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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.

The resolvent method, proposed by Sadovnichiy and Dubrovsky in the 1990s, is successfully applied in the direct spectral problem to calculate the asymptotics of eigenvalues of the perturbed operator, find formulas for the regularized trace, and recover perturbation. But the application of this method faces difficulties when the resolvent of the unperturbed operator is non-nuclear. Therefore, a number of physical problems could only be considered on the interval. This article describes a justification of the transition to the power of an operator in order to expand the area of possible applications of the resolvent method. Considering the problem of calculating the regularized trace of the Laplace operator on a parallelepiped of arbitrary dimension, we show that for every fixed dimension it is possible to choose the required power of the operator and to calculate the regularized traces. These studies are relevant due to the need to study important applied problems, particularly in hydrodynamics, electronics, elasticity theory, quantum mechanics, and other fields.

Получено приближённое решение обратной спектральной задачи для оператора Лапласа с кратным спектром.Ключевые слова: обратная спектральная задача, приближённое решение, оператор Лапласа, кратный спектр.The article is devoted to finding an approximated solution of the inverse spectral problem for the Laplace operator with multiple spectrum.