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Mathematics of Computation

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

Authors: Kendall E. Atkinson and Ian H. Sloan
Journal: Math. Comp. 56 (1991), 119-139
MSC: Primary 65R20; Secondary 31A10, 35C15
MathSciNet review: 1052084
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Abstract: Consider solving the Dirichlet problem

\begin{displaymath}\begin{array}{*{20}{c}} {\Delta u(P) = 0,} \hfill & {P \in {\... ...ll \\ {P \in {\mathbb{R}^2}} \hfill & {} \hfill \\ \end{array} \end{displaymath}

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

$\displaystyle \int_S {g(Q)\log \vert P - Q\vert 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.

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Article copyright: © Copyright 1991 American Mathematical Society

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