CAUCHY PROBLEM FOR A DEGENERATE HYPERBOLIC EQUATION OF THE SECOND KIND WITH THE TWO LINES AND IDENTICAL ORDER OF DEGENERACY

Abstract

In scientific literature, degenerate hyperbolic equations are usually divided into equations of the first and second kind. In the case of an equation of the first kind, the line of parabolic degeneracy is the locus of cusps of the equation's characteristics, and in the case of an equation of the second kind, it is simultaneously a special line (characteristic), i.e., it is the envelope of the family of characteristics. Therefore, degenerate hyperbolic equations of the second kind have been studied relatively little in all respects than equations of the first kind.  At present, a solution to the Cauchy problem for a degenerate hyperbolic equation of the second kind with two lines and different orders of degeneracy is known. Further studies have shown that if a hyperbolic equation of the second kind degenerates with identical order in two lines, then special studies are required to solve the Cauchy problem.  In this paper, using the Gauss hypergeometric function, new properties of the Riemann function are established for the named equation, due to which the unique regular solution to the Cauchy problem for a hyperbolic equation of the second kind with the identical order of degeneracy is constructed explicitly