Software engineering math for network applications

Software creation for cyber-physical systems is based on complex algorithms. A feature of the modern organization of production and commerce is the consolidation of participants in these activities into complex network structures. In this case, a scientifically based analysis of network processes of a physical and economic nature is necessary. In the present paper, the formalization of the problem is carried out using graph theory. A mathematical model based on partial differential equations, defined on a graph with one cycle is developed. The analysis comes down to the important problem of expanding a given function into eigenfunctions of the Sturm-Liouville tasks with measure on the graph. The obtained results are fundamental for cyber-physical systems of a wide range of applications.