BibTex RIS Cite

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

Year 2020, Volume: 10 Issue: 2, 370 - 378, 01.03.2020

Abstract

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.

References

  • 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.
Year 2020, Volume: 10 Issue: 2, 370 - 378, 01.03.2020

Abstract

References

  • 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.
There are 24 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

I. Tekin This is me

Publication Date March 1, 2020
Published in Issue Year 2020 Volume: 10 Issue: 2

Cite