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The paper is devoted to solvability of inverse problem for one-dimensional wave equations with the help of Galerkin`n method. This theme attracts the interest of researchers due to many practical applications of such type problems: it is very actual task to find the wave speed in many acoustic problems. To solve the inverse problem it means to find (numerically or analytically) unknown coefficient of equations (in our case—to find the wave speed). In this directions the papers of Gorbachuk V. I. , Lions J.-L. , L. Pestov [8, 10], D. Strelnikov  can be mentioned.
In the paper the principal methods of investigations are based on methods of mentioned papers and the main task of the paper is to adapt and to use the Galerkin’s method for solving the Cauchy-Dirichlet inverse problem for wave equation.
As it is known, wave equation simulates the wave processes in the space, 3-dimensional wave equation simulates the acoustic wave. Solving inverse problems it is arising the necessarity to find unknown coefficient of equation, which in the case of wave equations is acoustic speed. This tasks are very interesting due to many applications in tomography, car-industry, where it is necessary to find signal, acoustic wave as inverse signal: car testing in limit speed, image quality of tomography, etc.The main goal of the paper is to prove the existence theorem for weak solution to inverse Cauchy-Dirichlet problem for one-dimensional wave equation with the help of Galerkin`s method and its adaptation for evolution equations. Let us note, that result of the existence theorem (see Theorem 1) coincide with the results of the work  in the case of the first boundary value problem and . But we did not use BC method, which can be applied only for concrete equations and concrete type of boundary conditions due to definition of corresponding to equation quadratic form. As alternative approach, we use Galerkin`s method, which is universal for all equations and all types of boundary conditions and which allows only to realize its steps and to prove convergence of approximating consequences.
Lions J.-L. (1968). Contr’ole Optimale de Syst‘emes Gouvernes par des Equations aux Derivees Partielles. Paris. Retrieved from https://books.google.com.ua/books/about/Contr%C3%B4le_optimal_de_syst%C3%A8mes_gouvern%C3%A9.html?id=tE7vAAAAMAAJ&redir_esc=y
Pestov L. (1999) On reconstruction of the speed of sound from a part of boundary. J. Inverse Ill-Posed Probl. 7. no. 5. 481–486. Retrieved from https://www.degruyter.com/document/doi/10.1515/jiip.1922.214.171.1241/html
Pestov L., Bolgova V., Kazarina O. Numerical recovering of a density by the BC-method. Inverse Probl. Imaging 4. 2010. no. 4. 701–712. Retrieved from https://www.researchgate.net/publication/45854968_Numerical_recovering_a_density_by_BC-method
Rassel D. L. (1971) Boudary value control theory of the higher-dimensioanal wave equation. SIAM J. Control Optim. 9. 29–42. Retrieved from https://link.springer.com/chapter/10.1007/978-3-642-46329-7_16
Belishev M. I. (2001) Dynamical systems with boundary control: models and characterization of inverse data. Inverse Problems 17. no. 4. 659–682. Retrieved from https://www.researchgate.net/publication/230900261_Dynamical_systems_with_boundary_control_Models_and_characterization_of_inverse_data
Belishev M. I. (1990) Equations of Gel’fand-Levitan type in a multidimensional inverse problem for the wave equation. J. Soviet Math. 50. No. 6. 1940-1944. Retrieved from https://iopscience.iop.org/article/10.1070/SM1992v072n02ABEH002141
Gorbachuk V. I., Gorbachuk M. L. (1991) Boundary value problems for operator differential equations. Mathematics and its applications. 48. Kluwer Academic Publishers Group. Dordrecht. 347. Retrieved from https://www.springer.com/de/book/9780792303817
Pestov L., Strelnikov D. (2019) Approximate controllability of the wave equation with mixed boundary conditions. Journal of Mathematical Sciences. Vol. 239. №. 1. Retrieved from https://link.springer.com/article/10.1007/s10958-019-04289-8
Shramenko V.M., Buriachenko K.O., Lamanskiy D.V. (2011) Application of nonlinear functional analysis to the theory of differential equations. Manual text. Donetsk: DonNU. 184. Retrieved from https://mph.kpi.ua/assets/img/Shramenko-V.M/MTDCHKAfinal.pdf
Pestov L. (2012) Inverse problem of determining absorption coefﬁcient in the wave equation by BC method. J. Inverse Ill-Posed Probl.20 DOI 10.1515/jip-2011-0015. 103–110. Retrieved from https://www.degruyter.com/document/doi/10.1515/jip-2011-0015/html