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Year 2018, Volume: 1 Issue: 3, 178 - 185, 30.09.2018
https://doi.org/10.32323/ujma.439662

Abstract

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.

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

Year 2018, Volume: 1 Issue: 3, 178 - 185, 30.09.2018
https://doi.org/10.32323/ujma.439662

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.

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.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

İbrahim Tekin 0000-0001-6725-5663

Publication Date September 30, 2018
Submission Date July 2, 2018
Acceptance Date August 16, 2018
Published in Issue Year 2018 Volume: 1 Issue: 3

Cite

APA Tekin, İ. (2018). Existence and uniqueness of an inverse problem for a second order hyperbolic equation. Universal Journal of Mathematics and Applications, 1(3), 178-185. https://doi.org/10.32323/ujma.439662
AMA Tekin İ. Existence and uniqueness of an inverse problem for a second order hyperbolic equation. Univ. J. Math. Appl. September 2018;1(3):178-185. doi:10.32323/ujma.439662
Chicago Tekin, İbrahim. “Existence and Uniqueness of an Inverse Problem for a Second Order Hyperbolic Equation”. Universal Journal of Mathematics and Applications 1, no. 3 (September 2018): 178-85. https://doi.org/10.32323/ujma.439662.
EndNote Tekin İ (September 1, 2018) Existence and uniqueness of an inverse problem for a second order hyperbolic equation. Universal Journal of Mathematics and Applications 1 3 178–185.
IEEE İ. Tekin, “Existence and uniqueness of an inverse problem for a second order hyperbolic equation”, Univ. J. Math. Appl., vol. 1, no. 3, pp. 178–185, 2018, doi: 10.32323/ujma.439662.
ISNAD Tekin, İbrahim. “Existence and Uniqueness of an Inverse Problem for a Second Order Hyperbolic Equation”. Universal Journal of Mathematics and Applications 1/3 (September 2018), 178-185. https://doi.org/10.32323/ujma.439662.
JAMA Tekin İ. Existence and uniqueness of an inverse problem for a second order hyperbolic equation. Univ. J. Math. Appl. 2018;1:178–185.
MLA Tekin, İbrahim. “Existence and Uniqueness of an Inverse Problem for a Second Order Hyperbolic Equation”. Universal Journal of Mathematics and Applications, vol. 1, no. 3, 2018, pp. 178-85, doi:10.32323/ujma.439662.
Vancouver Tekin İ. Existence and uniqueness of an inverse problem for a second order hyperbolic equation. Univ. J. Math. Appl. 2018;1(3):178-85.

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