*3.2. Exponential Enclosures of the Mittag-Leffler Function*

For the computation of exponential tube enclosures of time-dependent Mittag-Leffler functions, the following monotonicity theorem is employed. It is a simplified version of Theorem 4 published in [13], where also the fractional derivative order *ν* was considered as a (temporally constant) interval parameter.

**Theorem 2** (Box-type interval bounds for the Mittag-Leffler function)**.** *The range of function values for the Mittag-Leffler function in (4) with the uncertain real-valued parameter λ* ∈ *λ* ; *λ ,* *<sup>λ</sup>* <sup>&</sup>lt; <sup>0</sup> *and the non-negative time argument <sup>t</sup>* <sup>∈</sup> *t* ; *t , t* ≥ 0*, with* 0 < *ν* ≤ 1*, is bounded by the box-type interval enclosure*

$$E\_{\nu,1}\left(\lambda t^{\nu}\right) \in \left[E\_{\nu,1}^{\*}\right]\left([\mathfrak{A}^{\*}]\right) = \left[E\_{\nu,1}^{\*}\left(\inf\left([\mathfrak{A}^{\*}]\right)\right) \; ; \; E\_{\nu,1}^{\*}\left(\sup\left([\mathfrak{A}^{\*}]\right)\right)\right] \tag{21}$$

*with* [**X**] := [*λ*] · [*t*] [*ν*] *, where* sup [**X**] <sup>≤</sup> <sup>0</sup> *holds and E*∗ *ν*,1 *is obtained by the outward rounded interval extension of a floating-point evaluation of the two-parameter Mittag-Leffler function according to Equation (31) of [13]. Due to λ* < 0*, Equation (21) simplifies to*

$$E\_{\nu,1}(\lambda t^{\nu}) \in \left[ E^\*\_{\nu,1}\left(\underline{\lambda} \cdot \overline{t}^{\nu}\right) \; ; \; E^\*\_{\nu,1}\left(\overline{\lambda} \cdot \underline{t}^{\nu}\right) \right] \; . \tag{22}$$

**Proof.** For a detailed proof of this theorem, the reader is referred to the proof of Theorem 4 published in [13]. It is a direct consequence of the fact that the two-parameter Mittag-Leffler function is strictly monotonically decreasing for a growing time argument *t* with *λ* < 0. This property is also reflected by the so-called complete monotonicity of the Mittag-Leffler function that is reported, for example, in [16,17].

Furthermore, monotonicity with respect to the parameter *λ* is verified by differentiating *Eν*,1(*λt <sup>ν</sup>*) with respect to *λ* together with the change of variables *τ* := *λ*<sup>1</sup> *<sup>ν</sup>* · *t* < 0. This leads to

$$\frac{\partial E\_{\nu,1}(\lambda t^{\nu})}{\partial \lambda} = \frac{\partial E\_{\nu,1}(\tau^{\nu})}{\partial \tau} \cdot \frac{\partial \tau}{\partial \lambda} = \frac{\partial E\_{\nu,1}(\tau^{\nu})}{\partial \tau} \cdot \frac{t \cdot \lambda^{\left(\frac{1}{\nu} - 1\right)}}{\nu} \,. \tag{23}$$

In (23), the first factor is non-positive due to the complete monotonicity of the Mittag-Leffer function as shown in [16,17]; for any *t* ≥ 0 and *λ* ≤ 0, the second factor is also non-positive, leading to *<sup>∂</sup>Eν*,1(*λ<sup>t</sup> ν*) *∂λ* ≥ 0, which completes the proof.

**Theorem 3** (Exponential enclosures for the Mittag-Leffler function)**.** *The range of function values for the Mittag-Leffler function in (4) with the uncertain real-valued parameter λ* ∈ *λ* ; *λ , <sup>λ</sup>* <sup>&</sup>lt; <sup>0</sup> *and the non-negative time argument <sup>t</sup>* <sup>∈</sup> *t* ; *t , t* ≥ 0*, t* > *t, with* 0 < *ν* ≤ 1*, is bounded by the exponential enclosure*

$$E\_{\nu,1}\left(\lambda t^{\nu}\right) \in c\left[\mathbb{Z}:\overline{\eta}\right] \cdot \left[t:\overline{t}\right] \tag{24}$$

*where*

$$\underline{\eta} = \begin{cases} \inf \left( \frac{1}{\underline{t}^{\nu}} \cdot \ln \left( \left[ E^{\*}\_{\nu, 1} \right] \left( \underline{\Delta} \cdot \underline{t}^{\nu} \right) \right) \right) & \text{for } \underline{t} > 0 \\\\ \frac{\underline{\Delta}}{\Gamma(\nu + 1)} & \text{for } \underline{t} = 0 \end{cases} \tag{25}$$

*and*

$$\overline{\eta} = \sup \left( \frac{1}{\overline{\mathbf{r}}^{\nu}} \cdot \ln \left( \left[ E^{\*}\_{\nu, 1} \right] \left( \overline{\lambda} \cdot \overline{\mathbf{r}}^{\nu} \right) \right) \right) \tag{26}$$

*with E*∗ *ν*,1 *being the outward rounded interval extension of a floating-point evaluation of the two-parameter Mittag-Leffler function according to Equation (31) of [13].*

**Proof.** Consider the Mittag-Leffler function

$$f(\overline{t}) = E\_{\nu,1}(-\overline{t})\ . \tag{27}$$

Its derivative with respect to ˜*t* satisfies the following properties:

1. <sup>d</sup>*<sup>f</sup>* d˜*t* (0) = <sup>−</sup> <sup>1</sup> Γ(*ν* + 1) ; 2. lim˜*<sup>t</sup>*→<sup>∞</sup> d*f* d˜*t* (˜*t*) = 0; 3. <sup>d</sup>*<sup>f</sup>* d˜*t* (˜*t*) < 0 for ˜*t* > 0; and 4. d2 *<sup>f</sup>* d˜*t*<sup>2</sup> (˜*t*) <sup>&</sup>gt; 0 for ˜*<sup>t</sup>* <sup>&</sup>gt; 0.

Property 1 is a consequence of the series definition (5) of the Mittag-Leffler function, while the properties 2–4 result from its complete monotonicity according to [16,17].

**Case 1**: For a fixed positive point ˜*t* = ˜*t* ∗ > 0, determine the intersection of an exponential function *eη*˜*<sup>t</sup>* and the Mittag-Leffler function *f*(˜*t*) according to

$$E\_{\nu,1}(-\tilde{t}^\*) = \varepsilon^{\eta^\* \cdot I^\*} > 0 \quad \Longleftrightarrow \quad \eta^\* = \frac{1}{\tilde{t}^\*} \cdot \ln\left(E\_{\nu,1}(-\tilde{t}^\*)\right) < 0 \quad , \tag{28}$$

where *<sup>E</sup>ν*,1(0) = *<sup>e</sup>*<sup>0</sup> <sup>=</sup> 1 obviously holds according to (5) and 0 <sup>&</sup>lt; *<sup>E</sup>ν*,1(−˜*<sup>t</sup>* ∗) < 1 due to the property of complete monotonicity.

**Case 2**: In the case ˜*<sup>t</sup>* <sup>→</sup> 0, ˜*<sup>t</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> <sup>0</sup> , the limit value

$$\begin{split} \eta^\* &= \lim\_{I^\* \to 0} \left( \frac{1}{\bar{I}^\*} \cdot \ln \left( E\_{\nu, 1} (-\bar{I}^\*) \right) \right) \\ &= \lim\_{I^\* \to 0} \left( \frac{1}{E\_{\nu, 1} (-\bar{I}^\*)} \cdot \frac{\mathbf{d} \left( E\_{\nu, 1} (-\bar{I}^\*) \right)}{\mathbf{d} \bar{I}^\*} \right) = -\frac{1}{\Gamma(\nu + 1)} \end{split} \tag{29}$$

is obtained, where the second line results from the application of L'Hôpital's rule.

Moreover, the property *η* < *η*, necessary for the interval definition in (24), is obvious due to the fact that *η*∗ in (28) is defined as the quotient of a strictly monotonically decreasing numerator and a strictly monotonically increasing denominator.

Thus, the substitution ˜*<sup>t</sup>* := −*λ<sup>t</sup> <sup>ν</sup>* together with the monotonicity of the Mittag-Leffler function with respect to the parameter *λ*, as already also exploited in Theorem 2, cf. (23), concludes the proof.

The Figure 1a,b give a comparison of the box-type interval enclosures of Mittag-Leffler functions according to Theorem 2 with the exponential enclosures according to Theorem 3. According to the Figure 1c,d, it becomes obvious that for identical subdivisions of the time interval *t* ∈ [0; 1], the box-type enclosure is much more pessimistic at the beginning of the time horizon than at its end. Therefore, to obtain an identical degree of overestimation for both types of enclosures, a significantly larger number of subintervals would be required in the box-type case at the beginning of the considered time span. Moreover, the lower bound for the range of the Mittag-Leffler function is exactly represented at the beginning of each temporal subslice by the exponential enclosure, while the lower bound at the endpoint is represented exactly by the box-type enclosure, cf. Figure 1c. For the upper bound of the range, this property is reversed between both representations for the enclosure of the Mittag-Leffler function, cf. Figure 1d.

**Figure 1.** Comparison between box-type and exponential enclosures of the Mittag-Leffler function *Eν*,1(*λt <sup>ν</sup>*) with *<sup>ν</sup>* <sup>=</sup> 0.5 for *<sup>t</sup>* <sup>∈</sup> [0; 1], *<sup>t</sup>* <sup>=</sup> *<sup>k</sup>* · 0.1, *<sup>t</sup>* = (*<sup>k</sup>* <sup>+</sup> <sup>1</sup>) · 0.1, *<sup>k</sup>* ∈ {0, 1, ... , 9}, and *<sup>λ</sup>* <sup>∈</sup> [−2 ; <sup>−</sup>1]: (**a**) Box-type enclosure of the Mittag-Leffler function; (**b**) Exponential enclosure of the Mittag-Leffler function; (**c**) Overestimation of the lower enclosure bound; (**d**) Overestimation of the upper enclosure bound.

Furthermore, Figure 2 illustrates the property stated in the proof above that *η*∗ is a strictly monotonically increasing function for increasing values of the time argument ˜*t* ∗. To show that this property holds for all 0 < *ν* < 1, several values of this fractional differentiation order are depicted. Moreover, the limit case for ˜*<sup>t</sup>* → 0 according to (29) is also depicted in this graph by using a logarithmic time scale.

**Figure 2.** Evolution of the parameter *η*<sup>∗</sup> as a function of ˜*t* ∗ for different values of the fractional differentiation order *ν*.
