##
**Multiple-parameter Hamiltonian approach for higher accurate approximations of a nonlinear oscillator with discontinuity.**
*(English)*
Zbl 1239.34033

Summary: We apply a new approach to obtain the natural frequency of a nonlinear oscillator with discontinuity. He’s Hamiltonian approach is modified for a nonlinear oscillator with discontinuity for which the elastic force term is proportional to \(\text{sgn}(u)\). We employ this method for higher-order approximate solution of the nonlinear oscillator. This property is used to obtain an approximate frequency-amplitude relationship of the nonlinear oscillator with high accuracy. Many numerical results are given to prove the efficiency of the suggested technique.

### MSC:

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

34A26 | Geometric methods in ordinary differential equations |

34A45 | Theoretical approximation of solutions to ordinary differential equations |

### Keywords:

He’s Hamiltonian approach
PDF
BibTeX
XML
Cite

\textit{N. A. Khan} et al., Int. J. Differ. Equ. 2011, Article ID 649748, 7 p. (2011; Zbl 1239.34033)

Full Text:
DOI

### References:

[1] | M. Rafei, D. D. Ganji, H. Daniali, and H. Pashaei, “The variational iteration method for nonlinear oscillators with discontinuities,” Journal of Sound and Vibration, vol. 305, no. 4-5, pp. 614-620, 2007. · Zbl 1242.65154 |

[2] | L.-N. Zhang and J.-H. He, “Resonance in Sirospun yarn spinning using a variational iteration method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1064-1066, 2007. · Zbl 1141.65373 |

[3] | T. Özi\cs and A. Yıldırım, “A study of nonlinear oscillators with u1/3 force by He’s variational iteration method,” Journal of Sound and Vibration, vol. 306, no. 1-2, pp. 372-376, 2007. · Zbl 1242.74214 |

[4] | A. Beléndez, C. Pascual, M. Ortuño, T. Beléndez, and S. Gallego, “Application of a modified He’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 601-610, 2009. · Zbl 1167.34327 |

[5] | A. Beléndez, C. Pascual, S. Gallego, M. Ortuño, and C. Neipp, “Application of a modified He’s homotopy perturbation method to obtain higher-order approximations of an x1/3 force nonlinear oscillator,” Physics Letters A, vol. 371, no. 5-6, pp. 421-426, 2007. · Zbl 1209.65083 |

[6] | J.-H. He, “The homotopy perturbation method nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287-292, 2004. · Zbl 1039.65052 |

[7] | A. Beléndez, A. Hernández, T. Beléndez, E. Fernández, M. L. Álvarez, and C. Neipp, “Application of He’s homotopy perturbation method to the Duffin-harmonic oscillator,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 1, pp. 79-88, 2007. · Zbl 06942245 |

[8] | H.-M. Liu, “Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincare method,” Chaos, Solitons and Fractals, vol. 23, no. 2, pp. 577-579, 2005. · Zbl 1078.34509 |

[9] | J.-H. He, “Variational approach for nonlinear oscillators,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1430-1439, 2007. · Zbl 1152.34327 |

[10] | D.-H. Shou, “Variational approach to the nonlinear oscillator of a mass attached to a stretched wire,” Physica Scripta, vol. 77, no. 4, Article ID 045006, 2008. · Zbl 1336.65127 |

[11] | F. Ö. Zengin, M. O. Kaya, and S. A. Demirba\ug, “Application of parameter-expansion method to nonlinear oscillators with discontinuities,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 3, pp. 267-270, 2008. · Zbl 06942345 |

[12] | J.-H. He, “Max-min approach to nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 207-210, 2008. · Zbl 06942339 |

[13] | Z. Guo and A. Y. T. Leung, “The iterative homotopy harmonic balance method for conservative Helmholtz-Duffing oscillators,” Applied Mathematics and Computation, vol. 215, no. 9, pp. 3163-3169, 2010. · Zbl 1183.65083 |

[14] | J.-H. He, “Hamiltonian approach to nonlinear oscillators,” Physics Letters. A, vol. 374, no. 23, pp. 2312-2314, 2010. · Zbl 1237.70036 |

[15] | Z.-F. Ren and J.-H. He, “A simple approach to nonlinear oscillators,” Physics Letters. A, vol. 373, no. 41, pp. 3749-3752, 2009. · Zbl 1233.70009 |

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.