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The Optimal Homotopy Asymptotic Method for Solving Two Strongly Fractional-Order Nonlinear Benchmark Oscillatory Problems

Author

Listed:
  • Mohd Taib Shatnawi

    (Department of Basic Science, Al-Huson University College, Al-Balqa Applied University, Irbid 21510, Jordan)

  • Adel Ouannas

    (Laboratory of Dynamical Systems and Control, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria)

  • Ghenaiet Bahia

    (Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Larbi Tebessi, Tebessa 12002, Algeria)

  • Iqbal M. Batiha

    (Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid 21110, Jordan
    Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, United Arab Emirates)

  • Giuseppe Grassi

    (Dipartimento Ingegneria Innovazione, Universita del Salento, 73100 Lecce, Italy)

Abstract

This paper proceeds from the perspective that most strongly nonlinear oscillators of fractional-order do not enjoy exact analytical solutions. Undoubtedly, this is a good enough reason to employ one of the major recent approximate methods, namely an Optimal Homotopy Asymptotic Method (OHAM), to offer approximate analytic solutions for two strongly fractional-order nonlinear benchmark oscillatory problems, namely: the fractional-order Duffing-relativistic oscillator and the fractional-order stretched elastic wire oscillator (with a mass attached to its midpoint). In this work, a further modification has been proposed for such method and then carried out through establishing an optimal auxiliary linear operator, an auxiliary function, and an auxiliary control parameter. In view of the two aforesaid applications, it has been demonstrated that the OHAM is a reliable approach for controlling the convergence of approximate solutions and, hence, it is an effective tool for dealing with such problems. This assertion is completely confirmed by performing several graphical comparisons between the OHAM and the Homotopy Analysis Method (HAM).

Suggested Citation

  • Mohd Taib Shatnawi & Adel Ouannas & Ghenaiet Bahia & Iqbal M. Batiha & Giuseppe Grassi, 2021. "The Optimal Homotopy Asymptotic Method for Solving Two Strongly Fractional-Order Nonlinear Benchmark Oscillatory Problems," Mathematics, MDPI, vol. 9(18), pages 1-13, September.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:18:p:2218-:d:632586
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    References listed on IDEAS

    as
    1. Mahmoud, Gamal M., 1993. "On the generalized averaging method of a class of strongly nonlinear forced oscillators," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 199(1), pages 87-95.
    2. S. S. Motsa, 2014. "On the Optimal Auxiliary Linear Operator for the Spectral Homotopy Analysis Method Solution of Nonlinear Ordinary Differential Equations," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-15, August.
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