Abstract
Under study are the inverse problems of finding, together with a solution u(x,t) of the differential equation cut − Δu + a(x, t)u = f(x, t) describing the process of heat distribution, some real c characterizing the heat capacity of the medium (under the assumption that the medium is homogeneous). Not only the initial condition is imposed on u(x, t), but also the usual conditions of the first or second initial-boundary value problems as well as some special overdetermination conditions. We prove the theorems of existence of a solution (u(x, t), c) such that u(x, t) has all Sobolev generalized derivatives entered into the equation, while c is a positive number.
Similar content being viewed by others
References
A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics (Marcel Decker, New York, 1999).
V. Isakov, Inverse Problems for Partial Differential Equations (Springer, Heidelberg, 2009).
M. Ivanchov, Inverse Problems for Equations of Parabolic Type (VNTL Publishers, Lviv, 2003) [Math. Studies. Monograph Series 10].
Yu. E. Anikonov, Inverse Problems for Kinetic and Other Evolution Equations (VSP, Utrecht, 2001).
A. I. Kozhanov, “Nonlinear Loaded Equations and Inverse Problems,” Zh. Vychisl. Mat. Mat. Fiz. 44 (4), 694–716 (2004).
A. I. Kozhanov, “Parabolic Equations with Unknown Coefficients,” Zh. Vychisl. Mat. Mat. Fiz. 57 (6), 963–973 (2017).
A. I. Kozhanov, “A Nonlinear Loaded Parabolic Equation and a Related Inverse Problem,” Mat. Zametki 76 (6), 840–853 (2004)
A. I. Kozhanov, Math. Notes 76 (6), 784–795 (2004)].
A. I. Kozhanov, “Parabolic Equations with an Unknown Absorption Coefficient,” Dokl. Akad. Nauk 401 (6), 740–743 (2006).
A. I. Kozhanov and R. R. Safiullova, “Linear Inverse Problems for Parabolic and Hyperbolic Equations,” J. Inverse Ill-Posed Probl. 18 (1), 1–18 (2010).
A. Lorenzi, “Recovering Two Constants in a Linear Parabolic Equation. Inverse Problems in Applied Sciences,” J. Phys. Conf. Series. 73, 012014 (2007).
A. Lorenzi and G. Mola, “Identification of a Real Constant in Linear Evolution Equation in Hilbert Spaces,” Inverse Probl. Imaging. 5 (3), 695–714 (2011).
G. Mola, “Identification of the Diffusion Coefficient in a Linear Evolution Equation in a Hilbert Space,” J. Abstr. Differential Equations Appl. 2 (1), 18–28 (2011).
A. Lorenzi and G. Mola, “Recovering the Reaction and the Diffusion Coefficients in a Linear Parabolic Equation,” Inverse Probl. 28 (7), 075006 (2012).
A. Lorenzi and E. Paparoni, “Identifications of Two Unknown Coefficients in an Integro-Differential Hyperbolic Equation,” J. Inverse Ill-Posed Probl. 1 (2), 331–348 (1993).
A. S. Lyubanova, “Identification of a Constant Coefficient in an Elliptic Equation,” Appl. Anal. 87 (10–11), 1121–1128 (2008).
A. I. Kozhanov and R. R. Safiullova, “Definition of Parameters in a Telegraph Equation,” Ufimsk. Mat. Zh. 9 (1), 63–74 (2017).
V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1976) [in Russian].
S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (Nauka, Moscow, 1988) [in Russian].
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type (Nauka, Moscow, 1973) [in Russian].
S. Ya. Yakubov, Linear Differential-Operator Equations and Their Applications (Elm, Baku, 1985) [in Russian].
A. I. Kozhanov, Composite Type Equations and Inverse Problems (VSP, Utrecht, 1999).
A. G. Sveshnikov, A. B. Al’shin, M. O. Korpusov, and Yu. D. Pletner, Linear and Nonlinear Sobolev Type Equations (Fizmatlit, Moscow, 2007) [in Russian].
Funding
The author was supported by the Russian Foundation for Basic Research (project no. 18-01-00620).
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2020, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2020, Vol. 23, No. 1, pp. 93–106.
Rights and permissions
About this article
Cite this article
Kozhanov, A.I. The Heat Transfer Equation with an Unknown Heat Capacity Coefficient. J. Appl. Ind. Math. 14, 104–114 (2020). https://doi.org/10.1134/S1990478920010111
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478920010111