By Rainer Klages, Günter Radons, Igor M. Sokolov
This multi-author reference paintings presents a distinct advent to the at present rising, hugely interdisciplinary box of these delivery methods that can not be defined through the use of commonplace tools 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 ideas and strategies from many various disciplines whereas delivering updated assurance of this new box of study, contemplating such assorted purposes as plasma physics, glassy fabric, mobile technology, and socio-economic facets. The e-book can be of curiosity to either theorists and experimentalists in nonlinear dynamics, statistical physics and stochastic methods. It additionally kinds a terrific place to begin for graduate scholars getting into this sector. 18 chapters written through the world over famous specialists during this box supply in-depth introductions to primary elements of anomalous delivery.
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Additional resources for Anomalous transport: foundations and applications
13) showing that Euler’s fractional derivatives on the right-hand side differs from Liouville’s and Leibniz’ idea on the left. Similarly, Eq. 5) corresponds to x β +1 d−1 x β = = β+1 dx −1 x y β dy. 14) 0 On the other hand, Eq. 9) corresponds to x 1− β d−1 x − β = =− 1−β dx −1 ∞ x y x −β y− β dy. 15) 19 20 2 Threefold Introduction to Fractional Derivatives This shows that Euler’s and Liouville’s deﬁnitions differ with respect to their limits of integration. 4 Liouville It has already been mentioned that Liouville deﬁned fractional derivatives using Eq.
58) 0 for locally integrable functions f : R + → C. 59) by the convolution theorem for Laplace transforms. The Laplace transform of I0α− f leads to a more complicated operator. 60) a holds. The formula is known as fractional integration by parts . 61) −∞ holds for Weyl fractional integrals. These formulae provide a second method of generalizing fractional integration to distributions. 62) for a distribution f and a test function ϕ. It shows that right- and left-sided fractional integrals are adjoint operators.
For 0 < Re α < 1 this becomes Iαa+ Dαa+ f ( x ) = f ( x ) − −α f )( a) (I1a+ ( x − a ) α −1 . 78b) for all k = 0, 1, 2, . . , n − 1. Note that the existence of g( x ) = Dαa+ f ( x ) in Eq. 77) does not imply that f ( x ) can be written as (Iαa+ g)( x ) for some integrable function g . 78) are satisﬁed. As an example where one of them fails, consider the function f ( x ) = ( x − a)α−1 −α for 0 < α < 1. Then Dαa+ ( x − a)α−1 = 0 exists. 78b) fails. There does not exist an integrable g such that Iαa+ g = ( x − a)α−1 .
Anomalous transport: foundations and applications by Rainer Klages, Günter Radons, Igor M. Sokolov