Derivatives fractional calculus

With the help of fractional calculus, Dassas and Duby123 have worked on the problem of diffusion towards the fractal interfaces. They have proposed the following generalized diffusion equation involving a fractional derivative operator ... 

In this section we describe some of the essential features of fractal functions starting from the simple dynamical processes described by functions that are fractal (such as the Weierstrass function) and that are continuous everywhere but are nowhere differentiable. This idea of nondifferentiability leads to the introduction of the elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. We find that the relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, the changes in time of phenomena that are best described by fractal functions are probably best described by fractional equations of motion, as well. In any event, this latter perspective is the one we developed elsewhere [52] and discuss herein. Others have also made inquiries along these lines [70] ... 

It is interesting to investigate whether fractional calculus, which generates the operation of derivation and integration to fractional order, can provide a possible calculus to deal with fractals. In fact there has been a surge of activity in recent times which supports this point of view. This possible connection between fractals and fractional calculus gives rise to various interesting questions ... 

Of course, the fractional calculus does not in itself constitute a physical/ biological theory however, one requires such a theory in order to interpret the fractional derivatives and integrals in terms of physical/biological phenomena. We therefore follow a pedagogical approach and examine the simple relaxation process described by the rate equation... 

The fractional integral-differential operators (fractional calculus) present a generalization of integration and derivation to noninteger order (fractional) operators. Fust, one can generalize the differential and integral operators into one fundamental Df operator t, which is known as fractional calculus ... 

For lumped elements, e.g. resistors, capacitors or combinations of these elements, the differential equations, impedances and VSR are well-known [4]. Distributed elements, i.e. Warburg impedance. Constant Phase Element, or parallel connections like RCPE, also known as ZARC or Cole-Cole element, have non-integer exponents a of the complex frequency s in frequency domain. This corresponds to fractional differential equations in time domain and thus the calculation of the VSR requires fractional calculus, as can be seen in the following derivations. 

In order to derive the VSR of the RCPE step,RCPE(0 u time domain, the inverse Laplace transform of Equation (8) has to be derived using fractional calculus, see Equations (9)-(16). 

A method was proposed for the parameterization of impedance based models in the time domain, by deriving the corresponding time domain model equation with inverse Laplace transform of the frequency domain model equation assuming a current step excitation. This excitation signal has been chosen, since it can be easily applied to a Li-ion cell in an experiment, allows the analytical calculation of the time domain model equation and is included in the definition of the inner resistance. The voltage step responses of model elements were presented for lumped elements and derived for distributed model elements that have underlying fractional differential equations using fractional calculus. The determination of the inner resistance from an impedance spectrum was proposed as a possible application for this method. Tests on measurement data showed that this method works well for temperatures around room temperature and current excitation amplitudes up to 10 C. This technique can be used for comparisons of measured impedance spectra with conventionally determined inner resistances. 

Mittag-Lefelvre function [59] usage is one more method of a diagrams o - e description within the frameworks of the fractional derivatives mathematical calculus. A nonlinear dependences, similar to a diagrams o - e for pol5miers, are described with the aid of the following equation [65] ... 

The power law form was apparently first proposed by Nutting (1921). In recent years [for example Torvik and Bagley (1984) and earlier papers by those authors and Koeller (1984)] this form [specifically (1.6.46) or (1.6.52a) plus constant] has inspired the application of the fractional calculus (calculus of fractional derivatives) to the description of viscoelastic phenomena, a possibility which had been suggested by several authors, notably Rabotnov (1948, 1969,... 

Fractional calculus is referred to derivatives and integrals of order G K or more generally to complex order y = p + it], p K, t] G M. There are many different definitions of fractional operators such as Riemann-Liouville, Riesz, Marchaud, Caputo, etc. (see, e.g., Podlubny 1999 Samko et al. 1993). The various definitions differ with each another by intervals of integration or are simply adaptations of the Riemann-LiouvUle (Mies. In any case all the fractional operators share some common points ... 

Samko SG, Kilbas AA, Marichev OI (1993) Fractional calculus integrals and derivatives. Gordon and Breach Science Publishers, Amsterdam Shinozuka M, Deodatis G (1988) Stochastic process models for earthquake ground motion. Probab Eng Mech 3(3) 114-123... 

From a theoretical point of view, the Lion et al. model has the merit to approach the DSS effect by applying constitutive laws formulated on the basis of fractional calculus, in other terms by formulating the behavior of materials with respect to fractional time derivatives of stress and strain an approach that in principle requires only a small number of material constants to express the material properties in the time or the frequency domain. However, deriving model parameters from experimental data is not straightforward and, for instance Lion et al. had to use a stochastic Monte Carlo method to estimate the model parameters for a comparison with experimental data on 60 phr CB filled rubber compound. Moreover, mathematical handling of the above equations (see Appendix 5.5) shows that, like the Kraus model, this one exhibits also horizontal symmetry for the G curve and vertical symmetry for the G" curve, and is therefore not expected to perfectly meet experimental data, at least in its present state of development. 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

Thermodynamics is a quantitative subject. It allows us to derive relations between the values of numerous physical quantities. Some physical quantities, such as a mole fraction, are dimensionless the value of one of these quantities is a pure number. Most quantities, however, are not dimensionless and their values must include one or more units. This chapter reviews the SI system of units, which are the preferred units in science applications. The chapter then discusses some useful mathematical manipulations of physical quantities using quantity calculus, and certain general aspects of dimensional analysis. 

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