Stability of fractional-order systems with Prabhakar derivatives
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ORIGINAL PAPER
Stability of fractional-order systems with Prabhakar derivatives Roberto Garrappa · Eva Kaslik
Received: 12 June 2020 / Accepted: 12 August 2020 © Springer Nature B.V. 2020
Abstract Fractional derivatives of Prabhakar type are capturing an increasing interest since their ability to describe anomalous relaxation phenomena (in dielectrics and other fields) showing a simultaneous nonlocal and nonlinear behaviour. In this paper we study the asymptotic stability of systems of differential equations with the Prabhakar derivative, providing an exact characterization of the corresponding stability region. Asymptotic expansions (for small and large arguments) of the solution of linear differential equations of Prabhakar type and a numerical method for nonlinear systems are derived. Numerical experiments are hence presented to validate theoretical findings. Keywords Fractional calculus · Fractional Prabhakar derivative · Asymptotic stability · Stability region This research was funded by the COST Action CA 15225 “Fractional-order systems- analysis, synthesis and their importance for future design”. The work of R. Garrappa was also partially supported by a GNCS-INdAM 2020 Project. R. Garrappa Department of Mathematics, University of Bari, Via E. Orabona 4, 70126 Bari, Italy e-mail: [email protected] R. Garrappa INdAM Research Group GNCS, Rome, Italy E. Kaslik (B) Department of Mathematics and Computer Science, West University of Timi¸soara, Bd. V. Pârvan 4, 300223 Timi¸soara, Romania e-mail: [email protected]
Mathematics Subject Classification 34A08 · 34D20 · 26A33 · 65L07 · 45M10
1 Introduction The Prabhakar function is named after the Indian mathematician Tilak Raj Prabhakar who introduced in 1971 a generalization to three parameters of the Mittag– Leffler function [18] and studied a convolution integral operator with this function as kernel [36]. After their introduction, Prabhakar’s function and integral have been overlooked for a long time until, in the first years of the twenty-first century, the connections with the Havriliak–Negami (HN) dielectric model [21] have been put in light. The HN model was introduced to incorporate the asymmetry and broadness observed in the dielectric dispersion of some polymers and today it is recognized as manifestation of the simultaneous nonlocality and nonlinearity [33,38] in the response of complex and heterogeneous systems. For these reasons operators based on the Prabhakar function are employed to describe in the time-domain sophisticated relaxation models in several areas (see e.g. [2,4,5,12–16,19,22,30,37,41]). In 2002 the Prabhakar integral was studied in the context of weakly-singular Volterra integral equations and an interpretation in the framework of fractional calculus was provided [23], thus leading 2 years later to the proposition of a left-inverse operator of the Prabhakar fractional integral [24]. A regularization of this
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inverse, known as the fractional Prabhakar derivative, was introduced in [8], and 1 year la
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