Direct and inverse problems for a wave equation with discontinuous coefficients

Mathematical physics

The present article is devoted to the studies in solutions of partial differential equations with discontinuous coefficients for the highest derivatives. This line of investigation is not only of purely academic interest for mathematicians, but plays an important part in the theory of sounding of unknown media composed of various substances. The direct and inverse problems have been considered. The theorem of existence and of the solution-uniqueness was proved for the first of them. For inverse problems, the uniqueness of the solution was proved.The integro-differential equation, which is a consequence of the physical laws, was used for solving the direct problem in the derivation of formulae. The meaning of inverse problems lies in determination of a junction point of different materials and a wave velocity. The used nature of the proof allows us to construct an appropriate numerical algorithm.