# Using differentiation matrices for pseudospectral method solve Duffing Oscillator

Volume 11, Issue 12, pp 1331--1336 Publication Date: September 13, 2018       Article History
• 40 Views

### 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.

### Keywords

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

•  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.

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

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

• [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.

• [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.

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

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

• [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.

• [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.

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

• [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.

• [13] A. Pinelli, C. Benocci, M. Deville , A Chebyshev collocation algorithm for the solution of advection–diffusion equations, Comput. Methods Appl. Mech. Engrg., 116 (1997), 201–210.

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

• [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.

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