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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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Fractional models in hydromechanics

Uchaikin V. V., Ulyanovsk State University

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.    

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