**2. A Brief Introduction to Fractional Calculus**

The fractional differo-integral operators are symbolized by*aD<sup>α</sup> <sup>t</sup> f*(*t*), where *a* and *t*, are the bounds of the operation and *<sup>α</sup>* <sup>∈</sup> <sup>R</sup> is a generalization of the standard integration and differentiation to an arbitrary order, which can be rational, irrational or even complex. The basic continuous differo-integral operator is given by the following:

$$\_aD\_t^\kappa := \begin{cases} \frac{d^\kappa}{dt^\kappa} \,' & \text{for} \quad \kappa > 0, \\\ 1 & \text{for} \quad \kappa = 0, \\\ \int\_a^t (d\tau)^\kappa \,' & \text{for} \quad \kappa < 0. \end{cases} \tag{1}$$

In the literature, different definitions can be found concerning fractional order systems. The best most commonly used definitions of fractional order derivatives are:

The Riemann–Liouville (RL) definition [53]:

$$\_aD\_t^a f(t) = \frac{1}{\Gamma(m-a)} \left(\frac{d}{dt}\right)^m \int\_a^t \frac{f(\tau)}{(t-\tau)^{1-(m-a)}} d\tau. \tag{2}$$

The Grunwald–Letnikov (GL) definition [53]:

$$\,\_aD\_t^a f(t) = \lim\_{h \to 0} \frac{1}{h^a} \sum\_{k=0}^{(t-a)/h} (-1)^k \binom{a}{k} f(t - kh). \tag{3}$$

The Caputo definition of the fractional differ-integral operator for the function *f*(*t*) is adopted in this paper as the Caputo definition allows the initial values of classical integer-order derivatives with clear physical interpretation to be used as follows [51,53]:

$$\_aD\_t^a f(t) = \frac{1}{\Gamma(m-a)} \int\_a^t \frac{f^m(\tau)}{(t-\tau)^{1-(m-a)}} d\tau. \tag{4}$$

In these expressions *m* − 1 < *α* < *m*, and Γ(.) is the well-known Eulers gamma function:

$$
\Gamma(\mathbf{x}) = \int\_0^\infty e^{-t} t^{(\mathbf{x}-1)} dt, \qquad \mathbf{x} > 0. \tag{5}
$$
