Reconstruction of a rapidly oscillating lowest coefficient and the source of a hyperbolic equation from the partial asymptotics of the solution

V. B. Levenshtam

Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 3, pp. 111-122

Résumé

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.