Journal of Applied Mathematics

Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

Berna Bülbül and Mehmet Sezer

Full-text: Open access

Abstract

We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 691614, 6 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808018

Digital Object Identifier
doi:10.1155/2013/691614

Mathematical Reviews number (MathSciNet)
MR3064960

Zentralblatt MATH identifier
1275.65038

Citation

Bülbül, Berna; Sezer, Mehmet. Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method. J. Appl. Math. 2013 (2013), Article ID 691614, 6 pages. doi:10.1155/2013/691614. https://projecteuclid.org/euclid.jam/1394808018


Export citation

References

  • G. F. Corliss, Guarented Error Bounds for Ordinary Differential Equations, in Theory and Numeric of Ordinary and Partial Equations, Oxford University press, Oxford, UK, 1995.
  • H. Bulut and D. J. Evans, “On the solution of the Riccati equation by the decomposition method,” International Journal of Computer Mathematics, vol. 79, no. 1, pp. 103–109, 2002.
  • Y. Khan, M. Mada, A. Yildirim, M. A. Abdou, and N. Faraz, “A new approach to van der Pol's oscillator problem,” Zeitschrift für Naturforschung A, vol. 66, pp. 620–624, 2011.
  • Y. Khan, H. Vázquez-Leal, and N. Faraz, “An efficient new iterative method for oscillator differential equation,” Scientia Iranica, vol. 19, pp. 1473–1477, 2012.
  • Y. Khan, “Mehdi Akbarzade Dynamic analysis of non-linear oscillator equation arising in double-sided driven clamped microbeam-based electromechanical resonator,” Zeitschrift für Naturforschung A, vol. 67, pp. 435–440, 2012.
  • Y. Khan, M. Akbarzade, and A. Kargar, “Coupling of homotopy and variational approach for conservative oscillator with strong odd-nonlinearity,” Scientia Iranica, vol. 19, pp. 417–422, 2012.
  • M. Akbarzade and Y. Khan, “Dynamic model of large amplitude non-linear oscillations arising in the structural engineering: analytical solutions,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 480–489, 2012.
  • Y. Khan and Q. Wu, “Homotopy perturbation transform method for nonlinear equations using He's polynomials,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 1963–1967, 2011.
  • J. H. He, G. C. Wu, and F. Austin, “The variational iteration method which should be followed,” Nonlinear Science Letters A, vol. 1, pp. 1–30, 2010.
  • N. Jamshidi and D. D. Ganji, “Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire,” Current Applied Physics, vol. 10, pp. 484–486, 2010.
  • J.-H. He, “Hamiltonian approach to nonlinear oscillators,” Physics Letters A, vol. 374, no. 23, pp. 2312–2314, 2010.
  • M. Akbarzade and J. Langari, “Determination of natural frequencies by coupled method of homotopy perturbation and variational method for strongly nonlinear oscillators,” Journal of Mathematical Physics, vol. 52, no. 2, Article ID 023518, 10 pages, 2011.
  • J.-H. He, “Variational approach for nonlinear oscillators,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1430–1439, 2007.
  • D. D. Ganji and M. Akbarzade, “Approximate analytical solutions to non-linear oscillators using He's amplitude-frequency formulation,” International Journal of Mathematical Analysis, vol. 32, no. 4, pp. 1591–1597, 2010.
  • M. O. Donnagáin and O. Rasskazov, “Numerical modelling of an iron pendulum in a magnetic field,” Physica B, vol. 372, no. 1-2, pp. 37–39, 2006.
  • Z. Feng, “Duffing's equation and its applications to the Hirota equation,” Physics Letters A, vol. 317, no. 1-2, pp. 115–119, 2003.
  • Z. Feng, “Monotonous property of non-oscillations of the damped Duffing's equation,” Chaos, Solitons and Fractals, vol. 28, no. 2, pp. 463–471, 2006.
  • Y. Z. Chen, “Solution of the Duffing equation by using target function method,” Journal of Sound and Vibration, vol. 256, no. 3, pp. 573–578, 2002.
  • Z. Wang, “P-stable linear symmetric multistep methods for periodic initial-value problems,” Computer Physics Communications, vol. 171, no. 3, pp. 162–174, 2005.
  • E. Yusufoğlu, “Numerical solution of Duffing equation by the Laplace decomposition algorithm,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 572–580, 2006.
  • S. A. Khuri, “A Laplace decomposition algorithm applied to a class of nonlinear differential equations,” Journal of Applied Mathematics, vol. 1, no. 4, pp. 141–155, 2001.
  • C. Pezeshki and E. H. Dowell, “An examination of initial condition maps for the sinusoidally excited buckled beam modeled by the Duffing's equation,” Journal of Sound and Vibration, vol. 117, no. 2, pp. 219–232, 1987.
  • S. A. Khuri and S. Xie, “On the numerical verification of the asymptotic expansion of Duffing's equation,” International Journal of Computer Mathematics, vol. 372, no. 3, pp. 25–30, 1999.
  • Y. H. Ku and X. Sun, “CHAOS and limit cycle in Duffing's equation,” Journal of the Franklin Institute, vol. 327, no. 2, pp. 165–195, 1990.
  • F. Battelli and K. J. Palmer, “Chaos in the Duffing equation,” Journal of Differential Equations, vol. 101, no. 2, pp. 276–301, 1993.
  • A. Loria, E. Panteley, and H. Nijmeijer, “Control of the chaotic Duffing equation with uncertainty in all parameters,” IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 45, no. 12, pp. 1252–1255, 1998.
  • H. Nijmeijer and H. Berghuis, “On Lyapunov control of the Duffing equation,” IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 42, no. 8, pp. 473–477, 1995.
  • J. G. Byatt-Smith, “Regular and chaotic solutions of Duffing's equation for large forcing,” IMA Journal of Applied Mathematics, vol. 37, no. 2, pp. 113–145, 1986.
  • P. Habets and G. Metzen, “Existence of periodic solutions of Duffing equations,” Journal of Differential Equations, vol. 78, no. 1, pp. 1–32, 1989.