Inverse problems of recovering the boundary data with integral overdetermination conditions

S. G. Pyatkov   ; M. A. Verzhbitskii

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 10 (2018) no. 2, pp. 37-46

Résumé

In the present article we examine an inverse problem of recovering unknown functions being part of the Dirichlet boundary condition together solving an initial boundary problem for a parabolic second order equation. Such problems on recovering the boundary data arise in various tasks of mathematical physics: control of heat exchange prosesses and design of thermal protection systems, diagnostics and identification of heat transfer in supersonic heterogeneous flows, identification and modeling of heat transfer in heat-shielding materials and coatings, modeling of properties and heat regimes of reusable heat protection of spacecrafts, study of composite materials, etc. As the overdetrermination conditions we take the integrals of a solution over the spatial domain with weights. The problem is reduced to an operator equation of the Volterra-type. The existence and uniqueness theorem for solutions to this inverse problem is established in Sobolev spaces. A solution is regular, i. e., all generalized derivatives occuring into the equation exists and are summable to some power. The proof relies on the fixed point theorem and bootstrap arguments. Stability estimates for solutions are also given. The solvability conditions are close to necessary conditions.

Détail

Keywords: inverse problem, boundary and initial condition, Sobolev space, existence and uniqueness theorem, solvability.
Mots-clés : parabolic equation

Bibliographie

[1] Triebel H., Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, North-Holland Publishing, Amsterdam, 1978, 528 pp. | MR | Zbl

[2] Alifanov O. M., Artyukhin E. A., Nenarokomov A. V., Obratnye zadachi v issledovanii slozhnogo teploobmena, Yanus-K Publ., M., 2009, 299 pp. (in Russ.)

[3] M.N. Ozisik, H.A.B. Orlando, Inverse heat transfer, Taylor Francis, New-York, 2000, 352 pp. | MR

[4] Kostin A. B., Prilepko A. I., “On some problems of recovering a boundary condition for a parabolic equation, I”, Differ. Equations, 32:1 (1996), 113–122 (in Russ.) | MR | Zbl

[5] Borukhov V. T., Korzyuk V. I., “Application of nonclassical boundary value problems for recovering boundary regimes of transfer processes”, Bulletin of Belarussian University. Ser. 1, 1998, no. 3, 54–57 (in Russ.)

[6] Tryanin A. P., “Determination of heat-transfer coefficients at the inlet into a porous body and inside it by solving the inverse problem”, Journal of engineering physics, 52:3 (1987), 346–351 | DOI

[7] Borukhov V. T., Vabishchevich P. N., Korzyuk V. I., “Reduction of a class of inverse heatconduction problems to direct initial/boundary-value problems”, Journal of Engineering Physics and Thermophysics, 73:4 (2000), 730–734 | DOI

[8] Korotkii A. I., Kovtunov D. A., “Reconstruction of boundary regimes in an inverse heat convection problem for an incompressible fluid”, Tr. IMM DVO AN, 12:2 (2006), 88–97 (in Russ.)

[9] Abylkairov U. U., “Obratnaya zadacha integral'nogo nablyudeniya dlya obshchego parabolicheskogo uravneniya”, Matematicheskiy zhurnal, 3:4(10) (2003), 5–12 (in Russ.)

[10] Abylkairov U. U., Abiev A. A., Aitzhanov S. E., “Inverse problem for the system of thermal convection”, Proc. International Youth Scientific School-Conference “Theory and numerical methods for solving inverse and ill-posed problems”, IM SB RAS Publ., Novosibirsk, 2009, 10–11 (in Russ.)

[11] Kozhanov A. I., “Linear inverse problems for some lasses of nonstationary equations”, Proc. VI International scientific school-conference for young scientists “Theory and numerical methods for solving inverse and ill-posed problem”, 2015, 264–275 (in Russ.) | DOI

[12] A.D. Iskenderov, A.Ya. Akhundov, “Inverse problem for a linear system of parabolic equations”, Doklady Mathematics, 79:1 (2009), 73–75 | DOI | MR | Zbl

[13] M.I. Ismailov, F. Kanca, “Inverse problem of finding the time-dependent coefficient of heat equation from integral overdetermination condition data”, Inverse Problems in Science and Engineering, 20:24 (2012), 463–476 | DOI | MR | Zbl

[14] J. Li, Y. Xu, “An inverse coefficient problem with nonlinear parabolic equation”, J. Appl. Math. Comput., 34 (2010), 195–206 | DOI | MR | Zbl

[15] N.B. Kerimov, M.I. Ismailov, “An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions”, J. of Mathematical Analysis and Applications, 396:2 (2012), 546–554 | DOI | MR | Zbl

[16] A.I. Kozhanov, “Parabolic equations with unknown time-dependent coefficients”, Comput. Math. and Math. Phys., 57:6 (2017), 956–966 | DOI | MR | Zbl

[17] Pyatkov S. G., Safonov E. I., “Determination of the Source Function in the Mathematical Models of Convection-Diffusion”, Mathematical notes of NEFU, 21:2 (2014), 117–130 (in Russ.)

[18] Kriksin Yu.A., Plushchev S. N., Samarskaya E. A., Tishkin V. F., “The inverse problem of source reconstruction for convective diffusion equation”, Matem. Mod., 7:11 (1995), 95–108 (in Russ.) | Zbl

[19] A.I. Prilepko, D.G. Orlovsky, I.A. Vasin, Methods for solving inverse problems in Mathematical Physics, Marcel Dekker, Inc., New-York, 1999, 744 pp. | DOI | MR

[20] M. Ivancho, Inverse problems for equations of parabolic type, Mathematical Studies Monograph Series, 10, VNTL Publishers, Lviv, 2003, 238 pp. | MR | Zbl

[21] Verzhbitskii M. A., Pyatkov S. G., “On Some Inverse Problems of Determining Boundary Regimes”, Mathematical notes of NEFU, 23:2 (2016), 3–16 (in Russ.)

[22] O.A. Ladyženskaja, V.A. Solonnikov, N.N. Uralceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1968, 648 pp. | DOI | MR

[23] R. Denk, M. Hieber, J. Prüss, “Optimal $L_p$-$L_q$-estimates for parabolic boundary value problems with inhomogeneous data”, Math. Z., 257:1 (2007), 193–224 | DOI | MR | Zbl