Initial Boundary and Inverse Problems for the Inhomogeneous Equation of a Mixed Parabolic-Hyperbolic Equation

A problem with inhomogeneous boundary and initial conditions is studied for an inhomogeneous equation of mixed parabolic-hyperbolic type in a rectangular domain. The solution is constructed as the sum of an orthogonal series. A criterion for the uniqueness of the solution is established. It is shown that the uniqueness of the solution and the convergence of the series depend on the ratio of the sides of the rectangle from the hyperbolic part of the mixed domain. On the basis of this problem, inverse problems for finding the factors of the time-dependent right-hand sides of the original equation of mixed type are stated and studied for the first time. The corresponding uniqueness theorems and the existence of solutions are proved using the theory of integral equations for inverse problems.