Electronics ceramics grain boundaries and complex fractal dimension

Analysis of ceramic grain boundaries, esspecially for BaTiO3 , is also important for its dielectric and conductive properties. In this regard, the fractal analysis was highlighted. The grain contacts geometry based on intergranular contact surface fractal morphology was the subject of our long term research. A new approach based on complex dimension fractal geometry and correlation between microstructurenanostructure and rare-earth properties and other additives doped BaTiO3-ceramics and electronics properties, is applied . In addition to the continuous type of scaling typical for real standard fractal objects, complex objects are considered here, which also have a discrete scaling symmetry with logarithmic space period. That rely on their appearance on the various , micro and macro, electrical and other properties of BaTiO3-ceramics

Modeling of bioimpedance for human skin based on fractional distributed-order modified cole model

Eksperimentalni podaci otpornosti i računa necelobrojnog reda koriste se za modeliranje bioimpedansnih osobina ljudske kože. Uveli smo i predložili modifikovani Kole model koristeći pri tom operator distribuiranog necelog reda koji je zasnovan na Caputo-Weyl-ovim izvodima necelog reda.Naš predloženi model predstavlja izmenjen jedno-disperzijski Kole model, jer uvodi nove parametre k i σ u jedno-disperzijskoj Kole impedansnoj jednačini. Ovi parametri karakterišu širinu intervala oko frakcionog indeksa α i oni su važni za precizniji opis bioimpedansnih osobina ljudske kože. Predloženi modifikovani Kole model mnogo bolje daje fitovanje date eksperimentalne krive u datom frekventnom opsegu u poređenju sa sa postojećim Kole modelima. Fitovanje je urađeno primenom Levenberg-Marquardt algoritma nelinearnih najmanjih kvadrata.Electrical impedance measurement data and fractional calculus have been utilized for modeling bioimpedance properties of human skin. We introduced and proposed revisited Cole model using modified distributed order operator based on the Caputo-Weyl fractional derivatives. Our proposed model presents essentially modified single-dispersion Cole model, since it introduces a new parameters k and σ in single-dispersion Cole impedance equation. These parameters characterize the width of interval around fractional index α and they are important for more accurate describing bioimpedance properties of human skin. The impedance spectrum was measured in a finite frequency range up to 100 kHz. Our proposed modified Cole model fits much better to experimental curve in a given frequency range compared to existing Cole models. The fitting is done using the Levenberg-Marquardt nonlinear least squares

Fractional Calculus Model of Electrical Impedance Applied to Human Skin

Fractional calculus is a mathematical approach dealing with derivatives and integrals of arbitrary and complex orders. Therefore, it adds a new dimension to understand and describe basic nature and behavior of complex systems in an improved way. Here we use the fractional calculus for modeling electrical properties of biological systems. We derived a new class of generalized models for electrical impedance and applied them to human skin by experimental data fitting. The primary model introduces new generalizations of: 1) Weyl fractional derivative operator, 2) Cole equation, and 3) Constant Phase Element (CPE). These generalizations were described by the novel equation which presented parameter (beta) related to remnant memory and corrected four essential parameters (R-0, R-infinity, alpha, tau(alpha)). We further generalized single generalized element by introducing specific partial sum of Maclaurin series determined by parameters (beta(*), gamma,delta ...). We defined individual primary model elements and their serial combination models by the appropriate equations and electrical schemes. Cole equation is a special case of our generalized class of models for beta* = gamma = delta = ... = 0. Previous bioimpedance data analyses of living systems using basic Cole and serial Cole models show significant imprecisions. Our new class of models considerably improves the quality of fitting, evaluated by mean square errors, for bioimpedance data obtained from human skin. Our models with new parameters presented in specific partial sum of Maclaurin series also extend representation, understanding and description of complex systems electrical properties in terms of remnant memory effects

Modeling of bioimpedance for human skin based on fractional distributed-order modified cole model

Eksperimentalni podaci otpornosti i računa necelobrojnog reda koriste se za modeliranje bioimpedansnih osobina ljudske kože. Uveli smo i predložili modifikovani Kole model koristeći pri tom operator distribuiranog necelog reda koji je zasnovan na Caputo-Weyl-ovim izvodima necelog reda.Naš predloženi model predstavlja izmenjen jedno-disperzijski Kole model, jer uvodi nove parametre k i σ u jedno-disperzijskoj Kole impedansnoj jednačini. Ovi parametri karakterišu širinu intervala oko frakcionog indeksa α i oni su važni za precizniji opis bioimpedansnih osobina ljudske kože. Predloženi modifikovani Kole model mnogo bolje daje fitovanje date eksperimentalne krive u datom frekventnom opsegu u poređenju sa sa postojećim Kole modelima. Fitovanje je urađeno primenom Levenberg-Marquardt algoritma nelinearnih najmanjih kvadrata.Electrical impedance measurement data and fractional calculus have been utilized for modeling bioimpedance properties of human skin. We introduced and proposed revisited Cole model using modified distributed order operator based on the Caputo-Weyl fractional derivatives. Our proposed model presents essentially modified single-dispersion Cole model, since it introduces a new parameters k and σ in single-dispersion Cole impedance equation. These parameters characterize the width of interval around fractional index α and they are important for more accurate describing bioimpedance properties of human skin. The impedance spectrum was measured in a finite frequency range up to 100 kHz. Our proposed modified Cole model fits much better to experimental curve in a given frequency range compared to existing Cole models. The fitting is done using the Levenberg-Marquardt nonlinear least squares

Generalized Lorentz model description of electrical, dielectric, conductive and magnetic processes two-time relaxations in BaTiO3 ceramics with constitutive relations

In this study, generalized Lorentz model is considered in the framework of dielectric, conductive and/or magnetic responses of materials. Beside positive temperature coefficient of resistivity (PTCR) materials (current stabilizers, time delay circuits and current limiters for overvoltage or overcurrent protection, temperature sensors, self-heating, …), magnetic properties indicate to multifunctional or specific applications (for example, nanocubic technologies). AC conductivity studies of various BaTiO3 ceramics or similar ceramics produced equivalent circuits with impedance spectra, usually within the framework of RCPE elements serial connection (CPE - constant phase element) or Cole element. One of the first models that explains PTC effect is the Heywang model, in terms of grain boundaries potential barriers of the Shottky type. Dielectric frequency spectra can be described in similar relationships. However, magnetic features of BaTiO3 ceramics are not well described. In this presentation all three behaviors (dielectric, conductive and magnetic) of materials and their relationships are considered in the case of electric or magnetic alternate fields, which are the basis for experimental measurements

Characterization of ptc effect in batio(3)-ceramics as a special phase transition - fractal approach

The applications of BaTiO3-ceramics are very important and constantly increasing nowadays. In that sense, we analyzed some phenomena related to inter granular effects. We used experimental data based on Murata powders and processing technology. Our original contribution to Heywang-Jonker-Daniels inter-granular capacity model is based on thermodynamic fractal analysis applied on phase transition in ceramic structures. In this case, PTCR effect has a diffuse first-order phase transition character in a modified Landau theory-fractal approach. Its basic properties are considered. This is an original contribution as a bridge between theoretical aspects of BaTiO3-ceramics and experimental results

Complex fractal dimension and possible application in electronic ceramics

Considering the extremely growing exigency for further miniaturization and a higher level of packaging of electronic circuits and components, this paper is aimed at developing a more sophisticated application of fractals. In this sense, the progress in the development of the mathematical-physical tool in further upgrading of fractal microelectronics is presented here. Barium titanate samples with bayi yttrium samples are used as the experimental basis under conditions of using the highest levels of nanotechnology, especially grain deposition. In this regard, the ideas of complex fractal analysis will be elaborated in this paper. Examples of complex fractal dimensions are known in the literature. The relationship between fBm (Fractional Brown motion) and Bm is given by the left-sided Riemann-Liuville fractional integral When is H=0.5, in the above equation, fBm and Bm is matching.For H >0.5 the process is positive, and for H <0.5 negatively correlated. It shows that the imaginary part of the fractal dimension is translated into log-periodic modulation, which completes the behavior by leading a degree law, and is based on discrete fractal symmetry. In particular, complex Brownian motion can be generated based on 1d complex Brownian motion in matlab code. There is also a corresponding fractional calculus of complex order. Other parallels with electrical processes in BatiO3 ceramics are also possible

Generalized Lorentz model description-Caputo-Fabrizio fractional derivative approach, of electrical, dielectric, conductive and magnetic processes in materials

In this study, generalized Lorentz model is basic one-particle model in the framework of dielectric, conductive and/or magnetic responses of materials. AC conductivity studies of various BaTiO3 or similar ceramics produced equivalent circuits with impedance spectra, usually within the framework of RCPE elements serial connection (CPE - constant phase element) or Cole element. This element, in the generalized Lorentz model, corresponds to Caputo fractional derivative, who, as operator, contains a singular integral kernel in itself. However, in the literature, fractional derivatives with a non singular integral kernels have recently emerged. One of them is a Caputo-Fabrizio fractional derivative. In this work, physical basics and all three behaviors (dielectric, conductive and magnetic) of materials and their relationships are considered in the case of electric or magnetic alternate fields, which are the tools for experimental measurements

CHARACTERIZATION OF PTC EFFECT IN BATIO3-CERAMICS AS A SPECIAL PHASE TRANSITION – FRACTAL APPROACH

The applications of BaTiO3-ceramics are very important and constantly increasing nowadays. In that sense, we analyzed some phenomena related to inter-granular effects. We used experimental data based on Murata powders and processing technology. Our original contribution to Heywang-Jonker-Daniels inter-granular capacity model is based on thermodynamic fractal analysis applied on phase transition in ceramic structures. In this case, PTCR effect has a diffuse first-order phase transition character in a modified Landau theory-fractal approach. Its basic properties are considered. This is an original contribution as a bridge between theoretical aspects of BaTiO3-ceramics and experimental results

Multistep generalized transformation method applied to solving equations of discrete and continuous time-fractional enzyme kinetics

In this paper, Caputo based Michaelis–Menten kinetic model based on Time Scale Calculus (TSC) is proposed. The main reason for its consideration is a study of tumor cells population growth dynamics. In the particular case discrete-continuous time kinetics, Michaelis–Menten model is numerically treated, using a new algorithm proposed by authors, called multistep generalized difference transformation method (MSGDETM). In addition numerical simulations are performed and is shown that it represents the upgrade of the multi-step variant of generalized differential transformation method (MSGDTM). A possible conditions for its further development are discussed and possible experimental verification is described.This is the peer-reviewed version of the article: Vosika, Z., Mitić, V.V., Vasić, A., Lazović, G., Matija, L., Kocić, L.M., 2017. Multistep generalized transformation method applied to solving equations of discrete and continuous time-fractional enzyme kinetics. Communications in Nonlinear Science and Numerical Simulation 44, 373–389. [https://doi.org/10.1016/j.cnsns.2016.08.024