EXISTENCE AND UNIQUENESS OF AN INVERSE PROBLEM FOR A WAVE EQUATION WITH DYNAMIC BOUNDARY CONDITION

Yıl 2020, Cilt: 10 Sayı: 2, 370 - 378, 01.03.2020

I. Tekin

Öz

In this paper, an initial boundary value problem for a wave equation with dynamic boundary condition is considered. Giving an additional condition, a timedependent coefficient is determined and existence and uniqueness theorem for small times is proved.

Anahtar Kelimeler

Wave equation, Inverse problem, Fourier method

Kaynakça

• Aliev, Z. S. and Mehraliev, Y. T., (2014), An inverse boundary value problem for a second-order hyperbolic equation with nonclassical boundary conditions, Doklady Mathematics, 90(1), 513-517.
• Beilin, S., (2001), Existence of solutions for one-dimensional wave equations with nonlocal conditions, Electronic Journal of Differential Equations, 76, 1-8.
• Binding P. A., Brown P. J. and Seddeghi K., (1993), Sturm–Liouville problems with eigenparameter dependent boundary conditions, Proc. Edinburgh Math. Soc. 37, 57-72.
• Budak, B. M., Samarskii, A. A. and Tikhonov, A. N., (2013) A Collection of Problems on Mathemat- ical Physics: International Series of Monographs in Pure and Applied Mathematics, Elsevier.
• Freiling, G. and Yurko, V. A., (2008), Lectures on the Differential Equations of Mathematical Physics: A First Course, New York: Nova Science Publishers.
• Fulton C. T., (1977), Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. R. Soc. Edinburgh: Sect. A Math. 77, 293-388.
• Goldstein, G. R., (2006), Derivation and physical interpretation of general boundary conditions. Ad- vances in Differential Equations 11(4), 457-480.
• Hazanee, A., Lesnic, D., Ismailov, M. I. and Kerimov, N. B., (2015), An inverse time-dependent source problem for the heat equation with a non-classical boundary condition, Applied Mathematical Modelling, 39(20), 6258-6272.
• Imanuvilov, O. and Yamamoto, M., (2001), Global uniqueness and stability in determining coefficients of wave equations, Comm. Part. Diff. Equat., 26, 1409- 1425.
• Isakov, V., (2006), Inverse problems for partial differential equations. Applied mathematical sciences, New York (NY): Springer.
• Kapustin N. Y. and Moiseev E. I., (1997), Spectral problems with the spectral parameter in the boundary condition, Differ. Equ. 33, 115-119.
• Kerimov N. B. and Allakhverdiev T. I., (1993), On a certain boundary value problem I, Differ. Equ. 29, 54-60.
• Kerimov N. B. and Mirzoev V. S., (2003), On the basis properties of one spectral problem with a spectral parameter in a boundary condition, Sib. Math. J. 44, 813-816.
• Khudaverdiyev, K. I. and Alieva, A. G., (2010), On the global existence of solution to one-dimensional fourth order nonlinear Sobolev type equations, Appl. Math. Comput. 217(1), 347-354.
• Megraliev, Y. and Isgenderova, Q. N., (2016), Inverse boundary value problem for a second-order hyperbolic equation with integral condition of the first kind, Problemy Fiziki, Matematiki i Tekhniki (Problems of Physics, Mathematics and Technics) 1, 42-47.
• Namazov, G. K., (1984), Inverse Problems of the Theory of Equations of Mathematical Physics, Baku, Azerbaijan. (in Russian).
• Prilepko, A. I. Orlovsky, D. G. and Vasin, I. A., (2000), Methods for solving inverse problems in mathematical physics. Vol. 231, Pure and Applied Mathematics, New York (NY): Marcel Dekker.
• Pul’kina, L. S. , (2003), A mixed problem with integral condition for the hyperbolic equation, Math- ematical Notes, 74.3(4), 411-421.
• Rao, Singiresu S., (2007), Vibration of continuous systems, John Wileys.
• Romanov, V. G., (1987), Inverse Problems of Mathematical Physics, VNU Science Press BV, Utrecht, Netherlands.
• ˇSiˇskov´a, K. and Slodiˇcka, M., (2017), Recognition of a time-dependent source in a time-fractional wave equation, Applied Numerical Mathematics 121, 1-17.
• ˇSiˇskov´a, K. and Slodiˇcka, M., (2018), A source identification problem in a time-fractional wave equa- tion with a dynamical boundary condition. Computers and Mathematics with Applications 75(12), 4337-4354.
• Walter J., (1973), Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z. 133, 301-312.
• Zhongyan, L. and Gilbert, R. P., (2004), Numerical algorithm based on transmutation for solving inverse wave equation, Mathematical and Computer Modelling 39(13), 1467-1476.