Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation - Jun-Sheng Duan; Di-Chen Hu; Ming Li (Fractal and Fractional, 2021)
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 <semantics>α</semantics> satisfying <semantics>0≤α≤2</semantics>. 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, <semantics>α=</semantics> 0, 1 and 2.