Abstract
Very often an inverse problem for a partial differential equation consists of finding a coefficient or a physical parameter in the equation when some additional information about the solution is given. In a sense, such problems are “inverse” to the “direct” ones where the differential operator is specified, which explains the name. The important feature of inverse problems is their incorrectness in a classical sense. We focus on two inverse problems for a non-stationary wave equation in a non-homogeneous medium — one in a domain with boundary and the other in the entire space. By establishing an equivalence between the inverse spectral problem for the Schrodinger equation and the problems mentioned above in a non-spectral formulation we prove the uniqueness theorems. A constructive approach for the solution of the second inverse problem is also discussed.
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