Two inverse problems are considered for the heat equation with unknown thermal conductivity coefficient and unknown function in the source term. The unknown functions are time-dependent and need to be determined using additional information about the solution of the mixed initial-boundary value problem for the heat equation. The difference in the two inverse problems is determined by the additional information used for their solution. Iterative numerical methods for the inverse problems are proposed and implemented. Results of the application of the iterative methods for the determination of the unknown functions are reported.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. N. Tikhonov, “Uniqueness theorems for the heat equation,” DAN SSSR, 1, No. 5, 294–300 (1935).
R. Lattes and J.-L. Lions, The Method of Quasi-Reversibility [Russian translation], Mir, Moscow (1970).
M. M. Lavrent’ev, V. G. Romanov, and S. P. Shithatskii, Ill-Posed Problems of Mathematical Physics and Analysis [in Russian], Nauka, Novosibirsk (1980).
M. M. Lavrent’ev, V. G. Romanov, and V. G. Vasil’ev, Multidimensional Inverse Problems for Differential Equations [in Russian], Nauka, Novosibirsk (1969).
V. G. Romanov, Inverse Problems of Mathematical Physics [in Russian], Nauka, Moscow (1984).
O. M. Alifanov, Inverse Problems of Heat Transfer [in Russian], Mashinostroenie, Moscow (1988).
A. M. Denisov, Introduction to the Theory of Inverse Problems [in Russian], MGU, Moscow (1994).
A. I. Prilepko, D. G. Orlovsky, and I. V. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York (2000).
V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York (2006).
S. I. Kabanikhin, Inverse and Ill-Posed Problems [in Russian], Nauka, Novosibirsk (2008).
S. A. Samarskii and P. N. Vabishchev, Numerical Methods for Inverse Problems of Mathematical Physics [in Russian], Izd. LKI, Moscow (2009).
N. V. Muzylev, “Uniqueness of simultaneous determination of the thermal conductivity and heat capacity coefficients,” Zh. Vychisl. Matemat. i Mat. Fiz., 23, No. 1, 102–108 (1983).
M. V. Klibanov, “A class of inverse problems for nonlinear parabolic equations,” DAN SSSR, 280, No. 3, 533–536 (1985).
N. I. Ivanchov and N. V. Pabyrivska, “Determination of two time-dependent coefficients in a parabolic equation,” Sib. Mat. Zh., 43, No. 2, 406–413 (2002).
A. C. Fatullaev, N. Gasilov, and I. Yusubov, “Simultaneous determination of unknown coefficients in a parabolic equation,” Applicable Analysis, 87, No. 10, 1167–1177 (2008).
M. S. Hussein, D. Lesnic, and M. I. Ivanchov, “Simultaneous determination of time-dependent coefficients in the heat equation,” Computer and Mathematics with Applications, 67, No. 5, 1065–1091 (2014).
L. D. Su, P. N. Vabishechevish, and V. I. Vasil’ev, “The inverse problem of simultaneous determination of right-hand side and the lowest coefficient in parabolic equations,” in: I. Dimov, I. Farabo, and I. Vulkov (eds.), Numerical Analysis and Its Applications, Springer (2017), pp. 633–639.
V. L. Kamynin, “The inverse problem of simultaneous determination of two time-dependent lower coefficients in a nondivergent parabolic equation on a plane,” Mat. Zametki, 107, No. 1, 74–86 (2020).
A. M. Denisov and S. I. Solov’eva, “Determining the intensity of variation of the heat sources in the heat equation,” Computational Mathematics and Modeling, 33, No. 1, 1–8 (2022).
A. M. Denisov, “Iterative method for the determination of the coefficient and the source term in the heat equation,” Diff. Uravn., 58, No. 6, 756–762 (2022).
S. Sh. Bimuratov and S. I. Kabanikhin, “Solution of the one-dimensional inverse problem of electrodynamics by the Newton–Kantorovich method,” 32, No. 12, 1900–195 (1992).
L. Monch, “A Newton method for solving inverse scattering problem for a sound-hard obstacle,” Inverse Problems, 12, No. 3, 309–324 (1996).
S. I. Kabanikhin, O. Scherzer, and M. A. Shichlenin, “Iteration method for solving a two-dimensional inverse problem for hyperbolic equation,” J. Inverse and Ill-Posed Problems, 11, No. 1, 1–23 (2003).
Yan-Bo Ma, “Newton method for estimation of the Robin coefficient,” J. Nonlin. Sci. Appl., 8, No. 5, 660–669 (2015).
A. M. Denisov, “Iterative method for the coefficient inverse problem for a hyperbolic equation,” Diff. Uravn., 53, No. 7, 943–949 (2017).
A. V. Baev and S. V. Gavrilov, “Iterative method for solving the inverse scattering problem for a system of acoustics equations in a layered-nonhomogeneous medium with absorption,” Vestnik MGU, ser. 15, Vychisl. Matem. i Kibern., No. 2, 7–14 (2018).
S. V. Gavrilov and A. M. Denisov, “Numerical method for solving the nonlinear operator equation arising in the coefficient inverse problem,” Diff. Uravn., 57, No. 7, 900–906 (2021).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Prikladnaya Matematika i Informatika, Issue 72, 2023, pp. 4–15
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Denisov, A.M., Solov’eva, S.I. Numerical Methods for Determining the Coefficient and the Source Term in the Heat Equation. Comput Math Model 33, 389–400 (2022). https://doi.org/10.1007/s10598-023-09581-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10598-023-09581-6