Classical Solutions of Inhomogeneous Factorized Second-Order Hyperbolic Equation in a Quarter of the Plane with a Seminonstationary Second Directional Derivative in the Boundary Condition
Previously mixed problems for an inhomogeneous equation of oscillations of a semi-infinite strings were solved by the characteristics of just using some extensions of the right-hand side, the initial data and the boundary value outside of the regions for these problems. The aim of this work was to find the explicit form of classical solutions for the semi-infinite string oscillations of more general form with the initial conditions and the boundary condition without any extension of the original data and of this mixed problem outside of set G. In this mixed problem - continuous functions of the variable t and - real constants. We consider the case of the boundary condition with non-characteristic directions of the first directional derivatives into the factored second directional derivative at the end of the string, i.e. The posed mixed problem is solved by the method characteristics and the Duhamel's method. We derive two formulas of its classical solutions, one of which is a generalization of the well-known d'Alembert's formula under the characteristic x = a1t, and the second formula gives the solution of the corresponding boundary value problem over of the characteristic x = a1t. For the existence and uniqueness of classical solutions of the posed mixed problem we determine the following necessary and sufficient conditions of smoothness and consistency conditions on the right-hand side of equation initial data and boundary value.