Investigation of the solution of a boundary value problem with variable coefficients whose principal part is the Cauchy–Riemann equation

Abstract

This study is devoted to obtaining an analytical expression for the solution of a non-local boundary value problem for a linear inhomogeneous differential equation with variable coefficients, which principle part is the Cauchy–Riemann equation. Since the Cauchy–Riemann equation is a first-order elliptic equation, the problem formulated with a classical boundary condition in a finite domain is ill-posed. Defining a boundary condition for a first-order elliptic equation within a finite domain requires special investigation. For a firstorder elliptic equation in the x1ox2 plane, a new boundary condition is proposed within a bounded region that is concave in the x2 direction, and an expression for the solution is obtained. For this purpose, using the fundamental solution of the principal part of the equation, the main relation consisting of two parts is obtained, the first part yields an arbitrary solution to the equation, and the second part gives the boundary values of the solution representing the necessary conditions. Utilizing these necessary and specified boundary conditions, a system of Fredholm integral equations of the second kind with a singular kernel is constructed to find a solution, and a method for elimination the singularity in the solution is proposed.