Finite-dimensional analogues of transfer differential operators with carriers on spatial networks

The presented results provide justification for the applicability of numerical methods for analyzing initial-boundary value problems for evolutionary differential equations with a spatial variable changing on a network (graph), i.e., on a manifold of one-dimensional continua with a scalar variable. Similar results for -dimensional spatial variables ( ) changing on a network-like -dimensional domain are still in the stage of formation due to the incomparably high level of technical complexity that naturally arises when increasing the dimensionality of the spatial variable. Confirmation of the possibility of justifying numerical methods for analyzing initial-boundary value problems for cases is provided using the results of applying computational methods to solving a test problem with a spatial variable changing on a two-dimensional network-like carrier – a two-dimensional complex-structured domain. The presented example of numerical analysis opens prospects for extending the obtained results to differential operators defined on functions with an m-dimensional carrier. To simplify the representations of difference schemes, a method of semi-discretization with respect to the time variable is used (in a sense, numerous routine costs that arise as a direct consequence of the multidimensionality of the spatial variable are leveled). The obtained results are applied in constructing and numerically analyzing mathematical models of laminar and turbulent network-like processes in applied hydrodynamics.

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