Geodesic
An inverse problem of location type for a~hyperbolic system
Sibirskij žurnal industrialʹnoj matematiki, Tome 16 (2013) no. 4, pp. 3-20.

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We consider an inverse problem for a hyperbolic system of two first-order partial differential equations with two independent variables. The right-hand sides of the system are assumed discontinuous functions. The inverse problem consists in the determination of a certain hull which contains the discontinuity line of the right-hand sides. We first study the corresponding direct problem. The existence and uniqueness of a generalized solution to the direct problem are established, and the differential properties of this solution are studied. In particular, we prove that its first-order partial derivatives are unbounded near some rays directed along characteristics. This property is the base of the algorithm for solving the inverse problem. The inverse problem is considered in the two versions: in the first version the coefficients of the equations are given, and in the second, they are unknown.
Keywords: inverse problem, hyperbolic equation, discontinuous function, generalized solution, differential property. 
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D. S. Anikonov; S. G. Kazantsev; D. S. Konovalova. An inverse problem of location type for a~hyperbolic system. Sibirskij žurnal industrialʹnoj matematiki, Tome 16 (2013) no. 4, pp. 3-20. http://geodesic.mathdoc.fr/item/SJIM_2013_16_4_a0/

[1] Petrovskii I. G., Lektsii ob uravneniyakh s chastnymi proizvodnymi, Gostekhizdat, M., 1950

[2] Germogenova T. A., Lokalnye svoistva reshenii uravneniya perenosa, Nauka, M., 1986 | MR | Zbl

[3] Anikonov D. S., Kovtanyuk A. E., Prokhorov I. V., Ispolzovanie uravneniya perenosa v tomografii, Logos, M., 2000

[4] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR

[5] Konovalova D. S., “Nekotorye svoistva reshenii uravneniya perenosa”, Differentsialnye uravneniya, 42:5 (2006), 684–689 | MR | Zbl

[6] Konovalova D. S., Prokhorov I. V., “Chislennaya realizatsiya algoritma poetapnoi rekonstruktsii dlya zadachi rentgenovskoi tomografii”, Sib. zhurn. industr. matematiki, 11:4(36) (2008), 61–65 | MR | Zbl

[7] Konovalova D. S., “Poetapnoe reshenie obratnoi zadachi dlya uravneniya perenosa primenitelno k zadache tomografii”, Zhurn. vychisl. matematiki i mat. fiziki, 49:1 (2009), 189–199 | MR | Zbl

[8] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1976

[9] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1978 | MR

[10] Tikhonov A. N., Samarskii A. A., Uravneniya matematicheskoi fiziki, Nauka, M., 1977