Solving a mixed boundary value problem via an integral equation with adjoint generalized Neumann kernel in bounded multiply connected regions

In this paper, we solve the mixed boundary value problem on bounded multiply connected region by using the method of boundary integral equation. Our approach in this paper is to reformulate the mixed boundary value problem into the form of Riemann-Hilbert problem. The Riemann-Hilbert problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the adjoint generalized Neumann kernel. As an examination of the proposed method, some numerical examples for some different test regions are presented.