On the numerical study of fractional and non-fractional model of nonlinear Duffing oscillator: a comparison of integer and non-integer order approaches

Abstract

ABSTRACT This manuscript analyzes nonlinear Duffing oscillators via integer and non-integer order differential operators known as the Caputo, Caputo–Fabrizio, and the Atangana–Baleanu taken in the Caputo sense. Application of the fixed-point theory shows that the fractional-order nonlinear Duffing oscillator does have a unique solution subject to appropriate initial conditions. Furthermore, the fractional-order parameters α,β and γ∈]0,1] for the Caputo, Caputo–Fabrizio, and the Atangana–Baleanu operators, respectively, when kept at variation, revealed exciting changes in the amplitude and the frequency responses for the fractional displacement function of the nonlinear oscillator. Lastly, numerical simulations have been carried out and presented in graphical illustrations to observe the effects of each parametric value of the nonlinear Duffing oscillator under the classical, the Caputo, the Caputo–Fabrizio, and the Atangana–Baleanu fractional differential operators wherein the Markovian nature of the oscillator is retained when α=β=γ=1. Based upon the comparative mathematical analysis of the oscillator, it has been shown that the Atangana–Baleanu fractional-order operator possessing non-locality and non-singular convolution kernel has the better capability to retain the memory effects of the system.