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Recovery of rapidly oscillated right-hand side of the wave equation by the partial asymptotics of the solution. (Russian. English summary) Zbl 1474.35428

Summary: 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.

MSC:

35L05 Wave equation
35L20 Initial-boundary value problems for second-order hyperbolic equations
35R30 Inverse problems for PDEs
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