A GOOD INITIAL GUESS FOR APPROXIMATING NONLINEAR OSCILLATORS BY THE HOMOTOPY PERTURBATION METHOD

Ji-Huan He, Chun-Hui He, Abdulrahman Ali Alsolami

DOI Number
https://doi.org/10.22190/FUME230108006H
First page
021
Last page
029

Abstract


A good initial guess and an appropriate homotopy equation are two main factors in applications of the homotopy perturbation method. For a nonlinear oscillator, a cosine function is used in an initial guess. This article recommends a general approach to construction of the initial guess and the homotopy equation. Duffing oscillator is adopted as an example to elucidate the effectiveness of the method.


Keywords

Homotopy perturbation method, Nonlinear oscillator, Periodic solution

Full Text:

PDF

References


Ibrahim, R.A., 2008, Recent advances in nonlinear passive vibration isolators, Journal of Sound and Vibration, 314(3-5), pp. 371-452.

Ke, L.L., Yang, J., Kitipornchai, S., 2010 , Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams, Composite Structures, 92(3) , pp. 676-683.

Yang, Q., 2023, A mathematical control for the pseudo-pull-in stability arising in a micro-electromechanical system, Journal of Low Frequency Noise, Vibration and Active Control, doi: 10.1177/14613484221133603

Feng, G.Q., Niu, J.Y., 2023, The analysis for the dynamic pull-in of a micro-electromechanical system, Journal of Low Frequency Noise, Vibration and Active Control, doi: 10.1177/14613484221145588

He, J.H., 1999, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178(3-4), pp. 257-262.

Nayfeh, A.H., 1973, Perturbation methods, New York, Wiley & Sons.

Wang, S.Q., He, J.H., 2007, Variational iteration method for solving integro-differential equations, Physics letters A, 367(3), pp. 188-191.

Odibat, Z.M., Momani, S., 2006, Application of variational iteration method to Nonlinear differential equations of fractional order, International Journal of Nonlinear Sciences and Numerical Simulation, 7(1), pp. 27-34.

Deng, S.X., Ge, X.X., 2022, The variational iteration method for Whitham-Broer-Kaup system with local fractional derivatives, Thermal Science, 26(3), pp. 2419-2426.

Wang, S.Q., 2009, A variational approach to nonlinear two-point boundary value problems, Computers & Mathematics with Applications, 58(11), pp. 2452-2455.

Shen, Y.Y., Huang, X.X., Kwak, K., et al., 2016, Subcarrier-pairing-based resource optimization for OFDM wireless powered relay transmissions with time switching scheme, IEEE Transactions on Signal Processing, 65(5), pp. 1130-1145.

Khan, Y., Akbarzade, M., Kargar, A., 2012, Coupling of homotopy and the variational approach for a conservative oscillator with strong odd-nonlinearity, Scientia Iranica, 19(3), pp. 417-422.

Biazar, J., Ghanbari, B., Porshokouhi, M.G., Porshokouhi, M.G., 2011, He's homotopy perturbation method: A strongly promising method for solving non-linear systems of the mixed Volterra-Fredholm integral equations, Computers & Mathematics with Applications, 61(4) , pp. 1016-1023.

Saranya, K., Mohan, V., Kizek, R., Fernandez, C., Rajendran, L., 2018, Unprecedented homotopy perturbation method for solving nonlinear equations in the enzymatic reaction of glucose in a spherical matrix, Bioprocess and Biosystems Engineering, 41(2) , pp. 281-294.

Anjum, N., He, J.H., Ain, Q. T., Tian, D., 2021, Li-He’s modified homotopy perturbation method for doubly-clamped electrically actuated microbeams-based microelectromechanical system, Facta Universitatis Series Mechanical Engineering, 19(4), pp. 601-612.

Ji, Q.P., Wang, J., Lu, L.X., Ge, C.F., 2021, Li-He's modified homotopy perturbation method coupled with the energy method for the dropping shock response of a tangent nonlinear packaging system, Journal of Low Frequency Noise, Vibration and Active Control, 40(2) , pp.675-682.

Nadeem, M., Li, F.Q.,2019, He-Laplace method for nonlinear vibration systems and nonlinear wave equations Journal of Low Frequency Noise, Vibration and Active Control, 38(3-4) , pp. 1060-1074.

Adamu, M.Y., Ogenyi, P., 2018, New approach to parameterized homotopy perturbation method, Thermal Science, 22(4), pp. 1865-1870.

Peker, H.A., Cuha, F.A., 2022, Application of Kashuri Fundo transform and homotopy perturbation method for fractional heat transfer and porous media equations, Thermal Science, 26(4), pp. 2877-2884.

Alam, M.S., Sharif, N., Molla, M.H.U., 2022, Combination of modified Lindstedt-Poincare and homotopy perturbation methods, Journal of Low Frequency Noise, Vibration and Active Control, doi: 10.1177/14613484221148049

Li, Z.Y., Wang, M.C., Wang, Y.L., 2022, Solving a class of variable order nonlinear fractional integral differential equations by using reproducing kernel function, AIMS Mathematics, 7(7), pp. 12935-12951.

Li, Z.Y., Chen, Q.T., Wang, Y.L., Li, X.Y., 2022, Solving two-sided fractional super-diffusive partial differential equations with variable coefficients in a class of new reproducing kernel spaces, Fractal and Fractional, 6(9), 492.

Wang, K.L., 2023, Novel analytical approach to modified fractal gas dynamics model with the variable coefficients, ZAMM, https://doi.org/10.1002/zamm.202100391.

Wang, K.L., Wei, C.F., 2023, Fractal soliton solutions for the fractal-fractional shallow water wave equation arising in ocean engineering, Alexandria Engineering Journal, 65, pp. 859-865.

Wang, S.Q., Wang, X.Y., Shen, Y.Y., et. al, 2022, An ensemble-based densely-connected deep learning system for assessment of skeletal maturity, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 52(1), pp. 426-437.

Wang, S.Q., Shen, Y.Y., Shi, C.H., et. al, 2018, Skeletal maturity recognition using a fully automated system with convolutional neural networks, IEEE Access, 6, pp. 29979-29993.

Hu, S.Y., Yuan, J.P., Wang, S.Q., 2019, Cross-modality synthesis from MRI to PET using adversarial U-net with different normalization, 2019 International Conference on Medical Imaging Physics and Engineering, doi: 10.1109/ICMIPE47306.2019.9098219

Wu, K., Shen, Y.Y., Wang, S.Q., 2018, 3D convolutional neural network for regional precipitation nowcasting, Journal of Image and Signal Processing, 7(4), pp. 200-212.

Wang, S.Q., Li, X., Cui, J.L., et al., 2015, Prediction of myelopathic level in cervical spondylotic myelopathy using diffusion tensor imaging, Journal of Magnetic Resonance Imaging, 41(6), pp. 1682-1688.

Yu, W., Lei, B.Y., Shen, Y.Y., et. al, 2021, Morphological feature visualization of Alzheimer's disease via Multidirectional Perception GAN, IEEE Transactions on Neural Networks and Learning Systems, doi: 10.1109/TNNLS.2021.3118369

Hu, S.Y., Lei, B.Y., Wang, S.Q., et. al, 2021, Bidirectional mapping generative adversarial networks for brain MR to PET synthesis, IEEE Transactions on Medical Imaging, 41(1), pp. 145-157.

Yu, W., Lei, B.Y., Shen, Y.Y., et. al, 2021, Morphological feature visualization of Alzheimer's disease via Multidirectional Perception GAN, IEEE Transactions on Neural Networks and Learning Systems, doi: 10.1109/TNNLS.2021.3118369

Hu, S.Y., Yu, W., Chen, Z., et al., 2020, Medical image reconstruction using generative adversarial network for Alzheimer disease assessment with class-imbalance problem, IEEE 6th International Conference on Computer and Communications (ICCC), doi: 10.1109/ICCC51575.2020.9344912

Machina, A., Edwards, R., van den Driessche, P., 2013, Sensitive dependence on initial conditions in gene networks, Chaos, 23(2), 025101.

El-Dib, Y. O., Elgazery, N. S., Mady, A. A., Alyousef, H.A., 2022, On the modeling of a parametric cubic–quintic nonconservative Duffing oscillator via the modified homotopy perturbation method, Zeitschrift für Naturforschung A, 77(5), pp. 475-486.

Vahidi, A.R., Babolian, E., Azimzadeh, Z., 2018, An Improvement to the Homotopy Perturbation Method for Solving Nonlinear Duffing's Equations, Bulletin of the Malaysian Mathematical Sciences Society, 41(2) , pp. 1105-1117.

Azimzadeh, Z., Vahidi, A.R., Babolian, E., 2012, Exact solutions for non-linear Duffing's equations by He's homotopy perturbation method, Indian Journal of Physics, 86(8) , pp. 721-726.

El-Dib, Y.O., 2023, Properties of complex damping Helmholtz-Duffing oscillator arising in fluid mechanics, Journal of Low Frequency Noise, Vibration and Active Control, doi: 10.1177/14613484221138560

Aljahdaly, N.H., Alharbi, M.A ., 2022, Semi-analytical solution of non-homogeneous Duffing oscillator equation by the Pade differential transformation algorithm, Journal of Low Frequency Noise, Vibration and Active Control, 41(4), pp.1454-1465.

Belendez, T., Belendez, F.J., Martinez, C., et al., 2016, Exact solution for the unforced Duffing oscillator with cubic and quintic nonlinearities, Nonlinear Dynamics, 86(3), pp. 1687–1700,

Aljahdaly, N.H., Shah, R.S., Naeem, M., Arefin, M.A., 2022, A Comparative Analysis of Fractional Space-Time Advection-Dispersion Equation via Semi-Analytical Methods, Journal of Function Spaces, 2022, 4856002, doi:10.1155/2022/4856002

Dubey, V.P., Kumar, D., Singh, J., Alshehri, A.M., Dubey, S., 2022, Analysis of local fractional Klein-Gordon equations arising in relativistic fractal quantum mechanics, Waves in Random and Complex Media, 2022, doi:10.1080/17455030.2022.2112993

Qie, N., Houa, W.-F., He, J.-H., 2021, The fastest insight into the large amplitude vibration of a string, Reports in Mechanical Engineering, 2(1), pp. 1-5.

Tian, Y., 2022, Frequency formula for a class of fractal vibration system, Reports in Mechanical Engineering, 3(1), pp. 55-61.




DOI: https://doi.org/10.22190/FUME230108006H

Refbacks

  • There are currently no refbacks.


ISSN: 0354-2025 (Print)

ISSN: 2335-0164 (Online)

COBISS.SR-ID 98732551

ZDB-ID: 2766459-4