*3.1. Exponential Pseudo-State Enclosures*

To directly replace the Mittag-Leffler functions by exponential ones in the enclosure technique according to Theorem 1 and Corollary 1, it is necessary to determine the Caputo fractional derivative of order *ν* (initialized at *t* = 0) of a classical exponential function.

According to [26,27], the derivative of *eλ<sup>t</sup>* is given by the closed-form representation

$$\frac{\mathbf{d}^{\nu}\boldsymbol{\sigma}^{\lambda t}}{\mathbf{d}t^{\nu}} = \lambda \cdot \left(t^{1-\nu} \cdot E\_{1,2-\nu}(\lambda t)\right) \tag{20}$$

with *<sup>t</sup>* <sup>≥</sup> 0 and *<sup>λ</sup>* <sup>∈</sup> <sup>R</sup>. In (20), the right-hand side again depends on the two-parameter Mittag-Leffler function (5). Note that this two-parameter Mittag-Leffler function, if substituted into the first line of Equation (11), leads to the same difficulty already observed in Equation (19) of Corollary 1 that the arising quotients of functions cannot be simplified, even in the case of (quasi-)linear system models in which *aij* [**x***e*] *κ* [*t*] = 0 holds for at least one *i* = *j*.

Moreover, it has to be pointed out that an interval evaluation of this fractional derivative of the exponential function on the time interval *t* ∈ [0 ; *T*] always contains the value 0 at the left end point of this time interval, so that the division of the differential formulation of the Picard iteration according to (11) by the term in parentheses in (20) is undefined. Therefore, solution representations in the form *eλ<sup>t</sup>* are not useful to generalize the iteration scheme according to Theorem 1.

At least theoretically, one could try to resolve this second problem by changing the solution template from *eλ<sup>t</sup>* to *eλ<sup>t</sup> ν* with a non-integer power of the time variable *t*. Unfortunately, however, its Caputo derivative of order *ν* does not have a closed-form, exponential-type solution in the general case so the problem persists that the solution presented in (19) can still not be simplified further. For this reason, we are switching to the idea of the following subsection in which the interval box enclosures of Mittag-Leffler functions employed in (9) as well as in (19) are replaced by tubes parameterized in terms of exponential functions.
