### SPECTRAL PROBLEMS FOR ONE MATHEMATICAL MODEL OF HYDRODYNAMICS

#### Abstract

This paper is devoted to the investigation of two spectral problems: the eigenvalue problem and the inverse spectral problem for one mathematical model of hydrodynamics, namely the mathematical model for the evolution of the free filtered-fluid surface. The Galerkin method is chosen as the main method for solving the eigenvalue problem. A theorem on the convergence of Galerkin's method applied to this problem was given. For the given spectral problem the algorithm was developed. A program that allows calculating the eigenvalues of the perturbed operator was produced in Maple. For the inverse spectral problem, the resolvent method was chosen as the main one. For this spectral problem, an algorithm is also developed. A program that allows one to approximately reconstruct the potential from the known spectrum of the perturbed operator was created in Maple. The theoretical results were illustrated by numerical experiments for a model problem. Numerous experiments carried out have shown a high computational efficiency of the developed algorithms.

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