Home Analysis of Drude model using fractional derivatives without singular kernels
Article Open Access

Analysis of Drude model using fractional derivatives without singular kernels

  • Leonardo Martínez Jiménez , J. Juan Rosales García EMAIL logo , Abraham Ortega Contreras and Dumitru Baleanu
Published/Copyright: November 6, 2017
Download Article (PDF)
Open Physics
From the journal Open Physics Volume 15 Issue 1

Abstract

We report study exploring the fractional Drude model in the time domain, using fractional derivatives without singular kernels, Caputo-Fabrizio (CF), and fractional derivatives with a stretched Mittag-Leffler function. It is shown that the velocity and current density of electrons moving through a metal depend on both the time and the fractional order 0 < γ ≤ 1. Due to non-singular fractional kernels, it is possible to consider complete memory effects in the model, which appear neither in the ordinary model, nor in the fractional Drude model with Caputo fractional derivative. A comparison is also made between these two representations of the fractional derivatives, resulting a considered difference when γ < 0.8.

Keywords: Fractional Calculus; Drude model; Caputo-Fabrizio derivative; Atangana-Baleanu derivative
PACS: 45.10.Hj; 45.20.D; 66.70.Df

1 Introduction

Fractional calculus (FC), involving derivatives and integrals of non-integer order, is the natural generalization of classical calculus, which during recent decades has become a powerful and widely used tool for better modelling and control of processes in many areas of science and engineering [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The fractional derivatives are non-local operators because they are defined using integrals. Therefore, the fractional derivative in time contains information about the function at earlier points, thus it possesses a memory effect, and it includes non-local spatial effects. In other words, such derivatives consider the history and non-local distributed effects which are essential for better and more precise descriptions and understanding of complex and dynamic system behaviour. Due to the lack of a consistent geometric and physical interpretation of the fractional derivative, several definitions of fractional derivatives and integrals exist, see [12] for a review of definitions for fractional derivatives and integrals. These definitions include, Riemann-Liouville, Grunwald-Letnikov, Caputo, and Weyl, among others. The most used definitions are the Riemann-Liouville and the Caputo fractional derivatives. There are classical applications where FC has showed its great capabilities, such as the tautochrone problem [13], models based on memory mechanisms [14], fractional diffusion equations [15], new linear capacitor theory [16], the non-local description of quantum dynamics like Brownian motion and anomalous diffusion [17, 18]. Other interesting applications are given in viscoelastic materials [19, 20, 21, 22, 23, 24], anomalous non-Gaussian transport [25] and dielectric materials [26, 27, 28, 29].

The concept of a fractional curl operator and a fractional paradigm in electromagnetic theory was introduced in [30]. The application of the fractional curl operator to different electromagnetic problems is discussed in [31, 32, 33]. In [34, 35] it is shown that the electromagnetic fields and waves in a wide class of dielectric materials are described by fractional differential equations with derivatives of non-integer order with respect to time. The order of these derivatives is defined by exponentials of the universal response laws for frequency dependence of the dielectric susceptibility. In [36] proposed a systematic way to construct fractional differential equations and applied them to the propagation of electromagnetic waves in an infinitely extended homogeneous medium at rest.

Despite the accurate results obtained with the Riemann-Liouville and Caputo fractional derivatives, they have the disadvantage that their kernel has a singularity at the end point of the interval. To avoid this problem, [37] proposed the Capauto-Fabrizio (CF) derivative. This is a new fractional-order derivative that does not have any singularity. The main advantage of the new definition is that the singular power-law kernel, Caputo, is now replaced by a non-singular exponential kernel CF, which is easier to use in theoretical analysis, numerical calculations and real-world applications. Based on this new derivative, some interesting studies can be found in [38, 39, 40, 41, 42, 43, 44]. However, some researchers have concluded that the operator is not a derivative with fractional order, but instead a filter with fractional parameters. To correct this deficiency, two fractional derivatives in the Caputo and Riemann-Liouville sense were defined by Atangana-Baleanu [45], based on the generalized stretched Mittag-Leffler function. These new derivatives have been applied to different systems in [46, 47, 48].

Motivated by recent experimental results [49, 50, 51] showing that simple models are best described by fractional differential equations, in [52] the fractional Caputo derivative is applied to study of the Drude model. However, as was mentioned above, this definition of the derivative has a singular kernel and cannot accurately describe the full memory effect. To correct this disadvantage, in the present work we analyse the Drude model using a Caputo-Fabrizio fractional derivative [37], as well as the Atangana-Baleanu in the Caputo sense (ABC) derivative [45], for different sources. Numerical simulations of these models are given in order to compare them and evaluate their effectiveness.

This work is organized as follows: in section 2, some basic concepts about Caputo (C), Caputo-Fabrizio (CF) and ABC fractional derivatives are given; in section 3, the classical Drude model is reviewed; in section 4, a new fractional Drude model is proposed using the CF fractional derivative; in section 5, the ABC derivative is applied to the same problem; finally, in section 6, the obtained results are discussed and compared; the conclusion is presented in section 7.

2 Some fractional derivatives

The usual Caputo fractional derivative of order γ is defined by [14]

dγf(t)dtγ=aDtγf(t)=1Γ(1−γ)∫atf˙(τ)(t−τ)γdτ,(1)

with 0 < γ ≤ 1 and a ∈ [−∞, t], f ∈ H1(a, b), b > a. By changing the kernel (t – τ)−γ with the function e−γ1−γt and 1Γ(1−γ) withM(γ)1−γ, the Caputo-Fabrizio fractional derivative [37] is obtained

CFaDtγf(t)=M(γ)(1−γ)∫atf˙(τ)exp−γ(t−τ)1−γdτ,(2)

where M(γ)1−γ is a normalization function with the property M(0) = M(1) = 1. If f(t) is a constant function, then the Caputo-Fabrizio derivative (2) is zero. However, in contrast to definition (1), the kernel in (2) does not have singularity for t = τ. This property is of particular interest, because it can describe the full memory effect for a given system. The Laplace transform of the CF fractional derivative is given by

L[CFaDtγf(t)]=sF(s)−f(0)s+γ(1−s).(3)

In [45], two new fractional derivatives appeared. We will apply one of them, defined as: f ∈ H1(a, b), a < b, γ ∈[0, 1], then the Atangana-Baleanu (AB) fractional derivative in the Caputo sense (ABC) is

ABCaDtγ[f(t)]=B(γ)1−γ∫atf′(x)Eγ[−γ(t−x)γ1−γ]dx,(4)

where Eγ(⋅) and Eβ,γ(⋅) are the one- and two-parameter Mittag-Leffler functions defined in [9]

Eγ(z)=∑n=0∞znΓ(γn+1),γ>0,(5)
Eβ,γ(z)=∑n=0∞znΓ(βn+γ),β,γ>0.(6)

Expression (4) has a non-singular and non-local kernel. The Laplace transform for (4) is

L[ABCaDtγ]=B(γ)1−γ⋅sγF(s)−sγ−1f(0)sγ+γ1−γ,0<γ≤1,(7)

where B(γ) has the same properties as in the CF case. These formulae are necessary for our study.

3 Classical Drude models

The fundamental properties of materials play an extremely important role in developing new technologies in various areas. Among the important properties of various materials, the electromagnetic interaction with matter is of great importance.

The Drude model is known as the first realistic model to describe metals. Despite the fact that it is a very simple model, it can explain electrical conductivity, thermal conductivity, and optical properties of metals [53, 54]. This model regards metals as a classical gas of electrons executing diffusive motion. The one-dimensional equation, which describes the motion of a charged particle e with mass m affected by an external electric field E(t), is given by [55]

mdvdt+mτv=−eE(t).(8)

Regarding the left side, the first term express the acceleration of the charges induced by the electric field; the second term describes the damping factor due to electron scattering, where τ is the relaxation parameter.

For a constant electric field E(t) = E0, the solution of (8) is given by

v(t)=−eE0τm1−exp−tτ.(9)

If the electric field is defined by a pulse, like a delta Dirac distribution E = E0δ(t) [56], then the solution of (8) is

v(t)=−eE0mexp−tτ.(10)

In the case of an oscillating field E(t)= E0 cos(ωt), the solution of (8) is given by

v(t)=−eE0τm[1+(ωτ)2]⋅cos⁡(ωt)+ωτsin⁡(ωt)−exp−tτ.(11)

Typically, only the current density j→=−eNv→ is experimentally accessible [57]. Then, for the oscillating field the current density is

j(t)=σ01+(ωτ)2⋅cos⁡(ωt)+ωτsin⁡(ωt)−exp−tτE0,(12)

where σ0 = e2Nτm is the static Drude electric conductivity.

In the following sections, we analyse the Drude model from the point of view of the fractional derivatives of CF and ABC, to obtain the current density of electrons in metals for different sources.

4 Fractional Drude models with Caputo-Fabrizio derivative

To go from an ordinary differential equation to a fractional one, it is necessary to make it dimensionless. For this purpose, we make the redefinition [52]

u(t)=−meτv(t).(13)

Substituting (13) in (8), we obtain the dimensionless differential equation

dudt¯+u(t¯)=E(t¯),(14)

with dimensionless parameter t = t/τ. Therefore, we can proceed to apply a fractional derivative, either CF, ABC, or another fractional derivative.

We replace the Caputo-Fabrizio fractional derivative. Then, equation (14) takes the form

CF0Dt¯γu(t¯)+u(t¯)=E(t¯),0<γ≤1.(15)

This fractional equation could give better experimental results for the current density in metals, due to the parameter 0 < γ ≤ 1 [49]-[51]. Recall that, in principle, we assume that the materials are homogeneous and isotropic, however, this is only the case under certain conditions. In the real world, such conditions are rarely fulfilled. We will study this fractional differential equation by means of three different sources:

First Case. The source is a constant, E(t) = E0. So, applying the Laplace transform (17)

LCF0Dt¯γu(t¯)+Lu(t¯)=LE0,(16)

we have

s¯U(s¯)−u(0)s¯+γ(1−s¯)+U(s¯)=E0s¯,(17)

where s = τs. Taking the initial condition u(0) = 0, and applying the inverse Laplace transform in (17), we obtain

u(t¯)=[1−12−γexp−γ2−γt¯]E0.(18)

Considering (13), we have the fractional electron velocity

v(t;γ)=−eE0τm1−12−γexp−γ2−γ⋅tτ,0<γ≤1.(19)

In the ordinary Drude model, the current density of electrons is given by j→=−eNv→; in the fractional case, it has the form

jCF(t,γ)=σ0[1−12−γexp⁡(−γ2−γ⋅tτ)]E0,0<γ≤1,(20)

where σ0 = e2Nτm is the static Drude conductivity.

Second case. We have E0δ(t) as a source, where δ(t) is the delta Dirac function. The corresponding fractional differential equation is

CF0Dt¯γu(t¯)+u(t¯)=E0δ(t¯).(21)

The response to the impulse function was interesting, as any system can be conveniently characterized in the time domain using its impulse response, because it is the time domain version of the transfer function [56]. Applying the Laplace transform (3), we obtain

s¯U(s¯)−u(0)s¯+γ(1−s¯)+U(s¯)=E0.(22)

As above, we consider the initial condition u(0) = 0 and s = τs. Then,

u(t¯)=[1−γ2−γδ(t¯)+γ(2−γ)2exp−γ2−γt¯]E0.(23)

From (13), and taking into account j→=−env→, we have

jCF(t;γ)=Ne2m(2−γ)⋅(1−γ)δ(t)+γ2−γexp−γ2−γ⋅tτE0.(24)

Third case. An oscillatory source E(t)= E0 cos(ωt). We can write it as

CF0Dt¯γu(t¯)+u(t¯)=E0cos⁡(ωτt¯).(25)

Applying the Laplace transform (3) with the same initial condition, we have

U(s¯)=s¯[s¯+γ(1−s¯)][(s¯)2+(ωτ)2][2s¯+γ(1−s¯)]E0,(26)

where s = τs. Taking the inverse Laplace transform and considering (13), the current density is

jCF(t;γ)=(2−γ)σ0γ2+(ωτ)2(2−γ)2⋅Acos⁡(ωt)+Bsin⁡(ωt)+Cexp−γ2−γ⋅tτ(27)

where

A=(ωτ)2(1−γ)+γ22−γ,B=γ(ωτ)−γ(1−γ)ωτ,C=γ22−γ1−γ−12−γ.(28)

We can see that, in the particular case γ = 1, these expressions become A = 1; B = ωτ and C = −1, which is just the ordinary case.

5 Drude models with fractional derivatives with non-singular Mittag-Leffler kernel

Now, we will consider the same equation (14) and the same sources, but taking into account the Atangana-Baleanu fractional derivative in the Caputo sense (4). That is

ABC0Dt¯γu(t¯)+u(t¯)=E(t¯).(29)

First case. A constant source E(t) = E0. Then, applying the Laplace transform (7) in (29), we have

U(s¯)=s¯γ(1−γ)+γs¯[(2−γ)s¯γ+γ]E0.(30)

From here, we obtain the inverse Laplace transform u(t), and using expression (13), the current density is given as

jABC(t;γ)=σ0(1−12−γEγ[−γ2−γ(tτ)γ])E0,(31)

where Eγ(⋅) is the Mittag-Leffler function (5).

Second case. Delta Dirac as a source, E0δ(t). In this case, we have

ABC0Dt¯γu(t¯)+u(t¯)=E0δ(t¯).(32)

Applying the direct Laplace transform (7), we have

U(s¯)=(1−γ)s¯γ+γ(2−γ)s¯γ+γE0,(33)

taking its inverse Laplace transform and considering equation (13), we have the current density

jABC(t;γ)=Ne2m(2−γ)((1−γ)δ(t)+γ2−γ(tτ)γ−1Eγ,γ[−γ2−γ(tτ)γ])E0,(34)

where Eγ,γ is the two parametric Mitag-Leffler function (6).

Third case. An oscillatory source E(t) = E0 cos(ωt). Then,

ABC0Dt¯γu(t¯)+u(t¯)=E0cos⁡(ωτt¯).(35)

Applying the Laplace transform in (7), we obtain

U(s¯)=s¯[(1−γ)s¯γ+γ][s¯2+(ωτ)2][(2−γ)s¯γ+γ],(36)

where s = τs. We take the highest power of s as a common factor from the denominator and then expanding it in an alternating geometric series [59], we have

u(t;γ)=1−γ2−γ∑n,m=0∞(−1)n+m(ωτ)2nγm(2−γ)mΓ[2n+mγ+1](tτ)(2n+mγ)+γ2−γ∑n,m=0∞(−1)n+m(ωτ)2nγm(2−γ)mΓ[(m+1)γ+2n+1]⋅(tτ)([m+1]γ+2n),(37)

then, from (13), the current density due to an oscillating field is

jABC(t;γ)=σ0u(t;γ)E0.(38)

6 Results and discussion

Figure 1, show the current density j(t;γ) for an constant electric field E0, with some values of γ. The asymptotic approximations for expression (20) are: when t → 0, then jCF(t; γ) → σ0(1−12−γ)E0, and if t → ∞, then jCF(t;γ) → σ0E0, which is in full agreement with the ordinary case Figure 1 (a).

Figure 1 These plots correspond to equations (20) and (31), respectively, for a range of values of γ. If γ = 1, the CF and ABC derivatives give the same curve.
Figure 1

These plots correspond to equations (20) and (31), respectively, for a range of values of γ. If γ = 1, the CF and ABC derivatives give the same curve.

In the case jABC(t;γ), the asymptotic approximations to the Mittag-Leffler function for small t → 0 and larger t → ∞ times, in first approximation, are [58]

Eγ(−tγ)∼e−tγΓ(1−γ),t→0,(39)
Eγ(−tγ)∼t−γΓ(1−γ),t→∞.(40)

As a consequence, the Mittag-Leffler function interpolates for intermediate time t between the stretched exponential and the negative power law. The stretched exponential models the very fast decay for small time t, whereas, the asymptotic power law is due to very slow decay for large time t, as it can be seen in Figure 1 (b), showing the behaviour of (31) for different values of γ.

Figure 2 shows plots of the current density j(t;γ) when electrons are excited by a pulsed electric field. The plots include different values of γ. Plot (a) corresponds to jCF(t;γ) (24) and plot (b) to jABC(t;γ) (34).

Figure 2 These plots correspond to equations (24) and (34), respectively, for different values of γ.
Figure 2

These plots correspond to equations (24) and (34), respectively, for different values of γ.

Figure 3 shows graphs of the current density j(t; γ) when the electrons are excited by an oscillating electric field. The graphs have been plotted for different values of γ. In the case when the electric field is an oscillating field, the current density is an oscillatory signal affected by the fractional order γ. Two characteristics are influenced by γ: the phase and the amplitude, which are similar for the cases jCF and jABC.

Figure 3 These plots correspond to equations (27) and (38), respectively, for different values of γ. In this case ω τ = 5.0.
Figure 3

These plots correspond to equations (27) and (38), respectively, for different values of γ. In this case ω τ = 5.0.

The numerical solutions of the current density with constant E0, impulse δ(t) and alternating cos ωt field, given in Figures 1, 2 and 3, show that the current density solution of fractional derivative models in the Caputo-Fabrizio sense exhibits an exponential decay similar to the classical integer-order model (9) and (20). It is clear that the CF derivative has limitations in characterizing anomalous diffusion with non-exponential nature [43]. On the other hand, using the Atangana-Baleanu definition, the solution is given by a Mittag-Leffler function, which interpolates for intermediate time t between the stretched exponential and the negative power law. The stretched exponential (39) models the very fast decay at small t, whereas the asymptotic power law (40) is due to very slow decay at large t. Notable differences between CF and ABC occur when γ < 0.8.

In the following figures we have taken the current densities jCa and Eqs. (13) and (24) from [52], where a Caputo derivative was used, to compare the Caputo-Fabrizio and Atangana-Baleanu derivatives for the specified values of γ.

From Figures 4, 5 and 6, we can observe that, when γ = 1, the three results obtained using the derivatives of Caputo, Caputo-Fabrizio and Atangana-Baleanu are exactly the same as the ordinary case. However, as gamma takes values smaller than one, the results obtained become a little different, with notable differences when γ < 0.8. This is due to the kernel in the definitions of the fractional derivatives. The difference between the results obtained for current density using the Caputo and Atangana-Baleanu fractional derivatives for very short and very large time is the term γ2−γ.

Figure 4 Variation of current density j(t; γ) with a constant electric field using: Caputo, CF and ABC fractionals derivatives with the specified values of γ.
Figure 4

Variation of current density j(t; γ) with a constant electric field using: Caputo, CF and ABC fractionals derivatives with the specified values of γ.

Figure 5 Variation of current density j(t; γ) when the electric field is an impulse given by δ(t), using Caputo, CF and ABC fractionals derivatives and the specified values of γ.
Figure 5

Variation of current density j(t; γ) when the electric field is an impulse given by δ(t), using Caputo, CF and ABC fractionals derivatives and the specified values of γ.

Figure 6 Variation of the current density j(t; γ) when the electric field is oscillating using Caputo, CF and ABC fractionals derivatives and various values of γ.
Figure 6

Variation of the current density j(t; γ) when the electric field is oscillating using Caputo, CF and ABC fractionals derivatives and various values of γ.

The graphs in Figure 7 represent the differences between the different fractional derivatives used here. We can see again that the difference becomes notable for γ < 0.8. It can also be observed that for large times, there is no practical difference between the fractional derivatives of Caputo and ABC.

Figure 7 Graphs showing differences between the three fractional solutions and a DC electric field using several values of γ.
Figure 7

Graphs showing differences between the three fractional solutions and a DC electric field using several values of γ.

7 Conclusion

An analysis of the Drude model in the time domain was realized using fractional derivatives without singular kernels. The fractional derivatives were: Caputo-Fabrizio (CF) and Atangana-Baleanu in the Caputo sense (ABC). We considered the cases when electrons are induced by a constant electric field, a pulse described by a delta Dirac distribution, and an oscillating electric field. The obtained results in the interval γ ∈ (0, 1) show that the current density depends not only on the electric field or the fractional order derivative, but also on the definition of the fractional derivative. Also, based on the current densities jCa, Eqs. (13) and (24) of [52], we have compared fractional derivatives with singular and non-singular kernels. We conclude that the models described by fractional Atangana-Baleanu derivatives could describe a large class of complex physical problems, particularly those with non-exponential decay.

The study of these models in the frequency domain is of great interest, in order to analyse their optical properties. This work is under investigation and will be presented in the future.

Acknowledgement

The authors acknowledge their fruitful discussion with I. Lyanzuridi. Also, the authors are thankful to DICIS, University of Guanajuato, for given support. Abraham Ortega acknowledges the support provided by CONACyT under the program: Graduate Scholarships.

References

[1] Baleanu D., Fractional Calculus Models and Numerical Methods, World Scientific Publisher Company, 2012.10.1142/8180Search in Google Scholar

[2] Monje C.A., Chen Y.Q., Vinagre B.M., Xue D., Feliu V., Fractional-Order Systems and Controls, Series: Advances in Industrial control, Springer 2010.10.1007/978-1-84996-335-0Search in Google Scholar

[3] Caponetto R., Dongola G., Fortuna L., Petrás I., Fractional Order Systems: Modelling and Control Applications, World Scientific, Singapore, 2010.10.1142/7709Search in Google Scholar

[4] Baleanu D., Diethelm K., Salas E., Trujillo J.J., Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, 2012.10.1142/8180Search in Google Scholar

[5] Baleanu D., Günvenc Z.B., Tenreiro Machado J.A., (Eds) New Trends in Nanotechnology and Fractional Calculus Applications, Springer 2010.10.1007/978-90-481-3293-5Search in Google Scholar

[6] Oldham K.B., Spanier J., The fractional calculus, Academic Press, New York, 1974.Search in Google Scholar

[7] Miller K.S., Ross B., An introduction to the fractional calculus and fractional differential equations, John Wiley, NY, 1993.Search in Google Scholar

[8] Samko S.G., Kilbas A.A., Marichev O. I., Fractional Integrals and Derivatives: Theory and Applications, New York: Gordon and Breach, 1993.Search in Google Scholar

[9] Podlubny I., Fractional differential equations, Academic Press, New York, 1999.Search in Google Scholar

[10] Golmankhaneh Alireza K., Lambert L., Investigations in Dynamics: with Focus on Fractional Dynamics, Academic Publishing, 2012.Search in Google Scholar

[11] Uchaikin V., Fractional Derivatives for Physicists and Engineers, Springer 2013.10.1007/978-3-642-33911-0Search in Google Scholar

[12] Capelas de Oliveira E., Tenreiro Machado J. A., A review of definitions for fractional derivatives and integral, Mathematical Problems in Engineering, 2014 Article ID:238459.10.1155/2014/238459Search in Google Scholar

[13] Abel, N.H. Résolution d’un probléme de mécanique. Oeuvres Complétes (tomo premier, pp. 27-30). Gróndah: Chirstiana (1839a).Search in Google Scholar

[14] Caputo M., Mainardi F., A new dissipation model based on memory mechanism, Pure Appl. Geophys. 1971, 91, 134-147.10.1007/BF00879562Search in Google Scholar

[15] Wyss W., Fractional diffusion equation, J. Math. Phys., 1986, 27, 2782-2785.10.1063/1.527251Search in Google Scholar

[16] Westerlund S., Capacitor theory, IEEE Transactions on Dielectrics and Electrical Insulation, 1994, 1(5), 826-839.10.1109/94.326654Search in Google Scholar

[17] Hermann R., Fractional calculus, New Jersey: World Scientific, 2011.10.1142/8072Search in Google Scholar

[18] Metzler R., Klafter J., The random walk’s guide to anomalous diffusion a fractional dynamics approach, Phys. Reports, 2000, 339, 1-77.10.1016/S0370-1573(00)00070-3Search in Google Scholar

[19] Schiessel H., Metzler R., Blumen A., Nonnenmacher F., Generalized viscoelastic models: their fractional equations with solutions, J. Phys. A: Math. Gen. 1995, 28, 6567-6584.10.1088/0305-4470/28/23/012Search in Google Scholar

[20] Muller S., Kastner M., Brummund J., Ulbricht V., A nonlinear fractional viscoelastic material model polymers, Computational Materials Science, 2011, 50, 2938-2949.10.1016/j.commatsci.2011.05.011Search in Google Scholar

[21] Colombaro I., Giusti A., Mainardi F., A class of linear viscoelastic models based on Bessel functions, Mecanica, 2017, 52, 825-832.10.1007/s11012-016-0456-5Search in Google Scholar

[22] Giusti A., Mainardi F., A dynamic viscoelastic analogy for fluid-filled elastic tubes, Mecanica 2016, 51, 2321-2330.10.1007/s11012-016-0376-4Search in Google Scholar

[23] Meral F.C., Roytson T.J., Magin R., Fractional calculus in viscoelasticity: An experimental study, Commun. Nonlinear Sci. Numer. Simulat. 2010, 15, 939-945.10.1016/j.cnsns.2009.05.004Search in Google Scholar

[24] Giusti A., Colombaro I., Prabhakar-like fractional viscoelasticity, arXiv:1705.09246v2, 2017, math-ph.10.1016/j.cnsns.2017.08.002Search in Google Scholar

[25] Scher H., Montroll E.W., Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B. 1975, 12, 2455-2477.10.1103/PhysRevB.12.2455Search in Google Scholar

[26] Garrappa R., Mainardi F., Maione G., Models of dielectric relaxation based on completely monotone functions, Frac. Calc. and Appl. Analysis, 2016, 19, 1105-1160.10.1515/fca-2016-0060Search in Google Scholar

[27] Garrappa R., Maione G., Fractional Prabhakar derivative and applicatios in anomalous dielectrics: a numerical approach, Theory and Applications of Non-Integer Order System. Ed. Babiar, A., Czornik, A., Klamka, J., Niezabitowski, M., Springer, pp. 429-439, 2017.10.1007/978-3-319-45474-0_38Search in Google Scholar

[28] Garrappa R., Grunwald-Letnikov operators for fractional relaxation in Havriliak-Negami models, Commun. Nonlinear Sci. Numer. Simulat. 2016, 38, 178-191.10.1016/j.cnsns.2016.02.015Search in Google Scholar

[29] Mainardi F., Garrappa R., On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics, Journal of Computational Physics, 2015, 293, 70-80.10.1016/j.jcp.2014.08.006Search in Google Scholar

[30] Engheta N., On fractional calculus and fractional multipoles in electromagnetism, IEEE Trans. Antennas Propagat. 1996, 44, 554-566.10.1109/8.489308Search in Google Scholar

[31] Hussain A., Ishfaq S., Naqvi Q.A., Fractional curl operator and fractional waveguides, Progress in Electromagnetic Research, PIER. 2006, 63, 319-335.10.2528/PIER06060604Search in Google Scholar

[32] Hussain A., Faryad M., Naqvi Q.A., Fractional curl operator and fractional Chiro-waveguide, Journ. of Electromagnetic Waves and Application 2007, 21(8), 119-1129.10.1163/156939307781749696Search in Google Scholar

[33] Faryad M., Naqvi Q.A., Fractional rectangular waveguide, Progress in Electromagnetic Research, PIER, 2007, 75, 383-39610.2528/PIER07052803Search in Google Scholar

[34] Tarasov V.E., Fractional equations of curie-von Schweidler and Gauss laws, J. Phys. Condens. Matter 2008, 20, 145-212.10.1088/0953-8984/20/14/145212Search in Google Scholar

[35] Tarasov V.E., Universal electromagnetic waves in dielectric, J. Phys. Condens. Matter, 2008, 20, 175-223.10.1088/0953-8984/20/17/175223Search in Google Scholar

[36] Rosales J.J., Gómez J.F., Guía M., Tkach V.I., Fractional electromagnetic waves, LFNM, International Conference on Laser and Fiber-Optical Networks Modelling. 2011, 4-8 Sept. Kharkov, Ucraine.10.1109/LFNM.2011.6144969Search in Google Scholar

[37] Caputo M., Fabrizio M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 2015, 1, 73-85.Search in Google Scholar

[38] Losada J., Nieto J.J., Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 2015, 1, 87-92.Search in Google Scholar

[39] Atangana A., Badr Saad T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, Entropy, 2015, 17, 4439-4453, 10.3390/e17064439.Search in Google Scholar

[40] Gómez-Aguilar J.F., Córdova-Fraga T., Escalante-Martínez J.E., Calderón-Ramón C., Escobar-Jiménez R.F., Electrical circuits described by a fractional derivative with a regular kernel, Rev. Mex. Fís. 2016, 62, 144-154.Search in Google Scholar

[41] Gómez-Aguilar J.F., Escobar-Jiménez R.F., Lopez-Lopez M.G., Alvarado-Martínez V.M., Cordova-Fraga T., Electromagnetic waves in conducting media described by a fractional derivative with non-singular kernel, Journal of Electromagnetic Waves and Applications, 2016, 30, 1493-1503.10.1080/09205071.2016.1204252Search in Google Scholar

[42] Singh J., Kumar D., Al Qurashi M., Baleanu D., Analysis of a new fractional model for damped Berger’s equation, Open Phys. 2017, 15, 35-41.10.1515/phys-2017-0005Search in Google Scholar

[43] HonGuan Sun, Xiaoxiao Hao, Yong Zhang, Baleanu D., Relaxation and diffusion models with non-singular kernels, Physica 2017, A468, 590-596.10.1016/j.physa.2016.10.066Search in Google Scholar

[44] Gao F., Yang X.J., Fractional Maxwell fluid with fractional derivative without singular kernel, Thermal Science, 2016 20(3), S871-S877.10.2298/TSCI16S3871GSearch in Google Scholar

[45] Atangana A., Baleanu D., New fractional derivatives with non-local and non-singular kernel, theory and application to heat transfer model, Thermal Science, 2016, 20(2), 763-769.10.2298/TSCI160111018ASearch in Google Scholar

[46] Atangana A., Koca I., Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons and Fractals, 2016, 89, 447-454.10.1016/j.chaos.2016.02.012Search in Google Scholar

[47] Gómez-Aguilar J.F., Morales-Delgado V.F., Taneco-Hernández M.A., Baleanu D., Escobar-Jiménez R.F., Al Qurashi M.M., Analytical solutions of the electrical RLC circuit via Liouville-Caputo operators with local and non-local kernels, Entropy, 2016,18, 402.10.3390/e18080402Search in Google Scholar

[48] Saad B., Alkahtani T., Chua’s cicuit model with Atangana-Baleanu derivative with fractional order, Chaos, Solitons and Fractals 2016, 89, 547-551.10.1016/j.chaos.2016.03.020Search in Google Scholar

[49] Larisse B., Cloot H.J.J.A., Schoombie S.W., Slabbert J.P., A proposed fractional-order Gompertz model and its application to tumour growth data, Mathematical Medicine and Biology, 2015, 32, 187-207.10.1093/imammb/dqt024Search in Google Scholar PubMed

[50] Ertik, H., Calik, A.E., Sirin, H., Sen, M., Öder, B., Investigation of electrical RC circuit within the framework of fractional calculus, Rev. Mex. Fís. 2015, 61, 58-63.Search in Google Scholar

[51] Escalante-Martínez J.E., Gómez-Aguilar J.F., Calderón-Ramón C., Morales-Mendoza L.J., Cruz-Orduña I., Laguna-Camacho J.R., Experimental evaluation of viscous damping coefficient in the fractional underdamped oscillator, Advances in Mechanical Engineering 2016, 8(4), 1-12.10.1177/1687814016643068Search in Google Scholar

[52] Guía M., Rosales J.J., Martínez L., Álvarez J.A., Fractional Drude model of electrons in a metal, Rev. Mex. Fís. 2016, 62, 155-159.Search in Google Scholar

[53] Drude P., Zur elektronentheorie der metalle, Ann. der Physik, 1900, 306(3), 466-613.10.1002/andp.19003060312Search in Google Scholar

[54] Drude P., Zur elektronentheorie der metalle; II. Teil. galvanomagnetische und thermomagnetische effecte, Ann. der Physik, 1900, 308(11), 369-402.10.1002/andp.19003081102Search in Google Scholar

[55] Dressel M., Grüner G., Electrodynamics of solids: optical properties of electrons in matter, Cambridge University Press, 2002.10.1017/CBO9780511606168Search in Google Scholar

[56] Banchuin R., Novel expressions for time domain response of fractance devices, Cogent Engineering, 2017, 4. 1320823.10.1080/23311916.2017.1320823Search in Google Scholar

[57] Dressel M., Scheffler M., Verifying the Drude response, Ann. Phys. 2006, 15, 535-544.10.1002/andp.200510198Search in Google Scholar

[58] Mainardi F., On some properties of the Mittag-Leffler function Eα(−tα), completely monotone for t > 0, with 0 < α < 1, Discrete and Continuos Dynamical systems, Series B, 2014, 19, 2267-2278.10.3934/dcdsb.2014.19.2267Search in Google Scholar

[59] Alzoubi F.Y., Alqadi M.K., Al-Khateeb H.M., Saadeh S.M., Ayoub N.Y., Solution of a fractional undamped forced oscillator, Jordan Journal of Physics, 2012, 5, 129-134.Search in Google Scholar

Received: 2017-4-5
Accepted: 2017-8-4
Published Online: 2017-11-6

© 2017 L. Martínez et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

  1. Regular Articles
  2. Analysis of a New Fractional Model for Damped Bergers’ Equation
  3. Regular Articles
  4. Optimal homotopy perturbation method for nonlinear differential equations governing MHD Jeffery-Hamel flow with heat transfer problem
  5. Regular Articles
  6. Semi- analytic numerical method for solution of time-space fractional heat and wave type equations with variable coefficients
  7. Regular Articles
  8. Investigation of a curve using Frenet frame in the lightlike cone
  9. Regular Articles
  10. Construction of complex networks from time series based on the cross correlation interval
  11. Regular Articles
  12. Nonlinear Schrödinger approach to European option pricing
  13. Regular Articles
  14. A modified cubic B-spline differential quadrature method for three-dimensional non-linear diffusion equations
  15. Regular Articles
  16. A new miniaturized negative-index meta-atom for tri-band applications
  17. Regular Articles
  18. Seismic stability of the survey areas of potential sites for the deep geological repository of the spent nuclear fuel
  19. Regular Articles
  20. Distributed containment control of heterogeneous fractional-order multi-agent systems with communication delays
  21. Regular Articles
  22. Sensitivity analysis and economic optimization studies of inverted five-spot gas cycling in gas condensate reservoir
  23. Regular Articles
  24. Quantum mechanics with geometric constraints of Friedmann type
  25. Regular Articles
  26. Modeling and Simulation for an 8 kW Three-Phase Grid-Connected Photo-Voltaic Power System
  27. Regular Articles
  28. Application of the optimal homotopy asymptotic method to nonlinear Bingham fluid dampers
  29. Regular Articles
  30. Analysis of Drude model using fractional derivatives without singular kernels
  31. Regular Articles
  32. An unsteady MHD Maxwell nanofluid flow with convective boundary conditions using spectral local linearization method
  33. Regular Articles
  34. New analytical solutions for conformable fractional PDEs arising in mathematical physics by exp-function method
  35. Regular Articles
  36. Quantum mechanical calculation of electron spin
  37. Regular Articles
  38. CO2 capture by polymeric membranes composed of hyper-branched polymers with dense poly(oxyethylene) comb and poly(amidoamine)
  39. Regular Articles
  40. Chain on a cone
  41. Regular Articles
  42. Multi-task feature learning by using trace norm regularization
  43. Regular Articles
  44. Superluminal tunneling of a relativistic half-integer spin particle through a potential barrier
  45. Regular Articles
  46. Neutrosophic triplet normed space
  47. Regular Articles
  48. Lie algebraic discussion for affinity based information diffusion in social networks
  49. Regular Articles
  50. Radiation dose and cancer risk estimates in helical CT for pulmonary tuberculosis infections
  51. Regular Articles
  52. A comparison study of steady-state vibrations with single fractional-order and distributed-order derivatives
  53. Regular Articles
  54. Some new remarks on MHD Jeffery-Hamel fluid flow problem
  55. Regular Articles
  56. Numerical investigation of magnetohydrodynamic slip flow of power-law nanofluid with temperature dependent viscosity and thermal conductivity over a permeable surface
  57. Regular Articles
  58. Charge conservation in a gravitational field in the scalar ether theory
  59. Regular Articles
  60. Measurement problem and local hidden variables with entangled photons
  61. Regular Articles
  62. Compression of hyper-spectral images using an accelerated nonnegative tensor decomposition
  63. Regular Articles
  64. Fabrication and application of coaxial polyvinyl alcohol/chitosan nanofiber membranes
  65. Regular Articles
  66. Calculating degree-based topological indices of dominating David derived networks
  67. Regular Articles
  68. The structure and conductivity of polyelectrolyte based on MEH-PPV and potassium iodide (KI) for dye-sensitized solar cells
  69. Regular Articles
  70. Chiral symmetry restoration and the critical end point in QCD
  71. Regular Articles
  72. Numerical solution for fractional Bratu’s initial value problem
  73. Regular Articles
  74. Structure and optical properties of TiO2 thin films deposited by ALD method
  75. Regular Articles
  76. Quadruple multi-wavelength conversion for access network scalability based on cross-phase modulation in an SOA-MZI
  77. Regular Articles
  78. Application of ANNs approach for wave-like and heat-like equations
  79. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  80. Study on node importance evaluation of the high-speed passenger traffic complex network based on the Structural Hole Theory
  81. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  82. A mathematical/physics model to measure the role of information and communication technology in some economies: the Chinese case
  83. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  84. Numerical modeling of the thermoelectric cooler with a complementary equation for heat circulation in air gaps
  85. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  86. On the libration collinear points in the restricted three – body problem
  87. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  88. Research on Critical Nodes Algorithm in Social Complex Networks
  89. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  90. A simulation based research on chance constrained programming in robust facility location problem
  91. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  92. A mathematical/physics carbon emission reduction strategy for building supply chain network based on carbon tax policy
  93. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  94. Mathematical analysis of the impact mechanism of information platform on agro-product supply chain and agro-product competitiveness
  95. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  96. A real negative selection algorithm with evolutionary preference for anomaly detection
  97. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  98. A privacy-preserving parallel and homomorphic encryption scheme
  99. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  100. Random walk-based similarity measure method for patterns in complex object
  101. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  102. A Mathematical Study of Accessibility and Cohesion Degree in a High-Speed Rail Station Connected to an Urban Bus Transport Network
  103. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  104. Design and Simulation of the Integrated Navigation System based on Extended Kalman Filter
  105. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  106. Oil exploration oriented multi-sensor image fusion algorithm
  107. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  108. Analysis of Product Distribution Strategy in Digital Publishing Industry Based on Game-Theory
  109. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  110. Expanded Study on the accumulation effect of tourism under the constraint of structure
  111. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  112. Unstructured P2P Network Load Balance Strategy Based on Multilevel Partitioning of Hypergraph
  113. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  114. Research on the method of information system risk state estimation based on clustering particle filter
  115. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  116. Demand forecasting and information platform in tourism
  117. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  118. Physical-chemical properties studying of molecular structures via topological index calculating
  119. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  120. Local kernel nonparametric discriminant analysis for adaptive extraction of complex structures
  121. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  122. City traffic flow breakdown prediction based on fuzzy rough set
  123. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  124. Conservation laws for a strongly damped wave equation
  125. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  126. Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution
  127. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  128. Computing the Ediz eccentric connectivity index of discrete dynamic structures
  129. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  130. A discrete epidemic model for bovine Babesiosis disease and tick populations
  131. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  132. Study on maintaining formations during satellite formation flying based on SDRE and LQR
  133. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  134. Relationship between solitary pulmonary nodule lung cancer and CT image features based on gradual clustering
  135. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  136. A novel fast target tracking method for UAV aerial image
  137. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  138. Fuzzy comprehensive evaluation model of interuniversity collaborative learning based on network
  139. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  140. Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation
  141. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  142. After notes on self-similarity exponent for fractal structures
  143. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  144. Excitation probability and effective temperature in the stationary regime of conductivity for Coulomb Glasses
  145. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  146. Comparisons of feature extraction algorithm based on unmanned aerial vehicle image
  147. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  148. Research on identification method of heavy vehicle rollover based on hidden Markov model
  149. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  150. Classifying BCI signals from novice users with extreme learning machine
  151. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  152. Topics on data transmission problem in software definition network
  153. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  154. Statistical inferences with jointly type-II censored samples from two Pareto distributions
  155. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  156. Estimation for coefficient of variation of an extension of the exponential distribution under type-II censoring scheme
  157. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  158. Analysis on trust influencing factors and trust model from multiple perspectives of online Auction
  159. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  160. Coupling of two-phase flow in fractured-vuggy reservoir with filling medium
  161. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  162. Production decline type curves analysis of a finite conductivity fractured well in coalbed methane reservoirs
  163. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  164. Flow Characteristic and Heat Transfer for Non-Newtonian Nanofluid in Rectangular Microchannels with Teardrop Dimples/Protrusions
  165. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  166. The size prediction of potential inclusions embedded in the sub-surface of fused silica by damage morphology
  167. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  168. Research on carbonate reservoir interwell connectivity based on a modified diffusivity filter model
  169. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  170. The method of the spatial locating of macroscopic throats based-on the inversion of dynamic interwell connectivity
  171. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  172. Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method
  173. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  174. A volumetric ablation model of EPDM considering complex physicochemical process in porous structure of char layer
  175. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  176. Numerical simulation on ferrofluid flow in fractured porous media based on discrete-fracture model
  177. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  178. Macroscopic lattice Boltzmann model for heat and moisture transfer process with phase transformation in unsaturated porous media during freezing process
  179. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  180. Modelling of intermittent microwave convective drying: parameter sensitivity
  181. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  182. Simulating gas-water relative permeabilities for nanoscale porous media with interfacial effects
  183. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  184. Simulation of counter-current imbibition in water-wet fractured reservoirs based on discrete-fracture model
  185. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  186. Investigation effect of wettability and heterogeneity in water flooding and on microscopic residual oil distribution in tight sandstone cores with NMR technique
  187. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  188. Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs
  189. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  190. Special Issue: Ever New "Loopholes" in Bell’s Argument and Experimental Tests
  191. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  192. The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  193. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  194. Erratum to: The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  195. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  196. Rhetoric, logic, and experiment in the quantum nonlocality debate
  197. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  198. What If Quantum Theory Violates All Mathematics?
  199. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  200. Relativity, anomalies and objectivity loophole in recent tests of local realism
  201. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  202. The photon identification loophole in EPRB experiments: computer models with single-wing selection
  203. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  204. Bohr against Bell: complementarity versus nonlocality
  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  206. Is Einsteinian no-signalling violated in Bell tests?
  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  208. Bell’s “Theorem”: loopholes vs. conceptual flaws
  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  210. Nonrecurrence and Bell-like inequalities
  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  212. Three-dimensional computer models of electrospinning systems
  213. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  214. Electric field computation and measurements in the electroporation of inhomogeneous samples
  215. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  216. Modelling of magnetostriction of transformer magnetic core for vibration analysis
  217. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  218. Comparison of the fractional power motor with cores made of various magnetic materials
  219. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  220. Dynamics of the line-start reluctance motor with rotor made of SMC material
  221. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  222. Inhomogeneous dielectrics: conformal mapping and finite-element models
  223. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  224. Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
  225. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  226. Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  228. Current superimposition variable flux reluctance motor with 8 salient poles
  229. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  230. Modelling axial vibration in windings of power transformers
  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
https://doi.org/10.1515/phys-2017-0073
Keywords for this article
Fractional Calculus; Drude model; Caputo-Fabrizio derivative; Atangana-Baleanu derivative
Creative Commons
BY-NC-ND 4.0

Articles in the same Issue

  1. Regular Articles
  2. Analysis of a New Fractional Model for Damped Bergers’ Equation
  3. Regular Articles
  4. Optimal homotopy perturbation method for nonlinear differential equations governing MHD Jeffery-Hamel flow with heat transfer problem
  5. Regular Articles
  6. Semi- analytic numerical method for solution of time-space fractional heat and wave type equations with variable coefficients
  7. Regular Articles
  8. Investigation of a curve using Frenet frame in the lightlike cone
  9. Regular Articles
  10. Construction of complex networks from time series based on the cross correlation interval
  11. Regular Articles
  12. Nonlinear Schrödinger approach to European option pricing
  13. Regular Articles
  14. A modified cubic B-spline differential quadrature method for three-dimensional non-linear diffusion equations
  15. Regular Articles
  16. A new miniaturized negative-index meta-atom for tri-band applications
  17. Regular Articles
  18. Seismic stability of the survey areas of potential sites for the deep geological repository of the spent nuclear fuel
  19. Regular Articles
  20. Distributed containment control of heterogeneous fractional-order multi-agent systems with communication delays
  21. Regular Articles
  22. Sensitivity analysis and economic optimization studies of inverted five-spot gas cycling in gas condensate reservoir
  23. Regular Articles
  24. Quantum mechanics with geometric constraints of Friedmann type
  25. Regular Articles
  26. Modeling and Simulation for an 8 kW Three-Phase Grid-Connected Photo-Voltaic Power System
  27. Regular Articles
  28. Application of the optimal homotopy asymptotic method to nonlinear Bingham fluid dampers
  29. Regular Articles
  30. Analysis of Drude model using fractional derivatives without singular kernels
  31. Regular Articles
  32. An unsteady MHD Maxwell nanofluid flow with convective boundary conditions using spectral local linearization method
  33. Regular Articles
  34. New analytical solutions for conformable fractional PDEs arising in mathematical physics by exp-function method
  35. Regular Articles
  36. Quantum mechanical calculation of electron spin
  37. Regular Articles
  38. CO2 capture by polymeric membranes composed of hyper-branched polymers with dense poly(oxyethylene) comb and poly(amidoamine)
  39. Regular Articles
  40. Chain on a cone
  41. Regular Articles
  42. Multi-task feature learning by using trace norm regularization
  43. Regular Articles
  44. Superluminal tunneling of a relativistic half-integer spin particle through a potential barrier
  45. Regular Articles
  46. Neutrosophic triplet normed space
  47. Regular Articles
  48. Lie algebraic discussion for affinity based information diffusion in social networks
  49. Regular Articles
  50. Radiation dose and cancer risk estimates in helical CT for pulmonary tuberculosis infections
  51. Regular Articles
  52. A comparison study of steady-state vibrations with single fractional-order and distributed-order derivatives
  53. Regular Articles
  54. Some new remarks on MHD Jeffery-Hamel fluid flow problem
  55. Regular Articles
  56. Numerical investigation of magnetohydrodynamic slip flow of power-law nanofluid with temperature dependent viscosity and thermal conductivity over a permeable surface
  57. Regular Articles
  58. Charge conservation in a gravitational field in the scalar ether theory
  59. Regular Articles
  60. Measurement problem and local hidden variables with entangled photons
  61. Regular Articles
  62. Compression of hyper-spectral images using an accelerated nonnegative tensor decomposition
  63. Regular Articles
  64. Fabrication and application of coaxial polyvinyl alcohol/chitosan nanofiber membranes
  65. Regular Articles
  66. Calculating degree-based topological indices of dominating David derived networks
  67. Regular Articles
  68. The structure and conductivity of polyelectrolyte based on MEH-PPV and potassium iodide (KI) for dye-sensitized solar cells
  69. Regular Articles
  70. Chiral symmetry restoration and the critical end point in QCD
  71. Regular Articles
  72. Numerical solution for fractional Bratu’s initial value problem
  73. Regular Articles
  74. Structure and optical properties of TiO2 thin films deposited by ALD method
  75. Regular Articles
  76. Quadruple multi-wavelength conversion for access network scalability based on cross-phase modulation in an SOA-MZI
  77. Regular Articles
  78. Application of ANNs approach for wave-like and heat-like equations
  79. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  80. Study on node importance evaluation of the high-speed passenger traffic complex network based on the Structural Hole Theory
  81. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  82. A mathematical/physics model to measure the role of information and communication technology in some economies: the Chinese case
  83. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  84. Numerical modeling of the thermoelectric cooler with a complementary equation for heat circulation in air gaps
  85. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  86. On the libration collinear points in the restricted three – body problem
  87. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  88. Research on Critical Nodes Algorithm in Social Complex Networks
  89. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  90. A simulation based research on chance constrained programming in robust facility location problem
  91. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  92. A mathematical/physics carbon emission reduction strategy for building supply chain network based on carbon tax policy
  93. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  94. Mathematical analysis of the impact mechanism of information platform on agro-product supply chain and agro-product competitiveness
  95. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  96. A real negative selection algorithm with evolutionary preference for anomaly detection
  97. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  98. A privacy-preserving parallel and homomorphic encryption scheme
  99. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  100. Random walk-based similarity measure method for patterns in complex object
  101. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  102. A Mathematical Study of Accessibility and Cohesion Degree in a High-Speed Rail Station Connected to an Urban Bus Transport Network
  103. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  104. Design and Simulation of the Integrated Navigation System based on Extended Kalman Filter
  105. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  106. Oil exploration oriented multi-sensor image fusion algorithm
  107. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  108. Analysis of Product Distribution Strategy in Digital Publishing Industry Based on Game-Theory
  109. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  110. Expanded Study on the accumulation effect of tourism under the constraint of structure
  111. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  112. Unstructured P2P Network Load Balance Strategy Based on Multilevel Partitioning of Hypergraph
  113. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  114. Research on the method of information system risk state estimation based on clustering particle filter
  115. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  116. Demand forecasting and information platform in tourism
  117. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  118. Physical-chemical properties studying of molecular structures via topological index calculating
  119. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  120. Local kernel nonparametric discriminant analysis for adaptive extraction of complex structures
  121. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  122. City traffic flow breakdown prediction based on fuzzy rough set
  123. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  124. Conservation laws for a strongly damped wave equation
  125. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  126. Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution
  127. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  128. Computing the Ediz eccentric connectivity index of discrete dynamic structures
  129. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  130. A discrete epidemic model for bovine Babesiosis disease and tick populations
  131. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  132. Study on maintaining formations during satellite formation flying based on SDRE and LQR
  133. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  134. Relationship between solitary pulmonary nodule lung cancer and CT image features based on gradual clustering
  135. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  136. A novel fast target tracking method for UAV aerial image
  137. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  138. Fuzzy comprehensive evaluation model of interuniversity collaborative learning based on network
  139. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  140. Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation
  141. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  142. After notes on self-similarity exponent for fractal structures
  143. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  144. Excitation probability and effective temperature in the stationary regime of conductivity for Coulomb Glasses
  145. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  146. Comparisons of feature extraction algorithm based on unmanned aerial vehicle image
  147. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  148. Research on identification method of heavy vehicle rollover based on hidden Markov model
  149. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  150. Classifying BCI signals from novice users with extreme learning machine
  151. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  152. Topics on data transmission problem in software definition network
  153. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  154. Statistical inferences with jointly type-II censored samples from two Pareto distributions
  155. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  156. Estimation for coefficient of variation of an extension of the exponential distribution under type-II censoring scheme
  157. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  158. Analysis on trust influencing factors and trust model from multiple perspectives of online Auction
  159. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  160. Coupling of two-phase flow in fractured-vuggy reservoir with filling medium
  161. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  162. Production decline type curves analysis of a finite conductivity fractured well in coalbed methane reservoirs
  163. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  164. Flow Characteristic and Heat Transfer for Non-Newtonian Nanofluid in Rectangular Microchannels with Teardrop Dimples/Protrusions
  165. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  166. The size prediction of potential inclusions embedded in the sub-surface of fused silica by damage morphology
  167. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  168. Research on carbonate reservoir interwell connectivity based on a modified diffusivity filter model
  169. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  170. The method of the spatial locating of macroscopic throats based-on the inversion of dynamic interwell connectivity
  171. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  172. Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method
  173. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  174. A volumetric ablation model of EPDM considering complex physicochemical process in porous structure of char layer
  175. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  176. Numerical simulation on ferrofluid flow in fractured porous media based on discrete-fracture model
  177. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  178. Macroscopic lattice Boltzmann model for heat and moisture transfer process with phase transformation in unsaturated porous media during freezing process
  179. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  180. Modelling of intermittent microwave convective drying: parameter sensitivity
  181. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  182. Simulating gas-water relative permeabilities for nanoscale porous media with interfacial effects
  183. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  184. Simulation of counter-current imbibition in water-wet fractured reservoirs based on discrete-fracture model
  185. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  186. Investigation effect of wettability and heterogeneity in water flooding and on microscopic residual oil distribution in tight sandstone cores with NMR technique
  187. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
  188. Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs
  189. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  190. Special Issue: Ever New "Loopholes" in Bell’s Argument and Experimental Tests
  191. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  192. The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  193. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  194. Erratum to: The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  195. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  196. Rhetoric, logic, and experiment in the quantum nonlocality debate
  197. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  198. What If Quantum Theory Violates All Mathematics?
  199. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  200. Relativity, anomalies and objectivity loophole in recent tests of local realism
  201. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  202. The photon identification loophole in EPRB experiments: computer models with single-wing selection
  203. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  204. Bohr against Bell: complementarity versus nonlocality
  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  206. Is Einsteinian no-signalling violated in Bell tests?
  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  208. Bell’s “Theorem”: loopholes vs. conceptual flaws
  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  210. Nonrecurrence and Bell-like inequalities
  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  212. Three-dimensional computer models of electrospinning systems
  213. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  214. Electric field computation and measurements in the electroporation of inhomogeneous samples
  215. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  216. Modelling of magnetostriction of transformer magnetic core for vibration analysis
  217. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  218. Comparison of the fractional power motor with cores made of various magnetic materials
  219. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  220. Dynamics of the line-start reluctance motor with rotor made of SMC material
  221. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  222. Inhomogeneous dielectrics: conformal mapping and finite-element models
  223. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  224. Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
  225. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  226. Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  228. Current superimposition variable flux reluctance motor with 8 salient poles
  229. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  230. Modelling axial vibration in windings of power transformers
  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 8.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/phys-2017-0073/html
Scroll to top button