Composition of fundamental solution for an odd order equation

Abstract

In recent years an enormous attention has been given to examining non-classical equations in quotient derivatives. On the one hand, this is explained by the lack of studying of these equations in theoretical way. On the other hand, often their enclosure to various problems of mechanics, physics and technics has been found out. To less studied non classical equations of high orders belongs the equation in this form: Dp xu (x, y) + aDq yu (x, y) = F 􀀀x, y,D1 xu,D1 yu, . . . ,Dp−1 x u,Dq−1 y u , Ds t u = @su @ts , p, q 2 N, p > q, a = const. This equation has only one system of characteristics y = const which is a p-multiple characteristics. This equation with p = 3, q = 1 involves a famous equation of Cortevega de Vrize, and when p = 3, q = 2 it describes stationary flow of transonic gas. In this paper in the area of = {(x, y) : 0 < x < p, 0 < y < q} fundamental solution U (x, y, , ) for an equation L[u] (−1)n D2n+1 x u (x, y) − (−1)m D2m y u (x, y) = 0, (2n + 1, 2m) = 1, n,m 2 N, n m was built.