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Homotopy perturbation method for strongly nonlinear oscillators

Author

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  • He, Ji-Huan
  • Jiao, Man-Li
  • Gepreel, Khaled A.
  • Khan, Yasir

Abstract

This paper reveals the effectiveness of the homotopy perturbation method for strongly nonlinear oscillators. A generalized Duffing oscillator is adopted to elucidate the solving process step by step, and a nonlinear frequency–amplitude relationship is obtained with a relative error of 0.91% when the amplitude tends to infinity, the solution morphology is also discussed, and the zero-th approximate solution is enough for conservative nonlinear oscillators, while the accuracy of the frequency can be improved if the iteration continues.

Suggested Citation

  • He, Ji-Huan & Jiao, Man-Li & Gepreel, Khaled A. & Khan, Yasir, 2023. "Homotopy perturbation method for strongly nonlinear oscillators," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 243-258.
    • Handle: RePEc:eee:matcom:v:204:y:2023:i:c:p:243-258
    DOI: 10.1016/j.matcom.2022.08.005
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    References listed on IDEAS

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    1. Ghaleb, A.F. & Abou-Dina, M.S. & Moatimid, G.M. & Zekry, M.H., 2021. "Analytic approximate solutions of the cubic–quintic Duffing–van​ der Pol equation with two-external periodic forcing terms: Stability analysis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 180(C), pages 129-151.
    2. Che Han & Yu-Lan Wang & Zhi-Yuan Li, 2021. "NUMERICAL SOLUTIONS OF SPACE FRACTIONAL VARIABLE-COEFFICIENT KdV–MODIFIED KdV EQUATION BY FOURIER SPECTRAL METHOD," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(08), pages 1-19, December.
    3. Dan Tian & Qura-Tul Ain & Naveed Anjum & Chun-Hui He & Bin Cheng, 2021. "Fractal N/Mems: From Pull-In Instability To Pull-In Stability," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(02), pages 1-8, March.
    4. Chun-Hui He & Chao Liu, 2022. "A Modified Frequency–Amplitude Formulation For Fractal Vibration Systems," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(03), pages 1-8, May.
    5. Muhammad Suleman & Qingbiao Wu, 2015. "Comparative Solution of Nonlinear Quintic Cubic Oscillator Using Modified Homotopy Perturbation Method," Advances in Mathematical Physics, Hindawi, vol. 2015, pages 1-5, June.
    6. Zhou, Liangqiang & Chen, Fangqi, 2022. "Chaos of the Rayleigh–Duffing oscillator with a non-smooth periodic perturbation and harmonic excitation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 1-18.
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    Cited by:

    1. Remus-Daniel Ene & Nicolina Pop, 2023. "Optimal Homotopy Asymptotic Method for an Anharmonic Oscillator: Application to the Chen System," Mathematics, MDPI, vol. 11(5), pages 1-14, February.
    2. Roy, Tapas & Maiti, Dilip K., 2024. "General approach on the best fitted linear operator and basis function for homotopy methods and application to strongly nonlinear oscillators," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 44-64.
    3. Shirazian, Mohammad, 2023. "A new acceleration of variational iteration method for initial value problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 246-259.
    4. Alim, Md. Abdul & Kawser, M. Abul, 2023. "Illustration of the homotopy perturbation method to the modified nonlinear single degree of freedom system," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).

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