Geodesic
Reconstruction of a rapidly oscillating lowest coefficient and the source of a hyperbolic equation from the partial asymptotics of the solution
Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 3, pp. 111-122 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The Cauchy problem is considered for a one-dimensional hyperbolic equation, the junior coefficient and the right part of which oscillate in time with a high frequency, and the amplitude of the junior coefficient is small. The question of reconstructing the cofactors of these rapidly oscillating functions independent of the spatial variable according to the partial asymptotics of the solution given at some point in space is investigated. For various evolutionary equations, numerous problems of determining unknown sources and coefficients without assuming their rapid oscillations are studied in the classical theory of inverse problems, where the exact solution of the direct problem appears in the additional condition (redefinition condition). At the same time, equations with rapidly oscillating data are often encountered in modeling physical, chemical, and other processes occurring in media subjected to high-frequency electromagnetic, acoustic, vibrational, etc. effects fields. This testifies to the topicality of perturbation theory problems on the reconstruction of unknown functions in high-frequency equations. The paper uses a non-classical algorithm for solving such problems, which lies at the intersection of two disciplines — asymptotic methods and inverse problems. In this case, the redefinition condition involves not the (exact) solution, as in the classics, but only its partial asymptotics of a certain length. 
@article{VMJ_2023_25_3_a9,
     author = {V. B. Levenshtam},
     title = {Reconstruction of a rapidly oscillating lowest coefficient and the source of a hyperbolic equation from the partial asymptotics of the solution},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {111--122},
     year = {2023},
     volume = {25},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2023_25_3_a9/}
}
TY  - JOUR
AU  - V. B. Levenshtam
TI  - Reconstruction of a rapidly oscillating lowest coefficient and the source of a hyperbolic equation from the partial asymptotics of the solution
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2023
SP  - 111
EP  - 122
VL  - 25
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VMJ_2023_25_3_a9/
LA  - ru
ID  - VMJ_2023_25_3_a9
ER  - 
%0 Journal Article
%A V. B. Levenshtam
%T Reconstruction of a rapidly oscillating lowest coefficient and the source of a hyperbolic equation from the partial asymptotics of the solution
%J Vladikavkazskij matematičeskij žurnal
%D 2023
%P 111-122
%V 25
%N 3
%U http://geodesic.mathdoc.fr/item/VMJ_2023_25_3_a9/
%G ru
%F VMJ_2023_25_3_a9
V. B. Levenshtam. Reconstruction of a rapidly oscillating lowest coefficient and the source of a hyperbolic equation from the partial asymptotics of the solution. Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 3, pp. 111-122. http://geodesic.mathdoc.fr/item/VMJ_2023_25_3_a9/

[1] Romanov, V. G., Inverse Problems of Mathematical Physics, Nauka, M., 1984, 262 pp. (in Russian) | MR

[2] Denisov, A. M., Introduction to the Theory of Inverse Problems, MSU Press, M., 1994, 208 pp. (in Russian)

[3] Kabanikhin, S. I., Inverse and Ill-Posed Problems, Sibirskoe Nauchnoe Izdatel'stvo, Novosibirsk, 2008 (in Russian)

[4] 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”, Mathematics of the USSR-Sbornik, 16:2 (1972), 245–263 | DOI | MR | Zbl

[5] Levenshtam, V. B., “Asymptotic Integration of a Vibration Convection Problem”, Differential Equations, 34:4 (1998), 522–531 (in Russian) | MR | Zbl

[6] Levenshtam, V. B., “Asymptotic Expansion of the Solution to the Problem of Vibrational Convection”, Computational Mathematics and Mathematical Physics, 40:9 (2000), 1357–1365 | MR | Zbl

[7] Babich, P. V. and Levenshtam, V. B., “Recovery of a Rapidly Oscillating Absolute Term in the Multidimensional Hyperbolic Equation”, Mathematical Notes, 104:4 (2018), 489–497 | DOI | DOI | MR | Zbl

[8] Babich, P. V., Levenshtam, V. B. and Prika, S. P., “Recovery of a Rapidly Oscillating Source in the Heat Equation from Solution Asymptotics”, Computational Mathematics and Mathematical Physics, 57 (2017), 1908–1918 | DOI | DOI | MR | Zbl

[9] Levenshtam, V. B., “Parabolic Equations with Large Parameter. Inverse Problems”, Mathematical Notes, 107:3 (2020), 452–463 | DOI | DOI | MR | Zbl

[10] Levenshtam, V. B., “Hyperbolic Equation with Rapidly Oscillating Data: Reconstruction of the Small Lowest Order Coefficient and the Right-Hand Side from Partial Asymptotics of the Solution”, Doklady Mathematics, 105 (2022), 28–30 | DOI | DOI | MR | Zbl