The Problem of Complex Heat Transfer with Cauchy-Type Conditions on a Part of the Boundary

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Abstract

The paper considers a boundary value problem for stationary equations of complex heat transfer with an undetermined boundary condition for the radiation intensity on a part of the boundary and an overdetermined condition on another part of the boundary. An optimization method for solving this problem is proposed, and an analysis of the corresponding problem of boundary optimal control is presented. It is shown that the sequence of solutions of extremum problems converges to the solution of a problem with Cauchy-type conditions. The efficiency of the algorithm is illustrated by numerical examples.

About the authors

P. R. Mesenev

Far Eastern Center for Mathematical Research, Far East Federal University

Email: cheb@iam.dvo.ru
690922, Vladivostok, Russia

A. Yu. Chebotarev

Institute of Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences; Amur State University

Author for correspondence.
Email: cheb@iam.dvo.ru
690041, Vladivostok, Russia; 675027, Blagoveshchensk, Amur oblast, Russia

References

  1. Kovtanyuk A.E., Chebotarev A.Yu., Botkin N.D., Hoffmann K.-H. The unique solvability of a complex 3D heat transfer problem // J. Math. Anal. Appl. 2014. V. 409. № 2. P. 808–815.
  2. Алифанов О.М., Ненарокомов А.В., Артюхин Е.А. Обратные задачи в исследовании сложного теплообмена. М., Янус-К. 2009. 297 с.
  3. Алифанов О.М. Определение тепловых нагрузок из решения нелинейной обратной задач // Теплофизика высоких температур. 1977. Т. 15. Вып. 3. С. 598–605.
  4. Алифанов О.М., Ненарокомов А.В. Трехмерная граничная обратная задача теплопроводности // Теплофизика высоких температур. 1999. Т. 37. № 2. С. 231–238.
  5. Ковтанюк А.Е., Чеботарев А.Ю. Стационарная задача сложного теплообмена // Ж. вычисл. матем. и матем. физ. 2014. Т. 54. № 4. С. 711–719.
  6. Kovtanyuk A.E., Chebotarev A.Y., Botkin N.D., Hoffmann K.-H. Solvability of P1 approximation of a conductive-radiative heat transfer problem // Appl. Math. Comput. 2014. V. 249. P. 247–252.
  7. Kovtanyuk A.E., Chebotarev A.Yu., Botkin N.D., Hoffmann K.-H. Unique solvability of a steady-state complex heat transfer model // Commun. Nonlinear Sci. Numer. Simul. 2015. V. 20. № 3. P. 776–784.
  8. Pinnau R. Analysis of optimal boundary control for radiative heat transfer modeled by -system // Commun. Math. Sci. 2007. V. 5. № 4. P. 951–969.
  9. Grenkin G.V., Chebotarev A.Yu., Kovtanyuk A.E., Botkin N.D., Hoffmann K.-H. Boundary optimal control problem of complex heat transfer model // J. Math. Anal. Appl. 2016. V. 433. № 2. P. 1243–1260.
  10. Kovtanyuk A.E., Chebotarev A.Yu., Botkin N.D., Hoffmann K.-H. Optimal boundary control of a steady-state heat transfer model accounting for radiative effects // J. Math. Anal. Appl. 2016. V. 439. № 2. P. 678–689.
  11. Chebotarev A.Yu., Kovtanyuk A.E., Grenkin G.V., Botkin N.D., Hoffmann K.-H. Nondegeneracy of optimality conditions in control problems for a radiative-conductive heat transfer model // Appl. Math. Comput. 2016. V. 289. P. 371–380.
  12. Chebotarev A.Yu., Grenkin G.V., Kovtanyuk A.E., Botkin N.D., Hoffmann K.-H. Inverse problem with finite overdetermination for steady-state equations of radiative heat exchange // J. Math. Anal. Appl. 2018. V. 460. № 2. P. 737–744.
  13. Chebotarev A.Y., Kovtanyuk A.E., Botkin N.D. Problem of radiation heat exchange with boundary conditions of the Cauchy type // Commun. Nonlinear Sci. and Numeric. Simulat. 2019. V. 75. P. 262–269.
  14. Chebotarev A.Yu., Pinnau R. An inverse problem for a quasi-static approximate model of radiative heat transfer // J. Math. Anal. Appl. 2019. V. 472. № 1. P. 314–327.
  15. Гренкин Г.В., Чеботарев А.Ю. Обратная задача для уравнений сложного теплообмена // Ж. вычисл. матем. и матем. физ. 2019. Т. 59. № 8. С. 1420–1430.
  16. Месенев П.Р., Чеботарев А.Ю. Граничная обратная задача для уравнений сложного теплообмена // Дальневост. матем. журн. 2018. Т. 18. № 1. С. 75–84.
  17. Месенев П.Р., Чеботарев А.Ю. Анализ оптимизационного метода решения задачи сложного теплообмена с граничными условиями типа Коши // Ж. вычисл. матем. и матем. физ. 2022. Т. 62. № 1. С. 36–44.
  18. Amosov A.A. Unique solvability of stationary radiative-conductive heat transfer problem in a system of semitransparent bodies // J. Math. Sci. 2017. V. 224. № 5. P. 618–646.
  19. Amosov A.A., Krymov N.E. On a nonstandard boundary value problem arising in homogenization of complex heat transfer problems // J. Math. Sci. 2020. V. 244. P. 357–377.
  20. Amosov A. Unique solvability of a stationary radiative-conductive heat transfer problem in a system consisting of an absolutely black body and several semitransparent bodies // Math. Meth. Appl. Sci. 2021. V. 44. № 13. P. 10703–10733.
  21. Amosov A.A. Unique solvability of the stationary complex heat transfer problem in a system of gray bodies with semitransparent inclusions // J. Math. Sci. (United States). 2021. V. 255. Iss. 4. P. 353–388.
  22. Amosov A. Nonstationary radiative-conductive heat transfer problem in a semitransparent body with absolutely black inclusions // Mathematics 2021. V. 9. № 13. P. 1471.
  23. Fučik S., Kufner A. Nonlinear differential equations. Amsterdam–Oxford–New York, Elsevier, 1980.
  24. Fursikov A.V. Optimal control of distributed systems // Theory and Appl., Am. Math. Soc., 2000.
  25. Alnaes M.S., Blechta J., Hake J., Johansson A., Kehlet B., Logg A., Richardson C., Ring J., Rognes M.E., Wells G.N. The FEniCS Project Version 1.5 Archive of Numerical Software, vol. 3, 2015.
  26. Logg A., Wells G.N. DOLFIN: Automated Finite Element Computing ACM Transactions on Mathematical Software, vol. 37, 2010.
  27. https://github.com/mesenev/articles_src

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Copyright (c) 2023 П.Р. Месенев, А.Ю. Чеботарев

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