ON GREEN’S FUNCTION OF DARBOUX PROBLEM FOR HYPERBOLIC EQUATION

Abstract

It is well known that the Darboux problem for the hyperbolic equation is correct, both in the sense
of classical and generalized solutions. An integral form of the solution of the Darboux problem in
a characteristic triangle for a general two-dimensional hyperbolic equation of the second order is
represented in the article. It is shown that the solution to this problem can be written in terms of
the Green function. It is also shown that the Riemann-Green function of the hyperbolic equation
is not defined in the entire domain. To construct the Riemann-Green function of this equation, it
is important to have the Riemann-Green function of this problem that was defined at all points
of the domain. For that, the coefficients of the general hyperbolic equation have been continued
odd. The definition of the Green function of the Darboux problem is given. To show that a Green
function exists and is unique, we divide the domain into several subdomains. Its existence and
uniqueness have been proven. An explicit form of the Green’s function is presented. It is shown
that the Green’s function can be represented by the Riemann–Green function. There is given a
method for constructing the Green function of such a problem. The main fundamental difference
of this paper is that it is devoted to the study of Green’s function for the hyperbolic problem. In
contrast to the (well-developed) theory of Green’s function for self-adjoint elliptic problems, this
theory has not been developed.