The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs

Abstract

Consider solving the Dirichlet problem \[ Δ u ( P ) = 0 , a m p ; P ∈ R 2 ∖ S , u ( P ) = h ( P ) , a m p ; P ∈ S , sup | u ( P ) | > ∞ , a m p ; P ∈ R 2 a m p ; \begin {array}{*{20}{c}} {\Delta u(P) = 0,} \hfill & {P \in {\mathbb {R}^2}\backslash S,} \hfill \\ {u(P) = h(P),} \hfill & {P \in S,} \hfill \\ {\sup |u(P)| > \infty ,} \hfill & {} \hfill \\ {P \in {\mathbb {R}^2}} \hfill & {} \hfill \\ \end {array} \] with S a smooth open curve in the plane. We use single-layer potentials to construct a solution u ( P ) u(P) . This leads to the solution of equations of the form \[ ∫ S g ( Q ) log ⁡ | P − Q | d S ( Q ) = h ( P ) , P ∈ S . \int _S {g(Q)\log |P - Q|dS(Q) = h(P),\quad P \in S.} \] 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.