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Research ArticleEquivalent Circuits Applied in ElectrochemicalImpedance Spectroscopy and Fractional Derivativeswith and without Singular Kernel

J. F. Gómez-Aguilar,1 J. E. Escalante-Martínez,2 C. Calderón-Ramón,2

L. J. Morales-Mendoza,3 M. Benavidez-Cruz,2 and M. Gonzalez-Lee3

1CONACYT-Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México,Interior Internado Palmira S/N, Colonia Palmira, 62490 Cuernavaca, MOR, Mexico2Facultad de Ingenieŕıa Mecánica y Eléctrica, Universidad Veracruzana, Avenida Venustiano Carranza S/N,Colonia Revolución, 93390 Poza Rica, VER, Mexico3Facultad de Ingenieŕıa Electrónica y Comunicaciones, Universidad Veracruzana, Avenida VenustianoCarranza S/N, Colonia Revolución, 93390 Poza Rica, VER, Mexico

Correspondence should be addressed to J. F. Gómez-Aguilar; [email protected]

Received 2 February 2016; Accepted 26 April 2016

Academic Editor: Alexander Iomin

Copyright © 2016 J. F. Gómez-Aguilar et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We present an alternative representation of integer and fractional electrical elements in the Laplace domain for modelingelectrochemical systems represented by equivalent electrical circuits. The fractional derivatives considered are of Caputo andCaputo-Fabrizio type. This representation includes distributed elements of the Cole model type. In addition to maintainingconsistency in adjusted electrical parameters, a detailed methodology is proposed to build the equivalent circuits. Illustrativeexamples are given and the Nyquist and Bode graphs are obtained from the numerical simulation of the corresponding transferfunctions using arbitrary electrical parameters in order to illustrate the methodology. The advantage of our representation appearsaccording to the comparison between our model and models presented in the paper, which are not physically acceptable due to thedimensional incompatibility. TheMarkovian nature of the models is recovered when the order of the fractional derivatives is equalto 1.

1. Introduction

Electrochemical Impedance Spectroscopy (EIS) is widelyused to investigate the interfacial and bulk properties ofmaterials, interfaces of electrode-electrolyte, and the inter-pretation of phenomena such as electrocatalysis, corrosion,or behavior of coatings onmetallic substrates.This techniquerelates directly measurements of impedance and phase angleas functions of frequency, voltage, or current applied. Thestimulus is an alternating current signal of low amplitudeintended to measure the electric field or potential differ-ence generated between different parts of the sample. Therelationship between the data of the applied stimulus andthe response obtained as a function of frequency provides

the impedance spectrum of samples studied [1]. Transferfunction analysis is a mathematical approach to relate aninput signal (or excitation) and the system’s response. Theratio formed by the pattern of the output and the input signalmakes it possible to find the zeros and poles, respectively. Toanalyze the behavior of the transfer function in the frequencydomain, several graphical methods were used, such as Bodeplots that provide a graphical representation of themagnitudeand phase versus frequency of the transfer function andthe Nyquist diagrams that are polar plots of impedancemodulus and phase lag. It is very common in the literatureto analyze impedance results by a physical model; this modelis expressed by a mathematical representation and usuallyis represented by equivalent electrical circuits composed of

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016, Article ID 9720181, 15 pageshttp://dx.doi.org/10.1155/2016/9720181

2 Advances in Mathematical Physics

elements such as resistors, capacitors, inductors, the ConstantPhase Element (CPE), and Warburg elements (which is aspecial case of the CPE). Although the resulting model isnot necessarily unique, it describes the system with greatprecision in the range of frequencies studied [1].

Fractional calculus (FC) is the investigation and treat-ment of mathematical models in terms of derivatives andintegrals of arbitrary order [2–5]. In the last years, theinterest in the field has considerably increased due to manypractical potential applications [6–13]. In the literature, anumber of definitions of the fractional derivatives have beenintroduced, namely, theHadamard, Erdelyi-Kober, Riemann-Liouville, Riesz, Weyl, Grünwald-Letnikov, Jumarie, and theCaputo representation [2–5]. For example, for the Caputorepresentation, the initial conditions are expressed in terms ofinteger-order derivatives having direct physical significance[14], this definition is mainly used to include memory effects.Recently, Caputo and Fabrizio in [15] present a new defini-tion of fractional derivative without a singular kernel; thisderivative possesses very interesting properties, for instance,the possibility to describe fluctuations and structures withdifferent scales. Furthermore, this definition allows for thedescription of mechanical properties related to damage,fatigue, and material heterogeneities. Properties of this newfractional derivative are reviewed in detail in [16].

Measurements of properties of materials, interfaces ofelectrode-electrolyte, corrosion, tissue properties of proteinfibers, semiconductors and solid-state devices, fuel cells,sensors, batteries, electrochemical capacitors, coatings, andelectrochromic materials have shown that their impedancebehavior can only be modeled by using Warburg elementsin conjunction with resistors, dispersive inductors, or CPEelements [17]. In this context, FC allows the investigationof the nonlocal response of electrochemical systems, thisbeing the main advantage when compared with classicalcalculus. Some researches concerning EIS introduce FC; forexample, the authors of [18] study the electrical impedance ofvegetables and fruits from a FC perspective; the experimentsare developed for measuring the impedance of botanicalelements and the results are analyzed using Bode andNyquist diagrams. In [19] a technique is presented to extractthe parameters that characterize a dispersion Cole-Coleimpedance model. Oldham in [20] describes how fractionaldifferential equations have influenced the electrochemistry.Other applications of fractional calculus in electrochemicalimpedance are given in [21–26].

Unlike thework of the authorsmentioned above, inwhichthe pass from an ordinary derivative to a fractional one isdirect, Gómez-Aguilar et al. in [12] analyze the ordinaryderivative operator and try to bring it to the fractional formin a consistent manner. Following this idea we present analternative representation of integer and fractional electricalelements in the Laplace domain formodeling electrochemicalsystems represented by equivalent electrical circuits; theorder of the fractional equation is 0 < 𝛾, 𝛽 ≤ 1. Inthis representation an auxiliary parameter 𝛼 is introduced;this parameter characterizes the existence of the fractionaltemporal components and relates the time constant of thesystem.

The paper is organized as follows: Section 2 explains thebasic concepts of the FC, Section 3 presents the examplesconsidered and the interpretation of typical diagrams, and theconclusions are given in Section 4.

2. Introduction to Fractional Calculus

The use of Caputo Fractional Derivative (CD) in Physics isgaining importance because of the specific properties: thederivative of a constant is zero and the initial conditions forthe fractional order differential equations can be given in thesamemanner as for the ordinary differential equations with aknown physical interpretation [4].

The CD is defined as follows [4]:

𝐶

0𝐷𝛾

𝑡𝑓 (𝑡) =

1

Γ (𝑛 − 𝛾)∫

𝑡

0

𝑓(𝑛)

(𝛼)

(𝑡 − 𝛼)𝛾−𝑛+1

𝑑𝛼, (1)

where 𝑑𝜑/𝑑𝑡𝜑 = 𝐶𝑎𝐷𝜑

𝑡is a CD with respect to 𝑡, 𝜑 ∈ 𝑅 is

the order of the fractional derivative, and Γ(⋅) represents thegamma function.

The Laplace transform of the CD has the following form[4]:

𝐿 [𝐶

0𝐷𝛾

𝑡𝑓 (𝑡)] = 𝑆

𝛾𝐹 (𝑆) −

𝑚−1

∑

𝑘=0

𝑆𝛾−𝑘−1

𝑓(𝑘)

(0) . (2)

The Caputo-Fabrizio fractional derivative (CF) is definedas follows [15, 16]:

CF0D𝛾

𝑡𝑓 (𝑡) =

𝑀(𝛾)

1 − 𝛾∫

𝑡

0

�̇� (𝛼) exp [−𝛾 (𝑡 − 𝛼)

1 − 𝛾] 𝑑𝛼, (3)

where 𝑑𝛾/𝑑𝑡𝛾 = CF0𝐷𝛾

𝑡is a CF with respect to 𝑡 and 𝑀(𝛾) is

a normalization function such that 𝑀(0) = 𝑀(1) = 1; inthis definition the derivative of a constant is equal to zero,but, unlike the usual Caputo definition (1), the kernel doesnot have a singularity at 𝑡 = 𝛼.

If 𝑛 ≥ 1 and 𝛾 ∈ [0, 1], the CF fractional derivative,CF0D(𝛾+𝑛)

𝑡𝑓(𝑡), of order (𝑛 + 𝛾), is defined by

CF0D(𝛾+𝑛)

𝑡𝑓 (𝑡) =

CF0D(𝛾)

𝑡(CF0D(𝑛)

𝑡𝑓 (𝑡)) . (4)

The Laplace transform of (3) is defined as follows [15, 16]:

𝐿 [CF0D(𝛾+𝑛)

𝑡𝑓 (𝑡)]

=1

1 − 𝛾𝐿 [𝑓(𝛾+𝑛)

𝑡] 𝐿 [exp(−𝛾

1 − 𝛾𝑡)]

=𝑠𝑛+1

𝐿 [𝑓 (𝑡)] − 𝑠𝑛𝑓 (0) − 𝑠

𝑛−1𝑓(0) ⋅ ⋅ ⋅ − 𝑓

(𝑛)(0)

𝑠 + 𝛾 (1 − 𝑠).

(5)

For this representation in the time domain it is suitable to usethe Laplace transform [15, 16].

Advances in Mathematical Physics 3

From this expression we have

𝐿 [CF0D𝛾

𝑡𝑓 (𝑡)] =

𝑠𝐿 [𝑓 (𝑡)] − 𝑓 (0)

𝑠 + 𝛾 (1 − 𝑠), 𝑛 = 0,

𝐿 [CF0D(𝛾+1)

𝑡𝑓 (𝑡)] =

𝑠2𝐿 [𝑓 (𝑡)] − 𝑠𝑓 (0) − �̇� (0)

𝑠 + 𝛾 (1 − 𝑠),

𝑛 = 1.

(6)

3. Electrochemical Impedance and theInterpretation of Typical Diagrams

Regarding the equivalent circuits, there is a diversity ofmodels used. The most common is to adjust the system to asimple model or one that includes a bilayer structure, eitherwith RC groups in parallel or in series model, although thereare cases where it is appropriate to include the Warburgimpedance element to consider possible diffusive processeson the surface. Generally, using a complex equivalent circuitis not necessary to obtain a good characterization of thereal system; commonly a simple electrical circuit is the firstchoice and increases its complexity when knowledge of theelectrochemical behavior of the system also increases.

The Cole impedance model is based on replacing theideal capacitor in the Debye model [27, 28]. Cole modelis represented by a series resistor 𝑅𝑠, a capacitor 𝐶𝑝, anda resistor in parallel 𝑅𝑝. In general it reflects the electricalresistance of the interface sample-electrode and maintains anegligible value with respect to𝑅𝑝; 𝛾 is the order of the powerthat best fits the model obtained, 𝛾 ∈ (0; 1), giving an idealcapacitor when it is 1. Using the algebraic representation ofthe circuit can be said to represent the total impedance as

𝑍𝑇 (𝑠) = 𝑅𝑠 +

𝑅𝑝

1 + (𝑠𝑅𝑝𝐶𝑝)𝛾 , (7)

where 𝑠 = 𝑗𝜔.On electric structures RC type, bias resistor 𝑅 represents

the charge transfer resistance and capacitance 𝐶 of doublelayer; the CPE is a component that models the behavior ofa double layer capacitor in actual electrochemical cells, thatis, an imperfect capacitor, and the impedance is representedas

𝑍𝑇= 𝐴. (8)

Equation (8) describes the deviation from ideal capacitors.Considering 𝛾 = 1 and the constant 𝐴 = 1/𝐶 (the inverse ofcapacitance), this equation describes a capacitor. For a CPE,the exponent 𝛾 is less than one [29].

In electrochemical systems the diffusion can createan impedance called Warburg impedance [26], commonlyused to describe phenomena such as diffusion, adsorp-tion, or desorption of electroactive substances at interfacesmetal/coating; this impedance depends on perturbationfrequency: at high frequency a small Warburg impedanceresults and at low frequency a higher Warburg impedanceis generated. The Warburg impedance parameter indicates

the existence of diffusive processes that can be related tothe release of the dissolved species. This parameter onlyreports the blocking ability of the passive layer, so it is notpossible to know the nature of the species which are dissolvedby electrochemical impedance spectroscopy technique. Theequation for the infinite thickness of theWarburg impedanceis given by

𝑍𝑇= (𝜔)

−1/2(1 − 𝑗) , (9)

where is aWarburg coefficient.On aNyquist plot the infiniteWarburg impedance appears as a diagonal line with a slope of0.5; on a Bode plot, the Warburg impedance exhibits a phaseshift of 45∘ [29].

Equation (9) is valid if the diffusion layer has an infinitethickness. Quite often this is not the case. If the diffusion layeris bounded, the impedance at lower frequencies no longerobeys (9). For the Warburg impedance with finite thickness,we get the form

𝑍𝑇 = (𝜔)−1/2

(1 − 𝑗) tanh [𝛿 (𝑗𝜔

𝐷)] , (10)

where𝛿 is theNernst diffusion layer thickness;𝐷 is an averagevalue of the diffusion coefficients of the diffusing species.Thisequation is more general and is called finite Warburg [29].

3.1. Equivalent Representation Based on Fractal Capacitors.One of the problems of the fractional representation is thecorrect sizing of the physical parameters involved in thedifferential equation, to be consistent with dimensionalityand following [12] we introduce an auxiliary parameter 𝛼 inthe following way:

𝑑

𝑑𝑡→

1

𝛼1−𝛾

𝐶

0𝐷𝛾

𝑡, 𝑛 − 1 < 𝛾 ≤ 𝑛, (11)

or

𝑑

𝑑𝑡→

1

𝛼1−𝛾

CF0D𝛾

𝑡, 𝑛 − 1 < 𝛾 ≤ 𝑛, (12)

where 𝑛 is an integer and when 𝛾 = 1. Expressions (11) and(12) become a classical derivative; the auxiliary parameter 𝛼has the dimension of time (seconds). This nonlocal time iscalled the cosmic time in the literature [30]. Another physicaland geometrical interpretation of the fractional operators isgiven in Moshrefi-Torbati and Hammond [31]. Parameter 𝛼characterizes the fractional temporal structures (componentsthat show an intermediate behavior between a conservativesystem and dissipative; such components change the timeconstant of the system) of the fractional temporal operator[32]. In the following we will apply this idea to construct thefractional equivalent circuits and examples are analyzed.

3.2. Polarizable Electrode. The model of polarizable elec-trode, also known as faradaic reaction, provides a simpledescription of the impedance of an electrochemical reactionon electrode surfaces.The equivalent circuit is represented byFigure 1.


CV(t)

i(t)

Rs

Rp

+

−

Figure 1: Equivalent electrical circuit for the polarizable electrodemodel.

Considering initial conditions equal to zero, the equiv-alent impedance is found by the following equation in thecomplex frequency domain:

𝑍𝑇 (𝑠) =𝑉 (𝑠)

𝐼 (𝑠). (13)

Applying Kirchhoff laws to the circuit of Figure 1, we have

𝑉 = 𝑅𝑠𝑖 + 𝑉𝐶, (14)

𝑖 = 𝑖𝑅+ 𝑖𝐶. (15)

Before applying the Laplace transform of (15) the followingconsiderations must be taken into account:

𝑖𝑅 =𝑉𝐶

𝑅𝑝

, (16)

𝑖𝐶= 𝐶

𝑑𝑉𝐶

𝑑𝑡. (17)

Consider (11); in the Caputo sense (1), (17) becomes

𝑖𝑅=

𝑉𝐶

𝑅𝑝

, (18)

𝑖𝐶 (𝑡) = 𝐶 +𝐶

𝛼1−𝛾

𝐶

0𝐷𝛾

𝑡𝑉𝐶 (𝑡) . (19)

Substituting (19) into (15), we obtain

𝑉 (𝑡) = 𝑅𝑠𝑖 (𝑡) + 𝑉𝐶 (𝑡) , (20)

𝑖 (𝑡) =𝑉𝐶

𝑅𝑝

+𝐶

𝛼1−𝛾

𝐶

0𝐷𝛾

𝑡𝑉𝐶 (𝑡) . (21)

Applying the Laplace transform (2) to (21), we obtain

𝑉 (𝑠) = 𝑅𝑠𝐼 (𝑠) + 𝑉𝐶 (𝑠) , (22)

𝐼 (𝑠) =𝑉𝐶 (𝑠)

𝑅𝑝

+𝐶

𝛼1−𝛾𝑠𝛾𝑉𝐶 (𝑠) . (23)

Finally from (23) we have the fractional impedance of thecircuit

𝑍𝑇 (𝑠) = 𝑅𝑠 +

𝑅𝑝

1 + (𝑅𝑝𝐶/𝛼1−𝛾) 𝑠𝛾

, (24)

where 𝑠 = 𝑗𝜔.Consider (12), in the Caputo-Fabrizio sense (3), (17)

becomes

𝑖𝑅 =𝑉𝐶

𝑅𝑝

, (25)

𝑖𝐶 (𝑡) = 𝐶 +𝐶

𝛼1−𝛾

CF0D𝛾

𝑡𝑉𝐶 (𝑡) . (26)

Substituting (26) into (15), we obtain

𝑉 (𝑡) = 𝑅𝑠𝑖 (𝑡) + 𝑉𝐶 (𝑡) , (27)

𝑖 (𝑡) =𝑉𝐶

𝑅𝑝

+𝐶

𝛼1−𝛾

CF0D𝛾

𝑡𝑉𝐶 (𝑡) . (28)

Applying the Laplace transform (5) to (28) we obtain

𝑉 (𝑠) = 𝑅𝑠𝐼 (𝑠) + 𝑉𝐶 (𝑠) , (29)

𝐼 (𝑠) =𝑉𝐶 (𝑠)

𝑅𝑝

+𝐶

𝛼1−𝛾[

𝑠

𝑠 + 𝛾 (1 − 𝑠)]𝑉𝐶 (𝑠) . (30)

Finally from (30) we have the fractional impedance of thecircuit

𝑍𝑇 (𝑠) = 𝑅𝑠 +

𝑅𝑝

1 + (𝑅𝑝𝐶/𝛼1−𝛾) (1/ (1 − 𝛾 + 𝛾/𝑠))

, (31)

where 𝑠 = 𝑗𝜔 and 𝛼1−𝛾 represent the fractional componentsof the system. Equations (24) and (31) are the result ofapplying the fractional temporal operator of Caputo andCaputo-Fabrizio type in (17) for the current in the capacitor;this general representation includes an arbitrary constant, 𝛼,which can be considered its own electrochemical parameter.In the particular case of 𝛼 = 𝑅

𝑝𝐶, (24) is reduced to the Cole

model (7). Equations (24) and (31) presented here preservethe dimensionality of the studied system for any value of theexponent of the fractional derivative. On the other hand, if𝛾 = 1 in (24) and (31), we obtain an ideal RC circuit and theMarkovian nature of the model is recovered.

Consider that the values of the parameters of the circuitshown in Figure 1 correspond to 𝑅

𝑠= 100Ω, 𝐶 = 1−4 F, and

𝑅𝑝= 200Ω and 𝑅

𝑝= 300Ω. Figures 2(a), 2(b), 2(c), and 2(d)

show the Nyquist and Bode plots for (24).Now consider that the values of the parameters of the

circuit shown in Figure 1 correspond to 𝑅𝑠 = 100Ω, 𝐶 =1−4 F, and 𝑅𝑝 = 200Ω and 𝑅𝑝 = 300Ω. Figures 3(a), 3(b),3(c), and 3(d) show the Nyquist and Bode plots for (31).

Several studies use equivalent circuits considering purecapacitances for adjusting the impedance spectra and thusdescribe phenomena as deterioration of materials due to itsporosity or the effects of exposure and surface preparation


Nyquist diagram

0

10

20

30

40

50

60

70

80

90

100

Imag

inar

y pa

rt

150 200 250 300100Real part

𝛾 = 1

𝛾 = 0.95

𝛾 = 0.9

𝛾 = 0.85

(a)

Bode diagram

40

45

50

Mag

nitu

de (d

B)

0

−10

−20

−30

Phas

e (de

g)

101 102 103 104100

Frequency (rad/s)

100 102 103 104101

Frequency (rad/s)

𝛾 = 1

𝛾 = 0.95

𝛾 = 0.9

𝛾 = 0.85

(b)

Nyquist diagram

0

50

100

150

Imag

inar

y pa

rt

150 200 250 300 350 400100Real part

𝛾 = 1

𝛾 = 0.95

𝛾 = 0.9

𝛾 = 0.85

(c)

Bode diagram

40

45

50

55M

agni

tude

(dB)

0

−10

−20

−30

−40

Phas

e (de

g)

101 102 103 104100

Frequency (rad/s)

101 102 103 104100

Frequency (rad/s)

𝛾 = 1

𝛾 = 0.95

𝛾 = 0.9

𝛾 = 0.85

(d)

Figure 2: Nyquist and Bode diagram for themodel of polarizable electrode, Caputo derivative approach, in (a) and (b):𝑅𝑠= 100Ω,𝐶 = 1−4 F,

and 𝑅𝑝= 200Ω for 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85; for (c) and (d), 𝑅

𝑠= 100Ω, 𝐶 = 1−4 F, and 𝑅

𝑝= 300Ω for 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9,

and 𝛾 = 0.85.

of substrates on the impedances [33–36]. The electricalcapacitance has information on the conductive properties ofmaterials and their chemical composition. Recently, severalauthors have replaced this pure capacitance by a CPE,because this has been considered an improvement in thesettings of the theoretical equations regarding experimentalresults [37–39], for example, in predicting the useful life ofcoatings and surface roughness heterogeneity resulting fromthe presence of impurities, fractality, dislocations, adsorp-tion of inhibitors, or formation of porous layers [40–47].Replacing the capacitor 𝐶 by a CPE adds a pseudocapacitiveconstant to the circuit and causes the attenuation of the

imaginary part of the impedance causing a flattening ofthe semicircles in the Nyquist diagrams (this is their mainfeature); see Figures 2(a) and 2(c) for the Caputo approachand Figures 3(a) and 3(c) for the Caputo-Fabrizio approach.These figures allow seeing the effect of varying the values ofthe pseudocapacitive constants: in the range 𝛾 ∈ (0.85; 1)we have the behavior of a CPE and when 𝛾 = 1 the valuescorrespond to pure capacitances. In this range, the orderof the derivative transforms the capacitor 𝐶

𝑝into a CPE.

When increasing the value of the pseudocapacitive constants,this also increases the imaginary part of the impedance andthis changes the time constants of the system [32], which


Nyquist diagram

𝛾 = 1

𝛾 = 0.999

𝛾 = 0.998

𝛾 = 0.997

0

10

20

30

40

50

60

70

80

90

100

Imag

inar

y pa

rt

150 200 250 300100Real part

(a)

Bode diagram

𝛾 = 1

𝛾 = 0.999

𝛾 = 0.998

𝛾 = 0.997

40

45

50

Mag

nitu

de (d

B)

0

−10

−20

−30

Phas

e (de

g)

101 102 103 104100

Frequency (rad/s)

100 102 103 104101

Frequency (rad/s)

(b)

Nyquist diagram

𝛾 = 1

𝛾 = 0.999

𝛾 = 0.998

𝛾 = 0.997

150 200 250 300 350 400100Real part

0

50

100

150

Imag

inar

y pa

rt

(c)

Bode diagram

𝛾 = 1

𝛾 = 0.999

𝛾 = 0.998

𝛾 = 0.997

101 102 103 104100

Frequency (rad/s)

40

45

50

55M

agni

tude

(dB)

0

−10

−20

−30

−40

Phas

e (de

g)

100 102 103 104101

Frequency (rad/s)

(d)

Figure 3: Nyquist and Bode diagram for the model of polarizable electrode, Caputo-Fabrizio derivative approach, in (a) and (b): 𝑅𝑠= 100Ω,

𝐶 = 1−4 F, and 𝑅

𝑝= 200Ω for 𝛾 = 1, 𝛾 = 0.999, 𝛾 = 0.998, and 𝛾 = 0.997; for (c) and (d), 𝑅

𝑠= 100Ω, 𝐶 = 1−4 F, and 𝑅

𝑝= 300Ω, for 𝛾 = 1,

𝛾 = 0.999, 𝛾 = 0.998, and 𝛾 = 0.997.

correspond to creating fractal structures for particular valuesof 𝛾. In theCaputo-Fabrizio approach it is noted that the valueof the resistance series is modified, showing the existence ofheterogeneities in this component. In Figures 2(b) and 2(d)for the Caputo approach and Figures 3(b) and 3(d) for theCaputo-Fabrizio approach, the Bode plots exhibit changes inthe cutoff frequency, hence the phase shift and the decrease ofthe magnitude. For the range of frequency 10 to 10000 rad/s ashift in themagnitude and phase is shown, which implies thatthe proposed circuit better defines these frequencies. We seethat, by increasing the frequency from the cutoff frequency,the magnitude decreases, while for frequencies below the

cutoff magnitude it is almost constant. This means that thecurrent through the circuit is very high with decreasingfrequency and is very low when the frequency increases. Thecurrent flowing through these resistors at low frequenciespresents dissipative effects that correspond to the nonlinearsituation of the physical process (realistic behavior that isnonlocal in time), for example, the ohmic friction, whichraises the temperature and therefore the kinetic energy ofthe molecules of the system [48]. Concerning the phase, wehave that by increasing frequency the displacement currentand polarization are very much increased. In this context,the need for CPE has been attributed by some authors to


i(t) R1

R2

R3

C1C2

Figure 4: Equivalent electrical circuit for the heterogeneous reac-tion model.

the presence of roughness, corrosion products, changes in themorphology of the material, or heterogeneous surfaces.

3.3. Heterogeneous Reaction. This model describes a hetero-geneous reaction which occurs in two stages with absorptionof intermediates products and absence of diffusion limita-tions.This circuit has been commonly used tomodel a porouselectrode coating or a defective electrolyte interface and hasrecently been applied in the assessment of passive metal-electrolyte or metal-electrolyte interfaces with hard coating[49, 50]. The equivalent circuit is represented by Figure 4.

This model has also been used in the study of coatedmetals. In this case, 𝐶

1represents the capacitance of the

coating; this value is smaller than a capacitance representedby a CPE;𝑅

2represents the pore resistance; it is the resistance

of ion conducting paths developed in the coating.These pathsmay be physical pores filled with electrolyte; this electrolytesolution can be very different than the bulk solution outsideof the coating; 𝐶2 (double layer capacitance) and 𝑅3 (chargetransfer reaction) represent the interface between this pocketof solution and the bare metal.

Following the samemethodology fromprevious example,the fractional impedance in the Caputo sense of this circuit is

𝑍𝑇 (𝑠) = 𝑅1

+𝑅3

(𝑅3𝐶1/𝛼1−𝛾

𝛾 ) 𝑠𝛾 + 1/ (𝑅

2/𝑅3+ 1/ ((𝑅

3𝐶2/𝛼1−𝛽

𝛽) 𝑠𝛽 + 1))

.(32)

In the Caputo-Fabrizio sense the fractional impedance isgiven by

𝑍𝑇 (𝑠) = 𝑅1 +

𝑅3

(𝑅3𝐶1/𝛼1−𝛾

𝛾 ) (1/ (1 − 𝛾 + 𝛾/𝑠)) + 1/ (𝑅2/𝑅3 + 1/ ((𝑅3𝐶2/𝛼1−𝛽

𝛽) (1/ (1 − 𝛽 + 𝛽/𝑠)) + 1))

, (33)

where 𝑠 = 𝑗𝜔, in (32) and (33), 𝛼1−𝛾𝛾

and 𝛼1−𝛽𝛽

representsthe fractional components of the system. This general rep-resentation includes an arbitrary constant, 𝛼, which can beconsidered its own electrochemical parameter. In the case of𝛼𝛾the physical parameters involved are 𝑅

3𝐶1and for 𝛼

𝛽they

are 𝑅3𝐶2, the time constant of the system. Equations (32)

and (33) presented here preserve the dimensionality of thestudied system for any value of the exponent of the fractionalderivative; when 𝛾 = 𝛽 = 1, we recover the Markovian natureof the model.

Consider that the values of the parameters of the circuitshown in Figure 4 correspond to 𝑅

1= 50Ω, 𝑅

2= 100Ω,

𝑅3= 200Ω, 𝐶

1= 1−3 F, 𝐶

2= 1−2 F, and 𝐶

2= 3−3 F. Figures

5(a), 5(b), 5(c), and 5(d) show the Nyquist and Bode plots for(32).

Now consider that the values of the parameters of thecircuit shown in Figure 4 correspond to 𝑅1 = 50Ω, 𝑅2 =100Ω, 𝑅

3= 200Ω, 𝐶

1= 1−3 F, 𝐶

2= 1−2 F, and 𝐶

2= 3−3 F.

Figures 6(a), 6(b), 6(c), and 6(d) show the Nyquist and Bodeplots for (33).

Some authors make changes to the ideal capacitorsincluding elements of Warburg [49, 50]; these elements arerepresented in our circuitmaintaining the order of the deriva-tives fixed at a value of 𝛾 = 𝛽 = 1/2. For Caputo and Caputo-Fabrizio approach, Figures 7(a) and 7(c), respectively, showthe resulting Nyquist diagrams considering the order of thederivative for both capacitors at 1/2. From these figures itcan be seen that for high frequency the impedance is smallbecause the reagents must move away from the surface.

The low-frequency disturbances allow the movement ofdistant molecules.

The circuit shown in Figure 4 has also been used withsuccess in the description of titanium alloys, considering car-bon coatings, silicon oxide, and titanium oxide; in such appli-cations it is generally considered that the coating presentsa porous outer sublayer and a denser internal sublayer[49, 51]. In Figure 7(c) it is observed that the resistance valuedecreases, which can provide valuable information on thecharacteristics of a coating film.

3.4. Limited Diffusion, Warburg Element. This model des-cribes the polarization of an electrode considering limitingthe diffusion. This circuit models a cell where polarization isdue to a combination of kinetic and diffusion processes, 𝑅𝑠 isthe resistance of the electrolyte solution, CPE is the imperfectcapacitor, 𝑅

𝑝is the electron-transfer resistance, and𝑊 is the

Warburg element due to diffusion of the redox couple to theinterface from the bulk of the electrolyte [52]. The equivalentcircuit is represented by Figure 8.

Following the samemethodology fromprevious example,the fractional impedance in the Caputo sense of this circuit is

𝑍𝑇 (𝑠)

= 𝑅𝑠

+

𝑅𝑝

(𝑅𝑝𝐶𝑝/𝛼1−𝛾

𝛾 ) 𝑠𝛾 + 1/ (1 + 1/ (𝑅

𝑝𝐶𝑠/𝛼1−𝛽

𝛽) 𝑠𝛽)

.

(34)


Nyquist diagram

100 150 200 250 300 35050Real part

0

20

40

60

80

100

120

Imag

inar

y pa

rt

𝛾 = 1

𝛾 = 0.95

𝛾 = 0.9

𝛾 = 0.85

(a)

Bode diagram

−30

−20

−10

0

Phas

e (de

g)

35

40

45

50

Mag

nitu

de (d

B)

𝛾 = 1

𝛾 = 0.95

𝛾 = 0.9

𝛾 = 0.85

100 102 103 104 105101

Frequency (rad/s)

100 102 103 104 105101

Frequency (rad/s)

(b)

Nyquist diagram

100 150 200 250 300 35050Real part

0

20

40

60

80

100

120

Imag

inar

y pa

rt

𝛾 = 1

𝛾 = 0.95

𝛾 = 0.9

𝛾 = 0.85

(c)

Bode diagram

−30

−20

−10

0

Phas

e (de

g)

35

40

45

50

Mag

nitu

de (d

B)

𝛾 = 1

𝛾 = 0.95

𝛾 = 0.9

𝛾 = 0.85

101 102 103 104 105100

Frequency (rad/s)

100 102 103 104 105101

Frequency (rad/s)

(d)

Figure 5:Nyquist andBode diagram for the heterogeneous reactionmodel, Caputo derivative approach, in (a) and (b):𝑅1= 50Ω,𝑅

2= 100Ω,

𝑅3= 200Ω, 𝐶

1= 1−3 F, and 𝐶

2= 1−2 F for 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85; for (c) and (d), 𝑅

1= 50Ω, 𝑅

2= 100Ω, 𝑅

3= 200Ω,

𝐶1= 1−3 F, and 𝐶

2= 3−3 F for 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85. The capacitor 𝐶

2is considered ideal, 𝛽 = 1.

In the Caputo-Fabrizio sense the fractional impedance isgiven by

𝑍𝑇 (𝑠) = 𝑅𝑠 +

𝑅𝑝

(𝑅𝑝𝐶𝑝/𝛼1−𝛾

𝛾 ) (1/ (1 − 𝛾 + 𝛾/𝑠)) + 1/ (1 + 1/ (𝑅𝑝𝐶𝑠/𝛼1−𝛽

𝛽) (1/ (1 − 𝛽 + 𝛽/𝑠)))

, (35)

where 𝑠 = 𝑗𝜔, in (34) and (35), and 𝛼1−𝛾𝛾

and 𝛼1−𝛽𝛽

representthe fractional components of the system. This general rep-resentation includes an arbitrary constant, 𝛼, which can be

considered its own electrochemical parameter. In the case of𝛼𝛾 the physical parameters involved are 𝑅𝑝𝐶𝑝 for 𝛼𝛾 and for𝛼𝛽 are 𝑅𝑝𝐶𝑠, the time constant of the system. For (34) and


Nyquist diagram

0

20

40

60

80

100

120

Imag

inar

y pa

rt

100 150 200 250 300 35050Real part

𝛾 = 1

𝛾 = 0.999

𝛾 = 0.998

𝛾 = 0.997

(a)

Bode diagram

35

40

45

50

55

Mag

nitu

de (d

B)

−30

−20

−10

0

Phas

e (de

g)

101 102 103 104 105100

Frequency (rad/s)

101 102 103 104 105100

Frequency (rad/s)

𝛾 = 1

𝛾 = 0.999

𝛾 = 0.998

𝛾 = 0.997

(b)

Nyquist diagram

100 150 200 250 300 35050Real part

0

20

40

60

80

100

120

Imag

inar

y pa

rt

𝛾 = 1

𝛾 = 0.999

𝛾 = 0.998

𝛾 = 0.997

(c)

Bode diagram

35

40

45

50

Mag

nitu

de (d

B)

−30

−20

−10

0

Phas

e (de

g)

101 102 103 104 105100

Frequency (rad/s)

101 102 103 104 105100

Frequency (rad/s)

𝛾 = 1

𝛾 = 0.999

𝛾 = 0.998

𝛾 = 0.997

(d)

Figure 6: Nyquist and Bode diagram for the heterogeneous reaction model, Caputo-Fabrizio derivative approach, in (a) and (b): 𝑅1= 50Ω,

𝑅2= 100Ω, 𝑅

3= 200Ω, 𝐶

1= 1−3 F, and 𝐶

2= 1−2 F for 𝛾 = 1, 𝛾 = 0.999, 𝛾 = 0.998, and 𝛾 = 0.997; for (c) and (d), 𝑅

1= 50Ω, 𝑅

2= 100Ω,

𝑅3= 200Ω, 𝐶

1= 1−3 F, and 𝐶

2= 3−3 F for 𝛾 = 1, 𝛾 = 0.999, 𝛾 = 0.998, and 𝛾 = 0.997. The capacitor 𝐶

2is considered ideal, 𝛽 = 1.

(35) when 𝛾 = 𝛽 = 1, we recover the Markovian nature of themodel.

The capacitor 𝐶 and Warburg element 𝑊 shown inFigure 8 are replaced by two fractional capacitors 𝐶

𝑝and

𝐶𝑠, respectively. The capacitor 𝐶

𝑠is assigned an exponent

𝛽 = 1/2, causing replacement of pure capacitance andcausing thus an impedance according to an infinite diffusionlayer or Warburg impedance. This element arises from one-dimensional diffusion of an ionic species to the electrode [18].The capacitance of fractional order 𝐶

𝑝can be modeled in

the range 0 < 𝛾 ≤ 1, causing that this pure capacitancerepresents a CPE or imperfect capacitance. Equations (34)and (35) presented here preserve the dimensionality of thestudied system for any value of the exponent of the fractionalderivative. The equivalent circuit shown in Figure 8 has beenused to model phenomena of corrosion stability in the phys-iological environment such as the simulated body fluid; 𝑅

𝑠

represents the electrolyte resistance,𝑅𝑝represents the coating

pore resistance, and the capacitances represent the frequency-dependent electrochemical phenomena, such as the coating


Nyquist diagram

0

10

20

30

40

50

60

Imag

inar

y pa

rt

100 150 200 250 300 35050Real part

(a)

Bode diagram

35

40

45

50

Mag

nitu

de (d

B)

−20

−15

−10

−5

0

Phas

e (de

g)

101 102 103 104 105 106100

Frequency (rad/s)

101 102 103 104 105 106100

Frequency (rad/s)

(b)

Nyquist diagram

0

10

20

30

40

50

60

70

80

90

100

110

Imag

inar

y pa

rt

150 200 250 300 350100Real part

(c)

Bode diagram

424446485052

Mag

nitu

de (d

B)

−20

−10

0

Phas

e (de

g)

101 102 103 104 105100

Frequency (rad/s)

101 102 103 104 105100

Frequency (rad/s)

(d)

Figure 7: Nyquist and Bode diagram for the heterogeneous reaction model, Caputo approach in (a) and (b) and Caputo-Fabrizio approachin (c) and (d). For (a) and (b), 𝑅

1= 50Ω, R

2= 100Ω, 𝑅

3= 200Ω, 𝐶

1= 1−3 F, 𝐶

2= 1−2 F (solid line), 𝐶

2= 3−3 F (dash line), 𝐶

2= 3−3 F (dot

line), and 𝐶2= 3−4 F (dash-dot line). For the capacitor 𝐶

1, 𝛾 = 1/2; for the capacitor 𝐶

2, 𝛽 = 1/2. For (c) and (d), 𝑅

1= 50Ω, 𝑅

2= 100Ω,

𝑅3= 200Ω, 𝐶

1= 1−3 F, 𝐶

2= 1−2 F (solid line), 𝐶

2= 3−3 F (dash line), 𝐶

2= 3−3 F (dot line), and 𝐶

2= 3−4 F (dash-dot line). For the capacitor

𝐶1, 𝛾 = 1/2; for the capacitor 𝐶

2, 𝛽 = 1/2.

WCPE

+ i i

−

+

−

Rs

Rp

iR iC

Cp

Rs

Rp

iR iC

Cs

Figure 8: Equivalent electrical circuit for the limited diffusionmodel.

impedance and passive oxide film capacitance. CPE is usedin these models to compensate the nonhomogeneity in thesystem [53].

Consider that the values of the parameters of the circuitshown in Figure 8 correspond to 𝑅𝑠 = 100Ω, 𝑅𝑝 = 5000Ω,𝐶𝑠 = 3

−6 F, 𝐶𝑠 = 3−7 F, and 𝐶𝑝 = 1

−3 F. Figures 9(a), 9(b),9(c), and 9(d) show the Nyquist and Bode plots for (34).

Now consider that the values of the parameters of thecircuit shown in Figure 8 correspond to 𝑅𝑠 = 100Ω, 𝑅𝑝 =5000Ω, 𝐶

𝑠= 3−6 F, 𝐶

𝑠= 3−7 F, and 𝐶

𝑝= 1−3 F. Figures 10(a),

10(b), 10(c), and 10(d) show the Nyquist and Bode plots for(35).

Replacing capacitor 𝐶 by a CPE introduces a pseudoca-pacitive constant at circuit and causes the attenuation of theimaginary part of the impedance causing a flattening of thesemicircles in the Nyquist diagrams; see Figure 9(a) for theCaputo approach and Figure 10(a) for the Caputo-Fabrizio


Nyquist diagram

2000 4000 6000 8000 100000Real part

0

1000

2000

3000

4000

5000

6000

Imag

inar

y pa

rt

𝛾 = 1

𝛾 = 0.95

𝛾 = 0.9

𝛾 = 0.85

(a)

Bode diagram

0

50

100

Mag

nitu

de (d

B)

−100

−50

0

Phas

e (de

g)

𝛾 = 1

𝛾 = 0.95

𝛾 = 0.9

𝛾 = 0.85

101 102 103 104 105 106100

Frequency (rad/s)

100 102 103 104 105 106101

Frequency (rad/s)

(b)

Nyquist diagram

2000 4000 6000 8000 100000Real part

0

1000

2000

3000

4000

5000

6000

Imag

inar

y pa

rt

𝛾 = 1

𝛾 = 0.95

𝛾 = 0.9

𝛾 = 0.85

(c)

Bode diagram

50

60

70

80

90

Mag

nitu

de (d

B)

−80

−60

−40

−20

0

Phas

e (de

g)

𝛾 = 1

𝛾 = 0.95

𝛾 = 0.9

𝛾 = 0.85

101 102 103 104 105 106100

Frequency (rad/s)

101 102 103 104 105 106100

Frequency (rad/s)

(d)

Figure 9: Nyquist and Bode diagram for the limited diffusion model, Caputo derivative approach, in (a) and (b): 𝑅𝑠= 100Ω, 𝑅

𝑝= 5000Ω,

𝐶𝑠= 3−6 F, and 𝐶

𝑝= 1−3 F for 𝛽 = 1/2 corresponding to 𝐶

𝑠, 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85; for (c) and (d), 𝑅

𝑝= 5000Ω, 𝐶

𝑠= 3−7 F,

and 𝐶𝑝= 1−3 F for 𝛽 = 0.4 corresponding to 𝐶

𝑠, 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85.

approach. Increasing the value of the pseudocapacitive con-stant implies an increase in the contribution of the imaginarypart of the impedance and this changes the system’s timeconstants [32], creating fractal structures which correspondto particular values of 𝛾. In Figures 9(b), 9(d), 10(b), and 10(d)for the Caputo and Caputo-Fabrizio approach, respectively,the Bode plots show variations in the cutoff frequency, hencethe phase shift and the decrease of the magnitude.

If 0 < 𝛽 < 0.5, then we have a model that geometricallydescribes the activation surface inhomogeneities or devia-tions from linear diffusion processes. In our example 𝛽 = 0.4.

This occurs naturally when the diffusion occurs in a dilutesolution or in the case that the spread does not obey the lawsof Fick; see Figures 9(c) and 10(c).

4. Conclusions

An impedance spectrum which is obtained in response tothe small amplitude signal excitation is often interpreted interms of an equivalent electrical circuit. This one is basedon a physical model that may represent and characterize


Nyquist diagram

0

500

1000

1500

2000

2500

3000

3500

4000

Imag

inar

y pa

rt

1000 2000 3000 4000 5000 60000Real part

𝛾 = 1

𝛾 = 0.999

𝛾 = 0.998

𝛾 = 0.997

(a)

Bode diagram

40

60

80

100

Mag

nitu

de (d

B)

−100

−50

0

Phas

e (de

g)

101 102 103 104 105 106100

Frequency (rad/s)

101 102 103 104 105 106100

Frequency (rad/s)

𝛾 = 1

𝛾 = 0.999

𝛾 = 0.998

𝛾 = 0.997

(b)

Nyquist diagram

𝛾 = 1

𝛾 = 0.999

𝛾 = 0.998

𝛾 = 0.997

1000 2000 3000 4000 5000 60000Real part

0

500

1000

1500

2000

2500

3000

3500

4000

Imag

inar

y pa

rt

(c)

Bode diagram

5060708090

100

Mag

nitu

de (d

B)

−80−60−40−20

0

Phas

e (de

g)

100 104 106102

Frequency (rad/s)

102 104 106100

Frequency (rad/s)

𝛾 = 1

𝛾 = 0.999

𝛾 = 0.998

𝛾 = 0.997

(d)

Figure 10: Nyquist and Bode diagram for the limited diffusion model, Caputo-Fabrizio derivative approach, in (a) and (b): 𝑅𝑠= 100Ω,

𝑅𝑝= 5000Ω, 𝐶

𝑠= 3−6 F, and 𝐶

𝑝= 1−3 F for 𝛽 = 1/2 corresponding to𝐶

𝑠, 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85; for (c) and (d), 𝑅

𝑝= 5000Ω,

𝐶𝑠= 3−7 F, and 𝐶

𝑝= 1−3 F for 𝛽 = 0.4 corresponding to 𝐶

𝑠, 𝛾 = 1, 𝛾 = 0.95, 𝛾 = 0.9, and 𝛾 = 0.85.

elements whose electrochemical properties and structuralfeatures or the physicochemical processes are taking place inthe studied being. The resulting spectrum described in awide frequency range might present one, two, or moretime constants, depending on the monolayer or multilayerstructure, porosity, or diffusive limitations caused by thecharge transfer process. The existence of a time constantindicates a homogeneous layer structure; if two differentconstants are presented, thismay indicate the existence of twosublayers.

In the field of biomaterials, the electrochemical impe-dance spectroscopy technique proves to be useful in char-acterizing roughness or heterogeneous surfaces, in coatingsanalysis, as cell suspensions, in studying the adsorption ofprotein, and in characterizing the performance of materials,mainly with regard to electrocatalysis and corrosion. In thiscontext, FC allows the investigation of the nonlocal responseof electrochemical systems. FC has been used successfullyto modify many existing models of physical processes;the representation of equivalent models in integer-order


derivatives provided a good approximation of the electro-chemical response of the model.

On the basis of Cole’s proposal to add an extra degree offreedom in order to solve the RC circuits for characterizationpurposes and improve the correlation in the adjustment toexperimental data, we have developed analytical argumentsto derive this result based on integration in weighted indi-vidual relaxation processes. However, the distributions ofrelaxation times involve complex functions and are diffi-cult to measure. This study has shown a pattern in whichthe Cole type behavior appears as a result of competitionbetween a capacitive and resistive behavior within the sam-ple, characterized by the fractional order derivative of theapplied voltage. This combination of stored and dissipatedenergy is conveniently based on the representation of linearviscoelastic behavior; this dissipation is known as internalfriction. In the literature it is common to characterize basedon least-squares fit of equivalent electrical circuit modelson experimental data, including Cole models. From thedescription of the fractional differential equation models itcan be noted that the representation of Colemodels is derivedas a particular solution to the RC circuit under FC.

Since some authors replace the derivative of a fractionalentire order in a purely mathematical context, the physicalparameters involved in the differential equation do not havethe dimensionality obtained in the laboratory and from thephysical point of view of engineering this is not entirelycorrect. In this representation an auxiliary parameter 𝛼 isintroduced; this parameter characterizes the existence ofthe fractional temporal components and related the timeconstant of the system; the solutions presented here preservethe dimensionality of the studied system for any value of theexponent of the fractional derivative. The advantage of thisalternative representation inwhen comparisonwith themod-els presented in the literature is the physical compatibility ofthe solutions.

TheCaputo representation has the disadvantage that theirkernel had singularity; this kernel includes memory effectsand therefore this definition cannot accurately describe thefull effect of the memory. Due to this inconvenience, Caputoand Fabrizio in [15] present a new definition of fractionalderivative without singular kernel, the Caputo-Fabrizio frac-tional derivative.The two definitions of fractional derivativesmust apply conveniently depending on the nature of thesystem and the choice of the fractional derivative dependsupon the problem studied and on the phenomenologicalbehavior of the system.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors would like to thank Mayra Mart́ınez for theinteresting discussions. J. F.Gómez-Aguilar acknowledges thesupport provided by CONACYT: catedras CONACYT parajovenes investigadores 2014.

References

[1] E. Barsoukov and J. R. Macdonald, Impedance Spectroscopy,Theory, Experiment, and Applications, John Wiley & Sons, NewYork, NY, USA, 2005.

[2] K. B. Oldham and J. Spanier,The Fractional Calculus, AcademicPress, New York, NY, USA, 1974.

[3] K. S. Miller and B. Ross, An Introduction to the FractionalCalculus and Fractional Differential Equations, John Wiley &Sons, New York, NY, USA, 1993.

[4] I. Podlubny, Fractional Differential Equations, Academic Press,New York, NY, USA, 1999.

[5] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, FractionalCalculus:Models andNumericalMethods, Series onComplexity,Nonlinearity and Chaos,World Scientific, River Edge, NJ, USA,2012.

[6] R. Hilfer, Applications of Fractional Calculus in Physics, WorldScientific, Singapore, 2000.

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