Boundary value problems with displacement for one mixed hyperbolic equation of the second order

Abstract

The paper studies two nonlocal problems with a displacement for the conjugation of two equations of secondorder hyperbolic type, with a wave equation in one part of the domain and a degenerate hyperbolic equation of the first kind in the other part. As a non-local boundary condition in the considered problems, a linear system of FDEs is specified with variable coefficients involving the first-order derivative and derivatives of fractional (in the sense of Riemann-Liouville) orders of the desired function on one of the characteristics and on the line of type changing. Using the integral equation method, the first problem is equivalently reduced to a question of the solvability for the Volterra integral equation of the second kind with a weak singularity; and a question of the solvability for the second problem is equivalently reduced to a question of the solvability for the Fredholm integral equation of the second kind with a weak singularity. For the first problem, we prove the uniform convergence of the resolvent kernel for the resulting Volterra integral equation of the second kind and we prove th a t its solution belongs to the required class. As to the second problem, sufficient conditions are found for the given functions th a t ensure the existence of a unique solution to the Fredholm integral equation of the second kind with a weak singularity of the required class. In some particular cases, the solutions are written out explicitly.