Necessary and sufficient conditions for the well-posed solvability of a boundary value problem for a linear loaded hyperbolic equation

Abstract

Problems for loaded hyperbolic equations have acquired particular relevance in connection with the study of the stability of vibrations of the wings of an aircraft loaded with masses, and in the calculation of the natural vibrations of antennas loaded with lumped capacities and self-inductions. Loaded differential equations have a number of features that must be taken into account when setting problems for these equations and creating methods for their solution. One of the features of loaded differential equations is that such equations can be undecidable without additional conditions. The main idea of the research work is to expand the class of solvable boundary value problems and develop methods that provide a numerical-analytical solution. The paper considers a boundary value problem for a linear hyperbolic equation with a mixed derivative, where the load points are set in terms of the spatial variable. By introducing unknown functions, the problem is reduced to an equivalent boundary value problem for a linear loaded hyperbolic equation of the first order. With the help of the well posed of the equivalent boundary value problem, the well posed of the original problem is established. The paper presents the necessary and sufficient conditions for the well-posedness of a periodic boundary value problem for a linear loaded hyperbolic equation with two independent variables.