Fractional integration

After breakdown into partial fractions, integration, and rearrangement, the final result in a number of different forms is... 

In the limit a —> 1, the Riemann-Liouville fractional integral oD a reduces to an ordinary integration so that lim i oL>,1- = JjJodr becomes the identity operator that is, Eqs. (15) and (19) simplify to the standard diffusion and Fokker-Planck equations, respectively. 

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] ... 

Consequently, we see that the fractional integral shifts the fractal dimension D D y. and the fractional derivative shifts the fractal dimension... 

S. G. Samko, A. A. Kilbas, and O. I. Marychev, Fractional, Integrals and Derivatives. Theory and Applications, Gordon and Breach, Amsterdam, 1993. 

Eq. (131) is defined in terms of the convolution (the Riemann Lionville fractional integral definition) [73]... 

In an analysis of the diffusion equations in magnetic fields, Dumarque et al. introduced a diffusivity tensor and an effectiveness parameter related to the difference between diffusivity in the presence and in the absence of a magnetic field. In a less general, but somewhat more amenable approach to numerical estimations, Lielmezs and Musbally derived a magnetic correction factor called the arithmetic mean average fractional integral diffusion coefficient... 

Samko, S.G., A.A. Kilbas, and O.I. Marichev. 2993. Fractional integrals and derivatives Theory and applications. Gordon and Breach Sci. Publ., London. 

Any rational function of x may be integrated, if the denominator is factored into linear and irreducible quadratic factors. The function may then be broken into partial fractions, and the individual partial fractions integrated by use of the appropriate formula from the integral table. See the section on partial fractions for further information. 

Nigmatullin, R. R. Fractional integral and its physical interpretation. Theoretical and Mathematical Physics, 1992,90(3), 354-367. 

Samko, S.G., Kilbas, A.A., Marichev, O.I. Fractional Integrals and Derivatives Theory and Applications. Gordon and Breach, London (1993)... 

Kozlov, G. V. Shustov, G. B. The simulation of limiting conversion degree at polycondensation by fractional integral. Proceedings of International interdisciplinary seminar Fracta/s and Applied Synergetics. Moscow, Publishers MSOU, 2001, 155-157. 

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 ... 

Another important property of the gamma function is that it has simple poles at the points z = —n, (n = 0, 1,2,...). For convenience, the Laplace domain is usually used to describe the fractional integral-differential operation for solving engineering problems. The formula for the Laplace transform of the RL fractional derivative has the form [37] ... 

In some cases, for numerical calculation of nonlinear equations, one can use a fact that fractional derivative is based on a convolution integral, the number of weights used in the numerical approximation to evaluate fractional derivatives. In addition, one can apply predictor-corrector formula for the solution of systems of nonlinear equations of lower order. This approach is based on rewriting the initial value problem (15.68) and (15.69) as an equivalent fractional integral equation (Volterra integral equation of the second kind)... 

Fractional integrated intensity of (111) peak with respect to total integrated intensity in X-ray spectra for unmilled and samples milled for 30, 300, and 1000min. (Reprinted with permission from Ref. 39. Copyright (2007) by Elsevier.)... 

Lately the mathematical apparatus of fractional integration and differentiation [58, 59] was used for fractal objects description, which is amorphous glassy polymers structure. It has been shown [60] that Kantor s set fractal dimension coincides with an integral fractional exponent, which indicates system states fraction, remaining during its entire evolution (in our case deformation). As it is known [61], Kantor s set ( dust ) is considered in onedimensional Euclidean space d = ) and therefore, its fractal dimension obey the condition d Euclidean spaces with d > 2 (d = 2, 3,. ..) the fractional part of fractal dimension should be taken as fractional exponent [62, 63] ... 

At present the tendency of polymers mechanics main principles revision is marked. One of the intensively developing trends is coimected with fractal conception using [34], The wide field of this conception in physics different branches is due to two features. The first is connected with using of notions of Hausdorff-Bezikovich fractional dimension geometry. This helped to describe adequately systems with complex spatial structure, what cannot be done within the frameworks of Euclidean geometry. The second feature is connected with fractional integration and differentiation calculus using [35],... 

For processes evolution with fractal time the mathematics of fractional integration and differentiation is used [39]. As it has been shown in Ref. [30], in this case the fractional exponent Vf. coincides with Cantor s set fractal dimension and indicates system states fraction, preserved during all evolution time t. Let us remind, that Cantor s set is considered in one-dimensional Euclidean space d = 1) and therefore, its fractal dimension < 1 in virtue of fractal definition [39], For fractal objects in Euclidean spaces with higher dimensions (d > 1) the fractional part of should be accepted as according to the Eq. (4.32) [40], Then the value characterizes the fractal (polymer structure) fraction, which is invariable in deformation process [31],... 

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