Beyond monofractional kinetics

  • We discuss generalized integro-differential diffusion equations whose integral kernels are not of a simple power law form, and thus these equations themselves do not belong to the family of fractional diffusion equations exhibiting a monoscaling behavior. They instead generate a broad class of anomalous nonscaling patterns, which correspond either to crossovers between different power laws, or to a non-power-law behavior as exemplified by the logarithmic growth of the width of the distribution. We consider normal and modified forms of these generalized diffusion equations and provide a brief discussion of three generic types of integral kernels for each form, namely, distributed order, truncated power law and truncated distributed order kernels. For each of the cases considered we prove the non-negativity of the solution of the corresponding generalized diffusion equation and calculate the mean squared displacement. (C) 2017 Elsevier Ltd. All rights reserved.

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Author:Trifce Sandev, Igor M. Sokolov, Ralf MetzlerORCiD, Aleksei Chechkin
Parent Title (English):Chaos, solitons & fractals : applications in science and engineering ; an interdisciplinary journal of nonlinear science
Place of publication:Oxford
Document Type:Article
Year of first Publication:2017
Year of Completion:2017
Release Date:2020/04/20
Tag:Complete Bernstein function; Completely monotone function; Distributed order diffusion-wave equations
First Page:210
Last Page:217
Funder:DFG - Deutsche Forschungsgemeinschaft project "Random search processes, Levy flights, and random walks on complex networks"
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie
Peer Review:Referiert