Non-integer Derivatives Research Articles

STUDY ON THE NEW MODEL OF THE ADVECTION-DISPERSION OF THE CONTAMINANT IN THE GROUNDWATER

Recently as a new governing equation of solute transport, a fractional advection dispersion equation (fADE) has been proposed, which includes non-integer derivatives for the mass-transport phenomena in heterogeneous geologic media by Benson et al.(1998, 2000). Using this equation, heterogeneitycan be taken into account effectively without scale dependency. We solved the fADE numerically by thefinitc difference method, and compare with new experimental data. The data were taken in a sand box with Tohoku silica sand and 3% of NaCl solution. Breakthrough curves are obtained from the experimental result, and we found that the experimental result is essentially different from the ordinary Gaussian profiles and the order of derivative is about 1.5 for our experiments.

The justified data-curve fitting approach: recognition of the new type of kinetic equations in fractional derivatives from analysis of raw dielectric data

Usually, for the description of dielectric spectra one uses the empirical Cole–Davidson (CD) and Havriliak–Negami (HN) equations each of which contains one relaxation time. However, the parameters figuring in the CD and HN equations (or the linear combination of several CD or HN equations) do not have any clear physical meaning. For the description of such asymmetric dielectric spectra, we suggest complex permittivity functions containing two or more characteristic relaxation times. These complex susceptibility functions correspond, in the time-domain, to a new type of kinetic equation, which contains non-integer (fractional) integrals and derivatives. The physical meaning of these operators is discussed in [1]. We suppose that these kinetic equations describe a wide class of dielectric relaxation phenomena taking place in heterogeneous substances. To support and justify this statement, a special recognition procedure has been developed that helps to identify this new kinetic equation from real dielectric data. This recognition procedure can be considered as the justified data-curve fitting (JDCF) approach, in contrast to the conventional ‘imposed’ data-curve fitting (IDCF) treatment invariably used in modern dielectric spectroscopy. The JDCF approach incorporates the ratio presentation (RP) format and a separation procedure. It is shown how this separation procedure can be helpful in the detection of the many relaxation processes (each process is described by a characteristic relaxation time), which are taking place in the dielectric material under consideration.

Signal processing and recognition of true kinetic equations containing non-integer derivatives from raw dielectric data

Signal processing in dielectric spectroscopy implies that it is necessary to find a ‘true’ fitting function (having a certain physical meaning), which describes well the complex permittivity and impedance data. In dielectric spectroscopy for description of complex permittivity/impedance data researches usually use the empirical Cole–Davidson (CD) and Havriliak–Negami (HN) equations that contains one relaxation time. But the parameters figuring in CD and HN equations do not have clear physical meaning as well as fitting parameters entering into linear combination of several CD or HN equations. For description of dielectric (especially asymmetric) spectra we suggest the complex permittivity functions containing two or more characteristic relaxation times. These complex susceptibility functions correspond in time domain to new type of kinetic equation containing non-integer (fractional) integrals and derivatives. We suppose that these kinetic equations describe a wide class of dielectric relaxation phenomena taking place in heterogeneous substances. To support and justify this statement the special recognition procedure has been developed that helps to identify this new kinetic equation from raw dielectric data. It incorporates the ratio presentation (or RP) format and separation procedure. Separation procedure was turned out to be helpful in detection of number of relaxation processes (each process is described by a characteristic relaxation time) taking place in the dielectric material under consideration. We suppose that this procedure can be applicable also for identification of fractal noises.

New approach in the description of dielectric relaxation phenomenon: correctdeduction and interpretation of the Vogel–Fulcher–Tamman equation

Based on the relationship between the power-law exponent and relaxation time ν(τ)recently established in Ryabov et al (2002 J. Chem. Phys. 116 8610) fornon-exponential relaxation in disordered systems and conventionalArrhenius temperature dependence for relaxation time, it becomespossible to derive the empirical Vogel–Fulcher–Tamman (VFT)equation ωp (T) = ω0 exp [−DTVF /(T − TVF)],connecting the maximum of the loss peak with temperature. The fitting parametersD andTVFof this equation are related accordingly with parameters (ν0, τs τ0),entering to ν(τ) = ν0 [ln (τ/τs)/ ln (τ/τ0)] and(τA, E)figuring in the Arrhenius formula τ(T) = τA exp (E/T). Ithas been shown that, in order to establish the loss peak VFT dependence, acomplex permittivity function should contain at least two relaxation timesobeying the Arrhenius formula with two different set of parameters τA1,A2and E1,2.It has been shown that (1) at a certain combination of initial parameters the parameterTVFcan be negative or even accept complex valued (2). The temperature dependenceof the minimum frequency formed by the two nearest peaks also obeys the VFTequation with another set of fitting parameters. The available experimental dataobtained for different substances confirm the validity and specific ‘universality’ ofthe VFT equation. It has been shown that the empirical VFT equation isapproximate and possible corrections to this equation are found. As amain consequence, which follows from the correct ‘reading’ of the VFTequation and interpretation of complex permittivity functions with two ormore characteristic relaxation times, we suggest a new type of kineticequation containing non-integer (fractional) integrals and derivatives. Wesuppose that this kinetic equation describes a wide class of dielectricrelaxation phenomena taking place in heterogeneous substances.

On the finite element approximation of functions with noninteger derivatives

The paper is devoted to the study of error estimates in the polynomial approxiamations of functions interpolation by affine families of finite elements treated in [4] is extended to functions from fractional Sobolev spaces .