• Bobomurod Khayitkulov Khayitovich
  • Muminjon Tukhtasinov National University of Uzbekistan, Tashkent, Uzbekistan


non-stationary problems, density, optimal choice, heat sources, implicit schemes, finite-dimensional approximation


In this paper, we study the problem of ensuring the temperature inside the field within the specified limits by choosing the optimal location of the heat sources in the parallelepiped. In this case, the optimal placement of heat sources on the area should be such that the total power of the consumed heat sources is minimal, so that the temperature is within the specified limits. By approximating the original problem, we obtain a difference equation. The construction of implicit difference schemes for the heat equation is given. From the difference equation it is reduced to a system of linear algebraic equations. The problem was solved using the M-method. In the parallelepiped, a new approach is proposed based on the numerical solution of the non-stationary problem of the optimal choice of the location of heat sources. Algorithms and software were developed for the numerical solution of the problem. A brief description of the software is provided. The results of a computational experiment are visualized.


A.G. Butkovskii. Nauka, Moscow, 1975.

Osipov O.V. Brusentsev, A.G. Approximate solution of the optimal choice problem of heat sources.Belgorod State University Scientific Bulletin, (5(124)):60–69,2012.

V.N. Akhmetzyanov, A.V. Kulibanov. Optimal placement of sources for stationary scalar fields. Automation and telemechanics, (6):50–58, 1999.

V.I. Mirskaya, S.Y. Sidelnikov. Efficient heating of the room as the optimal control problem. Technical and technological problems of service, (4(30)):75–78,2014.

T.M. Sabdenov, K.O. Baitasov. Optimal (energy efficient) heat supply to buildings in central heating system.Bulletin of the Tomsk Polytechnic University. GeoAssets Engineering, 326(8):53–60, 2015.

Y.V. Islamov, G.G. Kogan. The difference-differential problem of control by a diffusion process.The Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, pages 121–126, 2008.

Ershova A.A. Tanana, V.P. On the solution of an in-verse boundary value problem for composite materi-als.Vestnik Udmurtskogo Universiteta. Matematika.Mekhanika. Komp’yuternye Nauki, 28:474–488, 2018.

B.Kh. Khayitkulov. Numerical solution of the problem of the optimal choice of heat sources in a homogeneous stationary medium.Problems of computational and applied mathematics, (5(29)):141–146, 2020.

B.Kh. Khaitkulov. Homogeneous different schemes of the problem for optimum selection of the location of heat sources in a rectangular body.Solid State Technology, 65:583–592, 2020.

V.I. Agoshkov. Institute of Computational MathematicsRussian Academy of Sciences, Moscow, 2003.

Z.L. Lions. Dunod Gauthier-Villars, Paris, 1968.

R.P. Fedorenko. Nauka, Moscow, 1978.

A.N. Tikhonov.Inverse heat conduction problems. Journal of Engineering Physics and Thermophysics,29(1):7–12, 1975.

O.M Alifanov. Springer, Berlin, 1994.

Abduolimova G.M. Khayitkulov B.Kh. Tukhtasinov,M. Boundary control of heat propagation in a bounded body.Bulletin of the Institute of Mathematics, (1):1–10,2019.

Samarskii A.A. Tikhonov A.N. Nauka, Moscow, 7thedition, 2004.

V.N. Emelyanov. Yurayt Publishing House, Moscow,2th edition, 2018.[18] Tarasov D.V. Tarasov R.V. Bolotnikova, O.V. Publishing house of PSU, Penza, 2015.



How to Cite

Khayitkulov, B., & Tukhtasinov, M. (2021). NUMERICAL SOLUTION OF THE NON-STATIONARY PROBLEM OF CHOOSING THE OPTIMAL PLACEMENT OF HEAT SOURCES IN A PARALLELEPIPED. Acta of Turin Polytechnic University in Tashkent, 11(1), 29–34. Retrieved from