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.