Well posed boundary value problem in transonic gas dynamics. [Tricomi equation]

Abstract

A boundary value problem for the Tricomi equation is studied in connection with transonic gas dynamics. The transformed equation Au + jy u y = 0 in canonical coordinates is considered in the complex domain of two independent complex variables. A boundary value problem is then set by prescribing the real part of the solution on the boundary of the real unit circle. The Dirichlet problem in the upper unit semicircle with vanishing values of the solution at Y = 0 is solved explicitly in terms of the hypergeometric function for the more general Euler-Poisson-Darboux equation. An explicit representation of the solution is also given for a mixed Dirichlet and Neumann problem for the same equation and domain. Reflection rules are given for these solutions which permit one to extend them from the upper to the lower unit semicircle. The transonic boundary value problem is solved by expressing the solution as a linear combination of these two types of solutions. v