Approximation of an elliptic operator with a singularity in the space of functions specified on the graph

Was proposed an approach to approximation of an elliptic operator used in describing mathematical models of transfer processes of continuum and in problems of controlling elastic vibrations of network-like structures. To ease the problem of studying the presented material, i.e. to simplify the mathematical symbolism of grid functions, the space variable of functions of the domain of definition of the elliptic operator changes on the oriented geometric graph - star, which is not a restrictive circumstance, because an arbitrary graph (in applications – a network) is a collection of stars that differ from each other only in the quantity of edges. An algebraic system and its corresponding finite-dimensional operator are formed, the properties of this operator are established and examples of constructing (and analyzing) difference schemes for the heat transfer equation and the oscillation equation (wave equation) with a space variable changing on a graph (network) are given. In this case, the optimal control problem is reduced to a finite moment problem, which opens the way to obtaining a numerical analysis that does not depend on the dimension of the control vector, it is only necessary to know a limited number of grid eigenfunctions of the finite-difference analogue of the elliptic operator.

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