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).
Highlights
This section targets introducing the Optimal Homotopy Asymptotic Method (OHAM) for the purpose of generally establishing approximate solutions for the strongly fractional-order nonlinear oscillatory problems that can be expressed by the following form [35]: D α u(t) + f (u(t)) = 0, (11)
This section employs the OHAM to provide approximate analytic solutions for two strongly fractional-order nonlinear benchmark oscillatory problems, namely: the fractionalorder Duffing-relativistic oscillator, and the fractional-order stretched elastic wire oscillator
A further modification for an Optimal Homotopy Asymptotic Method (OHAM) has been successfully implemented to tsolvewo strongly fractional-order nonlinear benchmark oscillatory problems, namely: the fractional-order Duffing-relativistic oscillator and the fractional-order stretched elastic wire oscillator
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. In view of many of these studies, it was demonstrated that this method is a reliable, straightforward, and effective tool for offering accurate analytical approximate solutions to lots of strongly nonlinear problems [2,18,29]. This work employs this method to provide approximate analytic solution for two strongly fractional-order nonlinear benchmark oscillatory problems through establishing an optimal auxiliary linear operator, an auxiliary function, and an auxiliary control parameter. These two nonlinear oscillators are: the fractional-order Duffing-relativistic oscillator and the fractional-order stretched elastic wire oscillator (with a mass attached to its midpoint). The last section summarizes the main conclusions of this work
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