Inverse problems in the multidimensional hyperbolic equation with rapidly oscillating absolute term. (English) Zbl 1478.35030

Kusraev, Anatoly G. (ed.) et al., Operator theory and differential equations. Selected papers based on the presentations at the 15th conference on order analysis and related problems of mathematical modeling, Vladikavkaz, Russia, July 15–20, 2019. Cham: Birkhäuser. Trends Math., 7-23 (2021).
Summary: The paper is devoted to the development of the theory of inverse problems for evolution equations with terms rapidly oscillating in time. A new approach to setting such problems is developed for the case in which additional constraints are imposed only on several first terms of the asymptotics of the solution rather than on the whole solution. This approach is realized in the case of a multidimensional hyperbolic equation with unknown absolute term.
For the entire collection see [Zbl 1470.47003].


35B40 Asymptotic behavior of solutions to PDEs
35R30 Inverse problems for PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
34C29 Averaging method for ordinary differential equations
Full Text: DOI arXiv


[1] Lavret’ev, M.M., Reznitskaya, K.G., Yakhno, V.G.: One-Dimensional Inverse Problems of Mathematical Physics. Nauka, Novosibirsk (1982, in Russian)
[2] Romanov, V.G.: Inverse Problems of Mathematical Physics. Nauka, Moscow (1984, in Russian) · Zbl 0576.35001
[3] Denisov, A.M.: Introduction to the Theory of Inverse Problems. Nauka, Moscow (1994, in Russian)
[4] Anikonov, Jn.E.: Multidimensional Inverse and Ill-Posed Problems for Differential Equations. VSP, Utrecht (1995)
[5] Anikonov, Jn.E.: Formulas in Inverse and Ill-Posed Problems. VSP, Utrecht (1997)
[6] Anikonov, Jn.E., Bubhov, B.A., Erokhin, G.N.: Inverse and Ill-Posed Source Problems. VSP, Utrecht (1997)
[7] Prilepko, A.I., Orlovsky, D.G., Vasin, I.A.: Methods for Solving Inverse Problems in Mathematical Physics. Marcel Dekker, New York (1999) · Zbl 0947.35173
[8] Anikonov, Jn.E.: Inverse Problems for Kinetic and Other Evolution Equations. VSP, Utrecht (2001)
[9] Belov, Jn.Ya.: Inverse Problems for Partial Differential Equation. VSP, Utrecht (2002)
[10] Lavrentiev, M.M.: Inverse Problems of Mathematical Physics. VSP, Utrecht (2003)
[11] Megrabov, A.G.: Forward and Inverse Problems for Hyperbolic, Elliptic and Mixed Type Equations. VSP, Utrecht (2003) · Zbl 1104.35002
[12] Ivanchov, M.: Inverse Problems for Equations of Parabolic Type. VNTL, Lviv (2003) · Zbl 1147.35110
[13] Romanov, V.G.: Stability in Inverse Problems. Nauchnyi Mir, Moscow (2005, in Russian)
[14] Kabanikhin, S.I.: Inverse and Ill-Posed Problems. Sib. Nauchn. Izd., Novosibirsk (2008, in Russian) · Zbl 1170.35100
[15] 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(5), 744-752 (2013) [Comput. Math. Math. Phys. 53(5), 580-587 (2013)]
[16] 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(6), 795-806 (2014) [Differ. Equ. 50(6), 792-804 (2014)] · Zbl 1325.35280
[17] Denisov, A.M.: Problems of determining the unknown source in parabolic and hyperbolic equations. Zh. Vychisl. Mat. Mat. Fiz. 55(5), 830-835 (2015) [Comput. Math. Math. Phys. 55(5), 829-833 (2015)] · Zbl 06458254
[18] Babich, P.V., Levenshtam, V.B.: Recovery of a rapidly oscillating absolute term in the multidimensional hyperbolic equation. Math. Notes 104(4), 489-497 (2018) · Zbl 07035826
[19] 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)
[20] 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. Mat. Sb. 87(129)(2), 236-253 (1972) [Math. USSR-Sb. 16(2), 245-263 (1972)] · Zbl 0253.35049
[21] Levenshtam, V.B.: The averaging method in the convection problem with high-frequency oblique vibrations. Sibirsk. Mat. Zh. 37(5), 1103-1116 (1996) [Sib. Math. J. 37(5), 970-982 (1996)] · Zbl 0874.35094
[22] Levenshtam, V.B.: Asymptotic expansion of the solution to the problem of vibrational convection. Zh. Vychisl. Mat. Mat. Fiz. 40(9), 1416-1424 (2000) [Comput. Math. Math. Phys. 40(9), 1357-1365 (2000)] · Zbl 0997.76072
[23] Levenshtam, V.B.: Asymptotic integration of a problem of convection. Sibirsk. Mat. Zh. 30(4), 554-559 (1989) [Sibirsk. Mat. Zh. 30(4), 69-75 (1989)] · Zbl 0712.76083
[24] Levenshtam, V.B.: Justification of the averaging method for the convection problem with high-frequency vibrations. Sibirsk. Mat. Zh. 34(2), 280-296 (1993) [Sibirsk. Mat. Zh. 34(2), 92-109 (1993)] · Zbl 0834.35012
[25] Babich, P.V., Levenshtam, V.B.: Direct and inverse asymptotic problems high-frequency terms. Asymptot. Anal. 97, 329-336 (2016) · Zbl 1342.35387
[26] Babich, P.V., Levenshtam, V.B., Prika, S.P.: Recovery of a rapidly oscillating source in the heat equation from solution asymptotics. Comput. Math. Math. Phys. 57(12), 1908-1918 (2017) [57(12), 1955-1965 (2017)] · Zbl 06864318
[27] Il’in, V.A.: The solvability of mixed problems for hyperbolic and parabolic equations. Uspekhi Mat. Nauk 15(2(92)), 97-154 (1960) [Russian Math. Surv. 15(1), 85-142 (1960)]
[28] Krasnoselskii, M.A., Zabreiko, P.P., Pustylnik E.I., Sobolevskii, P.E.: Integral Operators in Spaces of Summable Functions. Noordhoff International Publishing, Leyden (1976)
[29] Agmon, S., Douglis, A., Nirenberg, L.: Estimates Near the Boundary for Solutions of Elliptic Partial Differential Equations Satisfying General Boundary Conditions. Interscience Publishers, New York (1959) · Zbl 0093.10401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.