SeMA Journal

pp 1–10

# Numerical solution of the Bagley–Torvik equation using Laguerre polynomials

• Tianfu Ji
• Jianhua Hou
Article

## Abstract

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.

## Keywords

Bagley–Torvik equation Laguerre polynomial Laplace transform Caputo derivative

## Mathematics Subject Classification

26A33 34A08 34K28

## References

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)
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)
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)
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)
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)
6. 6.
Diethelm, K., Ford, J.: Numerical solution of the Bagley–Torvik equation. BIT Numer. Math. 42(3), 490–507 (2002)
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)
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)
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)
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)
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)
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)
13. 13.
Machado, J., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)
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)
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)
16. 16.
Podlubny, I.: Fractional Differential Equations, vol. 198. Academic press, New York (1998)
17. 17.
Podlubny, I.: Matrix approach to discrete fractional calculus. Fract. Calculus Appl. Anal. 3(4), 359–386 (2000)
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)
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)
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)
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)
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)
24. 24.
Weeks, W.T.: Numerical inversion of Laplace transforms using Laguerre functions. J. ACM 13(3), 419–429 (1966)
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)
26. 26.
Yang, C., Ji, T.: Differential quadrature method for fractional Logistic differential equation. IAENG Int. J. Appl. Math. 48(3), 342–348 (2018)