Geodesic
Initial Boundary and Inverse Problems for the Inhomogeneous Equation of a Mixed Parabolic-Hyperbolic Equation
Matematičeskie zametki, Tome 102 (2017) no. 3, pp. 415-435.

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A problem with inhomogeneous boundary and initial conditions is studied for an inhomogeneous equation of mixed parabolic-hyperbolic type in a rectangular domain. The solution is constructed as the sum of an orthogonal series. A criterion for the uniqueness of the solution is established. It is shown that the uniqueness of the solution and the convergence of the series depend on the ratio of the sides of the rectangle from the hyperbolic part of the mixed domain. On the basis of this problem, inverse problems for finding the factors of the time-dependent right-hand sides of the original equation of mixed type are stated and studied for the first time. The corresponding uniqueness theorems and the existence of solutions are proved using the theory of integral equations for inverse problems.
Keywords: parabolic-hyperbolic equation, initial boundary-value problem, inverse problem, spectral analysis, small denominator, existence and uniqueness of solutions. 
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K. B. Sabitov. Initial Boundary and Inverse Problems for the Inhomogeneous Equation of a Mixed Parabolic-Hyperbolic Equation. Matematičeskie zametki, Tome 102 (2017) no. 3, pp. 415-435. http://geodesic.mathdoc.fr/item/MZM_2017_102_3_a7/

[1] K. B. Sabitov, “Zadacha Trikomi dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti”, Matem. zametki, 86:2 (2009), 273–279 | DOI | MR | Zbl

[2] I. M. Gelfand, “Nekotorye voprosy analiza i differentsialnykh uravnenii”, UMN, 14:3 (87) (1959), 3–19 | MR | Zbl

[3] K. B. Sabitov, E M. Safin, “Obratnaya zadacha dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti”, Izv. vuzov. Matem., 2010, no. 4, 55–62 | MR | Zbl

[4] K. B. Sabitov, E M. Safin, “Obratnaya zadacha dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa”, Matem. zametki, 87:6 (2010), 907–918 | DOI | MR | Zbl

[5] K. B. Sabitov, “Nelokalnaya zadacha dlya neodnorodnogo uravneniya smeshannogo tipa”, Trudy Sterlitamakskogo filiala AN RB. Ser. “Fiz.-matem. i tekhn. nauki”, 6, Gilem, Ufa, 2009, 85–93

[6] A. Ya. Khinchin, Tsepnye drobi, Nauka, M., 1978 | MR | Zbl

[7] A. N. Tikhonov, V. Ya. Arsenin, Metody resheniya nekorrektnykh zadach, Nauka, M., 1974 | MR | Zbl

[8] V. K. Ivanov, V. V. Vasin, V. P. Tanana, Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978 | MR | Zbl

[9] M. M. Lavrentev, V. G. Romanov, S. P. Shishatskii, Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980 | MR | Zbl

[10] M. M. Lavretev, K. G. Reznitskaya, V. G. Yakhno, Odnomernye obratnye zadachi matematicheskoi fiziki, Nauka, Novosibirsk, 1982 | MR | Zbl

[11] V. G. Romanov, Nekotorye obratnye zadachi dlya uravnenii giperbolicheskogo tipa, Nauka, Novosibirsk, 1972 | MR | Zbl

[12] V. G. Romanov, S. I. Kabanikhin, Obratnye zadachi geoelektriki, Nauka, M., 1991 | MR

[13] A. M. Denisov, Vvedenie v teoriyu obratnykh zadach, Izd-vo Mosk. un-ta, M., 1994

[14] A. I. Prilepko, D. G. Orlovsky, I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Monogr. Textbooks Pure Appl. Math., 231, Marcel Dekker, New York, 1999 | MR | Zbl

[15] S. I. Kabanikhin, Obratnye i nekorrektnye zadachi, Sibirskoe nauchnoe izd-vo, Novosibirsk, 2009

[16] A. I. Prilepko, V. V. Solovev, “Teoremy razreshimosti i metod Rote v obratnykh zadachakh dlya uravneniya parabolicheskogo tipa. I”, Differents. uravneniya, 23:10 (1987), 1791–1799 | MR | Zbl