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 second-order 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 that its solution belongs to the required class. As to the second problem, sufficient conditions are found for the given functions that 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.