A combined problem with local and nonlocal conditions for a class of mixed-type equations

Abstract

This paper investigates the issues of existence and uniqueness of a solution to a combined boundary value problem with local and nonlocal conditions for a specific class of mixed elliptic-hyperbolic-type equations with singular coefficients. A distinctive feature of the considered problem is that on one part of the boundary characteristic, the values of the desired function are specified, while on the other part, nonlocal conditions are imposed. These conditions establish pointwise connections between the values of the sought function on different parts of the boundary characteristics using the Riemann-Liouville fractional differentiation operator. At the same time, a portion of hyperbolic domain’s boundary remains free from boundary conditions. The proof of the solution’s uniqueness is based on the application of an analogue of A.V. Bitsadze’s extremum principle for mixed-type equations with singular coefficients. The existence of the solution is reduced to the analysis of a Tricomi singular integral equations’ system with a shift, containing a non-Fredholm operator with isolated first-order singularity in kernel. By applying the Carleman-Vekua regularization method, these equations are reduced to a Wiener-Hopf integral equation, for which it is proved that the index is equal to zero. This, in turn, reduces the problem to a Fredholm integral equation of the second kind, the uniqueness of whose solution ensures the well-posedness of the given problem.