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Reseach Article
A Comparison between New Iterative Solutions of Non-linear Oscillator Equation
| International Journal of Computer Applications |
| Foundation of Computer Science (FCS), NY, USA |
| Volume 128 - Number 4 |
| Year of Publication: 2015 |
| Authors: M. Khalid, Mariam Sultana, Uroosa Arshad, Muhammad Shoaib |
10.5120/ijca2015906501
|
M. Khalid, Mariam Sultana, Uroosa Arshad, Muhammad Shoaib . A Comparison between New Iterative Solutions of Non-linear Oscillator Equation. International Journal of Computer Applications. 128, 4 ( October 2015), 1-5. DOI=10.5120/ijca2015906501
Abstract
The prime focus is on a Van der Pol-Duffing oscillator in this paper. A newly proposed method, namely; the Perturbation Iteration Algorithm (PIA) and an Alternative Variation Iteration Method (AVIM) is used to solve governing equations. The study outlines the significant features of the two methods. The beauty of the paper lies in the error analysis between exact solutions and approximate solutions obtained by these two methods which proves that approximate solutions obtained by Alternative Variation Iteration Method converge very rapidly to the exact solutions. Both methods provide analytical solution in the form of a convergent series with components that are easily computable, requiring no linearization or small perturbation.
References
- Kolipanos, C.H.L., Kyprianidis, I.M., Stouboulos, I.N., Anagnostopoulos, A.N. and Magafas, L. (2003) Chaotic behaviour of a fourth-order autonomous electric circuit, Chaos Solitons Fractals, vol.16, pp. 173-182.
- Mickens, R.E. (1996) Oscillations in Planar Dynamics Systems, World Scientific,Singapore.
- Mickens, R.E., Gumel, A.B. (2002) Numerical study of a non-standard finite difference scheme for the Van der Pol equation, J. Sound Vib., vol.250, pp. 955-963.
- Hale, J. (1969) Ordinary Differential Equations, Wiley, New York.
- Guckenheimer, J., Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems,and Bifurcations of Vector Fields, Springer-Verlag, New York.
- Maccari, A. (2008) Vibration amplitude control for a Van der Pol-Duffing oscillator with time delay, J. Sound Vib., vol.317, pp. 20-29.
- Njah, A.N., Vincent, U.E. (2008) Chaos synchronization between single and double wells Duffing-Van der Pol oscillators using active control, Chaos Solitons Fractals, vol.37, pp. 1356-1361.
- Yamapi, R., Filatrella, G. (2008) Strange attractors and synchronization dynamics of coupled Van der Pol–Duffing oscillators, Commun. Nonlinear Sci. Numer. Simul, vol.13, pp. 1121-1130.
- Ji, J., Zhang, N. (2008) Additive resonances of a controlled Van der Pol-Duffing oscillator, J. Sound Vib., vol.315, pp. 22-33.
- Kao, Y.H., Wang, C.S. Analog study of bifurcation structures in a Van der Pol oscillator with a nonlinear restoring force, Phys. Rev. E, vol.48, pp. 2514-2520.
- Parlitz, U., Lauterborn, W. (1987) Period-doubling cascades and Peril’s staircases of the driven Van der-Pol oscillator, Phys. Rev. A, vol.36, pp. 1428-1434.
- Rafei, M., Ganji, D.D., Daniali, H., Pashaei, H. (2007) The variational iteration method for nonlinear oscillators white discontinuities, J. Sound Vib., vol.305, pp. 614-620.
- Momani, S., Ert¨urk, V.S. (2008) Solutions of non-linear oscillators by the modified differential transform method, Comput. Math. Appl., vol.55, pp. 833-842.
- O¨ zis, T., Yildirim, A. (2009) Generating the periodic solutions for forcing Van der-Pol oscillators by the iteration perturbation method, Nonlinear Anal. RWA, vol.10, pp. 1984-1989.
- Kimiaeifar, A., Saidi, A.R., Sohouli, A.R., Ganji, D.D. (2010) Analysis of modified Van der-Pol’s oscillator using He’s parameter-expanding methods, Curr. Appl. Phys., vol.10, pp. 279-283.
- Barari, A., Omidvar, M., Ghotbi, A.R., Ganji, D.D. (2008) Application of homotopy perturbation method and variational iteration method to nonlinear oscillator differential equations, Acta Appl. Math., vol.104, pp. 161-171.
- Khan, Y., Madani, M., Yildirim, A., Abdou, M.A., Faraz, N. (2011) A new approach to Van der-Pol’s oscillator problem, Z. Naturforsch., vol.66a, pp. 620-624.
- Nayfeh, A.H. (1973) Perturbation Methods, Wiley- Interscience, New York.
- Skorokhod, A.V., Hoppensteadt, F.C., Salehi, H. (2002) Random Perturbation Methods with Applications in Science and Engineering, Springer, New York.
- Pakdemirli, M., Aksoy, Y., H. Boyaci, H. (2011) A new perturbation-iteration approach for first order differential equations, Mathematical and Computational Applications, vol. 16, no. 4, pp. 890-899.
- Abassy, T.A., El-Tawil, M.A., El-Zoheiyr, H. (2007) Toward a modified variational iteration method, J. Comput. Appl. Math., vol.207, no.1, pp. 137-147.
- Abassy, T.A., El-Tawil, M.A., El-Zoheiyr, H. (2007) Solving nonlinear partial differential equations using the modified variational iteration Pad’e technique, J. Comput. Appl. Math., vol.207, no.1, pp. 73-91.
- Geng, F., Lin, Y., Cui, M. (2009) A piecewise variational iteration method for Riccati differential equations, Comput. Math. Appl., vol.58, no.11-12, pp. 2518-2522.
- Ghorbani, A., Saberi-Nadjafi, J. (2009) An effective modification of He’s variational iteration method, Nonlinear Anal. Real World Appl., vol.10, no.5, pp. 2828-2833. Odibat, Z.M. (2010) A study on the convergence of variational iteration method, Math. Comput. Model., vol.51, pp. 1181-1192.
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