ON GREEN’S FUNCTION OF DARBOUX PROBLEM FOR HYPERBOLIC EQUATION

Sadybekov M.A. Derbissaly B.O.
30 September 2021 al-Farabi Kazakh State National University

KazNU Bulletin. Mathematics, Mechanics, Computer Science Series
2021 #111 Issue 3 79 - 94 pp.

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.

a characteristic triangle , boundary condition , Darboux problem , Green function , Hyperbolic equation , initial-boundary value problem , Riemann–Green function

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Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Al-Farabi Kazakh National University, Almaty, Kazakhstan

Institute of Mathematics and Mathematical Modeling
Al-Farabi Kazakh National University

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