Geodesic
Problems of determining the unknown source in parabolic and hyperbolic equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 5, pp. 830-835 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Initial-boundary value problems for a parabolic and a hyperbolic equation with a source are considered. The hyperbolic equation involves the second time derivative multiplied by a positive parameter $\varepsilon$ and coincides with the parabolic equation when $\varepsilon$ is zero. The source is the sum of two unknown functions of spatial variables multiplied by exponentially decaying functions of time. The inverse problems of determining the unknown functions of spatial variable from additional solution information that represents a function of time are considered. It is proved that the inverse problem for the parabolic equation has an infinite set of solutions, while, for any positive $\varepsilon$, the inverse problem for the hyperbolic equation has a unique solution. 
@article{ZVMMF_2015_55_5_a8,
     author = {A. M. Denisov},
     title = {Problems of determining the unknown source in parabolic and hyperbolic equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {830--835},
     year = {2015},
     volume = {55},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_5_a8/}
}
TY  - JOUR
AU  - A. M. Denisov
TI  - Problems of determining the unknown source in parabolic and hyperbolic equations
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2015
SP  - 830
EP  - 835
VL  - 55
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_5_a8/
LA  - ru
ID  - ZVMMF_2015_55_5_a8
ER  - 
%0 Journal Article
%A A. M. Denisov
%T Problems of determining the unknown source in parabolic and hyperbolic equations
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2015
%P 830-835
%V 55
%N 5
%U http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_5_a8/
%G ru
%F ZVMMF_2015_55_5_a8
A. M. Denisov. Problems of determining the unknown source in parabolic and hyperbolic equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 5, pp. 830-835. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_5_a8/

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

[2] Isakov V., Inverse source problems, American Mathematical Society, Rhode Island, 1990 | MR | Zbl

[3] Denisov A. M., Elements of the theory of inverse problems, VSP, Utrecht, 1999 | MR | Zbl

[4] Prilepko A. I., Orlovsky D. G., Vasin I. A., Methods for solving inverse problems in mathematical physics, Marcel Dekker, New York, 2000 | MR | Zbl

[5] Kabanikhin S. I., Obratnye i nekorrektnye zadachi, Sibirskoe nauchnoe izdatelstvo, Novosibirsk, 2008

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