Boundary value problem for a third-order equation of mixed type in a rectangular domain

For a third-order equation of the parabolic-hyperbolic type, we suggest a method for studying a boundary value problem by solving the inverse problem for a second-order equation of the mixed parabolic-hyperbolic type with unknown right-hand side depending implicitly on time. We prove a criterion for the uniqueness of the solution of the boundary value problem constructed in the form of the sum of a series in the eigenfunctions of the corresponding one-dimensional Sturm-Liouville problem. We prove the stability of the solution with respect to the boundary data in the norms of the spaces W 2 [0, 1] and $$C\left( {\bar D} \right)$$ .

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