#### 519.6 Conservative difference schemes for the optimal selection of the location of heat sources in the rod

##### Khayitkulov B. K. (National University of Uzbekistan)

###### NONSTATIONARY PROBLEMS, OPTIMAL SELECTION, HEAT SOURCES, HEAT EQUATION, BALANCE EQUATION, CONSERVATION LAW, INTEGRO-INTERPOLATION METHOD, IMPLICIT SCHEMES, CONSERVATIVE SCHEMES, SIMPLEX METHOD

doi: 10.18698/2309-3684-2020-8598

In this paper, we have developed a method and an algorithm for solving the problem of the optimal selection of the density of heat sources on the rod in such a way that the temperature inside the considered area is within the specified limits is developed. In this case, the heat sources must provide the specified temperature regime of the minimum total power and temperature in the specified temperature corridor. Conservative finite-dimensional approximations of the original problem are constructed in the form of a linear programming problem. A method for constructing conservative difference schemes for solving the heat conduction equation with variable coefficients, a brief description of the developed software application for constructing computational grids and solving equations is presented. A new method for the numerical solution of non-stationary problems of optimal selection of heat sources in a rod is proposed and substantiated. A software application has been created for carrying out numerical experiments to solve the problem. The description of the based algorithm and the results of numerical experiments are given.

[1] Butkovskiy A.G. Metody upravlenija sistemami s raspredelennymi parametrami [Methods of controlling distributed parameter systems]. Moscow, Nauka Publ., 1975, 568 p.
[2] Akhmetzyanov A.V., Kulibanov V.N. Optimal source placement of stationary scalar fields. Automation and Remote Control, 1999, vol. 60, iss. 6, pp. 797–804.
[3] Mirskay S.Yu., Sidelnikov V.I. Efficient heating of the room as the optimal control problem. Tehniko-tehnologicheskie problemy servisa [Technical and Technological Problems of Service], 2014, no. 4(30), pp. 75–78.
[4] Sabdenov K.O., Baytasov T.M. Optimal (energy efficient) heat supply to buildings in central heating system. Izvestija Tomskogo politehnicheskogo universiteta. Inzhiniring georesursov [News of Tomsk Polytechnic University. Geo-Resource Engineering], 2015, vol. 326, no. 8, pp. 53–60.
[5] Moiseenko B.D., Fryazinov I.V. Conservative difference schemes for the equations of an incompressible viscous fluid in Euler variables. USSR Computational Mathematics and Mathematical Physics, 1981, vol. 21, iss. 5, pp. 108–120.
[6] Aristov V.V., Cheremisin F.G. The conservative splitting method for solving Boltzmann's equation. USSR Computational Mathematics and Mathematical Physics, 1980, vol. 20, iss. 1, pp. 208–225.
[7] Islamov G.G., Kogan Yu.V. The difference-differential problem of control by diffusion process. Vestnik Udmurtskogo universiteta. Matematika. Mehanika. Komp'juternye nauki [Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki], 2008, iss. 1, pp. 121–126.
[8] Tukhtasinov M., Abduolimova G.M., Khayitkulov B.Kh. Granichnoe upravlenie rasprostraneniem tepla v ogranichennom tele [Boundary control of heat propagation in a bounded body]. Bjulleten' Instituta matematiki [Bulletin of the Institute of Mathematics], 2019, no. 1, pp. 1–10.
[9] Agoshkov V.I. Metody optimal'nogo upravlenija i soprjazhjonnyh uravnenij v zadachah matematicheskoj fiziki [Optimal control methods and conjugate equations in problems of mathematical physics]. Moscow, IVM RAN Publ., 2003, 256 p.
[10] Egorov A.I. Optimal'noe upravlenie teplovymi i diffuzionnymi processami [Optimal control of thermal and diffusion processes]. Moscow, Nauka Publ., 1978, 464 p.
[11] Lions Zh.L. Optimal'noe upravlenie sistemami, opisyvaemymi uravnenijami s chast-nymi proizvodnymi [Optimal control of systems described by partial differential equations]. Moscow, Mir Publ., 1972, 412 p.
[12] Fedorenko R.P. Priblizhjonnoe reshenie zadach optimal'nogo upravlenija [An approximate solution of optimal control problems]. Moscow, Nauka Publ., 1978, 497 p.
[13] Vasil'eva M.V., Vasil'ev V.I., Tyrylgin A.A. Conservative difference scheme for filtering problems in fractured media. Mathematical Notes of NEFU, 2018, vol. 25, no. 4, pp. 84–101.
[14] Alifanov O.M. Obratnye zadachi teploobmena [Inverse heat transfer problems]. Moscow, Mashinostroenie Publ., 1988, 280 p.
[15] Khaitkulov B.Kh. Homogeneous different schemes of the problem for optimum selection of the location of heat sources in a rectangular body. Solid State Technology, 2020, vol. 63, iss. 4, pp. 583−592.
[16] Khayitkulov B.Kh. Numerical solution of the non-stationary problem of optimal selection of heat sources in a rod. Problems of Computational and Applied Mathematics, 2020, no. 5 (29), pp. 141−146.
[17] Tikhonov A.N., Samarskii A.A. Uravnenija matematicheskoj fiziki. 7-e izd. [Equations of Mathematical Physics. 7-th Edition]. Moscow, Nauka Publ., 2004, 798 p.
[18] Sigal I.Kh., Ivanova A.P. Metody optimizacii. Nachal'nyj kurs. Kurs lekcij po discipline “Metody optimizacii”. Chast 2. Simpleks-metod i smezhnye voprosy, jelementy teorii dvojstvennosti, mnogokriterial'naja optimizacija [Optimization methods. Initial course. A course of lectures on the discipline “Methods of optimization”. Part 2. Simplex method and related issues, elements of duality theory, multicriteria optimization]. Moscow, MIIT Publ., 2006, 104 p.

Хайиткулов Б.Х. Консервативные разностные схемы по оптимальному выбору местоположения источников тепла в стержне. Математическое моделирование и численные методы, 2020, № 3, с. 85–98.

Работа выполнена при финансовой поддержке Узбекского фонда фундаментальных исследований (проект ОТ-Ф4-33).

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