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 senseof classical and generalized solutions. An integral form of the solution of the Darboux problem ina characteristic triangle for a general two-dimensional hyperbolic equation of the second order isrepresented in the article. It is shown that the solution to this problem can be written in terms ofthe Green function. It is also shown that the Riemann-Green function of the hyperbolic equationis not defined in the entire domain. To construct the Riemann-Green function of this equation, itis important to have the Riemann-Green function of this problem that was defined at all pointsof the domain. For that, the coefficients of the general hyperbolic equation have been continuedodd. The definition of the Green function of the Darboux problem is given. To show that a Greenfunction exists and is unique, we divide the domain into several subdomains. Its existence anduniqueness have been proven. An explicit form of the Green’s function is presented. It is shownthat the Green’s function can be represented by the Riemann–Green function. There is given amethod for constructing the Green function of such a problem. The main fundamental differenceof this paper is that it is devoted to the study of Green’s function for the hyperbolic problem. Incontrast to the (well-developed) theory of Green’s function for self-adjoint elliptic problems, thistheory has not been developed.