The first mixed problem for the general telegraph equation with variable coefficients on the half-line

For the first time, an explicit formula is obtained for the unique and stable classical solution of the inhomogeneous model telegraph equation with variable velocity in the part of the first quarter of the plane, where the boundary and initial conditions are specified. The correctness of the first mixed problem for the general inhomogeneous telegraph equation in the first quarter of the plane is proved. The existence of a classical solution was established by the continuation method with respect to a parameter using theorems on increasing the smoothness of strong solutions. The uniqueness of this solution is derived from the energy inequality for strong solutions. The stability of the solution is established and necessary and sufficient smoothness conditions of the boundary and initial data and three their matching conditions with the right-hand side of the equation are derived. Sufficient smoothness requirements are indicated for the right-hand side of the equation