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Vestnik YuUrGU. Ser. Mat. Model. Progr., 2020, Volume 13, Issue 2, Pages 33–42 (Mi vyuru541)  

Mathematical Modelling

Stochastic mathematical model of internal waves

E. V. Bychkova, A. V. Bogomolovb, K. Yu. Kotlovanova

a South Ural State University, Chelyabinsk, Russian Federation
b 2Saint Petersburg Institute for Informatics and Automation of RAS, Saint Petersburg, Russian Federation

Abstract: The paper studies a mathematical model of internal gravitational waves with additive “white noise”, which models the fluctuations and random heterogeneity of the medium. The mathematical model is based on the Sobolev stochastic equation, Dirichlet boundary conditions and the initial Cauchy condition. The Sobolev equation is obtained from the assumption of the propagation of waves in a uniform incompressible rotation with a constant angular velocity of the fluid. The solution to this problem is called the inertial (gyroscopic) wave, since it arises due to the Archimedes's law and under the influence of inertia forces. By “white noise” we mean the Nelson–Gliklikh derivative of the Wiener process. The study was conducted in the framework of the theory of relatively bounded operators, the theory of stochastic equations of Sobolev type and the theory of (semi) groups of operators. It is shown that the relative spectrum of the operator is bounded, and the solution of the Cauchy–Dirichlet problem for the Sobolev stochastic equation is constructed in the operator form.

Keywords: relatively bounded operator, Sobolev equation, propagators, “white noise”, Nelson–Gliklikh derivative.

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation SSh-2553.2020.8
This work was supported by a grant from the President of the Russian Federation for state support of leading scientific schools of the Russian Federation (SSh-2553.2020.8).


DOI: https://doi.org/10.14529/mmp200203

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UDC: 517.9
MSC: 35C15, 60H30, 76B15
Received: 20.01.2020
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Citation: E. V. Bychkov, A. V. Bogomolov, K. Yu. Kotlovanov, “Stochastic mathematical model of internal waves”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 13:2 (2020), 33–42

Citation in format AMSBIB
\Bibitem{BycBogKot20}
\by E.~V.~Bychkov, A.~V.~Bogomolov, K.~Yu.~Kotlovanov
\paper Stochastic mathematical model of internal waves
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2020
\vol 13
\issue 2
\pages 33--42
\mathnet{http://mi.mathnet.ru/vyuru541}
\crossref{https://doi.org/10.14529/mmp200203}


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