Models of dielectric relaxation based on completely monotone functions

Roberto Garrappa; Francesco Mainardi; Maione Guido

doi:10.1515/fca-2016-0060

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Models of dielectric relaxation based on completely monotone functions

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Published/Copyright: November 8, 2016

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From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 5

Abstract

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.

MSC 2010: Primary 26A33; Secondary 33E12; 34A08; 26A48; 44A10; 91B74

Key Words and Phrases: fractional calculus; dielectric models; complete monotonicity; Mittag-Leffler functions; differential operators

Appendix A. Mittag-Leffler functions

The Mittag-Leffler (ML) function is a special function playing a key role in the solution and analysis of fractional differential equations. The first version with just one parameter was introduced in 1902 by the Swedish mathematician Magnus Gustaf Mittag–Leffler [89] but Wiman [133], few years later, proposed the generalization to two parameters

Eα,β(z)=∑k=0∞zkΓ(αk+β),z∈C.(A.1)

In most applications, it is preferable to deal with the Laplace transform of the ML function which has a very simple analytical representation

Ltβ−1Eα,β(tαz);s=sα−βsα−z,ℜ(s)>0and|zs−1|<1(A.2)

highlighting the relationship with the fractional calculus since the presence of fractional powers. For more details, we refer the reader to the recent treatise on functions of the ML type by Gorenflo, Kilbas, Mainardi and Rogosin [35].

In 1971, the Indian mathematician Tialk Raj Prabhakar [101] proposed a further generalization to three parameters of the ML function

Eα,βγ(z)=1Γ(γ)∑k=0∞Γ(γ+k)zkk!Γ(αk+β).(A.3)

and studied integral equations having this function as the kernel. Although Prabhakar considered his work only from a pure and theoretical mathematical point of view, nowadays it is of great importance for the time-domain analysis of the Havriliak-Negami model. Also in this case the Laplace transform has a very simple analytical formulation

Ltβ−1Eα,βγ(tαz);s=sαγ−β(sα−z)γ,ℜ(s)>0and|zs−1|<1.(A.4)

Important results on the asymptotic behaviour of the standard (one or two parameter) ML function are largely available in the literature (see, for instance, [35, 47, 79, 80, 96]). Asymptotic expansion of the three parameter ML function are instead less known. An expansion as t → +∞ has been recently presented in [81]

Eα,βγ(−tα)=tβ−αγ−1∑k=0∞−γkt−αkΓ(β−αγ−αk)forβ≠αγt−αγ∑k=1∞−γkt−αkΓ(−αk)forβ=αγ(A.5)

and the following asymptotic behaviour can be hence verified

Eα,βγ(−tα)∼1Γ(β−αγ)t−αγforβ≠αγ−γΓ(−α)t−αγ−αforβ=αγt→+∞(A.6)

As a special case (which is of interest in this paper), when β = 0 we have

Eα,0γ(−tα)=1Γ(γ)∑k=1∞Γ(γ+k)(−1)ktαkk!Γ(αk)∼−γΓ(α)tαt→0.(A.7)

Results on the complete monotonicity of the Prabhakar function have been recently discussed in [81, 118].

We consider now another type of three-parameter ML function which differs form the Prabhakar function and has important applications in some fractional differential equations related to phenomena of non standard relaxation studied in this survey paper. These further generalizations were introduced in 1995 by Kilbas and Saigo for studying the solutions of non-linear integral equations of Abel-Volterra type [63, 65, 64] and are therefore referred to as Kilbas and Saigo functions. The relations between these functions and fractional calculus was presented in [66] and their use for solving, in a closed form, a class of linear differential equations of fractional order was successively discussed in [67, 106]. Gorenflo et al. [36] presented recurrence relations for these functions and showed the connections with functions of hypergeometric type for a particular instance of the parameters. The properties of operators in fractional calculus associate with these generalized ML functions were finally investigated in [107].

In the complex plane ℂ we consider the ML type function introduced in [63] by means of the power series

Eα,m,ℓ(z)=∑n=0∞cnzn,cn=∏i=0n−1Γ[α(im+ℓ)+1]Γ[α(im+ℓ+1)+1],(A.8)

with α, m, l ∈ ℝ such that α > 0, m > 0 and α (im + ℓ) ≠ −1, −2, −3, … (an empty product is assumed always equal to one, so that c₀ = 1). Under the above assumptions for the parameters α, m and ℓ, E_{α, m,ℓ}(z) can be proved to be an entire function of order ρ = 1/α and type σ = m. As a consequence, for ϵ > 0 it is

|Eα,m,ℓ(z)|<exp1m+ϵz1/α,z∈C.(A.9)

Appendix B. Differential operators of non-integer order

In this section, we recall the fractional order operators used throughout the paper. These operators allow to formulate the evolution equations of the various models but also to represent the constitutive law (2.2) in the time domain.

This is obviously not a comprehensive treatment of the subject for which we refer to any of the available textbooks on fractional calculus [25, 68, 79, 88, 95, 97].

We preliminarily observe that in the following the symbol * will denote the convolution integral between two causal (locally integrable) functions f(t) and g(t), i.e.

f(t)∗g(t)≡∫0tf(t−u)g(u)du,

which for classical functions is commutative. Moreover, applying the Laplace transform leads to

f(t)∗g(t)=g(t)∗f(t)÷f~(s)⋅g~(s),

where ÷ denotes the juxtaposition between a time function and its image in the complex Laplace domain.

Riemann-Liouville and Caputo fractional derivatives

For a casual function f(t) which is assumed absolutely integrable on ℝ⁺, the Riemann-Liouville integral of order α>0 is defined as

0Jtαf(t)≡tα−1Γ(α)∗f(t)=1Γ(α)∫0t(t−u)α−1f(u)du,t≥0,(B.1)

Under the assumption 0 < α < 1 (which is reasonable for the models discussed in this paper), the left-inverse of the integral (B.1) is the Riemann-Liouville fractional derivative

0Dtαf(t)=Dt0Jt1−αf(t)=1Γ(1−α)ddt∫0t(t−u)−αf(u)du.(B.2)

An equivalent definition, which is known as the Grünwald-Letnikov derivative, allows to write fractional derivatives by means of fractional differences as

0Dtαf(t)=limh→01hα∑k=0∞ωk(α)f(t−kh),(B.3)

where h > 0 and ω^(α)_k—s are the binomial coefficients

ωk(α)=(−1)kαk=α(α−1)⋯(α−k+1)k!.(B.4)

The interchange of differentiation and integration in (B.2) leads to the so-called Caputo fractional derivative

0cDtαf(t)=0Jt1−αDtf(t)=1Γ(1−α)∫0t(t−t′)−αf′(u)du.(B.5)

To derive the relationship between RL and Caputo fractional derivatives, it is sufficient to preliminary observe that the Laplace transforms of (B.2) and (B.5) are respectively

L(0Dtαf(t);s)=sαL(f(t);s)−limt→0+0Jt1−αf(t)(B.6)

and

L(0Dtαf(t);s)=sαL(f(t);s)−sα−1f(0+);(B.7)

hence, after rewriting

L(0cDtαf(t);s)=sαL(f(t);s)−1sf(0+)=sαL(f(t)−f(0+);s),

by inverting back to the temporal domain we obtain the well-known relationship

0cDαtf(t)=0Dtα(f(t)−f(0+))(B.8)

which can be equivalently rewritten as

0cDαtf(t)=0Dtαf(t)−t−αΓ(1−α)f(0+).(B.9)

These operators, in particular the one of Caputo type, turn out to be useful in order to describe, in the time domain, the constitutive law (2.2) expressing the relationship between the electric and polarization field in some of the discussed models. For instance, for the CC model it is elementary to see that the inversion from the Fourier/Laplace domain, leads to

0cDtαPCC(t,x)=−1τ⋆αPCC(t,x)+Δετ⋆αE(t,x),PCC(0,x)=P0(x),(B.10)

where, for brevity, we denoted Δ ε = ε₀\bigl(ε_s−ε_∞\bigr) and P(t, x) and E(t, x) are the polarization and the electric field respectively, depending also on a space variable x. Thanks to the use of the Caputo’s derivative, an initial condition of Cauchy type, expressed in terms of a given initial polarization P₀(x) at the initial time t = 0, is coupled to (B.10).

Similarly, for the excess wing model (3.67) the inversion from the frequency to the time domain does not add particular difficulties because the relationship between the polarization and the electric field can be expressed in terms of the multi-term FDE

τ1DtPEW(t,x)+τ2α0cDtαPEW(t,x)+PEW(t,x)=τ2α0cDtαE(t,x)+E(t,x)

in which the standard derivative D_t is combined with the fractional order derivative 0cDtα. It is not always easy to compute analytic solutions of this equation; however, numerical methods for solving multi-term FDEs are nowadays available (e.g., see [24, 26]).

Derivatives of Prabhakar type

Finding suitable differential operators describing, in the time domain, the evolution equations or the constitutive law (2.2) for the DC, HN and JWS models can be less immediate than for CC or EW models.

A heuristic procedure which applies to the HN model (and hence to the special case of the DC model) has been presented by Nigmatullin and Ryabov in [93] and successively discussed in [14, 60]. After introducing the fractional pseudo-differential operator (0Dtα+λ)γ resulting from the inversion of the HN susceptibility (3.25), by using the binomial expansions, algebra of commutator operators and the Leibiniz formula for the RL derivative, it is possible to reformulate

(0Dtα+λ)γ=exp−tλα0Dt1−α0Dtαγexptλα0Dt1−α,(B.11)

where the exponentials must be understood as series of factional differential operators. This compound operator is surely helpful for understanding theoretical aspects of the HN model but its practical application for computational purposes appears rather doubtful.

The characterization proposed in (B.11) is however particularly useful, also from the practical point of view, in the special case α = 1 arising with the DC model because it reduces to

(Dt+λ)γ=e−tλ0Dtγetλ.(B.12)

Hanyga in [43, 44] proposed the use in (B.12) of the Caputo derivative instead of the RL derivative

C(Dt+λ)γ=e−tλ0cDtγetλ(B.13)

and illustrated the derivations necessary to obtain the Laplace transform of this operator

LC(Dt+λ)γ;s=(s+λ)γf~(s)−(s+λ)γ−1f(0+)(B.14)

(the reader should note that throughout the paper we use the symbols “C” and “C” to distinguish between different ways to regularize fractional operators in the Caputo sense; to this purpose we refer to Remark B.2).

The same approach described in [43, 44] can be applied, in a straightforward way, to derive also the Laplace transform of (B.12)

L(Dt+λ)γf(t);s=(s+λ)γf~(s)−limt→0+0Jt1−γ[etλf(t)].(B.15)

In light of (B.8) it is possible to verify that the following relationship between (B.12) and (B.13) holds

C(Dt+λ)γf(t)=(Dt+λ)γ(f(t)−e−tλf(0+)).(B.16)

In the time domain, the relationship (2.2) between the electric field and polarization in dielectric of DC type can be therefore expressed as

0Dtγet/τ⋆PDC(t,x)=Δετ⋆γet/τ⋆E(t,x)(B.17)

and, as expected, the standard ODE describing the relaxation of Debye type is returned when γ = 1.

In the more general case connected to the HN model, i.e. α≠ = 1, the operator (B.11) seems of little use for computation and presents the same difficulties of the alternative approach proposed in [94, 124] and consisting in expanding 0Dtα+τ⋆−αγ by means of an infinite binomial series of fractional RL derivatives

0Dtα+λγ=∑k=0∞γkλk0Dtα(γ−k).(B.18)

Although the truncation of (B.18) has been used for numerical computation (see [5]), it presents a major drawback since it is not clear when the above series must be truncated in order to obtain a prescribed accuracy.

An alternative way to introduce operators for the HN model can be devised on the basis of the work presented in [31] (successively studied also in [99]) and concerning the so-called Prabhakar integrals and derivatives. These operators are introduced in a similar way as the RL and Caputo operators, after replacing the standard kernel t^α−1/Γ(α) by the following generalization of the Prabhakar function

eα,βγ(t;λ)=tβ−1Eα,βγ(tαλ).

In particular, for a function f ∈ L¹([0, T]) the Prabhakar integral of orders α, γ>0 and parameter λ>0 can be defined for any t ∈ [0, T] as

(0Jtα+λ)γf(t)≡eα,αγγ(t;−λ)∗f(t)=∫0teα,αγγ(t−u;−λ)f(u)du,(B.19)

and, since (A.4), the corresponding Laplace transform is clearly given by

L(0Jtα+λ)γf(t);s≡1(sα+λ)γf~(s).

Under the assumption 0<αγ<1, the left-inverse of (B.19) is the special derivative

(0Dtα+λ)γf(t)≡ddteα,1−αγ−γ(t;−λ)∗f(t)=ddt∫0teα,1−αγ−γ(t−u;−λ)f(u)du,(B.20)

and it is a simple exercise to verify that the corresponding Laplace transform is

L(0Dtα+λ)γf(t);s=(sα+λ)γf~(s)−limt→0+Eα,1−αγ,−λ,0+−γf(t),(B.21)

where the operator Eρ,μ,ω,0+γ, introduced in [31], is the convolution integral whose kernel is tμ−1Eρ,μγ(ωtρ), i.e.

Eρ,μ,ω,0+γf(t)=∫0t(t−u)μ−1Eρ,μγ(ω(t−u)ρ)f(u)du.(B.22)

We must note that the definition of the derivative (0Dtα+λ)γ, on the basis of the integral (B.19) can appear a bit difficult to handle since the presence in the kernel of the Prabhakar function, whose evaluation is usually quite difficult (some methods have been recently proposed in [32, 111]). This kind of definition is however interesting because it allows to introduce a regularization of the same type of the regularization (B.8) introduced for the Caputo derivative (see also [71]). As proposed in [31], it is indeed possible to exchange integrals and derivatives in (B.20), thus to introduce, for an absolutely continuous function f, the operator

C(0Dtα+λ)γf(t)≡eα,1−αγ−γ(t;−λ)∗ddtf(t)=∫0teα,1−αγ−γ(t−u;−λ)f′(u)du,(B.23)

where the letter “C” indicates that (B.23) can be considered as the counterpart of the Caputo approach for the derivative (B.20). In this case, we observe that in the Laplace transform domain it is

LC(0Dtα+λ)γf(t);s=(sα+λ)γf~(s)−s−1(sα+λ)γf(0+)(B.24)

and hence by moving back to the temporal domain it is

C(0Dtα+λ)γf(t)=(0Dtα+λ)γf(t)−eα,1−αγ−γ(t;−λ)f(0+)(B.25)

or, equivalently,

C(0Dtα+λ)γf(t)=(0Dtα+λ)γf(t)−f(0+),(B.26)

thus establishing relationships between (0Dtα+λ)γ and its regularized version in the Caputo sense C(0Dtα+λ)γ, which are the analogous of the relationships (B.8) and (B.9) holding between RL and Caputo derivatives.

Remark B.1

The operator C(0Dtα+λ)γ cannot be intended as the γ power of the Caputo derivative shifted by λ, namely C(0Dtα+λ)γ≠(0cDtα+λ)γ. This difference is better perceived in the limit case γ = 1 if we observe, by applying first (B.24) and hence (B.7), that

C(0Dtα+λ)f(t)=0cDtαf(t)+λ(f(t)−f(0+))(B.27)

and hence C(0Dtα+λ)f(t)≠(0cDtα+λ)f(t). Actually, in light of (B.26) it is possible to conclude that the regularizing effect of C(0Dtα+λ) does not act just on the fractional derivative but also on the identity operator which returns f(t) − f (0⁺) instead of f(t). To treat the evolution equation (3.10) for the relaxation function of the CC model as the particular case for γ = 1 of the evolution equation (3.40) of the HN model, it would be necessary to introduce a Caputo regularization affecting only the fractional derivative and not the identity operator as well.

Remark B.2

The approach followed by Hanyga in [43, 44] to regularize (in the Caputo’s sense) the operator (D_t + λ\bigr)^γ, and consisting in replacing 0Dtαby0cDtα in (B.12), could be followed, at least hypothetically, also for (0Dtα+λ)γ in (B.11). Both the derivation process and the series expansion of the exponentials would however require very strong assumptions for the functions to which apply the operator, thus severely restricting its feasibility. It is clear that the Caputo’s regularization of operators for HN models is still an open problem which deserves further investigation. In this paper, we limit ourselves to denote with different symbols the operators obtained by Hanyga (for which the symbol “C” is indeed used) and the one (B.23) obtained by interchanging integration and derivation (for which the symbol “C” is instead used).

We also mention that in [33] it has been derived a representation of (0Dtα+λ)γ in terms of fractional differences of Grünwald-Letnikov type according to

(0Dtα+λ)γf(t)=limh→0(1+hαλ)γhαγ∑k=0∞Ωk(α,γ)f(t−kh),(B.28)

where the coefficients Ωk(γ) are given by

Ω0(α,γ)=1,Ωk(α,γ)=11+hαλ∑j=1kωj(α)(1+γ)jk−1Ωk−j(α,γ),(B.29)

with ωj(α) the binomial coefficients (B.4). As observed in [33], the coefficients Ωk(α,γ) are a generalization of the binomial coefficients ωk(α) and, indeed, it is Ωk(α,1)=ωk(α). Thus, for γ = 1 and λ = 0 the operator (B.28) gives back the differences (B.3) and hence it can be considered as a generalization of the Grünwald-Letnikov derivative.

By using the regularized derivative C(0Dtα+λ)γ it is now possible to express the constitutive law (2.2) for HN models as

C(0Dtα+τ⋆−α)γPHN(t,x)=Δετ⋆αγE(t,x)(B.30)

which can be completed by an initial condition PHN(t,x)=P0(x). Note that the use of (B.28), together with (B.26), provides a tool for the discretization of this equation. A similar approach can be followed for the JWS model and indeed it is easy to evaluate

C(0Dtα+τ⋆−α)γPJWS(t,x)=ΔεC(0Dtα+τ⋆−α)γE(t,x)−Δε0cDtαE(t,x).

Biographical notes

We conclude this survey by presenting brief biographical notes on some of the authors who distinguished in dielectric studies and introduced the models today named after them. Their names are surely familiar among physicists, chemists, engineers and applied mathematics but in some cases very few is known about them.

Donald West Davidson was born in 1925 and died on 2 August 1986 in Ottawa (Canada). He got BSc and MSc degrees at the University of New Brunswick (Canada). During the PhD at the Brown University of Providence in Rhode Island (USA) he conducted studies [19] on dielectric relaxation under the supervision of R.H. Cole. He joined in 1953 the Division of Applied Chemistry at the National Research Council in Ottawa (Canada) where he continued his dielectric studies on molecular motion in liquids [104].

Petrus (Peter) Josephus Wilhelmus Debye was born on March 24, 1884 at Maastricht (the Netherlands) and died on November 2, 1966 at Ithaca (USA). He got a degree in electrical engineering in 1905 at the Technische Hochschule in Aachen (Germany) and completed his doctoral program in Munich (Germany) in July 1908. He was appointed as Professor of Theoretical Physics at the University of Zurich in 1911 and at the University of Utrecht in 1912. Successively he worked at the the Physics Institute of Göttingen, at the Physics Laboratory of the Eidgenössische Technische Hochschul in Zurich, at the University of Leipzig, at the Max Planck Institute in Berlin-Dahlem and at the University of Berlin. In 1936 he was awarded of the Nobel Prize in Chemistry “for his contributions to our knowledge of molecular structure through his investigations on dipole moments and on the diffraction of X-rays and electrons in gases”. He moved to USA in 1940 to became Professor of Chemistry and, later, also chairman of the Department of Chemistry, at the Cornell University at Ithaca (New York, USA) by becoming emeritus in 1950 [132].

Kenneth Stewart Cole was born on July 10, 1900 at Ithaca in New York (USA) and died on April 18, 1984. He studied Physics at Oberlin College in Ohio (USA) and obtained the PhD at the Cornell University (New York, USA) under the supervision of F.K. Richtmyer in 1926. He obtained in 1926 a postdoctoral fellowship by the National Research Council to study the membrane capacity of sea-urchin eggs at Harvard. He joined in 1929 the Department of Physiology of the Columbia University of New York and in 1946 was appointed as Professor of Biophysics and Physiology and head of the Institute of Radiobiology and Biophysics at the University of Chicago. Successively directed laboratories of the Naval Medical Research and of the National Institutes of Health. Among other academic honours, Dr. Cole received in 1967 the U.S. Medal of Science and was honoured by foreign membership in the Royal Society of London (UK) in 1972 [54, 122].

Robert Hugh Cole was born on October 26, 1914 in Oberlin, Ohio (USA) and died in Providence, Rhode Island (USA), on November 17, 1990. As his brother Kenneth S. Cole (with which he conducted an intensive collaboration over the years), he graduated (in 1935) at the Oberlin College in Ohio (USA) and, after the PhD earned in 1940 at the Harvard University, he became an Instructor in Physics at the same University. In 1946 R.H. Cole became Associate Professor of Physics at the University of Missouri but one year later he accepted an Associate Professorship at the Brown University where in 1949 assumed the Chairmanship of the Chemistry Department. He received several prestigious awards (among them a Guggenheim Fellowship in 1956 and the Irving Langmuir Prize in 1975) and was appointed as John Howard Appleton Lecturer by the Brown Chemistry Department in 1975 [90, 122].

Stephen J. Havriliak was born on June 30, 1931 and passed away on February 11, 2010. He studied at the American International College of Springfield in Massachusetts (USA) and got his PhD in Chemistry from the Brown University of Providence in Rhode Island (USA). He worked at the Rohm and Haas Research Laboratories of Philadelphia in Pennsylvania (USA).

Andrzej Karol Jonscher was born in Warsaw (Poland) on 1921 and died in London (UK) in 2005. He graduated in Electrical Engineering at Queen Mary College (University of London) in 1949 and there obtained his PhD in 1952. In 1951 he joined the GEC Research Laboratories in Wembley, later named Hirst Research Center, where he worked on physical principles of semiconductor devices. In 1960 appeared his monograph “Principles of Semiconductors Device Operation”. He joined Chelsea College, University of London, in 1962 as reader and in 1965 Professor of Solid State Electronics in 1965, where interest in amorphous semiconductors gradually led him to studies of the dielectric properties of solids, with special emphasis on the “universality” of relaxation processes. In 1983 appeared his monograph “Dielectric Relaxation in Solids” and in 1996 the companion monograph “Universal Relaxation Law”. Under Jonscher’s guidance, the Chelsea Dielectric Group was started in the 1970's at Chelsea College: it grew up and became one of the leading research groups specialising in low frequency response down to milli-Hertz and in the corresponding time-domain behaviour. In 1987 Jonscher became Emeritus Professor at Royal Holloway, University of London, where he has been continuing his work up to his death.

Friedrich Wilhelm Georg Kohlrausch was born in Rinteln (Germany) on October 14, 1840 and died in Marburg (Germany) on January 17, 1910. He studied at the university of Göttingen where he become professor of physics from 1866 to 1870. Successively he worked at the universities of Darmstadt, Würzburg, Strassburg and Berlin [29].

Shinchi Negami get his B.S. at the Yokohama National University (Japan) in 1957 and moved to USA in 1960 where he received the MS degree from the Lehigh University of Bethlehem in Pennsylvania (USA) after discussing a thesis on “Dynamic Mechanical Properties of Synovial Fluid”. He worked at the Kent State University in Ohio (USA) and as a research chemist for Rohm and Haas Research Laboratories of Philadelphia in Pennsylvania (USA).

Acknowledgements

The work of RG has been partially supported by the INdAM-GNCS and partially by the COST Action CA15225. The work of FM has been carried out in the framework of the activities of the National Group of Mathematical Physics (INdAM-GNFM) and of the Interdepartmental Center “L. Galvani” for integrated studies of Bioinformatics, Biophysics and Biocomplexity of the University of Bologna. The work of GM is supported by the COST Action CA15225.

The authors are deeply indebted to Prof. Andrzej Hanyga, Prof. John Ripmeester and Prof. Karina Weron for providing useful material for the realization of this work and to Prof. Virginia Kiryakova for her useful suggestions and editorial assistance.

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Received: 2015-12-24

Revised: 2016-09-03

Published Online: 2016-11-08

Published in Print: 2016-10-01

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Articles in the same Issue

Frontmatter
Editorial
FCAA related news, events and books (FCAA–volume 19–5–2016)
Round Table Discussion
Fractional Calculus: D’où venons-nous? Que sommes-nous? Où allons-nous?
Survey Paper
Models of dielectric relaxation based on completely monotone functions
Survey Paper
Space-time fractional stochastic equations on regular bounded open domains
Discussion Paper
Geometric interpretation of fractional-order derivative
Survey Paper
Fractional calculus in image processing: a review
Research Paper
Structural derivative based on inverse Mittag-Leffler function for modeling ultraslow diffusion
Research Paper
On the regional controllability of the sub-diffusion process with Caputo fractional derivative
Discussion Survey
There’s plenty of fractional at the bottom, I: Brownian motors and swimming microrobots
Research Paper
On a fractional differential inclusion with “maxima”
Research Paper
On time-fractional representation of an open system response
Archive Paper
On the summation of Taylor’s series on the contour of the domain of summability

https://doi.org/10.1515/fca-2016-0060

Keywords for this article

fractional calculus; dielectric models; complete monotonicity; Mittag-Leffler functions; differential operators

Articles in the same Issue

Frontmatter
Editorial
FCAA related news, events and books (FCAA–volume 19–5–2016)
Round Table Discussion
Fractional Calculus: D’où venons-nous? Que sommes-nous? Où allons-nous?
Survey Paper
Models of dielectric relaxation based on completely monotone functions
Survey Paper
Space-time fractional stochastic equations on regular bounded open domains
Discussion Paper
Geometric interpretation of fractional-order derivative
Survey Paper
Fractional calculus in image processing: a review
Research Paper
Structural derivative based on inverse Mittag-Leffler function for modeling ultraslow diffusion
Research Paper
On the regional controllability of the sub-diffusion process with Caputo fractional derivative
Discussion Survey
There’s plenty of fractional at the bottom, I: Brownian motors and swimming microrobots
Research Paper
On a fractional differential inclusion with “maxima”
Research Paper
On time-fractional representation of an open system response
Archive Paper
On the summation of Taylor’s series on the contour of the domain of summability