Justification of collocation method for one class of curvilinear integral equations

In this work, we consider the curvilinear integral equation of the external Dirichlet boundary value problem for the Laplace equation. We divide the curve into elementary parts and establish a quadrature formula at some chosen control points for one class of curvilinear integrals. Then we replace the integral equation with the system of algebraic equations. We also establish the existence and uniqueness of the solution of this system, prove the convergence of its solution to the exact solution of the external Dirichlet boundary value problem for the Laplace equation at the control points, and determine the convergence rate of the method. Besides, we construct a sequence which converges to the solution of the external Dirichlet boundary value problem for the Laplace equation and provide an error estimate.