We consider the inverse scattering problem in a homogeneous nonstationary one-dimensional medium for a system of acoustic equations. A class of boundary sources is identified for which the problem of determining the time-dependent density is uniquely solvable. A method using integro-functional Volterra equations of first and third kind is proposed for the inverse problem. A regularized iterative algorithm is developed for the inverse problem in the framework of finite-difference theory. The results of a computational experiment are reported, applying the algorithm to various nonstationary media and boundary sources.
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References
V. G. Romanov, Inverse Problems of Mathematical Physics [in Russian], Nauka, Moscow (1984).
S. I. Kabanikhin, Projection-Difference Methods for the Determination of Coefficients of Hyperbolic Equations [in Russian], Nauka, Novosibirsk (1988).
M. I. Belishev and A. S. Blagoveshchenskii, Dynamic Inverse Problems in Wave Theory [in Russian], Izd. St. Peterburg University, St. Peterburg (1999).
S. I. Kabanikhin and M. A. Shishlenin, “Numerical algorithm for two-dimensional inverse acoustic problem based on Gel’fand-Levitan-Krein equation,” J. Inverse Ill-Posed Problems, 18, No. 9, 979–995 (2011).
S. I. Kabanikhin, Inverse and Ill-Posed Problems. Theory and Applications, De Gruyter (2011).
A. V. Baev, “Solution of an inverse scattering problem for the acoustic wave equation in three-dimensional media,” Comput. Math. and Math. Physics, 56, No. 12, 2043–2055 (2016).
A. V. Baev, “On local solvability of inverse dissipative scattering problems,” J. Inverse Ill-Posed Problems, 9, No. 4, 227–247 (2001).
A. V. Baev and S. V. Gavrilov, “An iterative way of solving the inverse scattering problem for an acoustic system of equations in an absorptive layered nonhomogeneous medium,” Moscow Univ. Comput. Math. and Cyber., 42, No. 2, 55–62 (2018).
L. D. Landau and E. M. Lifshits, Fluid Dynamics [in Russian], Nauka, Moscow (1988).
N. A. Magnitskii, “Volterra linear integral equations of the first and third kinds,” USSR Comput. Math. and Math. Physics, 19, No. 4, 182–200 (1979).
A. N. Tikhonov, A. V. Goncharskij, V. V. Stepanov, and A. G. Yagola, Numerical Methods for Ill-Posed Problems [in Russian], Nauka, Moscow (1990).
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Translated from Prikladnaya Matematika i Informatika, No. 60, 2019, pp. 38–50.
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Baev, A.V., Gavrilov, S.V. The Inverse Scattering Problem in a Nonstationary Medium. Comput Math Model 30, 218–229 (2019). https://doi.org/10.1007/s10598-019-09449-8
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DOI: https://doi.org/10.1007/s10598-019-09449-8