Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T14:35:13.823Z Has data issue: false hasContentIssue false
  • English
  • Français

The Multi-Step Differential Transform Method and Its Application to Determine the Solutions of Non-Linear Oscillators

Published online by Cambridge University Press:  03 June 2015

Vedat Suat Ertürk ,
Zaid M. Odibat  and
Shaher Momani
Vedat Suat Ertürk*
Affiliation:
Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayis University, 55139, Samsun, Turkey
Zaid M. Odibat*
Affiliation:
Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa Applied University, Al-Salt 19117, Jordan
Shaher Momani*
Affiliation:
The University of Jordan, Faculty of Science, Department of Mathermatics, Amman 1194, Jordan
*
Corresponding author. URL:http://www2.omu.edu.tr/akademikper.asp?id=156, Email: vserturk@yahoo.com
Email: z.odibat@gmail.com
Email: shahermm@yahoo.com
Get access

Abstract

In this paper, a reliable algorithm based on an adaptation of the standard differential transform method is presented, which is the multi-step differential transform method (MSDTM). The solutions of non-linear oscillators were obtained by MSDTM. Figurative comparisons between the MSDTM and the classical fourth-order Runge-Kutta method (RK4) reveal that the proposed technique is a promising tool to solve non-linear oscillators.

Keywords

74H1535E1537M05Non-linear oscillatory systemsdifferential transform methodnumerical solution
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 4 , August 2012 , pp. 422 - 438
Copyright
Copyright © Global-Science Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Nayef, A. H., Problems in Perturbation, John Wiley, New York, 1985.Google Scholar
[2]Yildirim, A., Application of He’s homotopy perturbation method for solving the Cauchy reaction-diffusion problem, Comput. Math. Appl., 57(4) (2009), pp. 612–618.CrossRefGoogle Scholar
[3]Fu, Y. M., Zhang, J. and Wan, L. J., Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS), Current. Appl. Phys., 11(3) (2011), pp. 482–485.Google Scholar
[4]Turkyilmazoglu, M., Numerical and analytical solutions for the flow and heat transfer near the equator of an MHD boundary layer over a porous rotating sphere, Int. J. Thermal. Sci., 50(5) (2011), pp. 831–842.Google Scholar
[5]Nourbakhsh, S. H., Pasha Zanoosi, A. A. and Shateri, A. R., Analytical solution for off-centered stagnation flow towards a rotating disc problem by homotopy analysis method with two auxiliary parameters, Commun. Nonlinear. Sci. Numer. Sim., 16(7) (2011), pp. 2772–2787.Google Scholar
[6]Odibat, Z., Bertelle, C., Aziz-Alaoui, M. A. and Duchamp, G., A multi-step differential transform method and application to non-chaotic or chaotic systems, Comput. Math. Appl., 59(4) (2010), pp. 1462–1472.Google Scholar
[7]Fidlin, A., Nonlinear Oscillations in Mechanical Engineering, Springer-Verlag, Berlin Heidelberg, 2006.Google Scholar
[8]Dimarogonas, A. D. and Haddad, S., Vibration for Engineers, Prentice-Hall, Engle-wood Cliffs, New Jersey, 1992.Google Scholar
[9]Özkan, O., Numerical implementation of differential transformations method for integro-differential equations, Int. J. Comput. Math., 87(12) (2010), pp. 2786–2797.Google Scholar
[10]Bert, C. W. and Zeng, H., Analysis of axial vibration of compound bars by differential transformation method, J. Sound. Vib., 275(3-5) (2004), pp. 641–647.Google Scholar
[11]Jang, B., Solving linear and nonlinear initial value problems by the projected differential transform method, Comput. Phys. Commun., 181(5) (2010), pp. 848–854.Google Scholar
[12]Arikoglu, A. and Ozkol, I., Solution of difference equations by using differential transform method, Appl. Math. Comput., 173(1) (2006), pp. 126–136.Google Scholar
[13]Ertürk, V. S., Solution of linear twelfth-order boundary value problems by using differential transform method, Int. J. Appl. Math. Stat., 13(8) (2008), pp. 57–63.Google Scholar
[14]Arikoglu, A. and Ozkol, I., Solution of boundary value problems for integro-differential equations by using differential transform method, Appl. Math. Comput., 168(2) (2005), pp. 1145–1158.Google Scholar
[15]Arikoglu, A. and Ozkol, I., Solution of differential difference equations by using differential transform method, Appl. Math. Comput., 181(1) (2006), pp. 153–162.Google Scholar
[16]Liu, H. and Song, Y., Differential transform method applied to high index differential-algebraic equations, Appl. Math. Comput., 184(2) (2007), pp. 748–753.Google Scholar
[17]Liu, H. and Song, Y., Differential transform method applied to high index differential-algebraic equations, Appl. Math. Comput., 184(2) (2007), pp. 748–753.Google Scholar
[18]Momani, S., Erjaee, G. H. and Alnasr, M. H., The modified homotopy perturbation method for solving strongly nonlinear oscillators, Comput. Math. Appl., 58 (2009), pp. 2209–2220.CrossRefGoogle Scholar
[19]Marinca, V. and Herisanu, N., Periodic solutions for some strongly nonlinear oscillations by He’s variational iteration method, Comput. Math. Appl., 54 (2007), pp. 1188–1196.Google Scholar
[20]Cai, X. C. and Wub, W. Y., He’s frequency formulation for the relativistic harmonic oscillator, Comput. Math. Appl., 58 (2009), pp. 2358–2359.CrossRefGoogle Scholar
[21]Lim, C. W. and Wub, B. S., A new analytical approach to the Duffing-harmonic oscillator, Phys. Lett. A., 311(4-5) (2003), pp. 365–373.Google Scholar