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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: [email protected]
Email: [email protected]
Email: [email protected]
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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

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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