zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The iterative homotopy harmonic balance method for conservative Helmholtz-Duffing oscillators. (English) Zbl 1183.65083
Summary: An approach that the iterative homotopy harmonic balance method which incorporates salient features of both the parameter-expansion and the harmonic balance is presented to solve conservative Helmholtz-Duffing oscillators. Since the behaviors of the solutions in the positive and negative directions are quite different, the asymmetric equation is separated into two auxiliary equations. The auxiliary equations are solved by the proposed method. The results show that it works very well for the whole range of initial amplitudes in a variety of cases, and the excellent agreement of the approximate periods and periodic solutions with the exact ones is demonstrated and discussed. The proposed method is very simple in its principle and has a great potential to be applied to other nonlinear oscillators.
MSC:
65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
References:
[1]He, J. H.: Some asymptotic methods for strongly nonlinear equations, International journal of modern physics B 20, No. 10, 1141-1199 (2006) · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[2]He, J. H.: Variational iteration method – a kind of non-linear analytical technique, International journal of nonlinear mechanics 34, No. 4, 699-708 (1999)
[3]He, J. H.: Homotopy perturbation technique, Computer methods in applied mechanics and engineering 178, 257-262 (1999)
[4]He, J. H.: New interpretation of homotopy perturbation method, International journal of modern physics B 20, No. 18, 2561-2568 (2006)
[5]He, J. H.: Homotopy perturbation method: a new nonlinear analytical technique, Applied mathematics and computation 135, No. 1, 73-79 (2003) · Zbl 1030.34013 · doi:10.1016/S0096-3003(01)00312-5
[6]He, J. H.: A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International journal of nonlinear mechanics 35, 37-43 (2000) · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7
[7]Geng, L.; Cai, X. C.: He’s formulation for nonlinear oscillators, European journal of physics 28, 923-931 (2007) · Zbl 1162.70019 · doi:10.1088/0143-0807/28/5/016
[8]Mickens, R. E.: Quadratic non-linear oscillators, Journal of sound and vibration 270, 427-432 (2004)
[9]Mickens, R. E.: Oscillations in planar dynamics systems, (1996)
[10]Hu, H.: Solution of a quadratic nonlinear oscillator by the method of harmonic balance, Journal of sound and vibration 293, 462-468 (2006)
[11]Hu, H.: Solution of mixed parity nonlinear oscillator: harmonic balance, Journal of sound and vibration 299, 331-338 (2002)
[12]Leung, A. Y. T.; Guo, Zhongjin: Homotopy perturbation for conservative Helmholtz – Duffing oscillators, Journal of sound and vibration 325, 287-296 (2009)
[13]He, Ji-Huan: The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied mathematics and computation 151, 287-292 (2004) · Zbl 1039.65052 · doi:10.1016/S0096-3003(03)00341-2
[14]He, J. H.: Asymptotology by homotopy perturbation method, Applied mathematics and computation 156, No. 3, 591-596 (2004) · Zbl 1061.65040 · doi:10.1016/j.amc.2003.08.011
[15]Song, Lina; Zhang, Hongqing: Application of the extended homotopy perturbation method to a kind of nonlinear evolution equations, Applied mathematics and computation 197, 87-95 (2008) · Zbl 1135.65387 · doi:10.1016/j.amc.2007.07.035
[16]Momani, S.; Abuasad, S.: Application of he’s variational iteration method to Helmholtz equation, Chaos, solitons & fractals 27, No. 5, 1119-1123 (2006) · Zbl 1086.65113 · doi:10.1016/j.chaos.2005.04.113
[17]Wu, B. S.; Sun, W. P.; Lim, C. W.: An analytical approximate technique for a class of strongly non-linear oscillators, International journal of nonlinear mechanics 41, 766-774 (2006) · Zbl 1160.70340 · doi:10.1016/j.ijnonlinmec.2006.01.006
[18]Metter, E.: Dynamic buckling, Handbook of engineering mechanics (1992)
[19]Lenci, Stefano; Rega, Giuseppe: Global optimal control and system-dependent solutions in the hardening Helmholtz – Duffing oscillator, Chaos, solitons & fractals 21, No. 5, 1031-1046 (2004) · Zbl 1060.93527 · doi:10.1016/S0960-0779(03)00387-4
[20]Gravador, E.; Thylwe, K. -E.; Hökback, A.: Stability transitions of certain exact periodic responses in undamped Helmholtz and Duffing oscillators, Journal of sound and vibration 182, No. 2, 209-220 (1995)