Recovery of a rapidly oscillating absolute term in the multidimensional hyperbolic equation. (English. Russian original) Zbl 07035826

Math. Notes 104, No. 4, 489-497 (2018); translation from Mat. Zametki 104, No. 4, 505-515 (2018).
Summary: The paper is devoted to the development of the theory of inverse problems for evolution equations with summands rapidly oscillating in time. A new approach to setting such problems is developed for the case in which additional constraints (overdetermination conditions) are imposed only on several first terms of the asymptotics of the solution rather that on the whole solution. This approach is realized in the case of a multidimensional hyperbolic equation with unknown absolute term.


35Lxx Hyperbolic equations and hyperbolic systems
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[1] Babich, P. V.; Levenshtam, V. B., Direct and inverse asymptotic problems with high-frequency terms, Asymptot. Anal., 97, 329-336, (2016) · Zbl 1342.35387
[2] Babich, P. V.; Levenshtam, V. B.; Prika, S. P., Recovery of a rapidly oscillating source in the heat equation from solution asymptotics, Zh. Vychisl. Mat. Mat. Fiz., 57, 1955-1965, (2017) · Zbl 06864318
[3] M.M. Lavret’ev, K.G. Reznitskaya, and V. G. Yakhno, One-Dimensional Inverse Problems ofMathematical Physics (Nauka, Novosibirsk, 1982) [in Russian].
[4] V. G. Romanov, Inverse Problems of Mathematical Physics (Nauka, Moscow, 1984) [in Russian]. · Zbl 0576.35001
[5] A.M. Denisov, Introduction to the Theory of Inverse Problems (Nauka, Moscow, 1994) [in Russian].
[6] Yu. E. Anikonov, Multidimentional Inverse and Ill-Posed Problems for Diffentional Equations (VSP, Utrecht, 1995).
[7] Yu. E. Anikonov, Formulas in Inverse and Ill-Posed Problems (VSP, Utrecht, 1997). · Zbl 0883.35124
[8] Yu. E. Anikonov, B. A. Bubhov, and G. N. Erokhin, Inverse and Ill-posed Source Problems (VSP, Utrecht, 1997). · Zbl 0905.35095
[9] A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics (Marcel Dekker, New York, 1999). · Zbl 0947.35173
[10] Yu. E. Anikonov, Inverse Problems for Kinetic and Other Evolution Equations (VSP, Utrecht, 2001).
[11] Yu. Ya. Belov, Inverse Problems for Partial Differential Equation (VSP, Utrecht, 2002). · Zbl 1045.35001
[12] M. M. Lavrentiev, Inverse Problems ofMathematical Physics (VSP, Utrecht, 2003).
[13] A. G. Megrabov, Forward and Inverse Problems for Hyperbolic, Elliptic and Mixed Type Equations (VSP, Utrecht, 2003). · Zbl 1104.35002
[14] M. Ivanchov, Inverse Problems for Equations of Parabolic Type (VNTL Publ., L’viv, 2003). · Zbl 1147.35110
[15] V. G. Romanov, Stability in Inverse Problems (Nauchnyi Mir,Moscow, 2005) [in Russian]. · Zbl 1171.35125
[16] S. I. Kabanikhin, Inverse and Ill-Posed Problems (Sib. Nauchn. Izd., Novosibirsk, 2008) [in Russian]. · Zbl 1170.35100
[17] Denisov, A. M., Asymptotic expansions of solutions to inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative, Zh. Vychisl.Mat.Mat. Fiz., 53, 744-752, (2013) · Zbl 1299.35313
[18] Kamynin, V. L., Inverse problem of simultaneously determining the right-hand side and the coefficient of a lower order derivative for a parabolic equation on the plane, Differ. Uravn., 50, 795-806, (2014) · Zbl 1325.35280
[19] Denisov, A. M., Problems of determining the unknown source in parabolic and hyperbolic equations, Zh. Vychisl. Mat. Mat. Fiz., 55, 830-835, (2015)
[20] Zen’kovskaya, S. M.; Simonenko, I. B., On the influence of a high-frequency vibration on the origin of convection, Izv. Akad. Nauk SSSR Ser. Mekh. Zhidk. Gaza, 5, 51-55, (1966)
[21] Simonenko, I. B., A justification of the averaging method for a problem of convection in a field of rapidly oscillating forces and for other parabolic equations, 236-253, (1972) · Zbl 0253.35049
[22] Levenshtam, V. B., The averaging method in the convection problem with high-frequency oblique vibrations, Sibirsk. Mat. Zh., 37, 1103-1116, (1996) · Zbl 0874.35094
[23] Levenshtam, V. B., Asymptotic expansion of the solution to the problem of vibrational convection, Zh. Vychisl. Mat. Mat. Fiz., 40, 1416-1424, (2000)
[24] Il’in, V. A., The solvability of mixed problems for hyperbolic and parabolic equations, 97-154, (1960)
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