Recovery of Rapidly Oscillated Right-Hand Side of the Wave Equation by the Partial Asymptotics of the Solution

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/s0301-1959-8380-s Recovery of Rapidly Oscillated Right-Hand Side of the Wave Equation by the Partial Asymptotics of the Solution Babich, P. V. , Levenshtam, V. B. Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 4. Abstract: The initial-boundary problem for the one-dimensional wave equation with unknown rapidly oscillated right-handside is considered in the paper. The latter is represented as a product of two functions; the first function depends on the spatial variable and the second one depends on time and fast time variables. Four different cases are considered: in two cases one of the factor-functions is known and in two other cases both factor-functions are unknown. In each of these cases, the inverse problems of recovering unknown functions from some information about partial asymptotics of the original problem with known data are posed and solved. This specified information consists, in general, in setting values for certain asymptotics coefficients in some spatial and/or time points. The use of some additional conditions (overdetermination conditions) in this form speaks of a fundamental difference between the above statements of inverse problems and the classics, where additional conditions are imposed on exact solutions. The construction of solutions asymptotics of the original problem with this approach act as direct problem. This approach to inverse problems with rapidly oscillated data in time is developed by the author over the past few years. Keywords: one-dimensional wave equation, rapidly oscillating absolute term, asymptotics of solution, inverse problem Language: Russian Download the full text For citation: Babich, P. V. and Levenshtam, V. B. Recovery of Rapidly Oscillated Right-Hand Side of the Wave Equation by the Partial Asymptotics of the Solution, Vladikavkaz Math. J., 2020, vol. 22, no. 4, pp.28-44 (in Russian). DOI 10.46698/s0301-1959-8380-s + References 1. Babich, P. V. and Levenshtam, V. B. Direct and Inverse Asymptotic Problems High-frequency Terms, Asymptotic Analysis, 2016, vol. 97, pp. 329-336. 2. 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, 2017, vol. 57, no. 12, pp. 1908-1918. DOI: 10.1134/S0965542517120065. 3. Lavret’ev, M. M., Reznitskaya, K. G. and Yakhno, V. G. Odnomernye obratnye zadachi matematicheskoj fiziki [One-Dimensional Inverse Problems of Mathematical Physics], Novosibirsk, Nauka, 1982 (in Russian). 4. Romanov, V. G. Obratnye zadachi matematicheskoj fiziki [Inverse Problems of Mathematical Physics], Moscow, Nauka, 1984 (in Russian). 5. Denisov, A. M. Vvedenie v teoriyu obratnyh zadach [Introduction to the Theory of Inverse Problems], Moscow, Nauka, 1994 (in Russian). 6. Kabanikhin, S. I. Obratnye i nekorrektnye zadachi [Inverse and Ill-Posed Problems], Novosibirsk, Sibirskoe Nauchnoe Izdatel'stvo, 2008 (in Russian). 7. Denisov, A. M. Asymptotic Expansions of Solutions to Inverse Problems for a Hyperbolic Equation with a Small Parameter Multiplying the Highest Derivative, Computational Mathematics and Mathematical Physics, 2013, vol. 53, no. 5, pp. 580-587. DOI: 10.1134/S0965542513050047. 8. Denisov, A. M. Problems of Determining the Unknown Source in Parabolic and Hyperbolic Equations, Computational Mathematics and Mathematical Physics, 2015, vol. 55, no. 5, pp. 829-833. DOI: 10.1134/S0965542515050085. 9. 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, Differential Equations, 2014, vol. 50, no. 6, pp. 792-804. 10. Babich, P. V. and Levenshtam, V. B. Recovery of a Rapidly Oscillating Absolute Term in the Multidimensional Hyperbolic Equation, Mathematical Notes, 2018, vol. 104, no. 4, pp. 489-497. DOI: 10.1134/S000143461809016X. 11. Ilin, V. A. The Solvability of Mixed Problems for Hyperbolic and Parabolic Equations, Russian Mathematical Surveys, 1960, vol. 15, no. 1, pp. 85-142. DOI: 10.1070/RM1960v015n02ABEH004217. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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