The existence of a unique solution to a nonlocal conjugation problem for a third-order partial differential equation of mixed parabolic-hyperbolic type is established. In the upper half-plane, the characteristic equation has a triple root, while in the lower half-plane, it has one simple root and two multiple ones. By applying the order reduction method, Green’s and Riemann’s functions, and the method of integral equations, the problem is equivalently reduced to a nonlocal problem with an integral condition imposed on the trace of the unknown function along the type-change line of the equation. This, in turn, is reduced to solving a Fredholm integral equation of the second kind, the solvability of which is proven using the method of successive approximations. The solution in the parabolic part of the domain is constructed using Green’s function, whereas in the hyperbolic part, the Riemann function method is employed, reducing the problem to a two-dimensional Volterra integral equation of the second kind. Examples are provided.
Apakov et al. (Sun,) studied this question.