Geodesic
A method for finding coefficients of a quasilinear hyperbolic equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 5, pp. 813-833 
A. Yu. Shcheglov. A method for finding coefficients of a quasilinear hyperbolic equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 5, pp. 813-833. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_5_a4/
@article{ZVMMF_2006_46_5_a4,
     author = {A. Yu. Shcheglov},
     title = {A~method for finding coefficients of a~quasilinear hyperbolic equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {813--833},
     year = {2006},
     volume = {46},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_5_a4/}
}
TY  - JOUR
AU  - A. Yu. Shcheglov
TI  - A method for finding coefficients of a quasilinear hyperbolic equation
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2006
SP  - 813
EP  - 833
VL  - 46
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_5_a4/
LA  - ru
ID  - ZVMMF_2006_46_5_a4
ER  - 
%0 Journal Article
%A A. Yu. Shcheglov
%T A method for finding coefficients of a quasilinear hyperbolic equation
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2006
%P 813-833
%V 46
%N 5
%U http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_5_a4/
%G ru
%F ZVMMF_2006_46_5_a4

Voir la notice de l'article provenant de la source Math-Net.Ru

The inverse problem of finding the coefficients $q(s)$ and $p(s)$ in the equation $u_{tt}=a^2u_{xx}+q(u)u_t-p(u)u_x$ is investigated. As overdetermination required in the inverse setting, two additional conditions are set: a boundary condition and a condition with a fixed value of the timelike variable. An iteration method for solving the inverse problem is proposed based on an equivalent system of integral equations of the second kind. A uniqueness theorem and an existence theorem in a small domain are proved for the inverse problem to substantiate the convergence of the algorithm.

[1] Denisov A. M., Vvedenie v teoriyu obratnykh zadach, Izd-vo MGU, M., 1994

[2] Prilepko A. I., Orlovsky D. G., Vasin I. A., Methods for solving inverse problems in mathematical physics, Marcel Dekker, Inc., N.Y, Basel, 1999 | MR

[3] Ishmukhametov A. Z., Voprosy ustoichivosti i approksimatsii zadach optimalnogo upravleniya, VTs RAN, M., 2000

[4] Iskenderov A. D., “Ob odnoi obratnoi zadache dlya kvazilineinykh parabolicheskikh uravnenii”, Differents. ur-niya, 10:5 (1974), 890–898 | MR | Zbl

[5] Muzylev N. V., “O edinstvennosti odnovremennogo opredeleniya koeffitsientov teploprovodnosti i ob'emnoi teploemkosti”, Zh. vychisl. matem. i matem. fiz., 23:1 (1983), 102–108 | MR

[6] Scheglov A. Yu., “Obratnaya zadacha dlya kvazilineinogo uravneniya teploprovodnosti”, Vestn. MGU. Ser. 15. Vychisl. matem. i kibernetika, 1987, no. 2, 8–11

[7] Drozhzhina O. V., “Metod chislennogo vosstanovleniya dvukh koeffitsientov v nelineinom neodnorodnom uravnenii teploprovodnosti”, Vestn. MGU. Ser. 15. Vychisl. matem. i kibernetika, 2003, no. 2, 6–12 | MR

[8] Glushkova E. S., “Über Die Eindeutigkeit einiger inverser Probleme fur die Telegraphengleichung”, Matem. probl. geofiz., 6, Novosibirsk, 1975, 130–144 | Zbl

[9] Cavaterra C., “An inverse problem for a semilinear wave equation”, Boll. Univ. Mat. Ital. (B). Ser. 7, 2:3 (1988), 695–711 | MR | Zbl

[10] Guo Bao Qi, Wang Jian Tao, “An inverse problem for a one-dimensional semilinear hyperbolic equation with an unknown source”, J. Harbin Inst. Techn., 4 (1989), 14–20 | MR | Zbl

[11] Grasselli M., “Local existence and uniqueness for a quasilinear hyperbolic inverse problem”, J. Appl. Analys., 32:1 (1989), 15–30 | DOI | MR | Zbl

[12] Denisov A. M., “Determination of a nonlinear coefficient in a hyperbolic equation for the Goursat problem”, J. Inverse. Ill-Posed Problems, 6:4 (1998), 327–334 | DOI | MR | Zbl

[13] Scheglov A. Yu., “Metod priblizhennogo resheniya obratnoi zadachi dlya polulineinogo uravneniya giperbolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 43:1 (2003), 111–126 | MR

[14] Scheglov A. Yu., “Metod priblizhennogo resheniya v $C^2$ uravneniya giperbolicheskogo tipa s lipshitsevoi nelineinostyu”, Zh. vychisl. matem. i matem. fiz., 41:3 (2001), 420–435