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Analytical Solutions of Undamped and Autonomous Cubic-Quintic Duffing Equation
American Journal of Physics and Applications
Volume 3, Issue 5, September 2015, Pages: 159-165
Received: Jul. 8, 2015; Accepted: Jul. 16, 2015; Published: Jul. 25, 2015
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Serge Bruno Yamgoué, Department of Physics, Higher Teacher Training College-Bambili, The University of Bamenda, Bamenda, Cameroon
Jules Hilaire Kamga, Laboratoire de Mécanique et de Modélisation des Systèmes Physiques (L2MSP), Département de Physique, Université de Dschang, Dschang, Cameroun
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In this paper, based on a combination of homogenous balance and the rational expansion method, the exact analytical and closed-form solutions of the Duffing equation with cubic and quintic nonlinearities are derived. We focus on heteroclinic and homoclinic solutions which are relevant for the prediction of chaos in forced mechanical systems. The conditions of existence of these solutions which also represent solitons of some wave equations are carefully analyzed.
Cubic-Quintic Duffing Equation, Heteroclinic and the Homoclinic Solutions, Soliton
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Serge Bruno Yamgoué, Jules Hilaire Kamga, Analytical Solutions of Undamped and Autonomous Cubic-Quintic Duffing Equation, American Journal of Physics and Applications. Vol. 3, No. 5, 2015, pp. 159-165. doi: 10.11648/j.ajpa.20150305.11
Camassa, R., Holm, D.: An integrable shallow wave equation with peaked solitons. Phys. Rev. Lett., Vol.71, pp.1661–1664, 1993.
Tian, L., Liang, S.: Global well-posedness and limit behaviorof the solutions to the viscous Degasperis–Procesi equation. J. Math. Phys., Vol.50, 033503, 2009.
Tian, L., Chen, Y., Jiang, X., Xia, L.: Low-regularity solutions of the periodic Fornberg–Whitham equation. J. Math. Phys., Vol. 50, 073507, 2009.
Cao, C., Geng, X., Wang, H.: Algebro-geometric solution of the (2+1)-dimensional Burgers equation with a discrete variable. J. Math. Phys., Vol.43, pp. 621–643, 2002.
Geng, X., Xue, B.: An extension of integrable Peakon equations with cubic nonlinearity. Nonlinearity, Vol.22, pp.1847–1856 , 2009.
Aiyong Chen • Jibin Li •Wentao Huang “The monotonicity and critical periods of periodic waves of the φ6 field model” Nonlinear Dyn, Vol.63, pp. 205–215, 2011.
I. A. Zeid Al-Muhiameed and A. B. Emad Abdel-Salam, “Generalized Jacobi Elliptic Function Solution to a Class of Nonlinear Schrödinger-Type Equations”, Mathematical Problems in Engineering, vol. 2011, Article ID 575679, 11 pages, 2011
F. Kenmogne and D. Yemele “Bright and peaklike pulse solitary waves and analogy with modulational instability in an extended nonlinear Schr¨odinger equation” Physical Review E, Vol.88, 043204 , 2013.
A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, New York, 1979.
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag , 1983.
D.Younesian, H.Askari, Z.Saadatnia, and M. K. Yazdi, ‘‘Periodic solutions for nonlinear oscillation of a centrifugal governor system using the He’s frequency–amplitude formulation and He’s energy balance method’’, Nonlinear Science Letters A, Vol.2, pp. 143–148, 2011.
M.T. Ahmadian, M. Mojahedi, and H. Moeenfard, ‘‘Free vibration analysis of a nonlinear beam using homotopy and modified Lindstedt–Poincare methods’’, Journal of Solid Mechanics, Vol.1, pp. 29–36, 2009. 368 S. Nourazar, A. Mirzabeigy / Scientia Iranica, Transactions B: Mechanical Engineering ,Vol.20, pp 364–368, 2013
F. Bakhtiari-Nejad, and M.Nazari, ‘‘Nonlinear vibration analysis of isotropic cantilever plate with viscoelastic laminate’’, Nonlinear Dynamics, Vol.56, pp. 325–356, 2009.
N. Srinil, and H. Zanganeh, ‘‘Modelling of coupled cross-flow/in-line vortex-induced vibrations using double Duffing and van der Pol oscillators’’, Ocean Engineering, Vol. 53, pp. 83–97, 2012.
S. K. Lai, C. W. Lim, B. S. Wu, C. Wang, C. Q. Zeng, X. F. He: Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators. Applied Mathematical Modelling, Vol. 33, pp. 852-866, 2009
Alex Elias-Zuniga: Exact solution of the cubic-quintic Duffing oscillator, Applied Mathematical Modelling, Vol. 37, pp. 2574-2579, 2013
Alex Elias-Zuniga: Solution of the damped cubic-quintic Duffing oscillator by using Jacobi elliptic functions. Applied Mathematics and Computation, Vol. 246, pp. 474-481, 2014
D. Baldwin, Ü. Göktas, W. Hereman, L. Hong, R. S. Martino, J. C. Miller: Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for PDEs”, Journal of Symbolic Computation, Vol. 37, pp. 669-705, 2004.
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