Tricomi Problem for a Second-Kind Mixed-Type Equation in a Domain Whose Elliptic Part Is a Vertical Half-Strip

In this paper, the Tricomi problem for a second-kind mixed-type equation with a lower-order term is studied in an unbounded domain. The elliptic part of the domain is a vertical half-strip, while the hyperbolic part is bounded by characteristics. Homogeneous Dirichlet conditions are imposed on the walls of the half-strip, gluing conditions are given on the parabolic degeneracy line, and the trace of the desired solution is prescribed on one of the characteristics. The uniqueness of the solution is proved using the extremum principle and the Zaremba–Giraud principle. The existence of the solution is established by Green’s function method: in the elliptic part, Green’s function of the mixed problem is constructed in the form of a rapidly convergent series; in the hyperbolic part, a generalized solution of the Cauchy problem of a special class is used. The functional relations on the degeneracy line lead to a singular integral equation, which is regularized by the Carleman–Vekua method into a Fredholm integral equation of the second kind with a weak singularity. Explicit formulas for the trace of the solution and its normal derivative are obtained. For a specific set of parameters, a numerical visualization of the solution is performed, the gluing conditions are verified, and a physical interpretation of the obtained graphs is given in the context of transonic gas dynamics. The results can be useful for mathematical modeling of flows in Laval nozzles and other problems of mechanics.

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