Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation

Jun-Sheng Duan,Ming Li,Di-Chen Hu

doi:10.3390/fractalfract5040188

Abstract

The impulse response of the fractional oscillation equation was investigated, where the damping term was characterized by means of the Riemann–Liouville fractional derivative with the order α satisfying 0≤α≤2. Two different analytical forms of the response were obtained by using the two different methods of inverse Laplace transform. The first analytical form is a series composed of positive powers of t, which converges rapidly for a small t. The second form is a sum of a damped harmonic oscillation with negative exponential amplitude and a decayed function in the form of an infinite integral, where the infinite integral converges rapidly for a large t. Furthermore, the Gauss–Laguerre quadrature formula was used for numerical calculation of the infinite integral to generate an analytical approximation to the response. The asymptotic behaviours for a small t and large t were obtained from the two forms of response. The second form provides more details for the response and is applicable for a larger range of t. The results include that of the integer-order cases, α= 0, 1 and 2.

Highlights

IntroductionPublisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
In [17,18,19], the Bromwich’s integral formula of inverse Laplace transform and the residue theorem were used for the solutions of fractional oscillation equations
By using two different methods of inversion Laplace transform, we consider the impulse response of the fractional oscillation equation m x 00 (t) + c Dtα x (t) + k x (t) = δ(t), m, c, k > 0, (4)

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. In [17], oscillator equations derived from the relaxation kernel and the creep kernel were considered based on the fractional calculus Kelvin–Voight model, Maxwell model, and standard linear solid model, respectively. In [17,18,19], the Bromwich’s integral formula of inverse Laplace transform and the residue theorem were used for the solutions of fractional oscillation equations. By using two different methods of inversion Laplace transform, we consider the impulse response of the fractional oscillation equation m x 00 (t) + c Dtα x (t) + k x (t) = δ(t), m, c, k > 0,. For the integer-order cases α = 0 and α = 2, making use of the Laplace transform table, the responses are obtained as the periodic oscillations.

Solution Rapidly Convergent for Small t

Solution Rapidly Convergent for Large t

Conclusions