A periodic solution of the fractional sine-Gordon equation arising in architectural engineering

Abstract

Many nonlinear vibrations arising in the engineering of architecture include noise and uncertain properties, and this paper suggests a fractional model to elucidate the properties. The fractional sine-Gordon equation with the Riemann-Liouville fractional derivative is used as an example to solve its periodic solution by the homotopy perturbation method. The frequency-amplitude relationship is obtained, and the effect of the fractional derivative order on the vibration property is discussed. Additionally, the harmonic resonance is also discussed. This preliminary research can be further extended to real applications.

The reducing rank method to solve third‐order Duffing equation with the homotopy perturbation

“…Recently, the HPM for the singular Duffing oscillator has been studied by He and El-Dib [24]. A modification to the HPM with the so-called auxiliary equivalence technique has been demonstrated by Shen and El-Dib [35]. El-Dib and Elgazery [13] applied the method of the so-called reducing rank technique, through the integrating of the fraction auxiliary operator.…”

Optimization of a fractal electrode-level charge transport model

“…Bao and Bessler [1] applied the Adomian decomposition method to the above system, some alternative approaches to the problem include the variational iteration method [5,6], the homotopy perturbation method [7,8], and others, see a complete review on various analytical methods in Refs. [9][10][11][12].…”

The Vibrational Motion of a Dynamical System Using Homotopy Perturbation Technique

“…HPT is used in [26] to obtain the periodic solution of the fractional sine-Gordon equation beside the Riemann-Liouville fractional derivative. The authors obtained a relationship between the frequency and amplitude, and the impact of the order of fractional derivative on the vibration property is investigated.…”

“…tained by HPT when they are compared with the numerical results of the governing equation of motion (4) using the fourth order Runge-Kutta method[26].Figures 2-4 are calculated at 1 3 kg,5 kg m = and 1 7 kg m = respectively for different values of 0.1 m, 0.6 m l = and 1 m, in which their parts (a), (b) and (c) are plotted when 2 0.2 kg, 0.4 kg m = and .6 kg m = respectively. It is…”