Grünwald–Letnikov operators for fractional relaxation in Havriliak–Negami models

• Operators for the representation in the time-domain of systems with relaxation of Havriliak–Negami type are studied.
• Systems of this type are encountered in physics, mechanical engineering, electric circuits, electromagnetism, etc.
• It is provided an explicit representation in terms of fractional differences of Grunwald–Letnikov type.
• The main properties of these operators are studied.
• The discretization of the operators is also shown. The comparison with an alternative approach is numerically tested.

Several classes of differential and integral operators of non integer order have been proposed in the past to model systems exhibiting anomalous and hereditary properties. A wide range of complex and heterogeneous systems are described in terms of laws of Havriliak–Negami type involving a special fractional relaxation whose behavior in the time-domain can not be represented by any of the existing operators. In this work we introduce new integral and differential operators for the description of Havriliak–Negami models in the time-domain. In particular we propose a formulation of Grünwald–Letnikov type which turns out to be effective not only to provide a theoretical characterization of the operators associated to Havriliak–Negami systems but also for computational purposes. We study some properties of the new operators and, by means of some numerical experiments, we present their use in practical computation and we show the superiority with respect to the few other approaches previously proposed in literature.