Research article

Numerical solution of full fractional Duffing equations with Cubic-Quintic-Heptic nonlinearities

  • Received: 21 September 2019 Accepted: 19 December 2019 Published: 06 February 2020
  • MSC : 26A33, 97N40, 34A08, 34C15

  • In this article, based on the operational matrix of fractional order integration, we introduce a method for the numerical solution of fractional strongly nonlinear Duffing oscillators with cubic-quintic-heptic nonlinear restoring force and then use it in some cases. For this purpose, concerning the Caputo sense, we implement the block-pulse wavelets matrix of fractional order integration. To reach this aim, we analyse the errors. The approach has been examined by some numerical examples and changes in coefficients as well as in the derivative of the equation too. It is shown that this method works well for all the parameters and order of the fractional derivative. Results indicate the precision and computational performance of the suggested algorithm.

    Citation: P. Pirmohabbati, A. H. Refahi Sheikhani, H. Saberi Najafi, A. Abdolahzadeh Ziabari. Numerical solution of full fractional Duffing equations with Cubic-Quintic-Heptic nonlinearities[J]. AIMS Mathematics, 2020, 5(2): 1621-1641. doi: 10.3934/math.2020110

    Related Papers:

  • In this article, based on the operational matrix of fractional order integration, we introduce a method for the numerical solution of fractional strongly nonlinear Duffing oscillators with cubic-quintic-heptic nonlinear restoring force and then use it in some cases. For this purpose, concerning the Caputo sense, we implement the block-pulse wavelets matrix of fractional order integration. To reach this aim, we analyse the errors. The approach has been examined by some numerical examples and changes in coefficients as well as in the derivative of the equation too. It is shown that this method works well for all the parameters and order of the fractional derivative. Results indicate the precision and computational performance of the suggested algorithm.


    加载中


    [1] H. Aminikhah, A. R. Sheikhani, H. Rezazadeh, Stability analysis of linear distributed order system with multiple time delays,, U. P. B. Sci. Bull., Series A, 77 (2015), 207-218.
    [2] M. Mashoof, A. H. R. Sheikhani, H. S. Najafi, Stability analysis of distributed-order HilferPrabhakar systems based on Inertia theory, Math. Notes, 104 (2018), 74-85. doi: 10.1134/S000143461807009X
    [3] M. Mashoof, A. H. R. Sheikhani, H. S. Najafi, Stability analysis of distributed order Hilfer-Prabhakar differential equations, Hacettepe J. Math. Stat., 47 (2018), 299-315.
    [4] A. Ansari, A. R. sheikhani, Approximate analytical solutions of distributed order fractional Riccati differential equation, Ain Shams Eng. J., 49 (2018), 581-58.
    [5] A. Ansari, A. R. sheikhani, New identities for the wright and the Mittag-Leffler functions using the Laplace transform, Asian-Eur. J. Math., 11 (2014), 1019-1032.
    [6] I. Kovacic, M. J. Brennan, Nonlinear oscillators and their behavior, First Edition. John Wiley & Sons, 2011, Ltd. ISBN: 978-0-470-71549-9.
    [7] A. Chatterjee, A brief introduction to nonlinear vibrations, Mechanical Engineering, Indian Institute of Science, Bangalore, February 2009.
    [8] H. Aminikhah, A. R. Sheikhani, H. Rezazadeh, Stability analysis of distributed order fractional Chen system, Sci. World J., 2013, Article ID 645080, 13.
    [9] H. Aminikhah, A. R. Sheikhani, H. Rezazadeh, Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method, Boletim da Sociedade Paranaense de Matemtica, 34 (2016), 213-229. doi: 10.5269/bspm.v34i1.26193
    [10] J. Alvarez-Ramirez, G. Espinosa-Paredes, H. Puebla, Chaos control using small-amplitude damping signals, Phys. Lett., 316 (2003), 196-205. doi: 10.1016/S0375-9601(03)01147-2
    [11] H. Vahedi, G. B. Gharehpetian, M. Karrari, Application of Duffing oscillators for passive Islanding detection of inverter-based distributed generation units, IEEE Trans. Power Delivery, 27 (2012), 1973-1983. doi: 10.1109/TPWRD.2012.2212251
    [12] M. Taylan, The effect of nonlinear damping and restoring in ship rolling, Jocean Eng., 27 (2000), 921-932.
    [13] H. Wagner, Large-Amplitude free vibrations of a beam, J. Appl. Mech., 32 (1965), 887-892 doi: 10.1115/1.3627331
    [14] S. Lenci, G. Menditto, A. M. Tarantino, Homoclinic and Heteroclinic Bifurcations in the non-linear dynamics of a beam resting on an elastic substrate, Int. J. Non-Linear Mech., 34 (1999), 615-632. doi: 10.1016/S0020-7462(98)00001-8
    [15] A. I. Maimistov, Propagation of an ultimately short electromagnetic pulsein a nonlinear medium described by the fifth-Order Duffing model, Optics and Spectroscopy, 94 (2003), 251-257. doi: 10.1134/1.1555186
    [16] R. Meucci, S. Euzzor, E. Pugliese, et al., Anosov flows with stable and unstable differentiable distributions, Optimal Phase-Control Strategy for Damped-Driven Duffing Oscillators, 116 (2016), 044-101.
    [17] J. F. Rhoads, S. W. Shaw, K. L. Turner, Nonlinear dynamics and its applications in Micro- and Nano resonators, J. Dyn. Syst. Meas. Control, 132 (2010), 034001.
    [18] R. Srebro, The Duffing oscillator: A model for the dynamics of the neuronal groupscomprising the transient evoked potential, Electroencephalogr. Clin. Neurophysiol., 96 (1995), 561-573. doi: 10.1016/0013-4694(95)00088-G
    [19] L. Accardi, W. Freudenberg, Quantum Probability and White Noise Analysis, Proceedings Quantum Bio-Informatics, 2011.
    [20] A. H. Nayfeh, D. T. Mook, Nonlinear Oscillations, John Wiley & Sons, 1979.
    [21] X. Y. Deng, B. Liu, T. Long, A new complex Duffing oscillator used in complex signal detection, Chin. Sci. Bull., 57 (2012), 2185-2191. doi: 10.1007/s11434-012-5145-8
    [22] Y. O. El Dib, Stability analysis of a strongly displacement time delayed Duffing oscillator using multiple scales homotopy perturbation method, ournal of Applied and Computational Mechanics, 4 (2018), 260-274.
    [23] A. H. Salas, E. Jairo, H. Castillo, Exact solutions to cubic Duffing equation for a nonlinear electrical circuit, J. Amer. Math. Soc., 7 (2014), 46-53.
    [24] M. A. Latif, J. C. Chedjou, K. Kyamakya, The paradigm of non-linear oscillators in image processing, Transp. Inf. Group, 2009, 1-5.
    [25] K. Tabatabaei, E. Gunerhan, Numerical solution of Duffing equation by the differential transform method, Appl. Math. Inf. Sci. Lett., 2 (2014), 1-6.
    [26] S. Nourazar, A. Mirzabeigy, Approximate solution for nonlinear Duffing oscillator with damping effect using the modified differential transform method, Scientia Iranica B, 20 (2013), 364-368.
    [27] R. E. Mickens, Mathematical and numerical study of the Duffing harmonic oscillator, J. Sound Vib., 244 (2001), 563-567. doi: 10.1006/jsvi.2000.3502
    [28] Md. A. Razzak, A new analytical approach to investigate the strongly nonlinear oscillators, Alexandria Eng. J., 55 (2016), 1827-1834. doi: 10.1016/j.aej.2016.04.001
    [29] M. O. Oyesanya, J. I. Nwamba, Duffing oscillator with heptic nonlinearity under single periodic forcing, Int. J. Mech. Appl., 3 (2013), 35-43.
    [30] S. J. Liao, A. T. Chwang, Application of homotopy analysis method in nonlinear oscillations, J. Appl. Mech., 65 (1998), 914-922. doi: 10.1115/1.2791935
    [31] J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178 (1999), 257-262. doi: 10.1016/S0045-7825(99)00018-3
    [32] A. Elias-Zuniga, Exact solution of the Cubic-Quintic duffing oscillator,, Appl. Math. Model., 37 (2013), 2574-2579. doi: 10.1016/j.apm.2012.04.005
    [33] F. Geng, Numerical solutions of Duffing equations involving both integral and non integral forcing terms,, Comput. Math. Appl., 61 (2011), 1935-1938. doi: 10.1016/j.camwa.2010.07.053
    [34] R. Novin, Z. S. Dastjerd, Solving Duffing equation using an improved semi-analytical method, J. Amer. Math. Soc. Commun. Adv. Comput. Sci. Appl., 2 (2015), 54-58.
    [35] S. Durmaz, S. A. Demirbag, M. O. Kaya, High order He's energy balance method based on collocation method, Int. J. Nonlinear Sci. Numer. Simul., 11 (2010), 1-5.
    [36] D. D. Ganji, M. Gorji, S. Soleimani, et al., Solution of nonlinear Cubic-Quintic Duffing oscillators using hes energy balance method,, J. Zhejiang University-Sci. A, 10 (2009), 1263-1268. doi: 10.1631/jzus.A0820651
    [37] E. Yusufoglu, Numerical solution of Duffing equation by the laplace decomposition algorithm, Appl. Math. Comput., 177 (2006), 572-580.
    [38] J. Sunday, The Duffing oscillator: Applications and computational simulations, Asian Res. J. Math., 2 (2017), 1-13.
    [39] J. J. Stoker, Nonlinear vibrations in mechanical and electrical systems, J. Amer. Math. Soc., 2 1950.
    [40] M. Li, Three classes of fractional oscillators, Symmetry, 10 (2018), 40.
    [41] H. Rezazadeh, H. Aminikhah, A. H. R. Sheikhani, Stability analysis of Hilfer fractional differential systems, Math. Commun., 21 (2016), 45-64.
    [42] K. S. Alghafri, H. Rezazadeh, Solitons and other solutions of (3 + 1) dimensional spacetime fractional modified KdVZakharovKuznetsov equation, Appl. Math. Nonlinear Sci., 4 (2019), 289-304. doi: 10.2478/AMNS.2019.2.00026
    [43] M. S. H. Chowdhuryz, Md. A. Hosen, K. Ahmad, et al., High-order approximate solutions of strongly nonlinear cubic-quintic Duffing oscillator based on the harmonic balance method, Results Phys., 7 (2017), 3962-3967. doi: 10.1016/j.rinp.2017.10.008
    [44] D. W. Brzezinski, Comparison of fractional order derivatives computational accuracy-right hand vs left hand definition, Appl. Math. Nonlinear Sci., 2 (2017), 237-248. doi: 10.21042/AMNS.2017.1.00020
    [45] I. Podlubny, Fractional differential equations, Academic Press, 1990.
    [46] A. Refahi, S. Kordrostami, Solution of the space-fractional BenjaminOno equation: An operational approach, Rendiconti del Circolo Matematico di Palermo Series 2, 66 (2017), 471-476.
    [47] I. K. Youssef, M. H. El Dewaik, Solving Poissons equations with fractional order using Haar wavelet, Appl. Math. Nonlinear Sci., 2 (2017), 271-284. doi: 10.21042/AMNS.2017.1.00023
    [48] E. C. L. Oyesanya, M. O. Agbebaku, D. F. Okofu, et al., Solution to nonlinear Duffing oscillator with fractional derivatives using homotopy analysis method (HAM), Global J. Pure Appl. Math., 14 (2018), 1363-1388.
    [49] A. Sonmezoglu, Exact solutions for some fractional differential equations, Hindawi Publishing Corporation Advances in Mathematical Physics, 2015.
    [50] Y. Xu, Y. Li, D. Liu, et al., Responses of Duffing oscillator with fractional damping and random phase, Nonlinear Dyn., 74 (2013), 745-753. doi: 10.1007/s11071-013-1002-9
    [51] Y. J. Yang, S. Q. Wang, An improved homotopy Perturbation method for solving local fractional nonlinear oscillators, J. Low Freq. Noise Vib. Active Control, 0 (2019), 1-10.
    [52] M. Merdan, A. Gokdogan, A. Yildirim, On numerical solution to fractional non-linear oscillatory equations, J. Amer. Math. Soc., 48 (2013), 1201-1213.
    [53] V. Marinca, N. HeriÅanu, Explicit and exact solutions to cubic Duffing and double-well Duffing equations, Math. Comput. Model., 53 (2011), 604-609. doi: 10.1016/j.mcm.2010.09.011
    [54] M. Mashoof, A. H. R. Sheikhani, Numerical solution of fractional differential equation by wavelets and hybrid functions, Bulletim da Sociedade Paranaense de Matemtica, 36 (2018), 231-244. doi: 10.5269/bspm.v36i2.30904
    [55] K. Maleknejad, M. Shahrezaee, H. Khatami, Numerical solution of integral equations system of the second kind by Block-Pulse functions, Appl. Math. Comput., 66 (2005), 15-24.
    [56] H. S. Najafi, S. A. Edalatpanah, A. H. R. Sheikhani, Convergence analysis of modified iterative methods to solve linear systems, Mediterr. J. Math., 11 (2014), 1019-1032. doi: 10.1007/s00009-014-0412-3
    [57] M. Mashoof, A. H. R. Sheikhani, Simulating the solution of the distributed order fractional differential equations by block-pulse wavelets, UPB Sci. Bull., Ser. A: Appl. Math. Phys., 79 (2017), 193-206.
    [58] B. N. Datta, Numerical linear algebra and application, Brooks/Cole Publishing Company, Pacic Grove, CA, 1995 (Custom published byBrooks/Cole, 2003).
  • Reader Comments
  • © 2020 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3377) PDF downloads(611) Cited by(11)

Article outline

Figures and Tables

Figures(14)  /  Tables(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog