A Gr ¨ unwald–Letnikov scheme for fractional operators of Havriliak–Negami type ROBERTO GARRAPPA University of Bari Department of Mathematics Via E.Orabona 4, 70125 Bari ITALY roberto.garrappa@uniba.it Abstract: In this paper we propose the generalization of the Gr¨ unwald–Letnikov scheme to fractional differential operators of Havriliak–Negami type; these operators have important applications in the description and simulation of polarization processes in anomalous dielectrics with hereditary properties. We discuss in details the technique used for generalizing the proposed scheme, we provide a recursive relationship for the evaluation of the corre- sponding weights and we study some of their main properties. Key–Words: Fractional calculus, Havriliak–Negami, Gr¨ unwald–Letnikov, convolution quadrature, dielectrics 1 Intorduction During the last decades, integral and differential oper- ators of fractional (i.e., non–integer) order have been studied with an increasing interest motivated by the suitability in modeling systems exhibiting anomalous and/or memory preserving properties. The theoretical analysis and the numerical ap- proximation of factional derivatives and fractional dif- ferential equations is therefore an active area of re- search with important applications in a wide range of fields such as biology, chemistry, engineering, fi- nance, physics and so on. More recently, the observation of experimental data has showed that in some systems the return to the equilibrium after the action of an external excitation obeys to some laws of fractional type (due to memory effects) but can not be described in a satisfactory way by means of standard operators of fractional order; it is therefore necessary to introduce and study more so- phisticated operators. This is the case of the relaxation of Havriliak–Negami type [9] used to describe polar- ization processes in media with anomalous dielectric properties. Havriliak–Negami models are usually derived in the Fourier or Laplace transform domain but for their simulation in the time–domain some completely new operators (based on fractional differentiation) are in- volved. These operators are much more complicated than classical fractional derivatives and their use for simulation purposes is still a challenge since very few approaches have been so far studied for their approxi- mation. For this reason, an important task, which moti- vates the present work, is the development of ad–hoc numerical techniques in order to simulate, in the time– domain, Havriliak–Negami models. In this paper we consider a classical and widely used scheme for approximating derivatives and in- tegrals of fractional order, namely the Gr¨ unwald– Letnikov scheme, and we discuss an approach for the generalization to operators of Havriliak–Negami type. A recursive relation for the evaluation of the weights in the resulting scheme is presented and the main properties of the weights are also investigated. This work is organized as follows. In Section 2 we review some basic facts concerning derivatives and integrals of fractional order and in Section 3 we present the Gr ¨ unwald–Letnikov operators and the cor- responding approximated schemes. Section 4 is de- voted to the description of Havriliak–Negami opera- tors. We hence present, in Section 5, an alternative approach for deriving the Gr¨ unwald–Letnikov scheme for fractional integral and derivatives thanks to which we are able to generalize such scheme to operators of Havriliak–Negami type and study some of its main properties. Finally, we present some concluding re- marks in Section 6. 2 Derivatives and integrals of frac- tional order Several operators have been proposed throughout the years to define derivatives of non–integer order. Al- Recent Advances in Applied Mathematics, Modelling and Simulation ISBN: 978-960-474-398-8 70