IJPAM: Volume 110, No. 2 (2016)

Title

ANALYTICAL SOLUTION OF BAGLEY TORVIK EQUATION
BY GENERALIZE DIFFERENTIAL TRANSFORM

Authors

Manish Kumar Bansal$^1$, Rashmi Jain$^2$
$^{1,2}$Department of Mathematics
Malaviya National Institute of Technology
Jaipur, 302017, INDIA

Abstract

In the present paper, we use generalized differential transform method (GDTM) to derive solution of Bagley Torvik equation. The fractional derivative are described in the Caputo sense. As an example we have found the exact solution of two such Bagley-Torvik equations which demonstrates the effectiveness and efficiency of the proposed method.

History

Received: March 16, 2016
Revised: September 8, 2016
Published: November 5, 2016

AMS Classification, Key Words

AMS Subject Classification: 26A33, 35A22, 65L05, 34A08
Key Words and Phrases: Caputo fractional derivative, generalized differential transform method, Bagley-Torvik equation, fractional differential equation

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.

Bibliography

1
Agarwal, R. P., Belmekki, M. and Benchohra, M. (2009). A survey on semi linear differential equations and inclusions involving Riemann-Liouville fractional derivative.Advances in Difference Equations, 2009(3), 1-47.

2
Bagley, R. L. and Torvik, P. J.(1983). A different approach to the analysis of viscoelastically damped structures.AIAA Journal, 21(5), 741-748.

3
Caputo, M.(1967). Linear models of dissipation whose Q is almost frequency independent-II. Geophysical Journal International, 13(5), 529-539.

4
Das, S.(2010). Solution of extraordinary differential equations with physical reasoning by obtaining modal reaction series.Modelling and Simulation in Engineering, 2010, 1-19.

5
Diethelm, K. and Ford, N. J.(2002). Numerical solution of the Bagley-Torvik equation.BIT Numerical Mathematics, 42(3), 490-507.

6
El-Sayed, A. M. A., El-Kalla, I. L. and Ziada, E. A. A.(2010). Analytical and numerical solutions of multiterm nonlinear fractional orders differential equations, Applied Numerical Mathematics, 60(8), 788-797.

7
Enesiz, Y. C, Keskin, Y. and Kurnaz, A. (2010). The solution of the Bagley-Torvik equation with the generalized Taylor collocation method. Journal of the Franklin Institute Engineering and Applied Mathematics, 347(2), 452-466.

8
Erturk, V.S., Momani, S. and Odibat, Z.(2008). Application of generalized differential transform method to multi-order fractional differential equations.Communications in Nonlinear Science and Numerical Simulation, 13(8), 1642-1654.

9
Garg, M. and Manohar, P.(2015). Three-dimensional generalized differential transform method for space-time fractional diffusion equation, Palestine Journal of Mathematics, 4(1), 127-135.

10
Ghorbani, A. and Alavi, A. (2008). Application of He's variational iteration method to solve semidifferential equations of nth order. Mathematical Problems in Engineering, 2008, 1-9.

11
Hu, Y., Luo, Y. and Lu, Z.(2008). Analytical solution of the linear fractional differential equation by Adomian decomposition method. Journal of Computational and Applied Mathematics, 215(1), 220-229.

12
Manabe, S.(2002). A suggestion of fractional-order controller for flexible spacecraft attitude control.Nonlinear Dynamics, 29(14), 251-268.

13
Naber, M. (2010). Linear fractionally damped oscillator. International Journal of Differential Equations, 2010, 1-12.

14
Podlubny, I.(1999).Fractional Differential Equations.California, USA: Academic Press.

15
Podlubny, I., Skovranek, T. and Vinagre Jara, B. M.(2009). Matrix approach to discretization of fractional derivatives and to solution of fractional differential equations and their systems.Proceedings of the IEEE Conference on Emerging Technologies and Factory Automation, 1-6.

16
Rawashdeh, E.A.(2006). Numerical solution of semidifferential equations by collocation method.Applied Mathematics and Computation, 174(2), 869-876.

17
Ray, S. S. and Bera, R. K. (2005). Analytical solution of the Bagley Torvik equation by Adomian decomposition method.Applied Mathematics and Computation, 168(1), 398-410.

18
StevanoviĀ“c Hedrih, K.(2006). The transversal creeping vibrations of a fractional derivative order constitutive relation of nonhomogeneous beam.Mathematical Problems in Engineering, 2006, 1-18.

19
Tian, Y. and Chen, A. (2009). The existence of positive solution to three-point singular boundary value problem of fractional differential equation.Abstract and Applied Analysis, 2009, 1-18.

20
Torvik, P. J. and Bagley, R. L. (1984). On the appearance of the fractional derivative in the behavior of real materials.Journal of Applied Mechanics, 51(2), 294-298.

21
Zahoor, R. M. A. and Qureshi, I. M. (2009). A modified least mean square algorithm using fractional derivative and its application to system identification.European Journal of Scientific Research, 35(1), 14-21.

22
Zhou, JK.(1986).Differential transformation and its applications for electrical circuits. Wuhan, China: Huazhong University Press.

How to Cite?

DOI: 10.12732/ijpam.v110i2.3 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 110
Issue: 2
Pages: 265 - 273


Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).