ИСТИНА

In the study [1] B. Riemann suggested the Riemann’s principle of integrating that was applied to partial hyperbolic second-order equation with two independent variables. In order to apply the method it is necessary to construct the Riemann’s function that is a solution of special characteristic problem. General method for the Riemann’s function construction does not exist. In [2] an extensive analysis of six certain methods for creation the Riemann’s function of particular types of equations is done. Ibragimov recommended to find the Riemann’s function with the aid of equation symmetries [3]. In the present study the Riemann’s function of the linear hyperbolic second order equation with two independent variables relative to symmetries of fundamental solutions [4] is shown to be invariant. The method for Riemann’s function construction that is based on the usage of fundamental solutions symmetries is offered. An algorithm for constructing of the Riemann’s function is proposed. Examples use of the offered method are given. Basing on Ovsyannikov study result [5] in group classification of homogeneous hyperbolic second-order equations, it is formulated the condition when the proposed method of constructing the Riemann function is inapplicable. References [1] Riemann B. Proceedings. 1948, M.–L.: OGIZ (in Russian). [2] Copson E.T. On the Riemann–Green Function. Archive for Rational Mechanics and Analysis,1957/58, 1, 324–348. [3] Ibragimov N.Kh. Group analysis of ordinary differential equations and the invariance principle in mathematical physics (for the 150th anniversary of Sophus Lie), Uspekhi Matematicheskikh Nauk, 1992, 47, 83–144. [4] Aksenov A.V. Symmetries of linear partial differential equations and fundamental solutions, Doklady Mathematics, 1995, 51, 329–331. [5] Ovsiannikov L.V. Group Analysis of Differential Equations. New York: Academic Press, 1982.