By Rainer Klages, Günter Radons, Igor M. Sokolov
This multi-author reference paintings offers a distinct creation to the at the moment rising, hugely interdisciplinary box of these shipping techniques that can not be defined by utilizing general equipment of statistical mechanics. It comprehensively summarizes subject matters 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 various disciplines whereas supplying up to date insurance of this new box of analysis, contemplating such diversified purposes as plasma physics, glassy fabric, telephone technology, and socio-economic facets. The e-book could be of curiosity to either theorists and experimentalists in nonlinear dynamics, statistical physics and stochastic methods. It additionally types a great place to begin for graduate scholars entering into this region. 18 chapters written by means of the world over famous specialists during this box supply in-depth introductions to primary features of anomalous delivery.
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Additional info for Anomalous Transport: Foundations and Applications (Wiley 2008)
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.
A relation between fractional derivatives of the same order but different types was given in [52, p. 434]. 3 Marchaud–Hadamard Fractional Derivatives Marchaud’s approach  is based on Hadamard’s ﬁnite parts of divergent integrals . The strategy is to deﬁne fractional derivatives as analytic continuation of fractional integrals to negative orders (see [99, p. 225]). 11 Let −∞ < a < b < ∞ and 0 < α < 1. 2 Mathematical Introduction to Fractional Derivatives and the Marchaud fractional derivative of order α with upper limit b is deﬁned as ( Mαb− f (x) α f )( x ) = + Γ(1 − α)(b − x )α Γ (1 − α ) b x f ( x ) − f (y) dy.
The theorem was generalized to higher dimensions by Sobolev in 1938, and is also known as the Hardy–Littlewood–Sobolev inequality (see [37, 38, 63, 113]). 5 Let 0 < α < 1, 1 < p < 1/α, −∞ ≤ a < b ≤ ∞. , there exists a constant C ( p, q) independent of f such that Iαa+ f q ≤ C f p . 64) 0 was used. 65) also called additivity law. It holds for Riemann–Liouville, Weyl, and Riesz– Feller fractional integrals of functions. 1 Riemann–Liouville Fractional Derivatives Riemann [96, p. 341] suggested to deﬁne fractional derivatives as integer order derivatives of fractional integrals.
Anomalous Transport: Foundations and Applications (Wiley 2008) by Rainer Klages, Günter Radons, Igor M. Sokolov