UDC 517.927

O.R. Balaban

The paper considers evolutionary problems underlying the mathematical description of oscillatory and hydrodynamic processes in network-like objects (waveguides, hydraulic networks, etc.). The main attention is paid to the analysis of the properties of the elliptic operator (the one-dimensional Laplace operator) with distributed parameters on the network, establishing the spectral completeness of the system of eigenfunctions in the class of square-integrable functions. Conditions are obtained that guarantee Neumann stability (spectral stability) of difference schemes for evolutionary problems; a solution to the moment method control problem is presented. The methods for studying evolutionary problems are based on the properties of a positive definite elliptic operator: a system of eigenfunctions forms a basis in the space of functions summable with a square; series in the system of eigenfunctions admit a priori estimates of the solutions of the evolutionary problem; approximation of an elliptic operator reduces it to a finite-dimensional operator in a finite-dimensional space of grid functions with a natural Euclidean norm, which (a finite-dimensional operator) approximates the original with any predetermined accuracy in the sense of the norm of the space of functions summable squared. For evolutionary problems, an explicit first-order approximation scheme on the graph grid (parabolic system) and an explicit second-order approximation scheme (hyperbolic system) are used. The oscillatory properties of the obtained operators are established, similar to the classical oscillatory properties. For difference schemes of parabolic and hyperbolic systems of equations, conditions are obtained that guarantee countable spectral stability (stability in the sense of Neumann) and, therefore, the possibility of obtaining analogues of A.F. Filippova on the convergence of difference schemes in terms of approximation steps of a graph grid. To illustrate the applicability of the approach used, the control problem is considered – the translation of evolutionary systems of parabolic and hyperbolic types from given initial to given final states; conditions are obtained that guarantee the controllability of the systems under study.

Keywords: laplace operator on a graph, evolution problems, approximation, difference schemes, stability, convergence, method of moments.

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