A Backward Difference Formulation for Solving Doffing-Van Der Pol Type Oscillators

  • Authors

    • Ahmad FadlyNurullahRasedee
    • Mohammad Hasan Abdul Sathar
    • Hazizah Mohd Ijam
    • Khairil Iskandar Othman
    • Norizarina Ishak
    • Siti Raihana Hamzah
    • Nur Amalina Jamaludin
    2018-11-27
    https://doi.org/10.14419/ijet.v7i4.19.22009
  • Backward difference, ODEs, Duffing-Van Der Pol oscillators.
  • The study of chaotic motions in periodic self-excited oscillators are an area of interest in science and engineering. The current research proposes a numerical solution in backward difference form for solving these chaotic motions in periodic- self excited oscillators. The research conducted in this study focuses on chaotic motions in the form of Duffing-Van Der Pol Oscillators because of its various applications. A backward difference formulation in predictor-corrector (PeCe) mode is introduced for solving these Duffing-Van Der Pol directly. Numerical simulations provided will show the accuracy of the PeCe backward difference formulation compared against known viable methods. Results will also show that the PeCe backward formulation is a feasible alternative for solving Duffing-Van Der Pol oscillators.

     

     


  • References

    1. [1] C. W. Gear,“Chaotic motions of the Duffing-Van der Pol oscillator with external and parametric excitations,†Shock and Vibration vol. 2014, Article ID 131367, 5 pages, 2014.

      [2] G. Halland J. M. Watt,Modern numerical methods for ordinary differential equations. Oxford: Clarendon Press, 1976.

      [3] G. Halland M. B. Suleiman, “Stability of Adams-type formulae for second-order ordinarydifferential equations,†IMA: Journal of Numerical Analysis,vol. 1, no. 4, pp. 427–438, 1981.

      [4] M. B. Suleiman, Generalised multistep Adams and backward differentiation methods for the solution of stiff and non-stiff ordinary differential equations. Manchester: University ofManchester, 1979.

      [5] Ibrahim, Z. B., Othman, K. I and Suleiman, M. B. “2-point block predictor-corrector of backwarddifferentiation formulas for solving second order ordinary differential equations directly,â€Chiang Mai J.vol. 33, no. 3, pp. 502–510, 2012.

      [6] M. B.Suleiman, Z. B. Ibrahimand A. F. N. Rasedee, “Solution of higher-order ODEs using backward difference method,†Mathematical Problems in Engineering, vol. 2011 Article ID 810324, 18 pages, 2011.

      [7] F. N.Rasedee, M. B. Suleiman and Z. B.Ibrahim, “Solving nonstiff higher order odes using variable order step size backward difference directlyâ€Mathematical Problems in Engineering, vol. 2014, Article ID 565137, 10 pages, 2014.

      [8] F. N. Rasedee, M. H. A. Sathar, F. Deraman, H. M. Ijam, M. Suleiman, N Saaluddin and A. Rakhimov, “2 point block backward difference method for solving Riccati type differential problemsâ€, AIP Proceedings, 1775, p. 030005, 2016.

      [9] F. N. Rasedee, N. Ishak, S. R. Hamzah, H. M. Ijam, M. B. Suleiman, Z. B. Ibrahim, M. H. A. Sathar, N. A. Ramli, and N. S. Kamaruddin, “Variable order variable stepsize algorithm for solving nonlinear Duffing oscillatorâ€, Journal of Physics: Conference Series, 890, p. 012045, 2017.

      [10] F. N. Rasedee, H. M. Ijam, M. H. A. Sathar, N. Ishak, M. A. Nazri, N. A. Ramli, and N. S. Kamaruddin, “Block variable order step size method for solving higher order orbital problemsâ€, AIP Proceedings, 1905, p. 030028, 2017.

      [11] F. N. Rasedee, M. H. A. Sathar, N. Ishak, N. S. Kamaruddin, M. A. Nazri, N. A. Ramli, I. Ismail and M. Sahrim, “Solution for nonlinear Duffing oscillator using variable order variable stepsize block methodâ€, MATEMATIKA33, 165–176, 2017.

      [12] S. Mukherjee, B. Roy, and S. Dutta, “Solution of the Duffing–van der Pol oscillator equation by a differential transform methodâ€, PhysicaScripta83, p. 015006, 2010.

      [13] N. A. Khan, M. Jamil, S. A. Ali, and N. A. Khan, “Solutions of the force-free duffing-van der pol oscillator equationâ€, International Journal of Differential Equationsvol. 2011 Article ID 852919, 9 pages, 2011.

      [14] G. A. Cordshooli and A. Vahidi, “Solutions of Duffing-van der Pol equation using decomposition methodâ€, Journal of Physics, 3, 121–129, 2011.

      [15] Zhang and Y. Zeng, “A Simple Numerical Method For Van der Pol-Duffing Oscillator Equationâ€, International Conference on Mechatronics, Control and Electronic Engineering, Atlantis, 476-480, 2014.

      [16] F. N. Rasedee, M. B. Suleiman, and Z. B. Ibrahim, “Solving nonstiff higher order odes using variable order step size backward difference directlyâ€, Mathematical Problems in Engineering 2014, Article ID 565137, 10 pages, 2014.

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  • How to Cite

    FadlyNurullahRasedee, A., Hasan Abdul Sathar, M., Mohd Ijam, H., Iskandar Othman, K., Ishak, N., Raihana Hamzah, S., & Amalina Jamaludin, N. (2018). A Backward Difference Formulation for Solving Doffing-Van Der Pol Type Oscillators. International Journal of Engineering & Technology, 7(4.19), 36-39. https://doi.org/10.14419/ijet.v7i4.19.22009