Recovering a Rapidly Oscillating Lower-Order Coefficient and a Source in a Hyperbolic Equation from Partial Asymptotics of a Solution

We consider the Cauchy problem for a one-dimensional hyperbolic equation whose lower-order coefficient and right-hand sideoscillate in time with a high frequency and the amplitude of the lower-order coefficient is small.Under study is the reconstruction of the cofactors of these rapidly oscillating functions independentof the space variable from a partial asymptotics of a solution at some point of the space.The classical theory of inverse problems examines the numerous problems of determining unknown sources, and coefficients withoutrapid oscillations for various evolutionary equations, where the exact solutionto the direct problem appears in the additional overdetermination condition.Equations with rapidly oscillating data are often encountered in modeling the physical, chemical, andother processes that occur in media subjected to high-frequency electromagnetic, acoustic, vibrational, and others fields,which demonstrates the topicality of perturbation theory problems on the reconstruction of unknown functionsin high-frequency equations.We give some nonclassical algorithm for solving such problems that lies at the junction ofasymptotic methods and inverse problems. In this case the overdetermination condition involvesa partial asymptotics of solution of a certain lengthrather than the exact solution.