Volume 6, no. 4 Pages 73 - 86

# Inverse Problems for the Second Order Hyperbolic Equation with Unknown Time Depended Coefficient

R.R. Safiullova
We analyze the solvability of the inverse problem with an unknown time depended coefficient for a second-order hyperbolic equation. We also study uniqueness of the problem solution. The problem is stated as follows: it is required to find a solution and an unknown coefficient of the equation. Here the problem is considered in a rectangle area, with a set conditions being typical of the first boundary-value problem and an overdetermination condition being necessary of the unknown coefficient searching. To study solvability of the inverse problem, we realize a conversion from the initial problem to a some direct supplementary problem with trivial boundary conditions. We prove the solvability of the supplementary problem in the class of the functions considered above. Then we realize a conversion to the first problem again and as a result we receive the solvability of the inverse problem. To prove solvability of the problem, we use the method of continuation on a parameter, fixed point theorem, cut-off functions, and the method of regularization. In the article we prove the theorems of the existence and the uniqueness of the problem solution in the class of the functions considered above.
Full text
Keywords
inverse problem; hyperbolic equation; weighted equation; continuation method on parameter; method of a motionless point; regularization method.
References
1. Valitov I.R., Kozhanov A.I. Inverse Problems for Hyperbolic Equations: Unknown Time Depended Coefficients Case [Obratnye zadachi dlya giperbolicheskikh uravneniy: sluchay neizvestykh koeffitsientov, zavisyashchikh ot vremeni]. Vestnik NGU, Ser. Matematika, mekhanika, informatika, 2006, vol. 6, no. 1, pp. 3-18.
2. Valitov I.R. On Solvability Two Inverse Problems for Hyperbolic Equations [O razreshimosti dvukh obratnykh zadach dlya giperbolicheskikh uravneniy]. Trudy Sterlitamakskogo filiala Akademii nauk respubliki Bashkortostan, Ser. Fiziko-matematicheskie i tekhnicheskie nauki, 2006, no. 3, pp. 64-73.
3. Pavlov S.S. Nolinear Inverse Problems for Many-Dimensional Hyperbolic Equations with Integral Overdetermination [Nelineynye obratnye zadachi dlya mnogomernykh giperbolicheskikh uravneniy s integral'nym pereopredeleniem]. Matematicheskie zametki YaGU, 2011, vol. 19, no. 2, pp. 128-154.
4. Yakubov S.Ya. Lineynye differentsial'no-operatornye uravneniya i ikh prilozheniya [Linear Differential-Operated Equations and It's Applications]. Baku, ELM, 1985. 220 p.
5. Kozhanov A.I. Nonlinear Weighted Equations and Inverse Problems [Nelineynye nagruzhennye uravneniya i obratnye zadachi]. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki, 2004, vol. 44, no. 4, pp. 694-716.
6. Kozhanov A.I. On Solvability Some Nonlinear Inverse Problems for Composite Type Equations [O razreshimosti nekotorykh nelineynykh obratnykh zadach dlya uravneniy sostavnogo tipa]. Trudy mezhdunarodnoy konferentsii 'Nelokal'nye kraevye zadachi i rodstvennye problemy biologii, informatiki, fiziki'. Nalchik, 2006, no. 5, pp. 42-51.
7. Trenogin V.A. Funktsianal'nyy analiz [Functional Analysis]. Moscow, Nauka, 1980, 488 p.
8. Demidovich B.P. Lektsii po matematicheskoy teorii ustoychivosti [Lectures on the Mathematical Theory of Stability]. Moscow, Nauka, 1967, 472 p.
9. Ladyzhenskaya, О.А., Ural'tseva N.N. Lineynye i kvazilineynye uravneniya ellipticheskogo tipa [Linear and Quasilinear Elliptic Equations]. Moscow, Nauka, 1973, 578 p.