Double layer potential scheme for Dirichlet problems on smooth open arcs

In this paper, we present a new numerical scheme for the Dirichlet problem on the smooth open arcs in the plane. By using the double layer potential to construct the solution of the problem we induce a Fredholm integral equation of the second kind. Existence and uniqueness of the solution of this integral equation are proved in the case when the boundary arc is convex. A numerical example is given to show the validity of the present scheme.