On some classes of coefficient inverse problems of recovering thermophysical parameters in stratified media
https://doi.org/10.25587/SVFU.2023.94.19.005
Abstract
We examine the question of regular solvability in Sobolev spaces of parabolic inverse coefficient problems in stratified media with conjugation conditions of the diffraction type. A solution has all generalized the derivatives occurring in the equation summable with some power. The overdetermination conditions are the values of the solution at some collection of points lying inside the domain. The proof is based on a priori estimates and the fixed point theorem.
About the Authors
S. G. Pyatkov
Yugra State University; Academy of Sciences of the Republic of Sakha (Yakutia)
Russian Federation
Russian Federation
Sergey G. Pyatkov
16 Chekhov Street, Khanty-Mansiisk 628011, Russia
33 Lenin Avenue, Yakutsk 677007, Russia
O. I. Sokolkov
Yugra State University
Russian Federation
Russian Federation
Oleg I. Sokolkov
16 Chekhov Street, Khanty-Mansiisk 628011, Russia
References
1. Permyakov P. P., Identification of Parameters of the Mathematical Model of Heat and Moisture Transfer in Frozen Soils [in Russian], Nauka, Novosibirsk (1989).
2. Prilepko A. I., Orlovsky D. G., and Vasin I. A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York (1999).
3. Belov Ya. Ya., Inverse problems for Parabolic Equations, VSP, Utrecht (2002).
4. Isakov V., Inverse Problems for Partial Differential Equations, Springer, Berlin (2006) (Appl. Math. Sci.; vol. 127).
5. Kabanikhin S. I., Inverse and Ill-Posed Problems, Theory and Applications, Walter de Gruyter, Boston; Berlin (2012).
6. Klibanov M. V. and Li J., Inverse Problems and Carleman Estimates, Walter de Gruyter, Berlin; Boston (2021) (Inverse Ill-Posed Probl. Ser., vol. 63).
7. Hussein M. S. and Huntul M. J., “Simultaneous identification of thermal conductivity and heat source in the heat equation,” Iraqi J. Sci., 62, No. 6, 1968–1978 (2021).
8. Hussein M. S., Lesnic D., and Ivanchov M. I., “Simultaneous determination of time-dependent coefficients in the heat equation,” Comput. Math. Appl., 67, 1065–1091 (2014).
9. Ivanchov M., Inverse Problems for Equations of Parabolic Type, WNTL Publ., Lviv (2003) (Math. Stud. Monogr. Ser.; vol. 10).
10. Ivanchov N. I. and Pabyrivska N. V., “On determination of two time-dependent coefficients in a parabolic equation,” Sib. Math. J., 43, No. 2, 323–329 (2002).
11. Iskenderov A. L., “Multi-dimensional inverse problems for linear ans quasilinear parabolic equations [in Russian],” Dokl. AN SSSR, 225, No. 5, 1005–1008 (1975).
12. Iskenderov A. D. and Akhundov A. Ya., “Inverse problem for a linear system of parabolic equations,” Dokl. Math., 79, No. 1, 73–75 (2009).
13. Frolenkov I. V. and Romanenko G. V., “On a solution of an inverse problem for multidimansional parabolic equation [in Russian],” Sib. J. Ind. Math., 15, No. 2, 139–146 (2012).
14. Pyatkov S. G. and Samkov M. L., “On some classes of coefficient inverse problems for parabolic systems of equations,” Sib. Adv. Math., 22, No. 4, 287–302 (2012).
15. Pyatkov S. G. and Tsybikov B. N., “On some classes of inverse problems for parabolic and elliptic equations,” J. Evol. Equ., 11, No. 1, 155–186 (2011).
16. Pyatkov S. G., “On some classes of inverse problems for parabolic equations,” J. Inverse IllPosed Probl., 18, No. 8, 917–934 (2011).
17. Pyatkov S. G., “On some classes of inverse problems with overdetermination data on spatial manifolds,” Sib. Math. J., 57, No. 5, 870–880 (2016).
18. Pyatkov S. G. and Rotko V. V., “Inverse problems with pointwise overdetermination for some quasilinear parabolic systems,” Sib. Adv. Math., 30, No. 2, 124–142 (2020).
19. Pyatkov S. G. and Rotko V. V., “On some parabolic inverse problems with the pointwise overdetermination,” AIP Conf. Proc., 1907, 020008 (2017).
20. Egger H., Pietschmann J.-F., and Schlottbom M., “Identification of nonlinear heat conduction laws,” J. Inverse Ill-Posed Probl., 23, No. 5, 429–437 (2015).
21. Pyatkov S. G., “Identification of thermophysical parameters in mathematical models of heat and mass transfer,” J. Comput. Eng. Math., 9, No. 2, 52–66 (2022).
22. Samarskii A. A. and Vabishchevich P. N., Numerical Methods for Solving Inverse Problems of Mathematical Physics, Walter de Gruyter, Berlin; Boston (2007).
23. Alifanov O. M., Artyukhov E. A., and Nenarokom A. V., Inverse problems of Complex Heat Transfer [in Russian], Yanus-K, Moscow (2009).
24. Ozisik M. N. and Orlande H. R. B., Inverse Heat Transfer, Taylor & Francis, New York (2000).
25. Tkachenko V. N., Mathematical Modeling, Identification and Control of Technological Processes of Heat Treatment of Materials [in Russian], Naukova Dumka, Kiev (2008).
26. Huntul M. J. and Lesnic D., “An inverse problem of finding the time-dependent thermal conductivity from boundary data,” Int. Commun. Heat Mass Transfer, 85, 147–154 (2017).
27. Kabanikhin S. I., Hasanov A., and Penenko A. V., “A gradient descent method for solving an inverse coefficient heat conduction problem,” Numer. Anal. Appl., 1, No. 1, 34–45 (2008).
28. Mustonen L., “Numerical study of a parametric parabolic equation and a related inverse boundary value problem,” Inverse Probl., 32, 105008 (2016).
29. Bushuev A. Yu. and Timofeev V. N., “Two parameters approach to solving the problem of identification of thermophysical characteristics of multilayer plate,” Eng. J. Sci. Innov., 9 (2013). http://engjournal.ru/articles/963/963.pdf
30. Bykov N. Yu., Khvatov A. A., Kalyuzhnaya A. V., and Bukhanovskii A. V., “The method of recovering mathematical models of heat and mass transfer with space and temporal distributions of parameters [in Russian],” Lett. JTP, 47, No. 24, 9–12 (2021).
31. Triebel H., Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verl. Wiss., Berlin (1978).
32. Denk R., Hieber M., and Pr¨uss J., “Optimal Lp-Lq-estimates for parabolic boundary value problems with inhomogeneous data,” Math. Z., 257, No. 1, 193–224 (2007).
33. Denk R., Hieber M., and Pr¨uss J., “R-boundedness, Fourier multipliers and problems of elliptic and parabolic type,” Mem. Amer. Math. Soc., 166, No. 788 (2003).
34. Ladyzhenskaya O. A., Solonnikov V. A., and Uraltseva N. N., Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence (1968).
35. Amann H., “Compact embeddings of vector-valued Sobolev and Besov spaces,” Glasnik Mat., 35, 161–177 (2000).
36. Pr¨uss J. and Simonett G., Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkh¨auser, Basel (2016).
37. Agranovich M. S. and Vishik M. I., “Elliptic problems with a parameter and parabolic problems of general type,” Russ. Math. Surv., 19, 53–157 (1964).
38. Amann H., “Maximal regularity of parabolic transmission problems,” J. Evol. Equ., 21, 3375–3420 ( 2021).
39. Mikhailov V. P., Partial Differential Equations [in Russian], Nauka, Moscow (1976) .
40. Nikolskii S. M., Approximation of Functions of Several Variables and Imbedding Theorems, Springer, Berlin (1975).
Review
For citations:
Pyatkov S.G., Sokolkov O.I. On some classes of coefficient inverse problems of recovering thermophysical parameters in stratified media. Mathematical notes of NEFU. 2023;30(2):56-74. (In Russ.) https://doi.org/10.25587/SVFU.2023.94.19.005
Views:
6
JATS XML
Email this article 