Finite difference method for transfer equation with distributed parameters on the network

The issue of constructing a solution to an initial-boundary value problem for an evolutionary differential equation with a spatial variable varying on a network (geometric graph) has remained under review of researchers over the past few years. There were many practical reasons for this - a large number of mathematical models describing the transport processes of continuous media over network carriers use formalisms of partial differential equations and their corresponding initial-boundary value problems. In this paper, classical approaches were used for approximating differential equations on linear network fragments (graph edges), and the principles of constructing approximations of differential relations generated by generalized Kirchhoff conditions at the junction points of these fragments (at the graph nodes) were also indicated. The latter was a distinctive feature of differential equations concept with a spatial variable, changing on a network (graph) and its corresponding finite-difference analogue of the classical equation and finite-difference analogue. The problems of the elliptic operator approximation of the initial-boundary value problem were studied (the error of approximations was established), the stability of the two-layer difference scheme and detailed analysis of its stability was carried out. An algorithm for constructing a solution was developed, based on new numerical methods for analyzing transport problems in materials with complex structure with non-uniformly distributed properties of a continuous medium over a network carrier. A computer program has been developed and tested on test objectives targeted at applied problems. The obtained results can be used in the analysis of initial-boundary value problems for differential equations with distributed parameters on a multidimensional network having interesting analogies with multiphase problems in multidimensional hydrodynamics.

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