*2.2. Three Classes of Fractional Oscillators*

Denote by *dν dtν* = −<sup>∞</sup>*D<sup>ν</sup><sup>t</sup>* the Weyl fractional derivative of order *ν* > 0. Then (Uchaikin [38], Miller and Ross [69], Klafter et al. [70]),

$$\,\_{-\infty}D\_t^\nu f(t) = \frac{1}{\Gamma(-\nu)} \int\_{-\infty}^t \frac{f(u) du}{\left(t - u\right)^{1+\nu}}\,\tag{29}$$

where Γ(*ν*) is the Gamma function. The Weyl fractional derivative is used in this research because it is suitable for the Fourier transform in the domain of fractional calculus (Lavoie et al. ([71], p. 247)).

The Fourier transform of *dν f*(*t*) *dtν* , following Uchaikin ([72], Section 4.5.3), is given by

$$\int\_{-\infty}^{\infty} \frac{d^\nu f(t)}{dt^\nu} e^{-i\omega t} dt = (i\omega)^\nu F(\omega),\tag{30}$$

where *F*(*ω*) is the Fourier transform of *f*(*t*).

This article relates to three classes of fractional oscillators as follow. We denote the following oscillation equation as a fractional oscillator in Class I.

$$\begin{cases} m\frac{d^\alpha y\_1(t)}{dt^\alpha} + ky\_1(t) = \varepsilon(t) \\ y\_1(0) = y\_{10}, y'\_{11}(0) = y'\_{10} \end{cases}, 1 < \alpha \le 2. \tag{31}$$

The free response to (31) is in the form (Mainardi [25], Achar et al. [33], Uchaikin ([38], Chapter 7))

$$y\_1(t) = y\_{10} E\_{\mathfrak{u},1} \left[ - \left( \omega\_{\mathfrak{u}} t \right)^a \right] + y\_{10}' t E\_{\mathfrak{u},2} \left[ - \left( \omega\_{\mathfrak{u}} t \right)^a \right], 1 < \mathfrak{u} \le 2, t \ge 0,\tag{32}$$

where *Ea*,*<sup>b</sup>*(*z*) is the generalized Mittag-Leffler function given by

$$E\_{a,b}(z) = \sum\_{k=0}^{\infty} \frac{z^k}{\Gamma(ak+b)}, a, b \in \mathbb{C}, \text{Re}(a) > 0, \text{Re}(b) > 0. \tag{33}$$

The Mittag-Leffler function denoted by *Ea*(*t*) is in the form

$$E\_{\mathfrak{a}}(z) = \sum\_{k=0}^{\infty} \frac{z^k}{\Gamma(ak+1)}, a \in \mathbb{C}, \text{Re}(a) > 0,\tag{34}$$

referring Mathai and Haubold [73], or Gorenflo et al. [74], or Erdelyi et al. [75] for the Mittag-Leffler functions.

Denote by *hy*1(*t*) the impulse response to a fractional oscillator in Class I. Then (Uchaikin ([38], Chapter 7)),

$$h\_{y1}(t) = t^{a-1} E\_{a,a} \left[ - \left( \omega\_n t \right)^a \right], 1 < a \le 2, t \ge 0. \tag{35}$$

Let *gy*1(*t*) be the step response to a fractional oscillator of Class I type. Then,

> *n*

$$\log\_{1}(t) = t^{a} E\_{a,a+1} \left[ -\left(\omega\_{n}t\right)^{a} \right], 1 < a \le 2, t \ge 0. \tag{36}$$

For a fractional oscillator in Class I, its sinusoidal response driven by sin*ωt* is expressed by

$$y\_1(t) = A\_1 \sin(\omega t - \theta\_1) + A\_2 e^{-\beta t} \cos\left[\omega\_n t \sin\frac{\pi}{a} - \theta\_2\right] + \int\_0^\infty e^{-st} K\_a(s) ds,\tag{37}$$

where

$$\begin{aligned} A\_1 &= \frac{1}{\sqrt{\omega\_n^{2a} + \omega^{2a} + 2\omega\_n^a \omega^a \cos\frac{a\pi}{2}}}, \\ A\_2 &= \frac{2\omega}{a\omega\_n^{a-1}\sqrt{\omega\_n^4 + \omega^4 + 2\omega\_n^2 \omega^2 \cos\frac{2\pi}{a}}}, \end{aligned} \tag{38}$$

$$
\beta = -\omega\_{\text{fl}} \cos \frac{\pi}{\alpha'} \tag{39}
$$

$$\theta\_1 = \tan^{-1} \frac{\omega^a \sin \frac{\theta \pi}{2}}{\omega\_n^a + \omega^a \cos \frac{\pi \pi}{2}},$$

$$\pi \int\_{-\alpha^2 \sin \frac{(1+a)\pi}{2} - \alpha^2 \sin \frac{(1-a)\pi}{2}} \tag{40}$$

$$\theta\_2 = \tan^{-1} \left[ \frac{\omega\_n^2 \sin \frac{(1+a)\pi}{a} - \omega^2 \sin \frac{(1-a)\pi}{a}}{\omega\_n^2 \cos \frac{(1+a)\pi}{a} + \omega^2 \cos \frac{(1-a)\pi}{a}} \right],\tag{40}$$

$$K\_{\mathfrak{a}}(s) = \frac{\omega \sin(\pi a)}{\pi (s^2 + \omega^2)(s^{2a} + 2s^a \omega\_n^2 \cos(\pi a) + \omega\_n^{2a})}. \tag{41}$$

An oscillator that follows the oscillation equation below is called a fractional oscillator in Class II.

$$m\frac{d^2y\_2(t)}{dt^2} + c\frac{d^\beta y\_2(t)}{dt^\beta} + ky\_2(t) = 0,\\ 0 < \beta \le 1. \tag{42}$$

The equation below is called an oscillation equation of a fractional oscillator in Class III.

$$m\frac{d^a y\_3(t)}{dt^a} + c\frac{d^\beta y\_3(t)}{dt^\beta} + ky\_3(t) = 0,\\ 1 < a \le 2, \ 0 < \beta \le 1. \tag{43}$$

### *2.3. Equivalence of Functions in the Sense of Fourier Transform*

Denote by *<sup>F</sup>*1(*ω*) and *<sup>F</sup>*2(*ω*) the Fourier transforms of *f*1(*t*) and *f*2(*t*), respectively. Then, if

$$F\_1(\omega) = F\_2(\omega),\tag{44}$$

one says that

$$f\_1(t) = f\_2(t),\tag{45}$$

in the sense of Fourier transform (Gelfand and Vilenkin [67], Papoulis [76]), implying

$$\int\_{-\infty}^{\infty} [f\_1(t) - f\_2(t)]e^{-i\omega t}dt = 0.\tag{46}$$

The above implies that a null function as a difference between *f*1(*t*) and *f*2(*t*) is allowed for (45). An example relating to oscillation theory is the unit step function.

Denote by *<sup>u</sup>*1(*t*) in the form

$$u\_1(t) = \begin{cases} \ 1, t \ge 0 \\\ 0, t < 0 \end{cases} . \tag{47}$$

Let *<sup>u</sup>*2(*t*) be

$$\mu\_2(t) = \begin{cases} \ 1, t > 0 \\ \ 0, t \le 0 \end{cases}.\tag{48}$$

Clearly, either *<sup>u</sup>*1(*t*) or *<sup>u</sup>*2(*t*) is a unit step function. The difference between two is a null function given by 

$$
\mu\_1(t) - \mu\_2(t) = \begin{cases} 1, & t = 1 \\ 0, & \text{elsewhere} \end{cases}.\tag{49}
$$

Thus, *<sup>u</sup>*1(*t*) = *<sup>u</sup>*2(*t*). In fact, the Fourier transform of either *<sup>u</sup>*1(*t*) or *<sup>u</sup>*2(*t*) equals to the right side on (23). Similarly, if *f*1(*t*) = *f*2(*t*), we say that (44) holds in the sense of

$$\int\_{-\infty}^{\infty} [F\_1(\omega) - F\_2(\omega)] \epsilon^{i\omega t} d\omega = 0. \tag{50}$$
