On green's function of second darboux problem for hyperbolic equation

A definition and justify a method for constructing the Green & rsquo;s function of the second Darboux problem for a two-dimensional linear hyperbolic equation of the second order in a characteristic triangle is given. In contrast to the (well-developed) theory of the Green & rsquo;s function for self-adjoint elliptic problems, this theory has not yet been developed. And for the case of asymmetric boundary value problems such studies have not been carried out. It is shown that the Green & rsquo;s function for a hyperbolic equation of the general form can be constructed using the Riemann-Green function for some auxiliary hyperbolic equation. The notion of the Green & rsquo;s function is more completely developed for Sturm-Liouville problems for an ordinary differential equation, for Dirichlet boundary value problems for Poisson equation, for initial boundary value problems for a heat equation. For many particular cases, the Greens & rsquo; function has been constructed explicitly. However, many more problems require their consideration. In this paper, the problem of constructing the Green & rsquo;s function of the second Darboux problem for a hyperbolic equation was investigated. The Green & rsquo;s function for the hyperbolic problems differs significantly from the Green & rsquo;s function of problems for equations of elliptic and parabolic types. Key words: Hyperbolic equation, initial-boundary value problem, second Darboux problem, boundary condition, Green function, a characteristic triangle, Riemann & ndash;Green function.