2.1.4. Generalization of Linear Oscillators

Let us be beyond the scope of the conventionally physical quantities, such as displacement, velocity, acceleration in mechanics, or current, voltage in electronics. Then, we consider the response of the quantity *<sup>q</sup>*(*n*)(*t*), where *n* is a positive integer. Precisely, we consider the following oscillation equation

$$\begin{cases} \frac{d^2}{dt^2} \left[ \frac{d^n q(t)}{dt^n} \right] + 2\zeta\omega\nu\_n \frac{d}{dt} \left[ \frac{d^n q(t)}{dt^n} \right] + \omega\_n^2 \frac{d^n q(t)}{dt^n} = \frac{c(t)}{m} \\\ q^{(n)}(0) = q\_0, q^{(n+1)}(0) = v\_0. \end{cases} \tag{25}$$

The above may be taken as a generalization of the conventional oscillator described by (3). Another expression of the above may be given by

$$\begin{cases} \frac{d^n}{dt^n} \left[ \frac{d^2 q(t)}{dt^2} \right] + 2\zeta\omega\nu\_n \frac{d^n}{dt^n} \left[ \frac{dq(t)}{dt} \right] + \omega\nu\_n^2 \frac{d^n q(t)}{dt^n} = \frac{c(t)}{m} \\\ q^{(n)}(0) = q\_0, q^{(n+1)}(0) = v\_0. \end{cases} \tag{26}$$

Alternatively, we have a linear oscillation system described by

$$\begin{cases} \frac{d^{n+2}q(t)}{dt^{n+2}} + 2\zeta\omega\iota\_{\mathfrak{n}}\frac{d^{n+1}q(t)}{dt^{n+1}} + \omega\_{\mathfrak{n}}^{2}\frac{d^{n}q(t)}{dt^{n}} = \frac{\varepsilon(t)}{m} \\\ q^{(n)}(0) = q\_{0}, q^{(n+1)}(0) = v\_{0}. \end{cases} \tag{27}$$

Physically, the above item with *q*(*n*+<sup>2</sup>)(*t*) corresponds to inertia, the one with *q*(*n*)(*t*) to restoration, and the one with *q*(*n*+<sup>1</sup>)(*t*) damping.

Note that (27) remains a linear oscillator after all. Nevertheless, when generalizing *n* to be fractions, for instance, considering −1 < *ε*1 ≤ 0 and −1 < *ε*2 ≤ 0, we may generalize (27) to be

$$\begin{cases} \ m \frac{d^{\nu\_1+2}q(t)}{dt^{\nu\_1+2}} + c \frac{d^{\nu\_2+1}q(t)}{dt^{\nu\_2+1}} + kq(t) = c(t) \\\ q(0) = q\_0, q'(0) = v\_0. \end{cases} \tag{28}$$

Then, we go into the scope of fractional oscillations.
