Fractional models of anomalous relaxation based on the Kilbas and Saigo function

Oliveira, E. Capelas de; Mainardi, Francesco; Vaz, Jayme

doi:10.1007/s11012-014-9930-0

Cited by 49 publications

(55 citation statements)

References 26 publications

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“…To some extent, the CC model (together with the DC, HN, JWS and EW models) and the KWW model represent two large and distinct families of relaxation models for dielectrics [29]. Nevertheless, an attempt of creating a mathematical bridge between these two distinct families has been proposed by Capelas, Mainardi and Vax in [10], where a more general model combining CC and KWW models was introduced; we will outline also this model which is indicated as the • Capelas-Mainardi-Vaz (CMV) model.…”

Section: Main Models For Anomalous Dielectric Relaxationmentioning

confidence: 99%

Models of Dielectric Relaxation Based on Completely Monotone Functions

Garrappa

Mainardi

Maione

2016

FCAA

Self Cite

143

177

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The relaxation properties of dielectric materials are described, in the frequency domain, according to one of the several models proposed over the years: Kohlrausch-Williams-Watts, Cole-Cole, Cole-Davidson, Havriliak-Negami (with its modified version) and Excess wing model are among the most famous. Their description in the time domain involves some mathematical functions whose knowledge is of fundamental importance for a full understanding of the models. In this work, we survey the main dielectric models and we illustrate the corresponding time-domain functions. In particular, we stress the attention on the completely monotone character of the relaxation and response functions. We also provide a characterization of the models in terms of differential operators of fractional order

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Section: Main Models For Anomalous Dielectric Relaxationmentioning

confidence: 99%

Models of Dielectric Relaxation Based on Completely Monotone Functions

Garrappa

Mainardi

Maione

2016

FCAA

Self Cite

143

177

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“…Thus, function φ(t), now called temporal correlation function, has the following form in the approximation context [38]:…”

Section: Kinetic Equationsmentioning

confidence: 99%

Relaxation Equations: Fractional Models

E¹,

Rosa

2015

J Phys Math

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The relaxation functions introduced empirically by Debye, Cole-Cole, Cole-Davidson and Havriliak-Negami are, each of them, solutions to their respective kinetic equations. In this work, we propose a generalization of such equations by introducing a fractional differential operator written in terms of the Riemann-Liouville fractional derivative of order γ, 0<γ≤1. In order to solve the generalized equations, the Laplace transform methodology is introduced and the corresponding solutions are then presented, in terms of Mittag-Leffler functions. In the case in which the derivative's order is γ = 1, the traditional relaxation functions are recovered. Finally, we presented some 2D graphs of these function.

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“…[16][17][18][19][20]). The frequency dependent behavior of the dielec- * katarzyna.gorska@ifj.edu.pl † andrzej.horzela@ifj.edu.pl ‡ lukasz.bratek@ifj.edu.pl § penson@lptl.jussieu.fr ¶ giuseppe.dattoli@enea.it tric polarizability was also described by the excess wing model [1,2] or by the Kilbas-Saigo function [21]. The HN function, introduced in [3] to describe the frequency dependence of the absolute dielectric permittivitŷ ε(ω), is conventionally written down aŝ (1) where α and β, called respectively the width and the symmetry parameters, are experimentally fitted numbers having nothing in common with the previously mentioned α− and β-relaxations.…”

Section: Introductionmentioning

confidence: 99%

The Havriliak–Negami relaxation and its relatives: the response, relaxation and probability density functions

Górska

Horzela

Bratek

et al. 2018

J. Phys. A: Math. Theor.

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We study functions related to the experimentally observed Havriliak-Negami dielectric relaxation pattern in the frequency domain ∼ [1 + (i ωτ0) α ] −β with τ0 being some characteristic time. For α = l/k < 1 (l and k positive integers) and β > 0 we furnish exact and explicit expressions for response and relaxation functions in the time domain and suitable probability densities in their "dual" domain. All these functions are expressed as finite sums of generalized hypergeometric functions, convenient to handle analytically and numerically. Introducing a reparameterization β = (2 − q)/(q − 1) and τ0 = (q − 1) 1/α (1 < q < 2) we show that for 0 < α < 1 the response functions f α,β (t/τ0) go to the one-sided Lévy stable distributions when q tends to one. Moreover, applying the self-similarity property of the probability densities g α,β (u), we introduce two-variable densities and show that they satisfy the integral form of the evolution equation.

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Fractional models of anomalous relaxation based on the Kilbas and Saigo function

Cited by 49 publications

References 26 publications

Models of Dielectric Relaxation Based on Completely Monotone Functions

Models of Dielectric Relaxation Based on Completely Monotone Functions

Relaxation Equations: Fractional Models

The Havriliak–Negami relaxation and its relatives: the response, relaxation and probability density functions

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