Anomalous transport: foundations and applications by Rainer Klages, Günter Radons, Igor M. Sokolov

By Rainer Klages, Günter Radons, Igor M. Sokolov

This multi-author reference paintings presents a distinct advent to the presently rising, hugely interdisciplinary box of these delivery approaches that can't be defined by utilizing regular equipment of statistical mechanics. It comprehensively summarizes issues starting from mathematical foundations of anomalous dynamics to the newest experiments during this box. In so doing, this monograph extracts and emphasizes universal rules and strategies from many alternative disciplines whereas supplying up to date insurance of this new box of analysis, contemplating such assorted purposes as plasma physics, glassy fabric, mobilephone technological know-how, and socio-economic features. The booklet might be of curiosity to either theorists and experimentalists in nonlinear dynamics, statistical physics and stochastic techniques. It additionally kinds an excellent start line for graduate scholars stepping into this sector. 18 chapters written via the world over famous specialists during this box offer in-depth introductions to basic elements of anomalous shipping.

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Euler knew that he needed to generalize (or interpolate, as he calls it) the product 1 · 2 · . . · n = n! to noninteger values of n, and he proposed an integral 1 n ∏ k = n! 4) 0 for this purpose. 5) valid for integer and for noninteger α, β. 3 Paradoxa and Problems Generalizing Eq. 5) to all functions that can be expanded into a power series might seem a natural step, but this “natural” definition of fractional derivatives does not really resolve Leibniz’ paradox. 6) dx α by demanding dα h = hdx α for integer α.

9)). 2 Mathematical Introduction to Fractional Derivatives n times gives the fundamental formula x n −1 x x1 (Ina+ f )( x ) = ··· a a f ( xn ) dxn · · · dx2 dx1 a 1 = ( n − 1) ! 27) a where a < x < b and n ∈ N. This formula may be proved by induction. It reduces n-fold integration to a single convolution integral (Faltung). The subscript a+ indicates that the integration has a as its lower limit. An analogous formula holds with lower limit x and upper limit a. In that case the subscript a− will be used.

I for derivatives, to emphasize the similarity between fractional integration and differentiation. If this is done, the choice of Riesz and Feller, namely I, seems superior in the sense that fractional derivatives, similar to integrals, are nonlocal operators, while integer derivatives are local operators. 3) The notation ∼ indicates that the sum does not need to converge, and, if it converges, does not need to converge to f ( x ). 32) k =0 with c0 = 0. Recall the convolution formula [132, p. 33) ikt and g (t) ∼ ikt for two periodic functions f (t) ∼ ∑∞ ∑∞ k =−∞ f k e k =−∞ gk e .

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