Journal
CHAOS SOLITONS & FRACTALS
Volume 102, Issue -, Pages 210-217Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.chaos.2017.05.001
Keywords
Distributed order diffusion-wave equations; Complete Bernstein function; Completely monotone function
Categories
Funding
- DFG - Deutsche Forschungsgemeinschaft project Random search processes, Levy flights, and random walks on complex networks
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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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