Modeling unbalanced systems in network-like oil and gas processes

In the paper proposes a analog of the method E. Rote (the method of semi-discrete by time variable) to construction converging different schemes when analyzing the mathematical models of network-like oil and gas processes. The proposed method reduce the study of the input initial boundary value problem to study the boundary value problem in a weak setting for elliptical type equations with distributed parameters on the net-work. Thus, there is another possibility, besides the Faedo-Galerkin method, to construction approaches to the desired solution of the initial boundary value problem, to analyze its stability and the way to prove the theorem of the existence of a weak solution to the input problem. The approach is applied to finding sufficient conditions for the existence of weak solutions to other initial boundary value problems with more total boundary conditions -- in which elliptical equations are considered with the boundary conditions of the second or third type. Further analysis is possible on the way to finding the conditions of the Lyapunov stability. The approach can be used to analyze the optimal control problems, as well as the problems of stabilization and stability of differential systems with delay. Presented method of finite difference open new ways of approximation of the states of the parabolic system, analysis of their stability when numerical implementation and algorithmic of optimal control problems.