×

zbMATH — the first resource for mathematics

Duffing equations with cubic and quintic nonlinearities. (English) Zbl 1165.34310
Summary: In this study, an accurate analytical solution for Duffing equations with cubic and quintic nonlinearities is obtained using the Homotopy Analysis Method (HAM) and Homotopy Padé technique. Novel and accurate analytical solutions for the frequency and displacement are derived. Comparison between the obtained results and numerical solutions shows that only the first order approximation of the Homotopy Pade technique leads to accurate solution with a maximum relative error less than 0.4%.

MSC:
34A45 Theoretical approximation of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Nayfeh, A.H.; Mook, D.T., Nonlinear oscillations, (1979), Wiley New York
[2] Nayfeh, A.H., Perturbation methods, (1973), Wiley New York · Zbl 0375.35005
[3] Mickens, R.E., Oscillations in planar dynamic systems, (1996), World Scientific Singapore · Zbl 0840.34001
[4] Hamdan, M.N.; Shabaneh, N.H., On the large amplitude free vibration of a restrained uniform beam carrying an intermediate lumped mass, Journal of sound and vibration, 199, 711-736, (1997)
[5] Lin, J., A new approach to Duffing equation with strong and high order nonlinearity, Communications in nonlinear science and numerical simulation, 4, 132-135, (1999) · Zbl 0944.34006
[6] Wu, B.S.; Sun, W.P.; Lim, C.W., An analytical approximate technique for a class of strongly non-linear oscillators, International journal of non-linear mechanics, 41, 766-774, (2006) · Zbl 1160.70340
[7] Lai, S.K., Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators, Applied mathematical modelling, (2008)
[8] Liao, S.J.; Tan, Y., A general approach to obtain series solutions of nonlinear differential equations, Studies in applied mathematics, 119, 297-354, (2007)
[9] Pirbodaghi, T.; Ahmadian, M.T.; Fesanghary, M., On the homotopy analysis method for non-linear vibration of beams, Mechanics research communications, (2008) · Zbl 1258.74110
[10] Hoseini, S.H.; Pirbodaghi, T.; Asghari, M.; Farrahi, G.H.; Ahmadian, M.T., Nonlinear free vibration of conservative oscillators with inertia and static type cubic nonlinearities using homotopy analysis method, Journal of sound and vibration, 316, 1-5, 263-273, (2008)
[11] S.J. Liao, On the proposed Homotopy analysis techniques for nonlinear problems and its application, Ph.D. Thesis, Shanghai Jiao Tong University, 1992
[12] Liao, S.J., Beyond perturbation: introduction to homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton
[13] ()
[14] Liao, S.J.; Cheung, K.F., Homotopy analysis of nonlinear progressive waves in deep water, Journal of engineering mathematics, 45, 105-116, (2003) · Zbl 1112.76316
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.