Solvability of hyperbolic systems with distributed parameters on the graph in the weak formulation

Abstract

We investigate weak solvability of the initial-boundary value problem for the hyperbolic equation with distributed parameters on the oriented limited graph and regional conditions of the third type. The spatial variable is changed to the limited oriented graph. The differential ratio was determined on the edges of the graph without the end points. Internal differential ratio graph nodes are replaced by generic terms of Kirchhoff. The differential ratio in internal knots of the graph is replaced with the generalized conditions of Kirchhoff (in applications the balance relations). The space of the admission weak solutions consists of functions with bearer on the graph belonging the space of Sobolev and satisfy the generalized conditions in “limits” sense. Idea analysis of initial-boundary value problem remains the classic: selected functional space with a special base (system of generalized eigenfunctions of an elliptic operator task), which consider the initial-boundary value problem; for approximations to the weak problem solving (the Faedo-Galerkin approximations) establishes a priori estimates of the energy type inequalities; shows weak compactness the Faedo-Galerkin approximation family. Previously initial-boundary value problem is seen in the space of functions with the second generalized derivatives and for such a task is proved equivalent energy inequality. From this it follows: a) convergence of Faedo-Galerkin approximation to the weak solution;b) approximation of the original problem of a course-measuring system of ordinary differential equations. In the work point out the path approximation of the original problem-dimensional system, which allows you to get the theorem on approximation used in the tasks of an applied character. Presents light conditions on the original task data, guaranteeing the weak solvability task. These conditions often occur in applications. Considered by the task and the approach to its analysis of enough is frequently often used in the mathematical description of the oscillation processes in technical designs-net, also in examining wave phenomena in hydronet. The results are fundamental in the study of problems of optimum (of optimum control)of network industrial construction.