Nonlocal problems with displacement for matching two second order hyperbolic equations

Abstract

In this work we study two nonlocal problems with a displacement for two second order hyperbolic equations being a wave equation in one part of the domain and a degenerate hyperbolic equation of the first kind in the other. As a nonlocal boundary condition, in the considered problems we use a linear combination with variable coefficients of the first derivative and fractional derivative (in the Riemann-Liouville sense) of the unknown function on one of the characteristics and one the line of the type changing. By using the methods of integral equations, the solvability issue of the first problem is equivalently reduced to the solvability of a Volterra equation of the second kind with a weak singularity, while the solvability of the second problem is reduced to the solvability of aFredholm equation of the second kind with a weak singularity. For the first problem we prove a uniform convergence of the resolvent for the kernel of the obtained Volterra equation of the second kind and that it solution belongs to a needed class. For the second problem we find sufficient conditions for the given functions, which ensure the existence of the unique solution of the Fredholm equation of the second kind with a weak singularity in the needed class. In some particular cases the solutions of the problems are written explicitly.