Tricomi problem for a mixed-type equation of the second kind in a domain the elliptic part of which is the first quadrant of the plane

In the theory of mixed-type equations, most studies have been carried out for bounded domains with smooth boundaries and for equations of the first kind. In the present paper, for a mixed-type equation of the second kind $u_{xx}+signy|y|^mu_{yy}=0$, $0<m<1$, a Tricomi problem is studied in an unbounded domain whose elliptic part is the first quadrant of the plane. The uniqueness of the solution is proved using the extremum principle. The existence of the solution is established by the Green's function method in the elliptic part and by the integral equation method in the hyperbolic part. In constructing the Green's function, properties of the modified Bessel functions of the second kind and the Gaussian hypergeometric function are employed. A Fredholm integral equation of the second kind is derived for the trace of the solution on the degeneracy line; its solvability follows from the proven uniqueness. Numerical calculations are performed to visualize the solution, and the results are presented as three-dimensional surfaces and contour plots. A mathematical and physical interpretation of the solution is given for various values of the parameter m.