Geodesic
Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 10, pp. 1754-1759 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

An initial-boundary value problem for the two-dimensional heat equation with a source is considered. The source is the sum of two unknown functions of spatial variables multiplied by exponentially decaying functions of time. The inverse problem is stated of determining two unknown functions of spatial variables from additional information on the solution of the initial-boundary value problem, which is a function of time and one of the spatial variables. It is shown that, in the general case, this inverse problem has an infinite set of solutions. It is proved that the solution of the inverse problem is unique in the class of sufficiently smooth compactly supported functions such that the supports of the unknown functions do not intersect. This result is extended to the case of a source involving an arbitrary finite number of unknown functions of spatial variables multiplied by exponentially decaying functions of time. 
@article{ZVMMF_2016_56_10_a6,
     author = {A. M. Denisov},
     title = {Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1754--1759},
     year = {2016},
     volume = {56},
     number = {10},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_10_a6/}
}
TY  - JOUR
AU  - A. M. Denisov
TI  - Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2016
SP  - 1754
EP  - 1759
VL  - 56
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_10_a6/
LA  - ru
ID  - ZVMMF_2016_56_10_a6
ER  - 
%0 Journal Article
%A A. M. Denisov
%T Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2016
%P 1754-1759
%V 56
%N 10
%U http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_10_a6/
%G ru
%F ZVMMF_2016_56_10_a6
A. M. Denisov. Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 10, pp. 1754-1759. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_10_a6/

[1] Lavrentev M. M., Romanov V. G., Vasilev V. G., Mnogomernye obratnye zadachi dlya differentsialnykh uravnenii, Nauka, Novosibirsk, 1969

[2] Cannon J. R., Perez-Esteva S., “Uniqueness and stability of 3d heat sources”, Inverse Problems, 7:1 (1991), 57–62 | DOI | MR | Zbl

[3] Prilepko A. I., Kostin A. B., “O nekotorykh obratnykh zadachakh dlya parabolicheskikh uravnenii s finalnym i integralnym nablyudeniem”, Matem. sb., 183:4 (1992), 49–68 | Zbl

[4] Yamamoto M., “Conditional stability in determining of force terms of heat equations in a rectangle”, Math. and Comput. Modeling, 18:1 (1993), 79–88 | DOI | MR | Zbl

[5] Burykin A. A., Denisov A. M., “Determination of the unknown source in the heat equation”, Comput. Math. and Modeling, 8:4 (1997), 309–311 | DOI | MR

[6] Cannon J. R., Du Chateau P., “Structural identification of an unknown source term in a heat equation”, Inverse Problems, 14:3 (1998), 535–551 | DOI | MR | Zbl

[7] Prilepko A. I., Tkachenko D. S., “Svoistva reshenii parabolicheskogo uravneniya i edinstvennost resheniya obratnoi zadachi ob istochnike s integralnym pereopredeleniem”, Zh. vychisl. matem. i matem. fiz., 43:4 (2003), 562–570 | MR | Zbl

[8] Choulli M., Yamamoto M., “Conditional stability in determining a heat source”, J. Inverse and Ill-Posed Problems, 12:3 (2004), 233–243 | DOI | MR | Zbl

[9] Ling L., Yamamoto M., Hon Y. C., Takeuchi T., “Identification of source location in two-dimentisional heat equations”, Inverse Problems, 22:4 (2006), 1289–1305 | DOI | MR | Zbl

[10] Denisov A. M., “Zadachi opredeleniya neizvestnogo istochnika v parabolicheskom i giperbolicheskom uravneniyakh”, Zh. vychisl. matem. i matem. fiz., 55:5 (2015), 830–835 | DOI | Zbl

[11] Yi Z., Murio D. A., “Source term identification in 1-D IHCP”, Comput. Math. Appl., 47:12 (2004), 1921–1933 | DOI | MR | Zbl

[12] Trong D. D., Long N. T., Alain P. N. D., “Nonhomogeneous heat equation: Identification and regularization for the inhomogeneous term”, J. Math. Anal. Appl., 312:1 (2005), 93–104 | DOI | MR | Zbl

[13] Farcas A., Lesnic D., “The boundary-elements method for determination of a heat source dependent on one variable”, J. Engineering Mathematics, 54:4 (2006), 375–388 | DOI | MR | Zbl

[14] Dou F. F., Fu C. L., Yang F. L., “Optimal error bound and fourier regularization for identifying an unknown source in the heat equation”, J. Computat. and Appl. Math., 230:2 (2009), 728–737 | DOI | MR | Zbl

[15] Bers L., Dzhon F., Shekhter M., Uravneniya s chastnymi proizvodnymi, Mir, M., 1966