Verification and Reachability Analysis of Fractional-Order Differential Equations Using Interval Analysis

TL;DR

This paper introduces an iterative interval arithmetic method for reachability analysis of fractional-order differential equations, addressing the limitations of classical methods for systems with memory effects, with applications in electrochemical modeling.

Contribution

It presents a novel interval arithmetic extension of Mittag-Leffler functions for analyzing fractional-order systems, expanding the tools available for such complex models.

Findings

Effective iterative method demonstrated on battery system model

Addresses long-term memory effects in system dynamics

Potential applications in electrochemical system analysis

Abstract

Interval approaches for the reachability analysis of initial value problems for sets of classical ordinary differential equations have been investigated and implemented by many researchers during the last decades. However, there exist numerous applications in computational science and engineering, where continuous-time system dynamics cannot be described adequately by integer-order differential equations. Especially in cases in which long-term memory effects are observed, fractional-order system representations are promising to describe the dynamics, on the one hand, with sufficient accuracy and, on the other hand, to limit the number of required state variables and parameters to a reasonable amount. Real-life applications for such fractional-order models can, among others, be found in the field of electrochemistry, where methods for impedance spectroscopy are typically used to identify fractional-order models for the charging/discharging behavior of batteries or for the dynamic relation between voltage and current in fuel cell systems if operated in a non-stationary state. This paper aims at presenting an iterative method for reachability analysis of fractional-order systems that is based on an interval arithmetic extension of Mittag-Leffler functions. An illustrating example, inspired by a low-order model of battery systems concludes this contribution.