The Numerical Solution of First - Kind Logarithmic - Kernel Integral Equations on Smooth Open Arcs

Consider solving the Dirichlet problem Au(P) = o, Pem2\s, u(P) = h(P), Pes, sup \u(P)\ < CO, Pes2 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 j g(Q)\og\P-Q\dS(Q) = h(P), PeS. 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.