Simulation of exchange processes in multi-component environments with account of data uncertainty
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Abstract
The goal of the work. Proposals for methods of solving systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients that defined in interval form and intended for modeling exchange processes in multicomponent environments. Research subject: systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients defined in interval form. Research method: interval analysis. The obtained results. Systems of linear homogeneous and non-homogeneous differential equations, which are used in modeling exchange processes in multicomponent environments, are considered. Such systems can be considered, for example, in problems of chemical kinetics, materials science, and the theory of Markov processes. To obtain the solution of these equations, specialized calculators of analytical transformations were used and tested. The Matlab system (ode15s solver) was used for numerical analysis of systems of differential equations. It is shown that the application of interval methods of numerical analysis at the initial stage of system modeling has some advantages over probabilistic methods because they do not require knowledge of the laws of distribution of the results of the system state parameter measurements and their errors. It is shown that existing methods of solving systems of linear differential equations can be divided into two groups. Common to these groups is the use of interval expansion of classical methods for solving differential equations given in interval form. The difference between these two groups of methods is as follows. The methods of the first group can be used for all types of differential equations but require the creation of special software. The peculiarity of the methods of the second group is that they can be used to solve equations analytically or using numerical analysis packages. The application of the methods of the second group is shown on the example of solving a system of differential equations, the coefficients of which are determined in interval form. The system of these equations is intended for modeling the processes of exchange with the external environment of the elements of the model of a specific physical system. In the case when the coefficients of these equations are variables, their piecewise-constant approximation is applied and a criterion that determines the possibility of its application is given. The technique proposed in the paper can be applied to solve systems of linear homogeneous and non-homogeneous differential equations with constant and variable coefficients if they are given by slowly varying functions. In the case when the coefficients of the equations are determined in the interval form, the technique allows obtaining their solution also in the interval form and does not require the creation of special software.
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References
Dubovyk, V. P. and Yuryk, I. I. (2006), Higher mathematics: Study guide, A.S.K, Kyiv, 648 p., available at: https://djvu.online/file/TruthdHinPW3q
Reuter, L. G. (2006), Theoretical sections of general chemistry: a textbook for students of higher educational institutions, Karavela, Kyiv, 303 p., available at: https://discovery.kpi.ua/Record/000179467/Description
Korobov, V. I. (2004), Systems of computer mathematics in chemistry: the main means of organizing calculations, RVV DNU, Donetsk, 135 p. available at: https://z-lib.io/book/16572323
Dubnitskiy, V. Yu., Butenko, V. O. and Cherniavskiy, V. L. (2014), “Estimation of the sensitivity of a system of linear differential equations simulating the interaction of cement concrete with the external environment”, Information processing systems, No. 8 (124), pp. 24–29. available at: https://www.hups.mil.gov.ua/periodic-app/article/11927
Prystavka, P. O. and Tivodar, O. V. (2017), “Model based on Markov’s Lantz in the problem of making decisions about the purchase or sale of material assets”, Current problems of automation and information technologies, Vol. 21, pp. 132–142, available at: https://actualproblems.dp.ua/index.php/APAIT/article/view/116
Dotsenko, N., Chumachenko, I., Galkin, A., Kuchuk, H. and Chumachenko, D. (2023), “Modeling the Transformation of Configuration Management Processes in a Multi-Project Environment”, Sustainability (Switzerland), Vol. 15(19), 14308, doi: https://doi.org/10.3390/su151914308
Lennart, L. (1999), System Identification: Theory for User, Prentice HAU PTR, Linkoping University, Sweden, 609 p. available at: https://vdoc.pub/download/system-identification-theory-for-the-user-58301k4uol20
Kovalenko, A. and Kuchuk, H. (2022), “Methods to Manage Data in Self-healing Systems”, Studies in Systems, Decision and Control, Vol. 425, pp. 113–171, doi: https://doi.org/10.1007/978-3-030-96546-4_3
Gavrylenko, S. and Hornostal, O. (2021), “Development of a method for identification of the state of computer systems based on bagging classifiers”, Advanced Information Systems, Vol. 5(4), pp. 5–9, doi: https://doi.org/10.20998/2522-9052.2021.4.01
Zhukovska, O. A. (2009), Basics of interval analysis: textbook, Osvita Ukrajiny, Kyiv, 136 p., available at: https://b.eruditor.link/file/625215/
Gadetska, S., Dubnitskiy, V., Kushneruk, Yu. and Khodyrev, A. (2022), “Performance of basic arithmetic actions with complex numb”, Advanced Information Systems, pp. 104–113, doi: https://doi.org/10.20998/2522-9052.2022.1.17
Kvyetnyy, R. N., Bohach, I. V., Boyko, O. R., Sofyna, O. Yu. and Shushura, O. M. (2013), Modeling of systems and processes. Calculation methods, VNTU, Vinnytsya, vol. 2, 233 p., available at: https://docplayer.net/77951475-Komp-yuterne-modelyuvannya-sistem-ta-procesiv-metodi-obchislen.html
Levin V. I. (2015), “Interval derivative and interval-differential calculus”, Radio Electronics, Computer Science, Control, No. 3, pp. 22-29, available at: http://ric.zntu.edu.ua/article/view/52593/48631
Burd, V. (2007), Method of averaging for differential equations on an infinite interval: Theory and applications, Chapman and Hall/CRC, New York, 360 p., doi: https://doi.org/10.1201/9781584888758
Nickel, K. L. E. (1986), “Using Interval Methods for the Numerical Solution of ODE's”, Zeitscrift fur Angewandte Mathematik und Mechanic, Vol. 66, No. 11, pp. 513–523, doi: https://doi.org/10.1002/zamm.19860661102
Dubnitskiy, V., Kobylin, A., Kobylin, O., Kushneruk, Y. and Khodyrev, A. (2023), “Calculation of harrington function (desirability function) values under interval determination of its arguments”, Advanced Information Systems, Vol. 7(1), pp. 71–81, doi: https://doi.org/10.20998/2522-9052.2023.1.12
Dobronets, B. S. (2007), Interval mathematics, Textbook, SFU, Krasnoyarsk, 287 p. available at: https://bookskeeper.top/knigi/obrazovanie/137833-intervalnaya-matematika.html
Moore, R. E. (1966), Interval analysis, Englewood Cliis, Prentice-Hall, Englewood Cliffs, Prentice-Hall, N. J., 145 p., available at: https://deepblue.lib.umich.edu/handle/2027.42/33298
Alefeld, G. and Herzberger, J. (1983), Introduction to Interval Computations, Academic Press, New York, 1983, 352 p., available at: https://archive.org/details/introductiontoin0000alef
Mchedlov-Petrosyan, O. P., Dubnitsky, V. Yu. and Chernyavsky, V. L. (1981), “Study of the late stages of cement hydration on a simulation model”, Reports of the USSR Academy of Sciences, Vol. 256, No. 3, pp. 429–431, available at: https://jglobal.jst.go.jp/en/detail?JGLOBAL_ID=200902083686310935
Chernyavsky, V. L. (2008), “Adaptation of abiotic systems: concrete and reinforced concrete”, Dnepropetrovsk National University of Railway Transport, Dnepropetrovsk, 412 p., available at: https://search.rsl.ru/ru/record/01004408623
Gomathi, B., Saravana Balaji, B., Krishna Kumar, V., Abouhawwash, M., Aljahdali, S., Masud, M. and Kuchuk, N. (2022), “Multi-Objective Optimization of Energy Aware Virtual Machine Placement in Cloud Data Center”, Intelligent Automation and Soft Computing, Vol. 33(3), pp. 1771–1785, doi: http://dx.doi.org/10.32604/iasc.2022.024052
Shkil, M. I., Leifura, V. M. and Samusenko, P. F. (2003), Differential equations, Technika, Kyiv, 321 p. available at: https://www.twirpx.com/file/3262449/
Vaytiev, V. A., Stepashina, E. V. and Mustafina, S. A. (2013), “Identification of а Mathematical Model of the Reduced Scheme of α-methylstyrene Dimerization Reaction”, European Journal of Natural History, No. 6, pp. 30–32. available at: https://world-science.ru/en/article/view?id=33209
Yuan, M., Li, P. and Wu, C. (2023), “Semi-parametric inference on Gini indices of two semi-continuous populations under density ratio models”, Econometrics Journal, Vol. 26(2), pp. 174–188, doi: http://dx.doi.org/10.1093/ectj/utac028
Coutler, P. B. (2019), Measuring Inequality, A Methodological Handbook, Taylor & Francis Group, Routledge, New York, 216 p., doi: https://doi.org/10.4324/9780429042874