Mathematical and numerical optimality of non-singular fractional approaches on free and forced linear oscillator

Abstract

The prediction of oscillators is usually employed in various industrial and technological problems; such as car shock absorbers, bungee jumping, earthquake-proof buildings, musical instruments, metronome and the process of hearing. This manuscript investigates the effects of newly presented fractional operators on free and forced linear oscillators. The second order nonlinear classical governing differential equation of Duffing oscillator is reduced into second order linear classical governing differential equation of free and forced linear oscillators by invoking non-integer order differential operators namely Atangana-Baleanu and Caputo-Fabrizio. The fractionalized differential equation is solved by invoking Laplace transform method for finding the optimal solutions of displacement based on infinite series approach. The solutions of displacement are obtained via Atangana-Baleanu and Caputo-Fabrizio differential operators separately then expressed in terms of elementary and gamma functions. Finally the parametric analysis is depicted graphically on the basis of comparison of modern fractional operators subject to the emerging rheological parameters.

Keywords

• free and forced linear oscillator
• atangana-baleanu and caputo-fabrizio differential operators
• laplace transform
• gamma and elementary functions
• parametric analysis