Nonlocal problems for a mixed parabolic-hyperbolic equation of the third order

The existence of a unique solution of the conjugation nonlocal problems in a rectangular domain for a third-order partial differential equation is proved, when for $y>0$ the characteristic equation has 3 multiple roots, and for $y<0$ it has $1$ simple and $2$ multiple roots. Using the Green's functions and the method of integral equations, the solution of the problem is equivalently reduced to solving the boundary value problem for the trace of the desired function at $y=0$, and then to solving the Fredholm integral equation of the $2$nd kind, the solvability of which is proved by the method of successive approximations. The solution of the problem for $y>0$ is constructed by the Green's function method, and for $y<0$ by reducing the problem to a two-dimensional Volterra integral equation of the $2$nd kind.