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)
Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators. (English) Zbl 1142.65055

Summary: We apply J.-H. He’s homotopy perturbation method [Int. J. Mod. Phys. B 20, No. 10, 1141–1199 (2006; Zbl 1102.34039)] to find improved approximate solutions to conservative truly nonlinear oscillators. This approach gives us not only a truly periodic solution but also the period of the motion as a function of the amplitude of oscillation. We find that this method works very well for the whole range of parameters in the case of the cubic oscillator, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed.

For the second order approximation we show that the relative error in the analytical approximate frequency is approximately 0.03% for any parameter values involved. We also compare the analytical approximate solutions and the Fourier series expansion of the exact solution. This allows us to compare the coefficients for the different harmonic terms in these solutions.

The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values and the results reveal that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems.

MSC:
65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
34C15Nonlinear oscillations, coupled oscillators (ODE)
References:
[1]Campbell, D. K.: Nonlinear science: the next decade, (1992)
[2]Liao, S.: Beyond perturbation: introduction to the homotopy analysis method, (2004)
[3]He, J. H.: A review on some new recently developed nonlinear analytical techniques, Int J non-linear sci numer simul 1, 51-70 (2000) · Zbl 0966.65056 · doi:10.1515/IJNSNS.2000.1.1.51
[4]Nayfeh, A. H.: Problems in perturbations, (1985) · Zbl 0573.34001
[5]Mickens, R. E.: Oscillations in planar dynamics systems, (1996)
[6]He, J. H.: A new perturbation technique which is also valid for large parameters, J sound vibr 229, 1257-1263 (2000)
[7]He, J. H.: Modified Lindstedt – Poincarè methods for some non-linear oscillations. Part III: Double series expansion, Int J non-linear sci numer simul 2, 317-320 (2001) · Zbl 1072.34507 · doi:10.1515/IJNSNS.2001.2.4.317
[8]He, J. H.: Modified Lindstedt – Poincarè methods for some non-linear oscillations. Part I: Expansion of a constant, Int J non-linear mech 37, 309-314 (2002) · Zbl 1116.34320 · doi:10.1016/S0020-7462(00)00116-5
[9]He, J. H.: Modified Lindstedt – Poincarè methods for some non-linear oscillations. Part II: a new transformation, Int J non-linear mech 37, 315-320 (2002) · Zbl 1116.34321 · doi:10.1016/S0020-7462(00)00117-7
[10]Amore, P.; Aranda, A.: Improved Lindstedt – Poincaré method for the solution of nonlinear problems, J sound vibr 283, 1115-1136 (2005)
[11]Amore, P.; Fernández, F. M.: Exact and approximate expressions for the period of anharmonic oscillators, Eur J phys 26, 589-601 (2005)
[12]He, J. H.: Homotopy perturbation method for bifurcation on nonlinear problems, Int J non-linear sci numer simul 6, 207-208 (2005)
[13]Amore, P.; Raya, A.; Fernández, F. M.: Alternative perturbation approaches in classical mechanics, Eur J phys 26, 1057-1063 (2005) · Zbl 1080.70014 · doi:10.1088/0143-0807/26/6/013
[14]Amore, P.; Raya, A.; Fernández, F. M.: Comparison of alternative improved perturbative methods for nonlinear oscillations, Phys lett A 340, 201-208 (2005) · Zbl 1145.70323 · doi:10.1016/j.physleta.2005.04.004
[15]Mickens, R. E.: Comments on the method of harmonic-balance, J sound vibr 94, 456-460 (1984)
[16]Mickens, R. E.: Mathematical and numerical study of the Duffing-harmonic oscillator, J sound vibr 244, 563-567 (2001)
[17]Wu, B. S.; Lim, C. W.: Large amplitude nonlinear oscillations of a general conservative system, Int J non-linear mech 39, 859-870 (2004)
[18]Lim, C. W.; Wu, B. S.: Accurate higher-order approximations to frequencies of nonlinear oscillators with fractional powers, J sound vibr 281, 1157-1162 (2005)
[19]Beléndez, A.; Hernández, A.; Márquez, A.; Beléndez, T.; Neipp, C.: Analytical approximations for the period of a simple pendulum, Eur J phys 27, 539-551 (2006)
[20]Lim, C. W.; Wu, B. S.: A new analytical approach to the Duffing-harmonic oscillator, Phys lett A 311, 365-373 (2003) · Zbl 1055.70009 · doi:10.1016/S0375-9601(03)00513-9
[21]Hu, H.; Tang, J. H.: Solution of a Duffing-harmonic oscillator by the method of harmonic balance, J sound vibr 294, 637-639 (2006)
[22]Hu, H.: Solution of a Duffing-harmonic oscillator by an iteration procedure, J sound vibr 298, 446-452 (2006)
[23]He, J. H.: Some asymptotic methods for strongly nonlinear equations, Int J mod phys B 20, 1141-1199 (2006) · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[24]Mickens, R. E.: A generalized iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillators, J sound vibr 287, 1045-1051 (2005)
[25]He, J. H.: New interpretation of homotopy perturbation method, Int J mod phys B 20, 2561-2568 (2006)
[26]He, J. H.: Homotopy perturbation method for bifurcation of nonlinear problems, Int J non-linear sci numer simul 6, No. 2, 207-208 (2005)
[27]Cai, X. C.; Wu, W. Y.; Li, M. S.: Approximate period solution for a kind of nonlinear oscillator by he’s perturbation method, Int J non-linear sci numer simul 7, No. 1, 109-170 (2006)
[28]Abbasbandy, S.: Application of he’s homotopy perturbation method for Laplace transform, Chaos, solitons & fractals 30, 1206-1212 (2006) · Zbl 1142.65417 · doi:10.1016/j.chaos.2005.08.178
[29]Rafei, M.; Ganji, D. D.: Explicit solution of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method, Int J non-linear sci numer simul 7, No. 3, 321-328 (2006)
[30]Ganji, D. D.: The application of he’s homotopy perturbation method to nonlinear equations arising in heat transfer, Phys lett A 355, 337-341 (2006)
[31]Ariel, P. D.; Hayat, T.; Asghar, S.: Homotopy perturbation method and axisymmetric flow over a stretching sheet, Int J non-linear sci numer simul 7, No. 4, 399-406 (2006)
[32]Ganji, D. D.; Sadighi, A.: Application of he’s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations, Int J non-linear sci numer simul 7, No. 4, 411-418 (2006)
[33]He, J. H.: Application of homotopy perturbation method to nonlinear wave equations, Chaos, solitons & fractals 36, 695-700 (2005)
[34]El-Shaded, M.: Application of he’s homotopy perturbation method to Volterra’s integro-differential equation, Int J non-linear sci numer simul 6, No. 2, 163-168 (2005)
[35]Siddiqui, A.; Mahmood, R.; Ghori, Q.: Thin film flow of a third grade fluid on moving a belt by he’s homotopy perturbation method, Int J non-linear sci numer simul 7, No. 1, 15-26 (2006)
[36]Whineray, S.: A cube-law air track oscillator, Eur J phys 12, 90-95 (1991)
[37]Marion, J. B.: Classical dynamics of particles and systems, (1970)
[38]Milne-Thomson, L. M.: Elliptic integrals, Handbook of mathematical functions (1972)
[39]Milne-Thomson, L. M.: Jacobi elliptic functions and theta functions, Handbook of mathematical functions (1972)