A method for solution of a mixed boundary value problem for a hyperbolic type equation using the operators $\mathbb{AT}_{\lambda,j}$

A mixed boundary value problem with arbitrary continuous, not necessarily satisfying boundary conditions, functions in initial conditions and inhomogeneities of the equation is solved. A method is proposed for finding a generalized solution by a modification of the interpolation operators of functions constructed from solutions of Cauchy problems with second-order differential expression. Methods of finding the Fourier coefficients of auxiliary functions using the Stieltjes integral or the resolvent of the third-order Cauchy differential operator are proposed.