A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses

The fractional oscillator equation with the sinusoidal excitation mx & DPRIME;(t)+bDt alpha x(t)+kx(t)=Fsin(omega t), m,b,k,omega > 0 and 0 < 2 is comparatively considered for the Weyl fractional derivative and the Caputo fractional derivative. In the sense of Weyl, the fractional oscillator equation is solved to be a steady periodic oscillation xW(t). In the sense of Caputo, the fractional oscillator equation is solved and subjected to initial conditions. For the fractional case alpha & ISIN;(0,1)boolean OR(1,2), the response to excitation, S(t), is a superposition of three parts: the steady periodic oscillation xW(t), an exponentially decaying oscillation and a monotone recovery term in negative power law. For the two responses to initial values, S0(t) and S1(t), either of them is a superposition of an exponentially decaying oscillation and a monotone recovery term in negative power law. The monotone recovery terms come from the Hankel integrals which make the fractional case different from the integer-order case. The asymptotic behaviors of the solutions removing the steady periodic response are given for the four cases of the initial values. The Weyl fractional derivative is suitable for a describing steady-state problem, and can directly lead to a steady periodic solution. The Caputo fractional derivative is applied to an initial value problem and the steady component of the solution is just the solution in the corresponding Weyl sense.

Automated summary

This passage mainly compares the applications of the Weyl fractional derivative and the Caputo fractional derivative in the fractional oscillator equation. Under the two perspectives, the equation has different solutions and response modes. One of the characteristics of the fractional case is the presence of a monotone recovery term in negative power law.