POTENTIAL PROBLEMS
29
The fundamental solution for the isotropic two-dimensional case is
"*-έ'"(;)
,ll4
>
and has similar properties to the three-dimensional case.
If
we
assume that the weighting function is composed of a series of
fundamental solutions applied over the boundary we can work with
equation (2.4). Conversely if the approximating function is assumed
to be a combination of fundamental solutions, equation
(2.3)
could be
used. (Both functions can also
be
taken
to be
combinations
of
fundamental solutions as in the case w
=
Au.)
For the first case we can consider a particular point T for which
u*
is the fundamental solution due to an applied potential acting at that
point. This equation can be written as,
- bu*dQ + (V
2
w*)udß+ qu*dr+ qu*dr
Ja
JQ Jr
2
Jr
l
üq*dr+ uq*dr
(2.15)
where q*
=
du*/dn.
Taking into consideration that at T the governing equation for
u*
is
V
2
«*
=
S
t
(2.16)
Formula (2.15) for the point
'f
under consideration becomes,
ιι,.-h bu*dQ+ uq*dr+ uq*dr= qu*dr
Jß
Jr
2
Jr
l
Jr
2
+
qu*dr (2.17)
This equation relates the value of
u
at the point T with the values of
q
and
u
over the boundary Γ (Figure 2.1). It appplies when the point T is
in the interior domain Ω, but in order to formulate the problem in
terms
of
boundary integrals we need
to
take the point
T to the
boundary. We will now do this analytically for
a
smooth boundary
though we will later on see that we do not need to know the analytical
expression.