An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions. (English) Zbl 1248.35234
Summary: This paper investigates the inverse problem of finding a time-dependent coefficient in a heat equation with nonlocal boundary and integral overdetermination conditions. Under some regularity and consistency conditions on the input data, the existence, uniqueness and continuous dependence upon the data of the solution are shown by using the generalized Fourier method.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35K05 | Heat equation |
| 35N10 | Overdetermined systems of PDEs with variable coefficients |
References:
| [1] | Cannon, J. R.; Hoek, J., Diffusion subject to the specification of mass, J. Math. Anal. Appl., 115, 2, 517-529 (1986) · Zbl 0602.35048 |
| [2] | Kamynin, L. A., A boundary value problem in a theory of heat conduction with a non-classical boundary conditions, USSR Comput. Math. Phys., 4, 33-59 (1964) · Zbl 0206.39801 |
| [3] | Ionkin, N. I., Solution of a boundary-value problem in heat conduction with a non-classical boundary condition, Differential Equations, 13, 204-211 (1977) · Zbl 0403.35043 |
| [4] | Cannon, J. R.; Lin, Y.; Wang, S., Determination of a control parameter in a parabolic partial differential equation, J. Austral. Math. Soc. Ser. B., 33, 149-163 (1991) · Zbl 0767.93047 |
| [5] | Coddington, E. A.; Levinson, N., Theory of Differential Equations (1955), McGraw-Hill: McGraw-Hill New York · Zbl 0042.32602 |
| [6] | Naimark, M. A., Linear Differential Operators: Elementary Theory of Linear Differential Operators (1967), Frederick Ungar Publishing Co.: Frederick Ungar Publishing Co. New York · Zbl 0219.34001 |
| [7] | Namazov, G. K., Definition of the unknown coefficient of a parabolic equation with nonlocal boundary and complementary conditions, Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 19, 5, 113-117 (1999) · Zbl 1243.35175 |
| [8] | Ismailov, M. I.; Kanca, F., An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions, Math. Methods. Appl. Sci., 34, 6, 692-702 (2011) · Zbl 1213.35402 |
| [9] | Ivanchov, M. I.; Pabyrivska, N. V., Simultaneous determination of two coefficients of a parabolic equation in the case of nonlocal and integral conditions, Ukrainian Math. J., 53, 5, 674-684 (2001) · Zbl 0991.35102 |
| [10] | Cannon, J. R.; Lin, Y.; Wang, S., Determination of source parameter in parabolic equation, Meccanica, 27, 85-94 (1992) · Zbl 0767.35105 |
| [11] | Fatullayev, A. G.; Gasilov, N.; Yusubov, I., Simultaneous determination of unknown coefficients in a parabolic equation, Appl. Anal., 87, 10-11, 1167-1177 (2008) · Zbl 1151.65342 |
| [12] | Cannon, J. R.; Lin, Y., An inverse problem of finding a parameter in a semi-linear heat equation, J. Math. Anal. Appl., 145, 2, 470-484 (1990) · Zbl 0727.35137 |
| [13] | Prilepko, A. I.; Solovev, V. V., On the solvability of inverse boundary value problems for the determination of the coefficient preceding the lower derivative in a parabolic equation, Differential Equations, 23, 1, 101-107 (1987) · Zbl 0661.35082 |
| [14] | Ivanchov, M. I., Inverse Problems for Equations of Parabolic Type (2003), VNTL Publishers: VNTL Publishers Lviv · Zbl 1142.35625 |
| [15] | Nakhushev, A. M., Equations of Mathematical Biology (1995), Vysshaya Shkola: Vysshaya Shkola Moscow, (in Russian) · Zbl 0991.35500 |
| [16] | Isakov, V., Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math., 44, 2, 185-209 (1991) · Zbl 0729.35146 |
| [17] | Rundell, W., The determination of a parabolic equation from initial and final data, Proc. Amer. Math. Soc., 99, 637-642 (1987) · Zbl 0644.35093 |
| [18] | Prilepko, A. I.; Kostin, A. B., Some inverse problems for parabolic equations with final and integral observation, Mat. Sb., 183, 4, 49-68 (1992), Translation in Russian Acad. Sci. Sb. Math. 75 (2) (1993) 473-490 · Zbl 0802.35158 |
| [19] | Ramm, A. G., An inverse problem for the heat equation, J. Math. Anal. Appl., 264, 2, 691-697 (2001) · Zbl 0987.35164 |
| [20] | Yang, L.; Yu, J. N.; Deng, Z. C., An inverse problem of identifying the coefficient of parabolic equation, Appl. Math. Model., 32, 10, 1984-1995 (2008) · Zbl 1145.35468 |
| [21] | Deng, Z. C.; Yu, J. N.; Yang, L., Optimization method for an evolutional type inverse heat conduction problem, J. Phys. A, 41, 3, 20 pp. (2008), 035201 · Zbl 1149.35079 |
| [22] | Deng, Z. C.; Yang, L.; Yu, J. N.; Luo, G. W., Identifying the radiative coefficient of an evolutional type heat conduction equation by optimization method, J. Math. Anal. Appl., 362, 1, 210-223 (2010) · Zbl 1181.35324 |
| [23] | Deng, Z. C.; Yang, L.; Chen, N., Uniqueness and stability of the minimizer for a binary functional arising in an inverse heat conduction problem, J. Math. Anal. Appl., 382, 1, 474-486 (2011) · Zbl 1218.49030 |
| [24] | Shkalikov, A. A., On the basis problem of the eigenfunctions of an ordinary differential operator, Russ. Math. Surv., 34, 5, 249-250 (1979) · Zbl 0471.34014 |
| [25] | Shkalikov, A. A., Basis property of eigenfunctions of ordinary differential operators with integral boundary conditions, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 120, 6, 12-21 (1982) · Zbl 0565.34020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
