**1. Introduction**

Fractional differential equations have the property of an infinite-horizon memory of the previous evolution of the system dynamics [1–4]. As opposed to integer-order models, where only initial conditions for the state variables at a certain point of time and the external system inputs after that time instant are necessary for determining a unique solution for the evaluation of the system states, fractional-order models need to be initialized with the complete past behavior as the initialization function [5].

Due to this reason, the system state at a specific point in time is typically referred to as a pseudo state for fractional system models as the complete history of its evolution is required for a unique solution [5]. To solve this difficulty when initializing a simulation at a specific point in time, additive (interval-valued) correction terms of the right-hand sides of explicit fractional-order models have been employed in [6] to account for the past pseudo-state evolution. Simulation methods, which make use of further extensions on the basis of a (pseudo) state observer concept for tightening these additive corrections after the reset of fractional integrators, were developed in [7]. These extensions exploit a formula derived by Podlubny in [1] for shifting the reference point in time, associated with the definition of a fractional differentiation operator.

On the one hand, the infinite-horizon memory property of fractional differential equations allows to efficiently model dynamic systems with long-term memory effects. On the other hand, however, interval-based simulations, allowing for enclosing the domains of reachable (pseudo) states in a guaranteed way, are significantly complicated because classical Taylor series-based simulation approaches, such as those employed in tools such as AWA [8], VNODE-LP [9–11], or VSPODE [12], for systems of integer-order ordinary differential equations, can no longer be employed without modifications.

**Citation:** Rauh, A. Exponential Enclosures for the Verified Simulation of Fractional-Order Differential Equations. *Fractal Fract.* **2022**, *6*, 567. https://doi.org/10.3390/fractalfract 6100567

Academic Editor: Gani Stamov

Received: 5 September 2022 Accepted: 30 September 2022 Published: 5 October 2022

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Therefore, the author has developed an iterative pseudo-state enclosure approach in [6,13] which is based on the Picard iteration [14]. Under the assumption that the pseudo state at the point of time *t* = 0 corresponds also to the complete previous, temporally constant state evolution for *t* < 0, corresponding to Caputo's definition of fractional derivatives, the iterative solution makes use of Mittag-Leffler functions [15–19] to represent pseudo-state enclosures in a guaranteed manner. This kind of function represents the true pseudo-state trajectories of linear fractional differential equations with the aforementioned Caputo-type initialization [20,21]. For nonlinear models, the iteration procedure developed in [6,13] yields outer bounds for the actually reachable pseudo states.

It has to be noted that this iteration scheme is a natural generalization of a counterpart making use of classical exponential functions in the integer-order case, cf. [6]. However, the drawback is an increased complexity of the numerical evaluation because the typically arising quotients of Mittag-Leffler functions with different arguments cannot be simplified further in an analytic manner to reduce the overestimation that is well-known in the domain of interval methods [22,23]. This issue is further discussed in the current paper and resolved by outer exponential enclosures of Mittag-Leffler functions.

In this paper, Section 2 summarizes the Mittag-Leffler function representation of guaranteed solution enclosures for fractional-order system models as presented in [6,13]. It is extended in Section 3 toward exponential functions for the computation of guaranteed pseudo-state enclosures. The representative simulation results, focusing on the tightness of the resulting pseudo-state enclosures and the required computational effort, are presented in Section 4 for a close-to-life quasi-linear fractional model of the charging/discharging behavior of Lithium-ion batteries before conclusions and an outlook on future work are given in Section 5.
