AN EFFICIENT APPROACH FOR SOLVING THE FRACTAL, DAMPED CUBIC–QUINTIC DUFFING’S EQUATION

ELà AS-ZÚÑIGA Alex, Oscar Martã Nez-Romero (), Daniel Olvera Trejo () and Luis Manuel Palacios-Pineda
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ELà AS-ZÚÑIGA Alex: Mechanical Engineering and Advanced Materials Department, Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501, Monterrey 64849, Mexico
Oscar Martã Nez-Romero: Mechanical Engineering and Advanced Materials Department, Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501, Monterrey 64849, Mexico
Daniel Olvera Trejo: Mechanical Engineering and Advanced Materials Department, Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501, Monterrey 64849, Mexico
Luis Manuel Palacios-Pineda: Mechanical Engineering and Advanced Materials Department, Institute of Advanced Materials for Sustainable Manufacturing, Ave. Eugenio Garza Sada 2501, Monterrey 64849, Mexico

FRACTALS (fractals), 2024, vol. 32, issue 01, 1-10

Abstract: The main goal of this work is to focus on using He’s two-scale fractal dimension transform, the Caputo–Fabrizio fractional-order derivative, and the harmonic balance and the homotopy methods are applied for deriving the approximate solution of the fractal, damped cubic–quintic Duffing’s equation when the fractional derivative order of the inertia term is not twice of that of the damping term. Numerical results obtained from the derived expressions and the numerical integration solution show good agreement, especially at small values of the nonlinear parameters. Furthermore, when the fractal order of the damping term decreases, the damping oscillation frequency values increase with a decrease in the system wavelength values, which indicates a slower decay in the system oscillation amplitudes. Our solution procedures elucidate the applicability of He’s two-scale fractal dimension transform for solving nonlinear dynamic systems with inertia and damping fractal terms.

Keywords: Fractal Cubic-Quintic Duffing’s Equation; Damped System; Caputo-Fabrizio Fractional Order Derivative; He’s Two-scale Dimension Transform; Homotopy Perturbation Method; Fractal Media (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:wsi:fracta:v:32:y:2024:i:01:n:s0218348x24500117

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DOI: 10.1142/S0218348X24500117

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