Using differentiation matrices for pseudospectral method solve Duffing Oscillator

Volume 11, Issue 12, pp 1331--1336 http://dx.doi.org/10.22436/jnsa.011.12.04

Publication Date: September 13, 2018 Submission Date: June 17, 2018 Revision Date: August 05, 2018 Accteptance Date: August 19, 2018

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Authors

L. A. Nhat - PhD student of, RUDN University, Moscow 117198, Russia. - Lecture at Tan, Trao University, Tuyen Quang province, Vietnam.

Abstract

This article presents an approximate numerical solution for nonlinear Duffing Oscillators by pseudospectral (PS) method to compare boundary conditions on the interval [-1, 1]. In the PS method, we have been used differentiation matrix for Chebyshev points to calculate numerical results for nonlinear Duffing Oscillators. The results of the comparison show that this solution had the high degree of accuracy and very small errors. The software used for the calculations in this study was Mathematica V.10.4.

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Keywords

• Duffing oscillator
• pseudospectral methods
• differential matrix
• Duffing system
• Chebyshev points

MSC

•  34B15
•  41A50
•  65L10

References

• [1] M. A. Al-Jawary, S. G. Abd-Al-Razaq, Analytic and numerical solution for duffing equations, Int. J. Basic Appl. Sci. , 5 (2016), 115–119.
  • Google Scholar


• [2] B. Bulbul, M. Sezer, Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method, J. Appl. Math., 2013 (2013), 6 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] W. S. Don, A. Solomonoff, Accuracy and speed in computing the Chebyshev collocation derivative, SIAM J. Sci. Comput., 16 (1995), 1253–1268.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] A. Elias-Ziga, O. Martnez-Romero, R. K. Crdoba-Daz, Approximate Solution for the Duffing–Harmonic Oscillator by the Enhanced Cubication Method, Math. Probl. Eng., 2012 (2012), 12 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] A. O. El-Nady, M. M. A. Lashin, Approximate Solution of Nonlinear Duffing Oscillator Using Taylor Expansion, J. Mech. Engi. Auto., 6 (2016), 110–116.
  • View Article
  • Google Scholar


• [6] A. M. El-Naggar, G. M. Ismail , Analytical solution of strongly nonlinear Duffing Oscillators, Alex. Engi. Jour., 55 (2016), 1581–1585.
  • View Article
  • Google Scholar


• [7] R. H. Enns, G. C. McGuire, Nonlinear Physics with Mathematica for Scientists and Engineers, Birkhauser Basel, Boston (2001)
  • View Article
  • Google Scholar
  • MATH


• [8] M. Gorji-Bandpy, M. A. Azimi, M. M. Mostofi , Analytical methods to a generalized Duffing oscillator, Australian J. Basic Appl. Sci., 5 (2011), 788–796.


• [9] M. A. Hosen, M. S. H. Chowdhury, M. Y. Ali, A. F. Ismail, An analytical approximation technique for the duffing oscillator based on the energy balance method , Ital. J. Pure Appl. Math., 37 (2017), 455–466.
  • MathSciNet
  • Google Scholar
  • MATH


• [10] H.-Y. Lin, C.-C. Yen, K.-C. Jen, K. C. Jea, A Postverification Method for Solving Forced Duffing Oscillator Problems without Prescribed Periods, J. Appl. Math., 2014 (2014), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [11] J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, FL (2003)
  • MathSciNet


• [12] S. Nourazar, A. Mirzabeigy, Approximate solution for nonlinear Duffing oscillator with damping effect using the modified differential transform method, Scientia Iranica, 20 (2013), 364–368.
  • View Article


• [13] A. Pinelli, C. Benocci, M. Deville , A Chebyshev collocation algorithm for the solution of advection–diffusion equations, Comput. Methods Appl. Mech. Engrg., 116 (1994), 201–210.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] M. Razzaghi, G. Elnagar , Numerical solution of the controlled Duffing oscillator by the pseudospectral method, J. Comput. Appl. Math., 56 (1994), 253–261.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] A. Saadatmandi, F. Mashhadi-Fini , A pseudospectral method for nonlinear Duffing equation involving both integral and nonintegral forcing terms, Math. Methods Appl. Sci., 38 (2015), 1265–1272.
  • View Article
  • Google Scholar


• [16] A. H. Salas, J. E. Castillo H., Exact Solution to Duffing Equation and the Pendulum Equation, Appl. Math. Sci., 8 (2014), 8781–8789.
  • View Article
  • Google Scholar