The uniqueness of the solution of an initial boundary value problem for a hyperbolic equation with a mixed derivative and a formula for the solution

An initial boundary value problem for an inhomogeneous  second-order hyperbolic equation on a finite segment with constant  coefficients and a mixed derivative is investigated. The case of  fixed ends is considered. It is assumed that the roots of the  characteristic equation are simple and lie on the real axis on  different sides of the origin. The classical solution of the  initial boundary value problem is determined. The uniqueness  theorem of the classical solution is formulated and proved.  A formula is given for the solution in the form of a series whose  members are contour integrals containing the initial data of the  problem. The corresponding spectral problem for a quadratic beam is constructed and a theorem is formulated on the expansion of the  first component of a vector-function with respect to the   derivative chains corresponding to the eigenfunctions of the beam. This theorem is essentially used in proving  the uniqueness theorem  for the classical solution of the initial boundary value problem.

hyperbolic equation, second order, constant coefficients, mixed derivative in the equation, finite segment, initial boundary value problem, fixed ends, classic solution, uniqueness of the solution, formula for the solution, expansion of the first component of the vector-function in a series