ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Uchaikin V. V. Fractional models in hydromechanics. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 1, pp. 5-40. DOI: 10.18500/0869-6632-2019-27-1-5-40

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Language: 
Russian
Heading: 
Review of Actual Problems of Nonlinear Dynamics
Article type: 
Article
UDC: 
517.524
DOI: 
10.18500/0869-6632-2019-27-1-5-40

Fractional models in hydromechanics

Autors: 
Uchaikin V. V., Ulyanovsk State University
Abstract: 

Topic and purpose. The last two decades are marked by wide spreading fractional calculus in theoretical description of the natural processes. Replacement of the integer-order operators by their fractional (and even complex) counterparts opens up a continuous field of new differential equations in which the standard set of equations of theoretical physics (wave, diffusion, etc.) is represented by separate spikelets at points with integer coordinates. But what do the fractional-order derivatives mean physically? What are the common reasons for the appearance of fractional derivatives in the equations? Is it possible to predict in advance the appearance of fractional operators in a particular problem? These questions are not yet removed from the agenda and remain the focus of attention of each of the conferences devoted to the theory and application of fractional calculus. This topic is developing in this review. Models investigated. The fractional calculus is demonstrated in application to various problem of the most, if one may say so, classical field of theoretical physics-hydrodynamic including turbulent diffusion. Results. The review shows how fractional operators appear on the classical field of hydrodynamic problems under the pen of Heisenberg, Weizsacker, Kolmogorov, Obukhov, Monin – theoreticians who can not be suspected of being uncritical of the mathematical tools. Discussion. Actually, the whole review is a continuous discussion of the «inevitability of the strange world» of fractional calculus (Uchaikin V.V. The method of fractional. Ulyanovsk: «Artishok», 2008), and the fact that this is done within the framework of classical hydromechanics only strengthens the convincing conclusions.    

Key words: 
fractional Laplacian
nonlocality
turbulent diffusion
projection operators
open systems
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Received: 
25.06.2018
Accepted: 
20.09.2018
Published: 
28.02.2019
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