Fractional derivative technique equation

The fractional derivative technique is used for the description of diverse physical phenomena (e.g., Refs. 208-215). Apparently, Blumen et al. [189] were the first to use fractal concepts in the analysis of anomalous relaxation. The same problem was treated in Refs. 190,194,200-203, again using the fractional derivative approach. An excellent review of the use of fractional derivative operators for the analysis of various physical phenomena can be found in Ref. 208. Yet, however, there seems to be little understanding of the relationship between the fractional derivative operator and/or differential equations derived therefrom (which are used for the description of various transport phenomena, such as transport of a quantum particle through a potential barrier in fractal structures, or transmission of electromagnetic waves through a medium with a fractal-like profile of dielectric permittivity, etc.), and the fractal dimension of a medium. 

The direction of planes in a lattice is described in a manner that, at first sight, seems rather strange but which is, in fact, derived directly from standard techniques in 3-D geometry. In essence, the unit cell is drawn and the plane of interest translated until it intercepts all three axes within the unit cell but as far away from the origin as possible. The point of intersection of the plane with the axes then determines the label given to the plane if the intersection takes place at a fraction (1 jh) of the a-axis, jk) of the fc-axis and (1/0 of the c-axis, then the plane is referred to as the hkl) plane . As indicated, this apparently rather strange method arises because if the axes a, A, c of the unit cell are mutually perpendicular, and of equal length, then the equation of any point x, y, z in the (hkl) plane can always be written ... 

Euler-Euler models assume interpenetrating continua to derive averaged continuum equations for both phases. The probability that a phase exists at a certain position at a certain time is given by a phase indicator function, which, for steady-state processes, is equivalent to the volume of fraction of the correspondent phase (volume-of-fluid technique). The phase-averaging process introduces further unknowns into the basic conservation equations their description requires empirical and problem-dependent input (94). In principal, Euler-Euler models are applicable to all multiphase flows. Advantages and disadvantages of both methods are compared, e.g., in Refs. 95 and 96. 

The rate constant kC8 of the transition from the contact to the solvent separated ion pair is obtainable from Equation 14 since all other quantities on the right side of the equation can be determined by kinetic experiments and UrJt can be determined by the described fractionation technique. The kinetic measurements also provide the equilibrium constant Kcs between the two ion pairs (14), and so the reverse constant, ksc, may be derived from ... 

The spectrophotometric technique determines K whenever the fraction of free ions is very low. The concentration of the free ions may be reduced to an insignificant level by the addition of some readily dissociated salt sharing a common cation with the investigated radical anion. On the other hand, the potentiometric technique yields K, and its value can be used to calculate K if the necessary dissociation constants are known. These constants may be derived from conductometric data (5). For an anthracene and pyrene pair incorporating Na+ as the counterion and tetrahydrofuran (THF) as the solvent, the ratio of equation 8 is only 1.6, which is equivalent to 10 millivolts (mV). However, for an anthracene and naphthalene pair, the ratio is 30.3, which is equivalent to 90 mV. 

The modulation technique mentioned above has been used to identify triplet excimers in 1,2-benzanthracene and 1,2 3,4-dibenzanthracene at high solute concentrations167 and the differences between luminescence from naphthalene in fluid solution in the temperature range 353—173 and naphthalene in a rigid solution at 77 have been ascribed to phosphorescence from a triplet excimer.168 Excimer formation in solid poly-(2-vinylnaphthalene) and polystyrene is found to be dependent on the temperature at which the film is cast, and a statistical model based on the rotational isomeric state approximation has been used to formulate an expression for the fraction of excimer sites in the solid systems.168 Kinetic equations for dimer formation and decay, based on the statistical mechanics of ideal gases, have been obtained. These equations, derived from the N-atom von Neumann equation, take into account both bimolecular and termolecular equations.157 158 160... 

BET adsorption and desorption is usually performed with liquid N2 and is a widespread technique for determining the specific surface area of porous materials [28]. In fact, only surfaces that are accessible to the N2 molecules are detected. The Kelvin equation correlates the curvature of the liquid surface with the applied partial pressure and pore size distributions can be derived [29]. However, this method is successful only for pore structures below about 20 nm. Thus, in aerogels with typical pore sizes in the 1—100 nm range, only a fraction of the total available pore space is detected. For an aerogel with a den-... 

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. 

Montaudo and co-workers [8] reported on a SEC/MALDI study of molar mass distribution in two random copolyesters polybutylene adipate - polybutylene sebacate (PBA - PBSe) and polybutylene sebacate - polybutylene sccinate (PBSe - PBSu). MALDI allows desorption and when polydisperse polymer samples are fractionated by size exclusion chromatography, they yield fractions with very narrow distributions which, when analysed by MALDI give mass spectra with molar mass values in excellent agreement with those obtained by conventional techniques. These workers derived the following equation for the molar mass of the copolymer molecule ... 

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