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Exponential Enclosures for the Verified Simulation of Fractional-Order Differential Equations

Fractional-order differential equations are powerful tools for the representation of dynamic systems that exhibit long-term memory effects. The verified simulation of such system models with the help of interval tools allows for the computation of guaranteed enclosures of the domains of reachable pseudo states over a certain finite time horizon. In the previous work of the author, an iteration scheme—derived on the basis of the Picard iteration—was published that makes use of Mittag-Leffler functions to determine guaranteed pseudo-state enclosures. In this paper, the corresponding iteration is generalized toward the use of exponential functions during the evaluation of the iteration scheme. Such exponential functions are well-known from a verified solution of integer-order sets of differential equations. The aim of this work is to demonstrate that the use of exponential functions instead of pure box-type interval enclosures for Mittag-Leffler functions does not only improve the tightness of the computed pseudo-state enclosures but also reduces the required computational effort. These statements are demonstrated with the help of a close-to-life simulation model for the charging/discharging dynamics of Lithium-ion batteries.