Конечномерные аналоги дифференциальных операторов переноса с носителями на пространственных сетях
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Научный журнал Моделирование, оптимизация и информационные технологииThe scientific journal Modeling, Optimization and Information Technology
Online media
issn 2310-6018

Finite-dimensional analogues of transfer differential operators with carriers on spatial networks

idHoang V., Makhinova O.A.,  Timoshenko V.V. 

UDC 519.65
DOI: 10.26102/2310-6018/2023.41.2.030

  • Abstract
  • List of references
  • About authors

The presented results provide justification for the applicability of numerical methods for analyzing initial-boundary value problems for evolutionary differential equations with a spatial variable changing on a network (graph), i.e., on a manifold of one-dimensional continua with a scalar variable. Similar results for -dimensional spatial variables ( ) changing on a network-like -dimensional domain are still in the stage of formation due to the incomparably high level of technical complexity that naturally arises when increasing the dimensionality of the spatial variable. Confirmation of the possibility of justifying numerical methods for analyzing initial-boundary value problems for cases is provided using the results of applying computational methods to solving a test problem with a spatial variable changing on a two-dimensional network-like carrier – a two-dimensional complex-structured domain. The presented example of numerical analysis opens prospects for extending the obtained results to differential operators defined on functions with an m-dimensional carrier. To simplify the representations of difference schemes, a method of semi-discretization with respect to the time variable is used (in a sense, numerous routine costs that arise as a direct consequence of the multidimensionality of the spatial variable are leveled). The obtained results are applied in constructing and numerically analyzing mathematical models of laminar and turbulent network-like processes in applied hydrodynamics.

1. Hoang V.N., Part A.A., Perova I.V. Numerical analysis of the mathematical model of the turbulent flow dynamics of a multiphase medium in network-like objects. Modeling, Optimization and Information Technology. 2023;11(2). URL: https://moitvivt.ru/ru/journal/pdf?id=1326. DOI: 10.26102/2310-6018/2023.41.2.006. (In Russ.).

2. Yurko V.A. Vvedenie v teoriiu obratnykh spektral’nykh zadach. M.: Fizmatlit, 1984. 384 p. (In Russ.).

3. Sergeev S.M., Raijhelgauz L.B., Hoang V.N., Panteleev I.N. Modeling unbalanced systems in network-like oil and gas processes. Journal of Physics: Conference Series. 2020;1679(2):022015. URL: https://iopscience.iop.org/article/10.1088/1742-6596/1679/2/022015.

4. Provotorov V.V Sobstvennyye funktsii krayevykh zadach na grafakh i prilogheniya. Voronezh, Nauchnaya kniga; 2008. 247 p. (In Russ.).

5. Levitan B.M. Obratnye zadachi Shturma-Liuvillia. M.: Nauka; 1984. 239 p. (In Russ.).

6. Zhabko A.P., Nurtazina K.B., Provotorov V.V. About one approach to solving the inverse problem for parabolic equation. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaya matematika. Informatika. Protsessy upravleniya = Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes. 2019;15(3):323–336. URL: http://hdl.handle.net/11701/16384.

7. Yurko V. A. Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-posed Problems Series. Utrecht, VSP; 2002. 303 p.

8. Artemov M.A., Baranovskii E. S. Solvability of the Boussinesq approximation for water polymer solutions. Mathematics. 2019;7(7). URL: https://www.mdpi.com/2227-7390/7/7/611.

9. Baranovskii E.S. Steady flows of an Oldroyd fluid with threshold slip. Communications on Pure and Applied Analysis. 2019;18(2):735–750. DOI: 10.3934/cpaa.2019036.

10. Baranovskii E.S. Existence results for regularized equations of second-grade fluids with wall slip. Electronic Journal of Qualitative Theory of Differential Equations. 2015;(91):1–12. URL: http://real.mtak.hu/32263/.

Hoang Van Nguyen

ORCID |

Voronezh State University

Voronezh, The Russian Federation

Makhinova Olga Alekseevna
Candidate of Physical and Mathematical Sciences

The Air Force Academy named after N.E. Zhukovsky and Y.A. Gagarin

Voronezh, The Russian Federation

Timoshenko Victor Vladimirovich

Voronezh State University

Voronezh, The Russian Federation

Keywords: differential operators on network-like domains, finite-dimensional analogues, properties of finite-dimensional analogues, difference schemes, numerical analysis

For citation: Hoang V., Makhinova O.A., Timoshenko V.V. Finite-dimensional analogues of transfer differential operators with carriers on spatial networks. Modeling, Optimization and Information Technology. 2023;11(2). URL: https://moitvivt.ru/ru/journal/pdf?id=1363 DOI: 10.26102/2310-6018/2023.41.2.030 (In Russ).

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Full text in PDF

Received 06.05.2023

Revised 08.06.2023

Accepted 27.06.2023

Published 30.06.2023