АППРОКСИМАЦИЯ ЭВОЛЮЦИОННЫХ ПРОЦЕССОВ С РАСПРЕДЕЛЕННЫМИ ПАРАМЕТРАМИ НА СЕТИ
Работая с нашим сайтом, вы даете свое согласие на использование файлов cookie. Это необходимо для нормального функционирования сайта, показа целевой рекламы и анализа трафика. Статистика использования сайта отправляется в «Яндекс» и «Google»
Научный журнал Моделирование, оптимизация и информационные технологииThe scientific journal Modeling, Optimization and Information Technology
Online media
issn 2310-6018

Approximation of evolutionary processes with distributed parameters on a network

Balaban O.R.  

UDC 517.927
DOI: 10.26102/2310-6018/2020.30.3.003

  • Abstract
  • List of references
  • About authors

The paper considers the approximation of mathematical models of network-like evolutionary transport processes as applied to differential systems with distributed parameters on a network (graph). An approach is indicated that uses the application of the theory of classical computational methods, which consists in reducing the investigated problem to systems of algebraic equations (auxiliary finitedimensional problems) in which the values of the grid functions at the points of partition of the edges of the graph are unknown. At the same time, there is a fairly wide opportunity for choosing different types of convergent difference schemes that are significantly different from each other: explicit difference schemes, implicit difference schemes, analogs of Krank-Nicholson difference schemes (below, in order not to load the study with technical difficulties, explicit difference schemes are used). It should be noted a characteristic feature of the studied mathematical models inherited by the rheological structure of the graph — the presence of singular points of the graph at which the differential equation is not determined (nodes or vertices of the graph) and is replaced by generalized Kirchhoff conditions. The formalisms of the latter describe the laws of continuum transfer at these points and require a separate approach to approximation issues (in the work, for the sake of simplicity, classical difference relations are used). It should also be noted that the use of an implicit difference scheme or the Crank-Nicholson scheme for approximation requires additional analysis of auxiliary finite-dimensional problems (solvability, uniform boundedness of approximations to the solution of the original problem), but it significantly increases the accuracy of calculating approximations. The use of an explicit difference scheme is freed from studying some of these issues, however (and this, when analyzing some applied problems, can be a significant obstacle to use) gives a rather large error in determining the solution to the original problem. The given particular examples of applied character illustrate the ways of numerical analysis of differential systems with carriers on an arbitrary network (graph). The results obtained are quite simply transferred to the study of wave processes and oscillation phenomena in transport processes by numerical methods.

1. Balaban O.R. Approksimatsiya ehvolyutsionnykh differentsial'nykh sistem s raspredelennymi parametrami na seti i metod momentov. Modeling, optimization and information technology. 2019;7(3). Available at: https://moit.vivt.ru/wpcontent/uploads/2019/09/Balaban_3_19_1.pdf. DOI: 10.26102/2310- 6018/2019.26.3.040 (accessed: 29.05.2020).

2. Balaban O.R., Ivanov A.V. Approksimatsiya ehvolyutsionnykh uravnenii parabolicheskogo tipa na seti. Sistemy upravleniya i informatsionnye tekhnologii. 2018;4(74):4-7.

3. Marchuk G.I. Metody vychislitel'noi matematiki. M.:Nauka. 1977:456.

4. Provotorov V.V., Provotorova E.N. Synthesis of optimal boundary control of parabolic systems with delay and distributed parameters on the graph. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes. 2017;13(2):209- 224. Available at: https://doi.org/10.21638/11701/spbu10.2017.207. (accessed: 29.05.2020).

5. Podval'nyi S.L., Provotorov V.V. Opredelenie startovoi funktsii v zadache nablyudeniya parabolicheskoi sistemy s raspredelennymi parametrami na grafe. Vestnik Voronezhskogo gosudarstvennogo tekhnicheskogo universiteta. 2014;10(6):29-35.

6. Provotorov V.V. Metod momentov v zadache gasheniya kolebanii differentsial'noi sistemy na grafe. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes. 2010;2:60-69.

7. Provotorov V.V. Modelirovanie kolebatel'nykh protsessov «machta-rastyazhkI». Sistemy upravleniya i informatsionnye tekhnologii. 2008;1.2(31):272-277.

8. Provotorov V.V., Ryazhskikh V.I., Gnilitskaya Yu.A. Unique weak solvability of a nonlinear initial boundary value problem with distributed parameters in a netlike region. Vestnik of Saint Petersburg University.Applied mathematics. Computer sciense. Control processes. 2017;13(3):264-277. Available at: https://doi.org/10.21638/11701/spbu10.2017.304. (accessed: 29.05.2020).

9. Zhabko A.P., Provotorov V.V., Balaban O.R. Stabilization of weak solutions of parabolic systems with distributed parameters on the graph. Vestnik of Saint Petersburg University. Applied mathematics. Computer sciense. Control processes. 2019;15(2):187-198. Available at: https://doi.org/10.21638/11702/spbu10.2019.203. (accessed: 30.05.2020).

10. Balaban O.R. Approximation of elliptic operator of evolutionary systems with distributed parameters on a network. Modern informatization problems in simulation and social technologies (MIP-2019’SCT): Proceedings of the XXIV-th International Open Science Conference (Yelm, WA, USA, January 2019). 2019:148-154.

Balaban Olesya R.

Email: bal-olesya@mail.ru

Military Training And Scientific Center Of The Air Force "Air Force Academy Named After Professor N.E. Zhukovsky And Y.A. Gagarina "

Voronezh, Russian Federation

Keywords: evolutionary transport processes on networks, approximation, difference scheme, features at network nodes, numerical methods

For citation: Balaban O.R. Approximation of evolutionary processes with distributed parameters on a network. Modeling, Optimization and Information Technology. 2020;8(3). Available from: https://moit.vivt.ru/wp-content/uploads/2020/08/Balaban_3_20_1.pdf DOI: 10.26102/2310-6018/2020.30.3.003 (In Russ).

449

Full text in PDF