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.
This article is devoted to the problems of investigation of routing processes in the large distribution systems. The numerical decision systems of the nonlinear equations for a preset network, traffic and conditions of functioning allows to implement determination of a probability-time characteristics of a network, carry out a rating of used algorithms of routing, methods of streams control. The identification of parameters of model of the routing process close to best values is possible during a repetitive process of search of the solution of a system of nonlinear equations. The designed analytic model of processes of multiparameter routing in networks based the theory of finite Markov chains, queuing theory etc., allows to define probability-time characteristics of a networks. Structure and the methods of model construction provide its practical applicability for auto configuration of routing algorithms and traffic control for specific topological structures, channels characteristics and traffic between routers of a network.
The paper deals with the problem of gas filtration process in a porous medium using a mathematical model of an object; the model is described by a nonlinear partial differential equation and the corresponding boundary and internal conditions. The paper presents the main stages in construction of mathematical model of the process of gas filtration in porous media, taking into account the changes in hydrodynamic parameters of the object of study. To solve the above problem, the following numerical methods have been used: local-one-dimensional schemes and longitudinal-transverse direction schemes. To solve the problem of nonlinear gas filtration in a porous medium, several methods of constructing an iterative process have been checked. To study the responses of the principal parameters of the process, a series of computational experiments on a computer, their analyses and conclusions are given in the paper.
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