*5.1. Problem Formulation*

Consider the simplest FDE initial value problem

$$\begin{cases} D^n(\mathbf{x}(t)) = u(t) & 0 < n < 1 \\ \mathbf{x}(t) = \mathbf{x}(0) & \text{at } t = 0 \end{cases} \tag{33}$$

Consider the special function *u*(*t*), composed of two delayed Heaviside functions, *UH*(*t*) and −*UH*(*t* − *T*), with *u*(*t*) = *UH*(*t*) − *UH*(*t* − *T*).

Consequently, see Figure 2

$$u(t) = \begin{cases} \mathcal{U} & \text{for } 0 \le t < T \\ 0 & \text{for } t \ge T \end{cases} \tag{34}$$

Moreover, assume that the system is at rest at *t* = 0, i.e., *x*(0) = 0.

**Remark 3**: The interest of this example is to create a realistic initial condition at *t* = *T*, where the free response can be calculated with two approaches, the first one from *t* = 0 with no ambiguity using usual fractional calculus theory and the second one from *t* = *T*, using Equation (15) at *t*<sup>0</sup> = *T*.

**Figure 2.** True free response and Caputo derivative initialization for n = 0.5.
