Zamana bağlı potansiyeli olan dikdörtgen bir zarın zorlanmış çapraz titreşimi için bir ters problem

Bu çalışmasa, dikdörtgen bir zarın zorlanmış enine titreşimi için hareket denkleminde ortaya çıkan iki boyutlu bir dalga denklemi için başlangıç-sınır değer problemi ele alınmıştır. Verilmiş bir ek koşul ile zamana bağlı katsayı belirlenmiştir ve yeteri kadar küçük zaman değerleri için varlık ve teklik teoremi ispatlanmıştır. Ayrıca, koşullu kararlılığın karakterizasyonu verilmiş ve ters problemin sayısal çözümü sonlu farklar yöntemi kullanılarak incelenmiştir.

Anahtar Kelimeler:

Ters problem, Fourier yöntemi, İki boyutlu dalga denklemi, Sonlu farklar yöntemi

An inverse problem for the forced transverse vibration of a rectangular membrane with time dependent potential

In this paper, an initial-boundary value problem for a two-dimensional wave equation which arises in the equation of motion for the forced transverse vibration of a rectangular membrane is considered. Giving an additional condition, a time-dependent coefficient is determined and existence and uniqueness theorem for small times is proved. Moreover, characterization of the conditional stability is given and numerical solution of the inverse problem investigated by using finite difference method.

Keywords:

Inverse problem, Fourier method, Two dimensional wave equation, Finite difference method,

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Rao S.S. 2007. Vibration of continuous systems. John Wiley & Sons, 744 pp.
Zachmanoglou E. C., Thoe D. W. 1986. Introduction to partial differential equations with applications. Courier Corporation, 417pp.
Aliev, Z. S., Mehraliev, Y.T. 2014. An inverse boundary value problem for a second-order hyperbolic equation with nonclassical boundary conditions: Doklady Mathematics, vol. 90, issue. 1, pp. 513-517. DOI: 10.1134/S1064562414050135
Aliyev, S.J., Aliyeva, A.G. 2017. The study of multidimensional mixed problem for one class of third order semilinear psevdohyperbolic equations: European Journal of Pure and Applied Mathematics, vol. 10, issue. 5, pp. 1078-1091.
Dehghan M., Mohebbi A. 2008. The combination of collocation, finite difference, and multigrid methods for solution of the twodimensional wave equation: Numerical Methods for Partial Differential Equations: An International Journal, vol. 24, issue. 3, pp. 897-910. DOI:10.1002/num.20295
Isakov V. 2006. Inverse problems for partial differential equations. Applied mathematical sciences,New York (NY): Springer, 358 pp.
Namazov G. K. 1984. Inverse Problems of the Theory of Equations of Mathematical Physics, Baku, Azerbaijan. (in Russian).
Prilepko A. I., Orlovsky D. G., Vasin I. A., 2000. Methods for solving inverse problems in mathematical physics. Vol. 231, Pure and AppliedMathematics, New York (NY): Marcel Dekker, 723 pp.
Romanov V.G. 1987. Inverse Problems of Mathematical Physics, VNU Science Press BV, Utrecht, Netherlands, 239pp.
Megraliev Y., Isgenderova Q.N. 2016. Inverse boundary value problem for a second-order hyperbolic equation with integral condition of the first kind, Problemy Fiziki, Matematiki Tekhniki(Problems of Physics, Mathematics and Technics) vol. 1 , pp. 42-47.
Aliyev S. J., Aliyeva A. G., Abdullayeva G. Z. 2018. The study of a mixed problem for one class of third order differential equations: Advances in Difference Equations, vol. 218, issue. 1. DOI:10.1186/s13662-018-1657-0
Romanov V.G. 1989. Local solvability of some multidimensional inverse problems for hyperbolic equations: Diff. Equ., vol. 25, no. 2, pp. 203-209.
Romanov V. G. 2018. Regularization of a Solution to the Cauchy Problem with Data on a Timelike Plane: Siberian Mathematical Journal, vol. 59, issue. 4, pp. 694-704. DOI: 10.1134/S0037446618040110
Yamamoto M. 1999. Uniqueness and stability in multidimensional hyperbolic inverse problems: Journal de mathématiques pures et appliquées, vol. 78, issue. 1, pp. 65-98. DOI: 10.1016/S0021-7824(99)80010-5
Imanuvilov O., Yu., Yamamoto M. 2001. Global uniqueness and stability in determining coefficients of wave equations: Communications in Partial Differential Equations, vol. 26, issue. 7-8, pp. 1409- 1425. DOI: 10.1081/PDE-100106139
Fatone L., Maponi P., Pignotti C., Zirilli F. 1997. An inverse problem for the two-dimensional wave equation in a stratified medium. In Inverse problems of wave propagation and diffraction,Springer, Berlin, Heidelberg. pp. 263-274.
Zhang G., Zhang Y. 1998. An iterative method for the inversion of the two-dimensional wave equation with a potential. Journal of Computational Physics, vol. 147, issue. 2, pp. 485-506. DOI: 10.1006/jcph.1998.9996
Shivanian E., Jafarabadi A. 2017. Numerical solution of twodimensional inverse force function in the wave equation with nonlocal boundary conditions: Inverse Problems in Science and Engineering, vol.25, issue. 12, pp. 1743-1767. DOİ:10.1080/17415977.2017.1289194
Han B., Fu H. S., Li Z. 2005. A homotopy method for the inversion of a two-dimensional acousticwave equation: Inverse Problems in Science and Engineering, vol. 13, issue. 4, pp. 411-431. DOI: 10.1080/17415970500126393
Kabanikhin S.I., Sabelfeld K.K., Novikov N.S., Shishlenin M.A. 2015. Numerical solution of the multidimensional GelfandLevitan equation: Journal of Inverse and Ill-Posed Problems, vol. 23, issue. 5, pp. 439-450. DOI: 10.1515/jiip-2014-0018
Kuliev M. A. 2002. A multidimensional inverse boundary value problem for a linear hyperbolicequation in a bounded domain: Differential Equations, vol.38, issue. 1, pp. 104-108. DOI: 10.1023/A:1014863828368
Khudaverdiyev K.I., Alieva A.G. 2010. On the global existence of solution to one-dimensional fourth order nonlinear Sobolev type equations: Appl. Math. Comput. Vol.217, issue. 1, pp. 347-354. DOI: 10.1016/j.amc.2010.05.067