Existence and uniqueness of an inverse problem for a second order hyperbolic equation

Year 2018, Volume: 1 Issue: 3, 178 - 185, 30.09.2018

İbrahim Tekin

https://doi.org/10.32323/ujma.439662

Cited By: 1

Abstract

In this paper, an initial boundary value problem for a second order hyperbolic equation is considered. Giving an additional condition, a time-dependent coefficient multiplying a linear term is determined and existence and uniqueness theorem for small times is proved. The finite difference method is proposed for solving the inverse problem numerically.

Keywords

Second order hyperbolic equation, Inverse problem, Finite difference method, Finite difference method

References

• [1] L. Beilina, M. V. Klibanov, "A globally convergent numerical method for a coefficient inverse problem." SIAM Journal on Scientific Computing 31.1 (2008): 478-509.
• [2] J. R. Cannon, P. DuChateau, "An inverse problem for an unknown source term in a wave equation." SIAM Journal on Applied Mathematics 43.3 (1983): 553-564.
• [3] M. Dehghan, "On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation." Numerical Methods for Partial Differential Equations 21.1 (2005): 24-40.
• [4] S. O. Hussein, D. Lesnic, M. Yamamoto, "Reconstruction of space-dependent potential and/or damping coefficients in the wave equation." Computers \& Mathematics with Applications 74.6 (2017): 1435-1454.
• [5] O. Imanuvilov, M. Yamamoto,, "Global uniqueness and stability in determining coefficients of wave equations." Comm. Part. Diff. Equat., 26 (2001), 1409-- 1425.
• [6] N. I. Ionkin, "The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition", Differ. Uravn., 1977, Volume 13, Number 2, 294--304
• [7] V. Isakov,, Inverse problems for partial differential equations. Applied mathematical sciences. New York (NY): Springer; 2006.
• [8] K. I. Khudaverdiyev, A. G. Alieva, "On the global existence of solution to one-dimensional fourth order nonlinear Sobolev type equations." Appl. Math. Comput. 217 (2010), no. 1, 347-354.
• [9] D. Lesnic, S. O. Hussein, B. T. Johansson, "Inverse space-dependent force problems for the wave equation." Journal of Computational and Applied Mathematics 306 (2016): 10-39.
• [10] Z. Lin, R. P. Gilbert,, "Numerical algorithm based on transmutation for solving inverse wave equation." Mathematical and computer modelling 39.13 (2004): 1467-1476.
• [11] Y. Megraliev, Q. N. Isgenderova, "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 (2016): 42-47.
• [12] Y. T. Mehraliyev, "On the identification of a linear sourcenfor the second order elliptic equation with integral condition", Tr. Inst. Mat., 2013, Volume 21, Number 2, 128--141
• [13] D.A. Murio,, Mollification and space marching, in:K.A.Woodbury (Ed.), Inverse Engineering Handbook, CRC Press, Boca Raton,Florida, 2002, pp. 219-326.
• [14] G. K. Namazov,, Inverse Problems of the Theory of Equations of Mathematical Physics, Baku, Azerbaijan, 1984. (in Russian).
• [15] A. I. Prilepko, D. G. Orlovsky, I. A. Vasin, Methods for solving inverse problems in mathematical physics. Vol. 231, Pure and AppliedMathematics. New York (NY): Marcel Dekker; 2000.
• [16] V.G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press BV, Utrecht, Netherlands, 1987.
• [17] K. \v{S}i\v{s}kov\'{a}, M. Slodi\v{c}ka. "Recognition of a time-dependent source in a time-fractional wave equation." Applied Numerical Mathematics 121 (2017): 1-17.