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 main purpose of the paper is to prove the convergence of a numerical solution to a nonstationary Leontief-type system with an initial-final condition. Non-stationary Leontief-type systems are used in the study of dynamic balance models of the economy. Nonstationarity of systems is described using a scalar function, which is multiplied by one of the matrices of the system. The distinctive feature of Leontief-type systems is the matrix singularity at the derivative with time, which is due to the fact that some types of resources of economic systems cannot be stored. In the article, the initial-final condition is used instead of the standard initial condition, which for economic systems can be interpreted as taking into account indicators not only at the initial moment of time, but also indicators that will be achieved at the final moment of time. Previously, the solution of such a problem was studied and described using contour integrals. However, this type of solution is not very convenient for largedimensional systems, so this article proposes a description of the numerical solution without the use of contour integrals, and also examines the convergence of this numerical solution.
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