THE NUMERICAL-SOLUTION OF 1ST-KIND LOGARITHMIC-KERNEL INTEGRAL-EQUATIONS ON SMOOTH OPEN ARCS

Consider solving the Dirichlet problem [GRAPHICS] with S a smooth open curve in the plane. We use single-layer potentials to construct a solution u(P). This leads to the solution of equations of the form [GRAPHICS] This equation is reformulated using a special change of variable, leading to a new first-kind equation with a smooth solution function. This new equation is split into a principal part, which is explicitly invertible, and a compact perturbation. Then a discrete Galerkin method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. A complete convergence analysis is given; numerical examples conclude the paper.