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.