2  Statement of the general problem and the two boundary integral formulations

The particular class of exterior Laplace problems that are to be considered are stated, together with the most obvious direct and indirect boundary integral formulations. In both cases they are Fredholm integral equations of the second kind.

2.1  The Laplace problem

The two-dimensional exterior Laplace problem considered is to obtain j satisfying
Ñ2 j(p)  = 0    (p = (p1,p2) Î E),
where E is the region exterior to some closed boundary S and the Neumann boundary condition
j(p)
n
 = v(p) ,   p Î S,
with j(p) = O(log|p-p0| ) as p ® ¥ and p0 is a fixed origin.

2.2  Direct and indirect formulation

The direct formulation is derived by the application of Green's second theorem to the Laplace equation, and is
{ M j}S(p) - { L v }S(p) = c(p) j(p)
where L and M are integral operators defined as follows
{L m}S(p) = ó
õ

S 
 G(p,q) m(q) dSq ,   {M m}S(p) = ó
õ

S 
 G(p,q)
n
 m(q) dSq ,   (p Î D ÈS ÈE),
(1)
where the Greens function is taken to be G(p,q) = - [1/(2 p)] lnr, r=|p - q| and c(p) represents the angle subtended by the exterior region for points p on the boundary, for interior points c(p)=0 and for exterior points c(p)=1.
The surface element of integration is at q in (1), and nq is the unit outward normal there.
In the indirect formulation it is assumed that the potential j(p) can be defined in terms of a single-layer potential
j(p) = {L s}S(p) ,   (p Î E ÈS),
(2)
where s is a source density function defined on the boundary S.
Through differentiation of the above equation we may obtain the following
{ Mt s}S(p) - 1
2
 s(p) = v(p)
(3)
where
{Mt m}S(p) = ó
õ

S 
 G(p,q)
np
 m(q) dSq ,   (p Î D ÈS ÈE),
where np is the unit outward normal to the boundary at p.