Application of Homotopy Perturbation Method and Parameter Expanding Method to Fractional Van der Pol Damped Nonlinear Oscillator
Affiliation(s)
Department of Mathematics, Faculty of Science, Taif University, Taif, KSA. Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt.
Department of Mathematics, Faculty of Science, Taif University, Taif, KSA. Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt.
ABSTRACT
In this study, homotopy perturbation method and parameter expanding method are applied to the motion equations of two nonlinear oscillators. Our results show that both the (HPM) and (PEM) yield the same results for the nonlinear problems. In comparison with the exact solution, the results show that these methods are very convenient for solving nonlinear equations and also can be used for strong nonlinear oscillators.
Cite this paper
T. Nofal, G. Ismail and S. Abdel-Khalek, "Application of Homotopy Perturbation Method and Parameter Expanding Method to Fractional Van der Pol Damped Nonlinear Oscillator," Journal of Modern Physics, Vol. 4 No. 11, 2013, pp. 1490-1494. doi: 10.4236/jmp.2013.411179.
T. Nofal, G. Ismail and S. Abdel-Khalek, "Application of Homotopy Perturbation Method and Parameter Expanding Method to Fractional Van der Pol Damped Nonlinear Oscillator," Journal of Modern Physics, Vol. 4 No. 11, 2013, pp. 1490-1494. doi: 10.4236/jmp.2013.411179.
References
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[1] J. H. He, European Journal of Physics, Vol. 29, 2008, pp. 19-22. http://dx.doi.org/10.1088/0143-0807/29/4/L02 [2] J. H. He, International Journal of Nonlinear Science and Numerical Simulation, Vol. 9, 2008, pp. 211-212.
http://dx.doi.org/10.1515/IJNSNS.2008.9.2.211 [3] A. G. Davodi, D. D. Gangi, R. Azami and H. Babazadeh, Modern Physics Letters B, Vol. 23, 2009, pp. 3427-3436.
http://dx.doi.org/10.1142/S0217984909021466 [4] I. Mehdipour, D. D. Ganji and M. Mozaffari, Current Applied Physics, Vol. 10, 2010, pp. 104-112.
http://dx.doi.org/10.1016/j.cap.2009.05.016 [5] D. Younesian, H. Askari, Z. Saadatnia and M. Kalami Yazdi, Computers and Mathematics with Applications, Vol. 59, 2010, pp. 3222-3228.
http://dx.doi.org/10.1016/j.camwa.2010.03.013 [6] H. Askari, M. Kalami Yazdi and Z. Saadatnia, Nonlinear Science Letters A, Vol. 1, 2010, pp. 425-430. [7] J. H. He, G. C. Wu and F. Austin, Nonlinear Science Letters A, Vol. 1, 2010, pp. 1-30. [8] N. Herisanu and V. Marinca, Nonlinear Science Letters A, Vol. 1, 2010, pp. 183-192. [9] J. H. He, International Journal of Modern Physics B, Vol. 20, 2006, pp. 2561-2568.
http://dx.doi.org/10.1142/S0217979206034819 [10] J. H. He, International Journal of Non-Linear Mechanics, Vol. 35, 2000, pp. 37-43. [11] J. H. He, Chaos, Solitons and Fractals, Vol. 26, 2005, pp. 695-700. http://dx.doi.org/10.1016/j.chaos.2005.03.006 [12] D. D. Ganji, Physics Letters A, Vol. 355, 2006, pp. 337-341. http://dx.doi.org/10.1016/j.physleta.2006.02.056 [13] J. H. He, International Journal of Modern Physics B, Vol. 20, 2006, pp. 1141-1199.
http://dx.doi.org/10.1142/S0217979206033796 [14] L. Xu, Journal of Computational and Applied Mathematics, Vol. 207, 2007, pp. 148-154.
http://dx.doi.org/10.1016/j.cam.2006.07.020 [15] L. Xu, Journal of Sound and Vibration, Vol. 302, 2007, pp. 178-184. http://dx.doi.org/10.1016/j.jsv.2006.11.011 [16] J. H. He, Computer Methods in Applied Mechanics and Engineering, Vol. 178, 1999, pp. 257-262.
http://dx.doi.org/10.1016/S0045-7825(99)00018-3 [17] J. H. H Journal of Sound and Vibration, Vol. 229, 2000, pp. 1257-1263. http://dx.doi.org/10.1006/jsvi.1999.2509 [18] J. H. He, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 2, 2001, pp. 257-264.
http://dx.doi.org/10.1515/IJNSNS.2001.2.3.257 [19] R. E. Mickens, Journal of Sound and Vibration, Vol. 292, 2006, pp. 964-968.
http://dx.doi.org/10.1016/j.jsv.2005.08.020 [20] T. Ozis and A. Yildirm, Journal of Sound and Vibration, Vol. 306, 2007, pp. 372-376.
http://dx.doi.org/10.1016/j.jsv.2007.05.021 [21] D. D. Ganji, M. Esmaeilpour and S. Soleimani, International Journal of Computer Mathematics, Vol. 9, 2010, pp. 2014-2023. [22] I. S. Gradshteyn and I. S. Ryzhik, “Table of Integrals, Series and Products,” Academic Press, New York, 1980. [23] C. W. Lim and S. K. Lai, International Journal of Mechanical Sciences, Vol. 48, 2006, pp. 483-892.
http://dx.doi.org/10.1016/j.ijmecsci.2005.12.009 [24] J. J. Stoker, “Nonlinear Vibrations in Mechanical and Electrical Systems,” Wiley, New York, 1992. [25] D. W. Jordan and P. Smith, “Nonlinear Ordinary Differential Equations: An Introduction to Dynamical Systems,” Oxford University Press, New York, 1999.