Analytical solutions for autonomous conservative nonlinear oscillator

International Journal of Nonlinear Sciences and Numerical Simulation

Volumn 11, Issue 11, 2010, Pages 979-984

  Yazdi, Mohammad Kalami ^a   Khan, Yasir ^b   Madani, M ^c   Askari, H ^a   Saadatnia, Z ^a   Yildirim, Ahmet ^d  

^a IRAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Iran)

^b ZHEJIANG UNIVERSITY   (China)

^c AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

^d EGE UNIVERSITY   (Turkey)

Author keywords

Autonomous conservative nonlinear oscillator; Energy balance method; Variational approach (VA)

Indexed keywords

EID: 79251633504     PISSN: 15651339     EISSN: None     Source Type: Journal    
DOI: 10.1515/IJNSNS.2010.11.11.979     Document Type: Article

References (27)

1. A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, New York, 1979.
2. J.H. He, An elementary introduction to the homotopy perturbation method, Comput. Math. Appl. 57 (2009) 410-412.
3. A. Beléndez, C. Pascual, M.L. Alvarez, D.I. Méndez, M.S. Yebra, A. Hernández, Higher order analytical approximate solutions to the nonlinear pendulum by He's homotopy method, PHYS. SCR. 79 (2009)015009.
4. J.H. He, Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput. 135 (2003) 73-79.
5. J.H. He, Homotopy perturbation technique, Comput. Meth.Appl. Mech.Eng. 178 (1999) 257-262.
6. J. H. He, Homotopy perturbation method for bifurcation of non-linear problems, Int. J. Nonlinear Sci.Num. Simul, 6 (2) (2005) 207-208.
7. A. Yildirim, Analytical approach to Fokker-Planck equation with space- and time-fractional derivatives by means of the homotopy perturbation method, J. King. Saud. Uni. Sci. (2010), doi:10.1016/j.jksus.2010.05.008.
8. A. R. Ghotbi, A. Barari, D.D. Ganji, Solving Ratio-Dependent Predator-Prey System with Constant Effort Harvesting Using Homotopy Perturbation Method, Math. Prob. Eng. DOI: 10.1155/2008/945420.
9. S.S. Ganji, D.D. Ganji, M.G. Sfahani, S. Karimpour.Application of AFF and HPM to the systems of strongly nonlinear oscillation, Curr. Appl. Phy. 10 (2010) 1317-1325.
10. L.Zhao, He's frequency-amplitude formulation for nonlinear oscillators with an irrational force, Comput. Math Appl. 58 (2009) 2477-2479.
11. D. Younesian, H. Askari, Z. Saadatnia, M. Kalami Yazdi, Frequency analysis of strongly nonlinear generalized Duffing oscillators using He's frequency-amplitude formulation and He's energy balance method. Comput.Math Appl. 59 (2010) 3222-3228.
12. J. Fan, He's frequency-amplitude formulation for the Duffing harmonic oscillator, Comput. Math. Appl. 58 (2009) 2473-2476.
13. J.H.He, An improved amplitude-frequency formulation for nonlinear oscillators, Int. J. Nonlinear Sci. Num. simul.9 (2) (2008) 211-212.
14. H. Askari, M. Kalami Yazdi, Z. Saadatnia, Frequency analysis of nonlinear oscillators with rational restoring force via He's energy balance method and He's variational approach, Nonlinear Sci. Lett. A. 1(2010)425-430.
15. J.H. He, Preliminary report on the energy balance for nonlinear oscillations, Mech. Res.Commun. 29 (2002) 107-111.
16. T. Ozis, A. Yildinm, Determination of the frequency-amplitude relation for a Duffing-harmonic oscillator by the energy balance method, Comput. Math. Appl. 54 (2007)1184-1187.
17. J.H. He, G.C. Wu, F. Austin, The variational iteration method which should be followed, Nonlinear Sci. Lett. A. 1 (2010) 1-30.
18. N. Faraz, Y. Khan, A. Yildirim, Analytical approach to two-dimensional viscous flow with a shrinking sheet via variational iteration algorithm-II, J. King. Saud. Uni. Sci. DOI: 10.1016/j.jksus.2010.06.010.
19. J.H. He, Varational iteration method some recent results and new interpretations, J. Comput.Appl. Math. 207 (2007) 3-17.
20. J.H. He, Variational approach for nonlinear oscillators, Chaos Solitons Fractals. 34 (2007) 1430-1439.
21. D.D. Ganji, H. Tari, H. Babazadeh, The application of He's variational iteration method to nonlinear equations arising in heat transfer, Phy. Lett. A.363 (3) (2007) 213-217.
22. J.H. He, Hamiltonian approach to nonlinear oscillators, Phys. Lett. A. 374 (2010) 2312-2314.
23. D. Younesian, E. Esmailzadeh, R. Sedaghati, Existence of periodic solutions for the generalized form of Mathieu equation, Nonlinear Dyn. 39 (2005) 335-348.
24. N. Faraz, Y. Khan, H. Jafari, A. Yildirim, M. Madani, Fractional variational iteration method via modified Riemann-Liouville derivative, J. King. Saud. Uni. Sci.DOI: 10.1016/j.jksus.2010.07.025
25. Y. Khan, N. Faraz, Application of Modified Laplace Decomposition Method for Solving Boundary Layer Equation, J. King. Saud. Uni. Sci. DOI:10.1016/j.jksus.2010.06.018.
26. M.N. Hamden, N.H. Shabaneh, On the large amplitude free vibrations of a restrained uniform beam carrying an intermediate lumped mass, J. Sound Vib. 199 (1997) 711-736.
27. Ch.K. Chen, Sh.Sh. Chen, Application of the differential transformation method to a non-linear conservative system, Appl. Math. Comput.154 (2004) 431-441.