Fractional differentiation

Weight Fraction Differential Cumulative Molecular Weight Light Cumulative Lansing- Kraemer... 

Recently the new concept of fractional time evolution was introduced [45]. In addition to the usual equilibrium state (96), this concept leads to the possibility of the existence of an equilibrium state with power-law long-time behavior. Here the infinitesimal generator of time evolution is proportional to the Riemann-Liouville fractional differential operator oDvt. By definition of the Riemann-Liouville fractional differentiation operator [231,232] we have... 

Since 0 < a < 1 the exponent in Eq. (137) is 1 — a > 0. The mathematical implication is that M(p) (137) is a multivalued function of the complex variable p. In order to represent this function in the time domain, one should select the schlicht domain using supplementary physical reasons [135]. These computational constraints can be avoided by using the Riemann-Liouville fractional differential operator oDlt a [see definitions (97) and (98)]. Thus, one can easily see that the Laplace transform of... 

Equation (139) was already discussed elsewhere [22,23,31] as a phenomenological representation of the dynamic equation for the CC law. Thus, Eq. (139) shows that since the fractional differentiation and integration operators have a convolution form, it can be regarded as a consequence of the memory effect. A comprehensive discussion of the memory function (137) properties is presented in Refs. 22 and 23. Accordingly, Eq. (139) holds for some cooperative domain and describes the relaxation of an ensemble of microscopic units. Each unit has its own microscopic memory function m (t), which describes the interaction between this unit and the surroundings (interaction with the statistical reservoir). The main idea of such an interaction was introduced in Refs. 22 and 23 and suggests that mg(f) JT 8(f,- — t) (see Fig. 50). 

K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley Sons, New York, 1993. 

Figure 3.3 (a) standard positron lifetime set-up PM TUBE photomultiplier tube/ scintillator assembly CF DIFF DISC constant fraction differential discriminator DELAY delay box or fixed length of 50Q cable TAC time-to-amplitude converter MCA multichannel analyser, (b) digital version of (a) [Rytsola etal, preprint, 2001]. 

We can now generalize the fractional differential equation to include a random force L(t) and in this way obtain a fractional Langevin equation... 

Another approach to the problem of anomalous relaxations uses fractal concepts [187-189,200-203], Here the problem is analyzed using the mathematical language of fractional derivatives [194,200-203] based on the previously mentioned Riemann-Liouville fractional differentiation operator... 

Now we will try to summarize more of our results. Instead of (3.210), we consider yet another operator of fractional differentiation. 

Various aspects of such fractional differential equations have been studied in [205-208]. We shall now give an example of the solution of such an equation. 

I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. 

Substitution of Eq. (58) into Eq. (56) then yields the following fractional differential equation for the function g(t) ... 

From the differentiation theorem of Laplace transform, J f /(t) = uP u) —P t = 0), we infer that the left-hand side in (x,t) space corresponds to 0P(x, t)/dt, with initial condition P(x. 0) = 8(x). Similarly in the Gaussian limit a = 2, the right-hand side is Dd2P(x, f)/0x2, so that we recover the standard diffusion equation. For general a, the right-hand side defines a fractional differential operator in the Riesz-Weyl sense (see below) and we find the fractional diffusion equation [52-56]... 

Anomalous diffusion of a continuous concentration field can be modelled in terms of fractional differential equations. To see how they arise we can write Eq. (2.9) for normal diffusion in terms of the spatial Fourier transform of the concentration field C(k, t). This can be easily done under periodic boundary conditions or in unbounded space as... 

Consequently, if the denominator of any fractional differential can be resolved into factors, the differential can be integrated by one or other of these processes. The remainder of this chapter will be mainly taken up with practical illustrations of integration processes. A number of geometrical applications will also be given because the accompanying figures are so useful in helping one to form a mental picture of the operation in hand. 

The fact, that macromolecular coil in diluted solution is a fractal object, allows to use the mathematical calculus of fractional differentiation and integration for its parameters description [72-74]. Within the framework of this formalism there is the possibility for exact accounting of such nonlinear phenomena as, for example, spatial correlations [74]. In the last years the methods of ftactional differentiation and integration are applied successfully for pol5mier properties description as well [75-77]. The authors [78-81] used this approach for average distance between polymer chain ends calculation of polycarbonate (PC) in two different solvents. 

Thus, the stated above results allow one to make two main conclusions. Firstly, fractional differentiation and integration methods (the Eq. (68)) allow to calculate polymer chain molecular characteristics as precisely as other existing at present computational techniques. Secondly, accounting for the change of macromolecular coil structure, that is, its dynamics, at the external conditions variation, is needed for the correct calculation of the indicated characteristics. 

For the evolutionary processes with fractal time description the mathematical calculus of fractional differentiation and integration is used [34]. As it has been shown in Ref [35], in this case the fractional exponent... 

The mole fraction differential distribution of the property E is given in the form of a Gaussian function as... 

Podlubny, I. Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198. Academic, San Diego (1999)... 

The mathematical calculus of fractional differentiation and integration is used for the description of evolutionary processes with fractal time [81 ]. As it has been shown in paper [82], in this case the fractional exponent v coincides with fractal dimension of Kantor s set and indicates fraction of system states, maintaining during all evolution time t. Let us remind, that Kantor s set is considered in onedimensional Euchdean space (d = 1) and therefore its fractal dimension djfractal definition [52]. For fractal objects in Euclidean spaces with Mgher dimensions (d>l) dj,fractional part should be accepted as v or [83] ... 

The two definitions generally used for the fractional differential-integral are the Grunwald definition and the Riemann-Liouville (RL) definition. The Grunwald definition is given by... 

One may note that the fractional differential operator is not a local operator, that is, the derivative is not only dependent on the value at the point but also on the value of the function on the whole interval. 

Here, the capacitive and inductive elements, using fractional order p G (0, 1), enable formation of the fractional differential equation, that is, more flexible or general model of liquid-liquid interfaces behavior. Now, using again, for example, RL definition of fractional derivative and integral... 

So, in that way, one can obtain linear fractional differential equation with zeros as initial conditions as follows ... 

Nonlinear fractional differential equations have received rather less attention in the literature, partly because many of the model equations proposed have been linear. Here, a nonlinear integral-differential equation of the van der Pol type will be considered. This equation represents the droplet or droplet-film stmcture formation, breathing, or destruction processes and taking into account the particular frequency component, which is included in the driving force and given by Equation (15.12) ... 

Matignon, D., Stability results for fractional differential equations with application to control processing, in Computation Engineering in Systems Applications, IMACS, lEEE-SMC Lille, France, July, 1996. 

Diethelm, K. and Ford, N.J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 401-418, 2002. 

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