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

Mohd Taib Shatnawi, Adel Ouannas, Ghenaiet Bahia, Iqbal M. Batiha and Giuseppe Grassi
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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
Giuseppe Grassi: Dipartimento Ingegneria Innovazione, Universita del Salento, 73100 Lecce, Italy

Mathematics, 2021, vol. 9, issue 18, 1-13

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).

Keywords: strongly nonlinear oscillators; fractional-order derivatives; optimal homotopy asymptotic method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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