On some classes of coefficirnt inverse problems of recovering thermophysical parameters in stratified media
https://doi.org/10.25587/2411-9326-2024-2-31-45
Abstract
We examine the solvability questions in Sobolev spaces of parabolic inverse coefficient problems in stratified media with transmission conditions of the imperfect contact type. A solution has all generalized derivatives involved in the equation summable to some power. The overdetermination conditions are the values of a solution at some points lying in the domain. The proof relies a priori estimates and the fixed-point theorem.
About the Authors
A. A. Potapkov
Yugra State University
Russian Federation
Russian Federation
Sergey G. Pyatkov
16 Chekhov Street, 628012 Khanty-Mansiisk
S. G. Pyatkov
Yugra State University
Russian Federation
Russian Federation
Alexey A. Potapkov
16 Chekhov Street, 628012 Khanty-Mansiisk
References
1. Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow (1967).
2. Permyakov P. P. Identification of Parameters in the Mathematical Model of Heat and Moisture Transfer in Frozen Soils, Nauka, Novosibirsk (1989).
3. Prilepko A. I., Orlovsky D. G., and Vasin I. A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York (1999).
4. Belov Ya. Ya., Inverse Problems for Parabolic Equations, VSP, Utrecht (2002).
5. Isakov V., Inverse Problems for Partial Differential Equations, Springer, Berlin (2006).
6. Kabanikhin S. I., Inverse and Ill-Posed Problems, Theory and Applications, Walter de Gruyter, Boston; Berlin (2012).
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. Ivanchov M., Inverse Problems for Equations of Parabolic Type, WNTL Publ., Lviv (2003).
9. Iskenderov A. D., “Multidimensional inverse problems for linear and quasilinear parabolic equations [in Russian],” Dokl. Akad. Nauk, 225, No. 5, 1005–1008 (1975).
10. Iskenderov A. D. and Akhundov A. Ya., “Inverse problem for a linear system of parabolic equations,” Dokl. Math., 79, No. 1, 73–75 (2009).
11. Frolenkov I. V. and Romanenko G. V., “On the solution of an inverse problem for a multidimensional parabolic equation [in Russian],” Sib. Zhurn. Ind. Mat., 15, No. 2, 139–146 (2012).
12. 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).
13. 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).
14. Pyatkov S. G., “On some classes of inverse problems for parabolic equations,” J. Inverse IllPosed Probl., 18, No. 8, 917–934 (2011).
15. Pyatkov S. G., “On some classes of inverse problems with overdetermination data on spatial manifolds,” Sib. Math. J., 57, No. 5, 870–880 (2016).
16. Pyatkov S. G. and Rotko V. V., “On some parabolic inverse problems with the pointwise overdetermination,” Sib. Adv. Math., 30, No. 2, 124–142 (2020).
17. Pyatkov S. G. and Rotko V. V., “Inverse problems with pointwise overdetermination for some quasilinear parabolic systems,” AIP Conf. Proc., 1907, paper ID 020008 (2017).
18. 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).
19. Samarskii A. A. and Vabishchevich P. N., Numerical Methods for Solving Inverse Problems of Mathematical Physics, Walter de Gruyter, Berlin; Boston (2007).
20. Alifanov O. M., Artyukhin E. A., and Nenarokomov A. V., Inverse Problems in the Study of Complex Heat Transfer [in Russian], Yanus-K, Moscow (2009).
21. Ozisik M. N. and Orlande H. R. B., Inverse Heat Transfer, Taylor & Francis, New York (2000).
22. 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).
23. Kabanikhin S. I., Khasanov A. Kh., and Penenko A. V., “Gradient descent method for solving inverse coefficient problem of heat conduction [in Russian],” Sib. J. Comput. Math., 11, No. 1, 41–51 (2008).
24. 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).
25. Pyatkov S. G. and Sokolkov O. I., “On some classes of coefficient inverse problems for determining thermophysical parameters in stratified media [in Russian],” Mat. Zamet. SVFU, 30, No. 2, 56–74 (2023).
26. Triebel H., Interpolation Theory, Function Spaces, Differential Operators, VEB Deutsch. Verl. Wissensch., Berlin (1978).
27. 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, 93–224 (2007).
28. Denk R., Hieber M., and Pr¨uss J., “R-boundedness, Fourier multipliers, and problems of elliptic and parabolic type,” Mem. AMS, 166, No. 788 (2003).
29. Belonogov V. A. and Pyatkov S. G., “On the solvability of conjugation problems with conditions of the non-ideal contact type [in Russian],” Izv. Vuzov, Mat., No. 7, 18–32 (2020).
30. Mikhaylov V. P., Partial Differential Equations, Imp. Publ, Chicago (1978).
31. Amann H., “Compact embeddings of vector-valued Sobolev and Besov spaces,” Glasnik Mat., 35, 161–177 (2000).
32. Nikolsky S. M., Approximation of Functions of Several Variables and Imbedding Theorems, Springer, Berlin; Heidelberg; New York (1975).
33. Amann H., Linear and Quasilinear Parabolic Problems, Birkh¨auser, Basel (1995).
Review
For citations:
Potapkov A.A., Pyatkov S.G. On some classes of coefficirnt inverse problems of recovering thermophysical parameters in stratified media. Mathematical notes of NEFU. 2024;31(2):31-45. (In Russ.) https://doi.org/10.25587/2411-9326-2024-2-31-45
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