Growth of vehicle-to-population ratio is accompanied by environmental pollutants growth.This paper deals with health risk indicators of gaseous and dust components in environment result from vehicles operation. A quantitative assessment of road-cars operation impact at different levels of air pollution. is provided. Value of public health risk indicator at different values of air pollution caused by road-cars operation was determined. Dependence of public health risk on distance from the roads with different traffic intensity was determined. Growth of vehicle-to-population ratio resulted in high level of air pollution formation in megalopolises, caused by fuel combustion; road surface destruction and accumulation of braking system and tires wastes. Solid particles in atmospheric air samples by 70 % consists of aluminosilicates, which provide fibrogenic effect, contains heavy metal and substances with a pronounced carcinogenic effect (Benz-(a)-pyrene, dibenz-(a, h)-anthracene, Benz-(B)-fluoranthene and chrysene). It was shown that non-carcinogenic risk caused by vehicle emissions in case of traffic intensity constituting 2500–3000 cars per hour for population up to 30 years old was determined as negligible; up to 40 years — as moderate, at the age of 50 years — as high and for the age over 53 years as very high. Respiratory and cardiovascular systems were determined as the most susceptive. Dependence of the risk level on distance from the road is determined.

Weak solutions of the initial boundary value problem for the second order hyperbolic equation with distributed parameters on an arbitrary oriented graph are considered. Weak solutions are determined by an integral identity, replacing the equation, the initial and boundary conditions. This indicates the spaces in which the search a weak solution is proposed, the conditions of weak solvability of such a problem and the consequent analysis of the stability of the solution are given. The central idea that defined all the content of the work is to apply the Fourier method to analysis evolutionary initial boundary value problems. Namely, the presentation of the solution in the form of a generalized series of Fourier, followed by an analysis of the convergence of this series and series, obtained by his one-time member differentiation. A essential role will play the system of generalized eigenfunctions and spectral characteristics of the elliptical operator of the hyperbolic equation. This considers homogeneous boundary conditions of the first genus (Dirichle conditions), the analysis of stability for the boundary conditions of the second and third genus is similar. The results are fundamental in the study of the control problems of oscillation of network-like industrial constructions, first of oll under studying the question of damping antenna constructions of different types.

This work is a natural continuation of the authors’ research of flow phenomena in the direction of increasing the dimension of the network-like region of change in the spatial variable. The possibility of practical use of the stability analysis of the differential-difference scheme to solve the problem of stabilization of wave phenomena in the engineering of the process of transfer of continuous media through network-like carriers (pipelines, gas and oil pipelines, industrial carriers of gas and petroleum products) is shown. Namely, if the scheme is stable, then sufficiently small changes in the initial data of the mathematical model of the process under study entail small changes in the solution of the difference problem, i.e. in practice do not lead to undesirable consequences; if the scheme is unstable, then small changes in the initial data can lead to arbitrarily large changes in the solution. During the operation of the above-mentioned industrial structures, wave phenomena inevitably arise, the consequence of which are various kinds of instabilities, entailing destruction of one or another nature. The results obtained are used in the digitalization of modern technological processes for the movement of liquid media and gases.

Analytical methods for solving various problems of an applied nature (for example, non-stationary transfer problems over network hydro, gas and heat carriers), whose mathematical models use the formalisms of evolutionary differential systems, are possible with rare exceptions. That is why the construction of numerical and simulation models for the use of quantitative analysis methods becomes a universal research tool, if at the same time the implementation of these models on a computer is carried out – in other words, a complex of software engineering of the process under study is formed. The study uses the method of semidiscretization by a time variable of the mathematical model of the evolutionary non-equilibrium process of continuous medium transfer, which remains one of the most effective methods for analyzing applied problems. In this case, the elliptic operator of the mathematical model has a special basis (a system of eigenfunctions), which is why the analysis is reduced to the study of a boundary value problem for elliptic-type equations with a spatial variable changing on a network-like domain. The paper presents the conditions for unambiguous weak solvability of a differential-difference system, which is a difference analogue in the time variable of the original system, and the way of constructing an algorithm for finding an approximate solution is indicated. The study contains an analysis of the stability and convergence of difference schemes of evolutionary network-like nonequilibrium processes of continuous media transfer over network carriers and includes an analysis of the correctness of the mathematical model of this process. The results of the work are applicable in the framework of oil and gas engineering to the study of issues of stabilization and parametric optimization of the processes of transportation of liquid media through spatial networks.

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