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 a modified He’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities. (English) Zbl 1167.34327
Summary: A modified He’s homotopy perturbation method is used to calculate the periodic solutions of a nonlinear oscillator with discontinuities for which the elastic force term is proportional to sgn$(x)$. The He’s homotopy perturbation method is modified by truncating the infinite series corresponding to the first-order approximate solution before introducing this solution in the second-order linear differential equation. We find that this modified homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate period of less than 1.6% for all values of oscillation amplitude, while this relative error is 0.65% for the second iteration and 0.24% when the third-order approximation is considered. Comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient.

MSC:
34C15Nonlinear oscillations, coupled oscillators (ODE)
65L99Numerical methods for ODE
WorldCat.org
Full Text: DOI
References:
[1] Mickens, R. E.: Oscillations in planar dynamics systems. (1996) · Zbl 0840.34001
[2] He, J. H.: Modified Lindstedt--Poincarè methods for some non-linear oscillations. Part III: Double series expansion. Int. J. Nonlinear sci. Numer. simul. 2, 317-320 (2001) · Zbl 1072.34507
[3] He, J. H.: Modified Lindstedt--Poincarè methods for some non-linear oscillations. Part I: Expansion of a constant. Int. J. Nonlinear mech. 37, 309-314 (2002) · Zbl 1116.34320
[4] Amore, P.; Fernández, F. M.: Exact and approximate expressions for the period of anharmonic oscillators. Eur. J. Phys. 26, 589-601 (2005)
[5] Amore, P.; Raya, A.; Fernández, F. M.: Alternative perturbation approaches in classical mechanics. Eur. J. Phys. 26, 1057-1063 (2005) · Zbl 1080.70014
[6] He, J. H.: A new perturbation technique which is also valid for large parameters. J. sound vibration 229, 1257-1263 (2000) · Zbl 1235.70139
[7] Özis, T.; Yildirim, A.: Determination of periodic solution for a u1/3force by he’s modified Lindstedt--Poincaré method. J. sound vibration 301, 415-419 (2007)
[8] Gottlieb, H. P. W.: Harmonic balance approach to limit cycles for nonlinear Jerk equations. J. sound vibration 297, 243-250 (2006) · Zbl 1243.70020
[9] Lim, C. W.; Wu, B. S.; Sun, W. P.: Higher accuracy analytical approximations to the Duffing-harmonic oscillator. J. sound vibration 296, 1039-1045 (2006) · Zbl 1243.34021
[10] 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)
[11] Itovich, G. R.; Moiola, J. L.: On period doubling bifurcations of cycles and the harmonic balance method. Chaos solitons fractals 27, 647-665 (2005) · Zbl 1083.37041
[12] Beléndez, A.; Hernández, A.; Beléndez, T.; Álvarez, M. L.; Gallego, S.; Ortuño, M.; Neipp, C.: Application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire. J. sound vibration 302, 1018-1029 (2007) · Zbl 1242.34056
[13] He, J. H.: Application of parameter-expanding method to strongly nonlinear oscillators. Int. J. Nonlinear sci. Numer. simul. 8, No. 1, 121-124 (2007)
[14] Beléndez, A.; Hernández, A.; Beléndez, T.; Márquez, A.; Neipp, C.: An improved ’heuristic’ approximation for the period of a nonlinear pendulum: linear analysis of a classical nonlinear problem. Int. J. Nonlinear sci. Numer. simul. 8, No. 3, 329-334 (2007)
[15] Geng, L.; Cai, X. C.: He’s formulation for nonlinear oscillators. Eur. J. Phys. 28, 923-931 (2007) · Zbl 1162.70019
[16] He, J. H.; Wu, X. H.: Construction of solitary solution and compact on-like solution by variational iteration method. Chaos solitons fractals 29, 108-113 (2006) · Zbl 1147.35338
[17] Dehghan, M.; Tatari, M.: Te use of he’s variational iteration method for solving multipoint boundary value problems. Phys. scripta 72, 672-676 (2007)
[18] He, J. H.: Variational approach for nonlinear oscillators. Chaos solitons fractals 34, 1430-1439 (2007) · Zbl 1152.34327
[19] J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation, de-Verlag im Internet GmbH, Berlin, 2006
[20] He, J. H.: Some asymptotic methods for strongly nonlinear equations. Internat. J. Modern phys. B 20, 1141-1199 (2006) · Zbl 1102.34039
[21] Beléndez, A.; Pascual, C.; Gallego, S.; Ortuño, M.; Neipp, C.: Application of a modified he’s homotopy perturbation method to obtain higher-order approximations of a x1/3 force nonlinear oscillator. Phys. lett. A (2007) · Zbl 1209.65083
[22] Liu, H. M.: Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincaré method. Chaos solitons fractals 23, 577-579 (2005) · Zbl 1078.34509
[23] Özis, T.; Yildirim, A.: A comparative study of he’s homotopy perturbation method for determining frequency--amplitude relation of a nonlinear oscillator with discontinuities. Int. J. Nonlinear sci. Numer. simul. 8, No. 2, 243-248 (2007)
[24] He, J. H.: Homotopy perturbation method for bifurcation on nonlinear problems. Int. J. Nonlinear sci. Numer. simul. 6, 207-208 (2005)
[25] He, J. H.: The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl. math. Comput. 151, 287-292 (2004) · Zbl 1039.65052
[26] 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. Nonlinear sci. Numer. simul. 7, No. 1, 109-117 (2006)
[27] Cveticanin, L.: Homotopy-perturbation for pure nonlinear differential equation. Chaos solitons fractals 30, 1221-1230 (2006) · Zbl 1142.65418
[28] Gorji, M.; Ganji, D. D.; Soleimani, S.: New application of he’s homotopy perturbation method. Int. J. Nonlinear sci. Numer. simul. 8, No. 3, 319-328 (2007)
[29] Chowdhury, M. S. H.; Hashim, I.: Solutions of time-dependent Emden--Fowler type equations by homotopy-perturbation method. Phys. lett. A 368, 305-313 (2007) · Zbl 1209.65106
[30] He, J. H.: Homotopy perturbation method for solving boundary value problems. Phys. lett. A 350, 87-88 (2006) · Zbl 1195.65207
[31] Beléndez, A.; Hernández, A.; Beléndez, T.; Fernández, E.; Álvarez, M. L.; Neipp, C.: Application of he’s homotopy perturbation method to the Duffing-harmonic oscillator. Int. J. Nonlinear sci. Numer. simul. 8, No. 1, 79-88 (2007)
[32] Beléndez, A.; Hernández, A.; Beléndez, T.; Neipp, C.; Márquez, A.: Application of the homotopy perturbation method to the nonlinear pendulum. Eur. J. Phys. 28, 93-104 (2007) · Zbl 1119.70017
[33] 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, No. 4, 411-418 (2006)
[34] 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. Nonlinear sci. Numer. simul. 7, No. 1, 15-26 (2006) · Zbl 1187.76622
[35] 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, No. 3, 321-328 (2006) · Zbl 1160.35517
[36] Ariel, P. D.; Hayat, T.: Homotopy perturbation method and axisymmetric flow over a stretching sheet. Int. J. Nonlinear sci. Numer. simul. 7, No. 4, 399-406 (2006)
[37] Shakeri, F.; Dehghan, M.: Inverse problem of diffusion by he’s homotopy perturbation method. Phys. scripta 75, 551-556 (2007) · Zbl 1110.35354
[38] Dehghan, M.; Shakeri, F.: Solution of a partial differential equation subject to temperature overspecification by he’s homotopy perturbation method. Phys. scripta 75, 778-787 (2007) · Zbl 1117.35326
[39] Chowdhury, M. S. H.; Hashim, I.: Solutions of a class of singular second-order ivps by homotopy-perturbation method. Phys. lett. A 365, 439-447 (2007) · Zbl 1203.65124
[40] Chowdhury, M. S. H.; Hashim, I.: Application of homotopy-perturbation method to nonlinear population dynamics models. Phys. lett. A 368, 251-258 (2007) · Zbl 1209.65107
[41] Wu, B. S.; Sun, W. P.; Lim, C. W.: An analytical approximate technique for a class of strongly non-linear oscillators. Int. J. Nonlinear mech. 41, 766-774 (2006) · Zbl 1160.70340