64, 45–54 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Diethelm, K., Ford, J.: Numerical solution of the Bagley–Torvik equation. BIT Numer. Math. 42(3), 490–507 (2002)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 35(12), 5662–5672 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D’Amore, L., Campagna, R., Mele, V., Murli, A.: Relative. An ANSI C90 software package for the real Laplace transform inversion. Num. Algorithms. 63(1), 187–211 (2013)CrossRefzbMATHGoogle Scholar
  9. 9.
    Giunta, G., Laccetti, G., Rizzardi, M.R.: More on the weeks method for the numerical inversion of the Laplace transform. Numer. Math. 54(2), 193–200 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gülsu, M., Öztürk, Y., Anapali, A.: Numerical solution the fractional Bagley–Torvik equation arising in fluid mechanics. Int. J. Comput. Math. 11(7), 1–12 (2015)zbMATHGoogle Scholar
  11. 11.
    Li, Y., Zhao, W.: Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl. Math. Comput. 216(8), 2276–2285 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Luchko, Y., Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnam. 24(2), 207–233 (1999)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Machado, J., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mashayekhi, S., Razzaghi, M.: Numerical solution of the fractional Bagley–Torvik equation by using hybrid functions approximation. Math. Methods Appl. Sci. 39(3), 353–365 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mokhtary, P.: Numerical treatment of a well-posed Chebyshev tau method for Bagley–Torvik equation with high-order of accuracy. Numer. Algorithms 72(4), 875–891 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Podlubny, I.: Fractional Differential Equations, vol. 198. Academic press, New York (1998)zbMATHGoogle Scholar
  17. 17.
    Podlubny, I.: Matrix approach to discrete fractional calculus. Fract. Calculus Appl. Anal. 3(4), 359–386 (2000)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ray, S.S.: On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley–Torvik equation. Appl. Math. Comput. 218(9), 5239–5248 (2012)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ray, S.S., Bera, R.K.: Analytical solution of the Bagley–Torvik equation by Adomian decomposition method. Appl. Math. Comput. 168(1), 398–410 (2005)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Rehman, M.U., Idrees, A., Saeed, U.: A quadrature method for numerical solutions of fractional differential equations. Appl. Math. Comput. 307(15), 38–49 (2017)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Sakar, M.G., Saldır, O., Akgül, A.: A novel technique for fractional Bagley–Torvik equation. In: Proceedings of the National Academy of Sciences Sciences, India Section A: Physical. (2018).
  22. 22.
    Srivastava, H.M., Shah, F.A., Abass, R.: An application of the Gegenbauer wavelet method for the numerical solution of the fractional Bagley-Torvik equation. Russian J. Math. Phys. 26(1), 77–93 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51(2), 725–728 (1984)CrossRefzbMATHGoogle Scholar
  24. 24.
    Weeks, W.T.: Numerical inversion of Laplace transforms using Laguerre functions. J. ACM 13(3), 419–429 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Weideman, J.A.C.: Algorithms for parameter selection in the weeks method for inverting the Laplace transform. SIAM J. Sci. Comput. 21(1), 111–128 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yang, C., Ji, T.: Differential quadrature method for fractional Logistic differential equation. IAENG Int. J. Appl. Math. 48(3), 342–348 (2018)MathSciNetGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2019

Authors and Affiliations

  1. 1.Department of ScienceLiangyungang Technical CollegeLianyungangChina
  2. 2.Department of ScienceJiangsu Ocean UniversityLianyungangChina

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