A modified global error minimization method for solving nonlinear Duffing-harmonic oscillators

Gamal M. Ismail; Maha M. El-Moshneb; Mohra Zayed; Gamal M. Ismail; Maha M. El-Moshneb; Mohra Zayed

doi:10.3934/math.2023023

AIMS Mathematics

2023, Volume 8, Issue 1: 484-500. doi: 10.3934/math.2023023

Previous Article Next Article

Research article Special Issues

A modified global error minimization method for solving nonlinear Duffing-harmonic oscillators

1.
Department of Mathematics, Faculty of Science, Sohag University, Sohag, 82524, Egypt
2.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, 42351, Madinah, Saudi Arabia
3.
Mathematics Department, College of Science, King Khalid University, Abha, Saudi Arabia

Received: 24 July 2022 Revised: 10 September 2022 Accepted: 13 September 2022 Published: 08 October 2022
MSC : 34A34, 34C15, 34C25, 35L65, 37N30

In this paper, a third-order approximate solution of strongly nonlinear Duffing-harmonic oscillators is obtained by extending and improving an analytical technique called the global error minimization method (GEMM). We have made a comparison between our results, those obtained from the other analytical methods and the numerical solution. Consequently, we notice a better agreement with the numerical solution than other known analytical methods. The results are valid for both small and large oscillation amplitude. The obtained results demonstrate that the present method can be easily extended to strongly nonlinear problems, as indicated in the presented applications.
- global error minimization method,
- Duffing-harmonic oscillator,
- analytical and approximate solutions,
- periodic solution,
- numerical solution
Citation: Gamal M. Ismail, Maha M. El-Moshneb, Mohra Zayed. A modified global error minimization method for solving nonlinear Duffing-harmonic oscillators[J]. AIMS Mathematics, 2023, 8(1): 484-500. doi: 10.3934/math.2023023

Related Papers:

Abstract

In this paper, a third-order approximate solution of strongly nonlinear Duffing-harmonic oscillators is obtained by extending and improving an analytical technique called the global error minimization method (GEMM). We have made a comparison between our results, those obtained from the other analytical methods and the numerical solution. Consequently, we notice a better agreement with the numerical solution than other known analytical methods. The results are valid for both small and large oscillation amplitude. The obtained results demonstrate that the present method can be easily extended to strongly nonlinear problems, as indicated in the presented applications.

References

[1]	J. H. He, The simplest approach to nonlinear oscillators, Results Phys., 15 (2019), 102546. https://doi.org/10.1016/j.rinp.2019.102546 doi: 10.1016/j.rinp.2019.102546
[2]	S. Li, J. Niu, X. Li, Primary resonance of fractional-order Duffing-Van der Pol oscillator by harmonic balance method, Chinese Phys. B, 27 (2018), 120502. https://doi.org/10.1088/1674-1056/27/12/120502 doi: 10.1088/1674-1056/27/12/120502
[3]	B. Wu, W. Liu, X. Chen, C. W. Lim, Asymptotic analysis and accurate approximate solutions for strongly nonlinear conservative symmetric oscillators, Appl. Math. Model., 49 (2017), 243–254. https://doi.org/10.1016/j.apm.2017.05.004 doi: 10.1016/j.apm.2017.05.004
[4]	J. H. He, T. S. Amer, S. Elnaggar, A. A. Galal, Periodic property and instability of a rotating pendulum system, Axioms., 10 (2021), 191. https://doi.org/10.3390/axioms10030191 doi: 10.3390/axioms10030191
[5]	N. Qie, W. F. Houa, J. H. He, The fastest insight into the large amplitude vibration of a string, Rep. Mech. Eng., 2 (2021), 1–5. https://doi.org/10.31181/rme200102001q doi: 10.31181/rme200102001q
[6]	Y. Tian, Frequency formula for a class of fractal vibration system, Rep. Mech. Eng., 3 (2022), 55–61. https://doi.org/10.31181/rme200103055y doi: 10.31181/rme200103055y
[7]	J. B. Marion, Classical dynamics of particles and system, Harcourt Brace Jovanovich, San Diego, CA, 1970.
[8]	N. N. Krylov, N. N. Bogoliubov, Introduction to nonlinear mechanics, Princeton University Press, New Jersey, 1947.
[9]	N. N. Bogoliubov, Y. A. Mitropolskii, Asymptotic methods in the theory of nonlinear oscillations, Gordan and Breach, New York, 1961.
[10]	A. H. Nayfeh, D. T. Mook, Nonlinear oscillations, John Wiley & Sons, New York, 1979.
[11]	R. E. Mickens, Mathematical and numerical study of the Duffing-harmonic oscillator, J. Sound Vib., 244 (2000), 563–567. https://doi.org/10.1006/jsvi.2000.3502 doi: 10.1006/jsvi.2000.3502
[12]	J. Fan, He's frequency-amplitude formulation for the Duffing harmonic oscillator, Comput. Math. Appl., 58 (2009), 2473–2476. https://doi.org/10.1016/j.camwa.2009.03.049 doi: 10.1016/j.camwa.2009.03.049
[13]	C. W. Lim, B. S. Wu, A new analytical approach to the Duffing-harmonic oscillator, Phys. Lett. A, 311 (2003), 365–373. https://doi.org/10.1016/S0375-9601(03)00513-9 doi: 10.1016/S0375-9601(03)00513-9
[14]	A. Beléndez, D. I. Mendez, T. Beléndez, A. Hernández, M. L. Álvarez, Harmonic balance approaches to the nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable, J. Sound Vib., 314 (2008), 773–782. https://doi.org/10.1016/j.jsv.2008.01.021 doi: 10.1016/j.jsv.2008.01.021
[15]	J. H. He, Q. Yang, C. H. He, Y. Khan, A simple frequency formulation for the tangent oscillator, Axioms, 10 (2021), 320. https://doi.org/10.3390/axioms10040320 doi: 10.3390/axioms10040320
[16]	N. Sharif, A. Razzak, M. Z. Alam, Modified harmonic balance method for solving strongly nonlinear oscillators, J. Interdiscipl. Math., 22 (2019), 353–375. https://doi.org/10.1080/09720502.2019.1624304 doi: 10.1080/09720502.2019.1624304
[17]	M. A. Hosen, M. S. H. Chowdhury, G. M. Ismail, A. Yildirim, A modified harmonic balance method to obtain higher-order approximations to strongly nonlinear oscillators, J. Interdiscipl. Math., 23 (2020), 1325–1345. https://doi.org/10.1080/09720502.2020.1745385 doi: 10.1080/09720502.2020.1745385
[18]	M. Momeni, N. Jamshidi, A. Barari, D. D. Ganji, Application of He's energy balance method to Duffing-harmonic oscillators, Int. J. Comput. Math., 88 (2011), 135–144. https://doi.org/10.1080/00207160903337239 doi: 10.1080/00207160903337239
[19]	A. M. El-Naggar, G. M. Ismail, Periodic solutions of the Duffing harmonic oscillator by He's energy balance method, J. Appl. Comput. Mech., 2 (2016), 35–41.
[20]	D. D. Ganji, N. R. Malidarreh, M. Akbarzade, Comparison of energy balance period with exact period for arising nonlinear oscillator equations, Acta Appl. Math., 108 (2009), 353–362. https://doi.org/10.1007/s10440-008-9315-2 doi: 10.1007/s10440-008-9315-2
[21]	D. H. Shou, The homotopy perturbation method for nonlinear oscillators, Comput. Math. Appl., 58 (2009), 2456–2459. https://doi.org/10.1016/j.camwa.2009.03.034 doi: 10.1016/j.camwa.2009.03.034
[22]	J. H. He, Y. O. El-Dib, A. A. Mady, Homotopy perturbation method for the fractal toda oscillator, Fractal Fract., 5 (2021), 93. https://doi.org/10.3390/fractalfract5030093 doi: 10.3390/fractalfract5030093
[23]	J. H. He, T. S. Amer, S. Elnaggar, A. A. Galal, Periodic property and instability of a rotating pendulum system, Axioms, 10 (2021), 191. https://doi.org/10.3390/axioms10030191 doi: 10.3390/axioms10030191
[24]	S. Noeiaghdam, A. Dreglea, J. H. He, Z. Avazzadeh, M. Suleman, M. A. F. Araghi, et al., Error estimation of the homotopy perturbation method to solve second kind Volterra integral equations with piecewise smooth kernels: Application of the CADNA Library, Symmetry, 12 (2020), 1730. https://doi.org/10.3390/sym12101730 doi: 10.3390/sym12101730
[25]	M. Bayat, M. Head, L. Cveticanin, P. Ziehl, Nonlinear analysis of two-degree of freedom system with nonlinear springs, Mech. Syst. Signal Pr., 171 (2022), 108891. https://doi.org/10.1016/j.ymssp.2022.108891 doi: 10.1016/j.ymssp.2022.108891
[26]	A. Yildirim, Z. Saadatnia, H. Askari, Application of the Hamiltonian approach to nonlinear oscillators with rational and irrational elastic terms, Math. Comput. Model., 54 (2011), 697–703. https://doi.org/10.1016/j.mcm.2011.03.012 doi: 10.1016/j.mcm.2011.03.012
[27]	G. M. Ismail, L. Cveticanin, Higher order Hamiltonian approach for solving doubly clamped beam type N/MEMS subjected to the van der Waals attraction, Chinese J. Phys., 72 (2021), 69–77. https://doi.org/10.1016/j.cjph.2021.04.016 doi: 10.1016/j.cjph.2021.04.016
[28]	V. H. Dang, An approximate solution for a nonlinear Duffing-harmonic oscillator, Asian Res. J. Math., 15 (2019), 1–14. https://doi.org/10.9734/arjom/2019/v15i430154 doi: 10.9734/arjom/2019/v15i430154
[29]	P. Ju, Global residue harmonic balance method for Helmholtz-Duffing oscillator, Appl. Math. Model., 39 (2015), 2172–2179. https://doi.org/10.1016/j.apm.2014.10.029 doi: 10.1016/j.apm.2014.10.029
[30]	J. Lu, Global residue harmonic balance method for strongly nonlinear oscillator with cubic and harmonic restoring force, J. Low Freq. Noise V. A., 2022. https://doi.org/10.1177/14613484221097465 doi: 10.1177/14613484221097465
[31]	G. M. Ismail, M. Abul-Ez, N. M. Farea, N. Saad, Analytical approximations to nonlinear oscillation of nanoelectro-mechanical resonators, Eur. Phys. J. Plus, 134 (2019), 47. https://doi.org/10.1140/epjp/i2019-12399-2 doi: 10.1140/epjp/i2019-12399-2
[32]	S. S. Ganji, D. D. Ganji, A. G. Davodi, S. Karimpour, Analytical solution to nonlinear oscillation system of the motion of a rigid rod rocking back using max-min approach, Appl. Math. Model., 34 (2010), 2676–2684. https://doi.org/10.1016/j.apm.2009.12.002 doi: 10.1016/j.apm.2009.12.002
[33]	M. Bayat, M. Bayat, M. Kia, H. R. Ahmadi, I. Pakar, Nonlinear frequency analysis of beams resting on elastic foundation using max-min approach, Geomech. Eng., 16 (2018), 355–361.
[34]	C. W. Lim, B. S. Wub, W. P. Sun, Higher accuracy analytical approximations to the Duffing-harmonic oscillator, J. Sound Vib., 296 (2006), 1039–1045. https://doi.org/10.1016/j.jsv.2006.02.020 doi: 10.1016/j.jsv.2006.02.020
[35]	M. Fesanghary, T. Pirbodaghi, M. Asghari, H. Sojoudi, A new analytical approximation to the Duffing-harmonic oscillator, Chaos Soliton. Fract., 42 (2009), 571–576. https://doi.org/10.1016/j.chaos.2009.01.024 doi: 10.1016/j.chaos.2009.01.024
[36]	L. Xu, Application of He's parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire, Phys. Lett. A, 368 (2007), 259–262. https://doi.org/10.1016/j.physleta.2007.04.004 doi: 10.1016/j.physleta.2007.04.004
[37]	C. S. Liu, Y. W. Chen, A Simplified Lindstedt-Poincaré method for saving computational cost to determine higher order nonlinear free vibrations, Mathematics, 9 (2021) 3070. https://doi.org/10.3390/math9233070 doi: 10.3390/math9233070
[38]	Z. Li, J. Tang, High accurate homo-heteroclinic solutions of certain strongly nonlinear oscillators based on generalized Padé-Lindstedt-Poincaré method, J. Vib. Eng. Technol., 10 (2022), 1291–1308. https://doi.org/10.1007/s42417-022-00446-7 doi: 10.1007/s42417-022-00446-7
[39]	M. K. Yazdi, P. H. Tehrani, Frequency analysis of nonlinear oscillations via the global error minimization, Nonlinear Eng., 5 (2016). https://doi.org/10.1515/nleng-2015-0036 doi: 10.1515/nleng-2015-0036
[40]	Y. Farzaneh, A. A. Tootoonchi, Global error minimization method for solving strongly nonlinear oscillator differential equations, Comput. Math. Appl., 59 (2010), 2887–2895. https://doi.org/10.1016/j.camwa.2010.02.006 doi: 10.1016/j.camwa.2010.02.006
[41]	A. Mirzabeigy, M. K. Yazdi, A. Yildirim, Analytical approximations for a conservative nonlinear singular oscillator in plasma physics, J. Egypt. Math. Soc., 20 (2012), 163–166. https://doi.org/10.1016/j.joems.2012.05.001 doi: 10.1016/j.joems.2012.05.001
[42]	G. M. Ismail, H. Abu-Zinadah, Analytic approximations to non-linear third order jerk equations via modified global error minimization method, J. King Saud Univ. Sci., 33 (2021), 101219. https://doi.org/10.1016/j.jksus.2020.10.016 doi: 10.1016/j.jksus.2020.10.016
[43]	K. P. Badakhshan, A. V. Kamyad, A. Azemi, Using AVK method to solve nonlinear problems with uncertain parameters, Appl. Math. Comput., 189 (2007), 27–34. https://doi.org/10.1016/j.amc.2006.11.172 doi: 10.1016/j.amc.2006.11.172

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)