On a Boundary Value Problem for the Holmgren Equation with Singular Coefficients in a Half-Plane

In this paper, we study a nonlocal boundary value problem for Holmgren equation with singular coefficients in an unbounded domain. The uniqueness of the solution to the problem is proved using the energy integral method. The existence of a solution to the problem is proved by the method of integral equations. To establish the existence of a solution, we employ the solution of a generalized Neumann problem for the Holmgren equation with singular coefficients in the upper half-plane. The nonlocal problem under consideration involves a boundary condition containing a fractional-order Riemann–Liouville operator. Depending on the values of the fractional derivative’s order in this operator, the problem reduces to solving a Fredholm’s integral equation of the second kind and a singular integral equation with a Cauchy-type kernel. We find its solution using the Carleman–Vekua method.

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