ASYMPTOTICALLY STABLE SOLUTIONS WITH BOUNDARY AND INTERNAL LAYERS IN DIRECT AND INVERSE PROBLEMS FOR THE SINGULARLY PERTURBED HEAT EQUATION WITH A NONLINEAR THERMAL DIFFUSION

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

This paper proposes a new approach to the study of direct and inverse problems for a singularly perturbed heat equation with nonlinear temperature-dependent diffusion, based on the further development and use of asymptotic analysis methods in the nonlinear singularly perturbed reactiondiffusion-advection problems. The essence of the approach is presented using the example of a class of one-dimensional stationary problems with nonlinear boundary conditions, for which the case of applicability of asymptotic analysis is highlighted. Sufficient conditions for the existence of classical solutions of the boundary layer type and the type of contrast structures are formulated, asymptotic approximations of an arbitrary order of accuracy of such solutions are constructed, algorithms for constructing formal asymptotics are substantiated, and the Lyapunov asymptotic stability of stationary solutions with boundary and internal layers as solutions to the corresponding parabolic problems is investigated. A class of nonlinear problems that take into account lateral heat exchange with the environment according to Newton’s law is considered. A theorem on the existence and uniqueness of a classical solution with boundary layers in problems of this type is proven. As applications of the study, methods for solving specific direct and inverse problems of nonlinear heat transfer related to increasing the operating efficiency of rectilinear heating elements in the smelting furnaces — heat exchangers are presented: the calculation of thermal fields in the heating elements and the method for restoring the coefficients of thermal diffusion and heat transfer from modeling data.

About the authors

M. A. Davydova

Lomonosov Moscow State University

Email: m.davydova@physics.msu.ru
Russia

G. D. Rublev

A.M. Obukhov Institute of Atmospheric Physics of RAS

Email: rublev.gd15@physics.msu.ru
Moscow, Russia

References

  1. Галактионов, В.А. Процессы в открытых диссипативных системах / В.А. Галактионов, С.П. Курдюмов, А.А. Самарский. — М. : Знание, 1988. — 599 с.
  2. Маслов, В.П. Математическое моделирование процессов теплои массопереноса. Эволюция диссипативных структур / В.П. Маслов, В.Г. Данилов, К.Л. Волосов. — М. : Наука, 1987. — 351 с.
  3. Самарский, А.А. Вычислительная теплопередача / А.А. Самарский, П.Н. Вабищевич. — M. : Едиториал УРСС, 2003. — 784 с.
  4. Карташов, Э.М. Аналитические методы теории теплопроводности и её приложений / Э.М. Карташов, В.А. Кудинов. — 4-е изд., перераб. и сущ. доп. — М. : Ленанд, 2018. — 1078 с.
  5. Davydova, M.A. Multidimensional thermal structures in the singularly perturbed stationary models of heat and mass transfer with a nonlinear thermal diffusion coefficient / M.A. Davydova, S.A. Zakharova // J. Comput. Appl. Math. — 2022. — V. 400. — Art. 113731.
  6. Применение численно-асимптотического подхода в задаче восстановления параметров локального стационарного источника антропогенного загрязнения / Давыдова М.А., Н.Ф. Еланский, С.А. Захарова, О.В. Постыляков // Докл. РАН. Математика, информатика, процессы управления. — 2021. — Т. 496, № 1. — С. 34–39.
  7. Тепломассоперенос в теплозащитных композиционных материалах в условиях высокотемпературного нагружения / В.Ф. Формалев, С.А. Колесник, Е.Л. Кузнецова, Л.Н. Рабинский // Теплофизика высоких температур. — 2016. — Т. 54, № 3. — С. 415–422.
  8. Колмогоров, А.Н. Исследование уравнения диффузии, соединённой с возрастанием вещества, и его применение к одной биологической проблеме / А.Н. Колмогоров, И.Г. Петровский, Н.С. Пискунов // Бюлл. МГУ. Сер. А. Математика и механика. — 1937. — Т. 1, № 6. — С. 1–26.
  9. Crank, J. The Mathematics of Diffusion / J. Crank. — London : Oxford Univ. Press, 1956. — 347 p.
  10. Галактионов, В.А. Методы построения приближённых автомодельных решений нелинейных уравнений теплопроводности. IV / В.А. Галактионов, А.А. Самарский // Мат. сб. — 1983. — Т. 121, № 2. — С. 131–155.
  11. Cole, J.D. Perturbation Methods in Applied Mathematics / J.D. Cole. — New York : Springer-Verlag, 1981. — 558 p.
  12. Архитектура многомерных тепловых структур / С.П. Курдюмов, Е.С. Куркина, А.Б. Потапов, А.А. Самарский // Докл. АН СССР. — 1984. — Т. 274, № 5. — С. 1071–1074.
  13. Васильева, А.Б. Асимптотические разложения решений сингулярно возмущённых уравнений / А.Б. Васильева, В.Ф. Бутузов. М. : Наука, 1973. — 272 с.
  14. Васильева, А.Б. О контрастной структуре типа ступеньки для одного класса нелинейных сингулярно возмущённых уравнений второго порядка / А.Б. Васильева, М.А. Давыдова // Журн. вычислит. математики и мат. физики. — 1998. — Т. 38, № 6. — С. 938–947.
  15. Нефедов, Н.Н. Метод дифференциальных неравенств для некоторых сингулярно возмущённых задач в частных производных / Н.Н. Нефедов // Дифференц. уравнения. – 1995. — Т. 31, № 4. — С. 718–722.
  16. Inkmann, F. Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions / F. Inkmann // Indiana Univ. Math. J. — 1982. — V. 31, № 2. — P. 213–221.
  17. Wang, J. Monotone method for diffusion equations with nonlinear diffusion coefficients / J. Wang // Nonlinear Analysis. — 1998. — V. 34. — P. 113–142.
  18. Lukyanenko, D.V. Asymptotic analysis of solving an inverse boundary value problem for a nonlinear singularly perturbed time-periodic reaction–diffusion–advection equation / D.V. Lukyanenko, M.A. Shishlenin, V.T. Volkov // J. of Inverse and Ill-Posed Problems. — 2019. — V. 27, № 5. — P. 745–758.
  19. Применение асимптотического анализа для решения обратной задачи определения коэффициента линейного роста в уравнении типа Бюргерса / Д.В. Лукьяненко, В.Т. Волков, Н.Н. Нефедов, А.Г. Ягола // Вестн. Моск. ун-та. Сер. 3. Физика, астрономия. — 2019. — № 2. — С. 38–41.
  20. Numerical Methods for the Solution of Ill-Posed Problems / A.N. Tikhonov, A.V. Goncharsky, V.V. Stepanov, A.G. Yagola. — Dordrecht : Kluwer Academic Publishers, 1995. — 253 p.
  21. Давыдова, М.А. О новом подходе к задаче восстановления вертикального коэффициента турбулентной диффузии в пограничном слое атмосферы / М.А. Давыдова, Н.Ф. Еланский, С.А. Захарова // Докл. РАН. — 2020. — Т. 490, № 2. — С. 51–56.
  22. Zakharova, S.A. Use of asymptotic analysis for solving the inverse problem of source parameters determination of nitrogen oxide emission in the atmosphere / S.A. Zakharova, M.A. Davydova, D.V. Lukyanenko // Inverse Probl. Sci. Eng. — 2021. — V. 29, № 3. — P. 365–377.
  23. Давыдова, М.А. Существование и устойчивость решений с пограничными слоями в многомерных сингулярно возмущенных задачах реакция–диффузия–адвекция / М.А. Давыдова // Мат. заметки. — 2015. — Т. 98, № 6. — С. 853–864.
  24. Nefedov, N.N. On the existence and asymptotic stability of periodic contrast structures in quasilinear reaction–advection–diffusion equations / N.N. Nefedov, E.I. Nikulin, L. Recke // Russ. J. Math. Phys. — 2019. — V. 26, № 1. — P. 55–69.
  25. Курбатов, Ю.Л. Металлургические печи : учеб. пособие / Ю.Л. Курбатов, А.Б. Бирюков, Ю.Е. Рубан. М. ; Вологда : Инфра-Инженерия, 2022. — 384 с.
  26. Карбид кремния (Карборунд, SiC) [Электронный ресурс]. — Режим доступа: https://si-c.ru/informat/infosic.html. — Дата доступа: 20.11.2023.
  27. Литовский, Е.Я. Термофизические свойства огнеупоров / Е.Я. Литовский, Н.А. Пучкелевич. — М. : Металлургия, 1982. — 152 с.
  28. Калиткин, Н.Н. Численные методы / Н.Н. Калиткин. — М. : Наука, 1978. — 512 с.
  29. scipy.integrate.solve_ivp [Электронный ресурс]. — Режим доступа: https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html. — Дата доступа: 21.11.2023.
  30. Optimization (scipy.optimize) [Электронный ресурс]. — Режим доступа: https://docs.scipy.org/doc/scipy/tutorial/optimize.html#nelder-mead-simplex-algorithm-method-nelder-mead. — Дата доступа: 21.11.2023.
  31. Galaktionov, V.A., Kurdyumov, S.P., and Samarsky, A.A., Protsessy v otkrytykh dissipativnykh sistemakh (Processes in open dissipative systems), Moscow: Znanie, 1988.
  32. Maslov, V.P., Danilov, V.G., and Volosov, K.L., Matematicheskoye modelirovaniye protsessov teploi massoperenosa. Evolyutsiya dissipativnykh struktur (Mathematical modeling of heat and mass transfer processes. Evolution of dissipative structures), Moscow: Nauka, 1987.
  33. Samarsky, A.A. and Vabishchevich, P.N., Vychislitel’naya teploperedacha (Computational Thermoransfer). Moscow: Editorial URSS, 2003.
  34. Kartashov, E.M. and Kudinov, V.A., Analiticheskiye metody teorii teploprovodnosti i yeye prilozheniy (Analytical methods of the theory of thermotransfer and its applications), Moscow: Lenand, 2018.
  35. Davydova, M.A. and Zakharova, S.A., Multidimensional thermal structures in the singularly perturbed stationary models of heat and mass transfer with a nonlinear thermal diffusion coefficient, J. Comput. Appl. Math., 2022, vol. 400, art. 113731.
  36. Davydova, M.A., Elansky, N.F., Zakharova, S.A., and Postylyakov, O.V., Application of a numerical-asymptotic approach to the problem of restoring the parameters of a local stationary source of anthropogenic pollution, Doklady Mathematics, 2021, vol. 103, no. 1, pp. 26–31.
  37. Formalev, V.F., Kolesnik, S.A., Kuznetsova, E.L., and Rabinskii, L.N., Heat and mass transfer in thermal protection composite materials upon high temperature loading, High Temp., 2016, vol. 54, no. 3, pp. 390–396.
  38. Kolmogorov, A.N., Petrovsky, I.G., and Piskunov, N.S., The research of the equation of diffusion coupled with the increase of matter, and its application to a biological problem, Bulletin of Moscow State University. Ser. A. Mathematics and Mechanics, 1937, vol. 1, no. 6, pp. 1–26.
  39. Crank, J., The Mathematics of Diffusion, London: Oxford Univ. Press, 1956.
  40. Galaktionov, V.A. and Samarskii, A.A. Methods of constructing approximate self-similar solutions of nonlinear heat equations. IV, Math. USSR-Sb., 1984, vol. 49, no. 1, pp. 125–149.
  41. Cole, J.D., Perturbation Methods in Applied Mathematics, New York: Springer-Verlag, 1981.
  42. Kurdyumov, S.P., Kurkina, E.S., Potapov, A.B., and Samarskiy, A.A., The architecture of the multidimensional thermal structures, Dokl. AN USSR, 1984, vol. 274, no. 5, pp. 1071–1075.
  43. Vasilyeva, A.B. and Butuzov, V.F., Asimptoticheskiye razlozheniya resheniy singulyarno vozmushchennykh uravneniy (Asymptotic expansions of solutions to singularly perturbed equations), Moscow: Nauka, 1973.
  44. Vasil’eva, A.B. and Davydova, M.A., On a contrast steplike structure for a class of second-order nonlinear singularly perturbed equations, Comput. Math. Math. Phys., 1998, vol. 38, no. 6, pp. 900–910.
  45. Nefedov, N.N., The method of differential inequalities for some singularly perturbed partial differential equations, Different. Equations, 1995, vol. 31, no. 4, pp. 668–671.
  46. Inkmann, F., Existence and multiplicity theorems for semilinear elliptic equations with nonlinear boundary conditions, Indiana Univ. Math. J., 1982, vol. 31, no. 2, pp. 213–221.
  47. Wang, J., Monotone method for diffusion equations with nonlinear diffusion coefcients, Nonlinear Analysis, 1998, vol. 34, pp. 113–142.
  48. Lukyanenko, D.V., Shishlenin, M.A., and Volkov, V.T., Asymptotic analysis of solving an inverse boundary value problem for a nonlinear singularly perturbed time-periodic reaction–diffusion–advection equation, J. of Inverse and Ill-Posed Problems, 2019, vol. 27, no. 5, pp. 745–758.
  49. Lukyanenko, D.V., Volkov, V.T., Nefedov, N.N., and Yagola, A.G. Application of asymptotic analysis for solving the inverse problem of determining the coefcient of linear amplifcation in burgers’ equation, Moscow University Physics Bulletin, 2019, vol. 74, no. 2, pp. 131–136.
  50. Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., and Yagola, A.G., Numerical Methods for the Solution of Ill-Posed Problems, Dordrecht: Kluwer Academic Publishers, 1995.
  51. Davydova, M.A., Elanskii, N.F., and Zakharova, S.A., A new approach to the problem of reconstructing the vertical turbulent diffusion coefcient in the atmospheric boundary layer, Doklady Earth Sci., 2020, vol. 490, no. 2, pp. 92–96.
  52. Zakharova, S.A., Davydova, M.A., and Lukyanenko, D.V., Use of asymptotic analysis for solving the inverse problem of source parameters determination of nitrogen oxide emission in the atmosphere, Inverse Probl. Sci. Eng., 2021, vol. 29, no. 3, pp. 365–377.
  53. Davydova, M.A., Existence and stability of solutions with boundary layers in multidimensional singularly perturbed reaction–diffusion–advection problem, Math. Notes, 2015, vol. 98, pp. 909–919.
  54. Nefedov, N.N., Nikulin, E.I., and Recke, L., On the existence and asymptotic stability of periodic contrast structures in quasilinear reaction–advection–diffusion equations, Russ. J. Math. Phys., 2019, vol. 26. no. 1, pp. 55–69.
  55. Kurbatov, Yu.L., Biryukov, A.B., and Ruban, Yu.E., Metallurgicheskiye pechi (Metallurgical furnaces), Moscow; Vologda: Infra-Engineering, 2022.
  56. [Electronic resource] Silicon carbide (Carborundum, SiC). URL: https://si-c.ru/informat/infosic.html (date of the application: 20.11.2023).
  57. Litovsky, E.Ya. and Puchkelevich, N.A., Termofzicheskiye svoystva ogneuporov (Thermophysical properties of refractories), Moscow: Metallurgia, 1982.
  58. Kalitkin, N.N., Chislennyye metody (Numerical methods), Moscow: Nauka, 1978. 29. [Electronic resource] URL: https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve ivp.html (date of the application: 21.11.2023).
  59. [Electronic resource] URL: https://docs.scipy.org/doc/scipy/tutorial/optimize.html#nelder-mead-simplex-algorithm-method-nelder-mead (date of the application: 21.11.2023).

Copyright (c) 2024 Russian Academy of Sciences

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies