Abstract
In this paper, homotopy perturbation method (HPM) and variational iteration method (VIM) are applied to solve nonlinear oscillator differential equations. Illustrative examples reveal that these methods are very effective and convenient for solving nonlinear differential equations. Moreover, the methods do not require linearization or small perturbation. Comparisons are also made between the exact solutions and the results of the homotopy perturbation method and variational iteration method in order to prove the precision of the results obtained from both methods mentioned.
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Guckenheimer, J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Field. Springer, New York (1983)
Elabbay, E.M., El-Dessoky, M.M.: Synchronization of van der Pol oscillator and Chen Chaotic dynamical system. Chaos Solitions Fractals 36, 1425–1435 (2008)
He, J.-H.: A new approach to nonlinear partial differential equations. Commun. Nonlinear Sci. Numer. Simul. 2, 230–235 (1997)
He, J.-H.: Variational iteration method—a kind of non-linear analytical technique: some examples. Int. J. Non-Linear Mech. 34, 699–708 (1999)
He, J.-H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. 20, 1141–1199 (2006)
Ganji, D.D., Jannatabadi, M., Mohseni, E.: Application of He’s variational iteration method to nonlinear Jaulent–Miodek equations and comparing it with ADM. J. Comput. Appl. Math. 207, 35–45 (2007)
Ganji, D.D., Sadighi, A.: Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations. J. Comput. Appl. Math. 207, 24–34 (2007)
Momani, S., Odibat, Z.: Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations. Comput. Math. Appl. 54, 910–919 (2007)
Adomian, G.: Stochastic Systems. Academic Press, San Diego (1983)
Adomian, G.: Nonlinear Stochastic Operator Equations. Academic Press, San Diego (1986)
Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic, Boston (1994)
He, J.-H.: Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135, 73–79 (2003)
He, J.-H.: Addendum: New interpretation of homotopy perturbation method. Int. J. Mod. Phys. 20, 2561–2568 (2006)
He, J.-H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999)
Rafei, M., Ganji, D.D.: Explicit solutions of Helmholtz Equation and Fifth-order Kdv Equation using homotopy-perturbation method. Int. J. Nonlinear Sci. Numer. Simul. 7, 321–328 (2006)
Ganji, D.D., Sadighi, A.: Application of He’s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations. Int. J. Nonlinear Sci. Numer. Simul. 7, 411–418 (2006)
Barari, A., Omidvar, M., Gholitabar, S., Ganji, D.D.: Variational iteration method and homotopy-perturbation method for solving second-order non-linear wave equation. AIP Conf. Proc. 936, 81–85 (2007)
Rafei, M., Ganji, D.D., Daniali, H., Pashaei, H.: The variational iteration method for nonlinear oscillators with discontinuities. J. Sound Vib. 305, 614–620 (2007)
He, J.H., Wu, X.H.: Variational iteration method: new development and applications. Comput. Math. Appl. 54, 881–894 (2007)
Behiry, S.H., Hashish, H., El-Kalla, I.L., Elsaid, A.: A new algorithm for the decomposition solution of nonlinear differential equations. Comput. Math. Appl. 54, 459–466 (2007)
Gottlieb, H.P.W.: Velocity-dependent conservative nonlinear oscillators with exact harmonic solutions. J. Sound Vib. 230, 323–333 (2000)
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Barari, A., Omidvar, M., Ghotbi, A.R. et al. Application of Homotopy Perturbation Method and Variational Iteration Method to Nonlinear Oscillator Differential Equations. Acta Appl Math 104, 161–171 (2008). https://doi.org/10.1007/s10440-008-9248-9
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DOI: https://doi.org/10.1007/s10440-008-9248-9