CHEBYSHEV PSEUDOSPECTRAL METHOD FOR DUFFING NONLINEAR DIFFERENTIAL EQUATIONS

Authors

  • Le Anh Nhat Tan Trao University

DOI:

https://doi.org/10.51453/2354-1431/2022/839

Keywords:

Duffing oscillator, Duffing equation, pseudospectral methods, Duffing system, Chebyshev points.

Abstract

The system of Duffing nonlinear differential equations is often used in dynamics, which are known to describe many important oscillating phenomena in nonlinear engineering systems. This article presents the pseudospectral method to calculate numerical solutions for nonlinear Duffing differential equations on the interval [–1, 1]. This method is based on the differentiation matrix using the Chebyshev Gauss – Lobatto points. To find numerical solutions of the nonlinear Duffing differential equations, we have built an iterative procedure. The software used for the calculations in this study was Mathematica 10.4. The numerical results of the comparison show that this solution had a high degree of accuracy and very small errors.

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References

/[1]. Kovacic I., Brennan M. J., Background: On Georg Duffing and the Duffing equation, The Duffing Equation: Nonlinear Oscillators and their Behaviour, Wiley, 2011, pp. 1–23.

/[2]. Sibanda P., Khidir A. A new modification of the HPM for the Duffing equation with cubic nonlinearity, ICACM’11, Lanzarote, Spain, 2011, pp. 139–143. https://doi.org/10.24297/jap.v2i2.2099

/[3]. Salas A. H., Castillo J. E. Exact Solution to Duffing Equation and the Pendulum Equation, Appl. Math. Sci., 2014, vol. 8, no. 176. pp. 8781–8789. https://doi.org/10.12988/ams.2014.44243

/[4]. Korsch H. J., Jodl H.-J., Hartmann T., Chaos: A Program Collection for the PC, Springer, 2008.

/[5]. Enns R. H., McGuire G. C. Nonlinear Physics with Mathematica for Scientists and Engineers, Boston: Birkhauser, 2001.

/[6]. Kovacic I., Brennan M. J. The Duffing Equation: Nonlinear Oscillators and their Behaviour, ed. first, Wiley, 2011.

/[7]. Weisstein E. W. Duffing Differential Equation, MathWorld - A Wolfram Web Resource, https://mathworld.wolfram.com/DuffingDifferentialEquation.html

/[8]. Bashkirtseva, I. A. The impact of colored noise on the equilibria of nonlinear dynamic systems, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2018, vol. 28, no. 2, pp. 133–142. https://doi.org/10.20537/vm180201

/[9]. Longsuo, L. Suppressing Chaos of Duffing-Holmes System Using Random Phase, Math. Probl. Eng., 2011, vol. 2011, id. 538202, 8p. https://doi.org/10.1155/2011/5382028

/[10]. Tamaseviius A., Bumelien S., Kirvaitis R. et al. Autonomous Duffing-Holmes Type Chaotic Oscillator, Elektronika ir Elektrotechnika, 2009, vol. 5, no. 93. pp. 43–46.

/[11]. Tamaseviciute E., Tamasevicius A., Mykolaitis G. et al. Analogue Electrical Circuit for Simulation of the Duffing-Holmes Equation, Nonlinear Analysis: Modelling and Control, 2008, vol. 13, no. 2, pp. 241–252.

/[12]. Nourazar S., Mirzabeigy A. Approximate solution for nonlinear Duffing oscillator with damping effect using the modified differential transform method, Sci. Iran. B, 2013, vol. 20, no. 2, pp. 364–368. https://doi.org/10.1016/j.scient.2013.02.023

/[13]. Rad J. A., Kazem S., Parand K. A numerical solution of the nonlinear controlled Duffing oscillator by radial basis functions, Computers and Mathematics with Applications, 2012, vol. 64, no. 6, pp. 2049–2065.

/[14]. Al-Jawary M. A., S. G. Abd-Al-Razaq. Analytic and numerical solution for duffing equations, Int. J. Basic Appl. Sci., 2016, vol. 5, no. 2, pp. 115–119. https://doi.org/10.14419/IJBAS.V5I2.5838

/[15]. Gorji-Bandpy M., Azimi M., Mostofi M. Analytical methods to a generalized Duffing oscillator, Australian J. Basic Applied Sci., 2011, vol. 5, no. 11, pp. 788–796.

/[16]. El-Naggar A. M., Ismail G. M. Analytical solution of strongly nonlinear Duffing oscillators, Alex. Engg. J., 2016, vol. 55, pp. 1581–1585. https://doi.org/10.1016/j.aej.2015.07.017

/[17]. A. Okasha El-Nady, Maha M. A. Lashin. Approximate Solution of Nonlinear Duffing Oscillator Using Taylor Expansion, J. Mech. Engi. Auto., 2016, vol. 6, no. 5, pp. 110–116. https://doi.org/10.5923/j.jmea.20160605.03

/[18]. Razzaghi M., Elnagar G. N. Numerical solution of the controlled Duffing oscillator by the pseudospectral method, J. Comput. Appl. Math., 1994, vol. 56, no.3 pp. 253–261. https://doi.org/10.1016/0377-0427(94)90081-7

/[19]. Saadatmandi A, Mashhadi-Fini F. A pseudospectral method for nonlinear Duffing equation involving both integral and non-integral forcing terms, Math. Methods Appl. Sci., 2015, vol. 38, no. 7, pp. 1265–1272. https://doi.org/10.1002/mma.3142

/[20]. Elnagar G. N., Razzaghi M. A Chebyshev spectral method for the solution of nonlinear optimal control problems, Appl. Math. Modelling, 1997, vol. 21, no. 5, pp. 255–260. https://doi.org/10.1016/S0307-904X(97)00013-9

/[21]. Bulbul B., Sezer M. Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method, J. Appl. Math., 2013, vol. 2013, id. 691614, 7 p. https://doi.org/10.1155/2013/691614

/[22]. Lin H.-Y., Yen C.-C., Jen K.-C. et al. A Postverification Method for Solving Forced Duffing Oscillator Problems without Prescribed Periods, J. Appl. Math., 2014, vol. 2014, id. 317640, 11p. https://doi.org/10.1155/2014/317460

/[23]. Hosen M. A., Chowdhury M. S. H., Ali M. Y. et al. An analytical approximation technique for the duffing oscillator based on the energy balance method, Ital. J. Pure Appl. Math., 2017, vol. 37, pp. 455–466.

/[24]. Ganji D. D., Gorji M., Soleimani S. et al. Solution of nonlinear cubic-quintic Duffing oscillators using He’s Energy Balance Method, J Zhejiang Univ Sci A, 2009, vol. 10, no. 9, pp. 1263–1268. https://doi.org/10.1631/jzus.A0820651

/[25]. Rasedee A. F. N., Sathar M. H. A., Ishak N. et al. Solution for nonlinear Duffing oscillator using variable order variable stepsize block method, Matematika, 2017, vol. 33, no. 2, pp. 165–175. https://doi.org/10.11113/matematika.v33.n2.1015

/[26]. Mason J. C., Handscomb D. C. Chebyshev Polynomials, CRC Press LLC, 2003.

/[27]. Trefethen L. N. Spectral Methods in Matlab, SIAM, 2000.

/[28]. Don W.S., Solomonoff A. Accuracy and Speed in Computing the Chebyshev Collocation Derivative, SIAM J. of Sci. Comput., 1991, vol. 16, no 6, pp. 1253–1268. https://doi.org/10.1137/0916073

/[29]. Odeyemi T., Mohammadian A., Seidou O. Application of the Chebyshev pseudospectral method to van der Waals fluids, Commun Nonlinear Sci Numer Simulat, 2012, vol. 17, no. 9, pp. 3499–3507. https://doi.org/10.1016/j.cnsns.2011.12.025

/[30]. Jensen A. Lecture Notes on Spectra and Pseudospectra of Matrices and Operators, Aalborg University, 2009.

/[31]. Dang-Vu H., Delcarte C. Hopf Bifurcation and Strange Attractors in Chebyshev Spectral Solutions of the Burgers Equation, Appl. Math. Comput., 1995, vol. 73, no 2-3, pp. 99–113. https://doi.org/10.1016/0096-3003(94)00242-8

/[32]. Canuto C, Quarteroni A., Hussaini M. Y. et al. Spectral Methods: Fundamentals in Single Domains, Springer–Verlag Berlin Heidelberg, 2006. https://doi.org/10.1007/978-3-540-30726-6

/[33]. Nhat L. A. Using differentiation matrices for pseudospectral method solve Duffing Oscillator, J. Nonlinear Sci. Appl., 2018, vol. 11, no. 12 , pp. 1331–1336. http://doi.org/10.22436/jnsa.011.12.04

/[34]. Nhat L. A. Pseudospectral method for the second-order autonomous nonlinear differential equations, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2019, vol. 29, no. 1, pp. 61–72. https://doi.org/10.20537/vm190106

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Published

2023-03-13

How to Cite

Le, N. (2023). CHEBYSHEV PSEUDOSPECTRAL METHOD FOR DUFFING NONLINEAR DIFFERENTIAL EQUATIONS. SCIENTIFIC JOURNAL OF TAN TRAO UNIVERSITY, 9(1). https://doi.org/10.51453/2354-1431/2022/839

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Section

Natural Science and Technology