The boundary S is a circle of radius 4, centred at the origin.
The Neumann condition on the boundary is that produced by a point
source at (0,1). So that
v(q) = -
4 - cosq
2 p(17 - 8 cosq)
, q Î [0,2 p]
which has the exact solution on the boundary
j(q) = -
ln( 17 - 8 cosq)
2 p
where q is the angle that p subtends clockwise
about the centre of the circle from the point (0,4),
as shown in figure 3.
Figure 3.
4.2 Square boundary
The boundary is a square with length of side 4 and with vertices (0,0),
(4,0), (4,4), and (0,4). The Neumann boundary condition is that
which is produced by a point source at (2,2).
v(d) =
1
p
(d2 + 4)-1 d Î [-2,2]
where d is the distance from the center of each side. This has the
exact solution
