By approximating the operators in the boundary integral equations
in the way described in Section 3.3, they can each be
reduced to an eigenvalue problem of the
form (6.4).
The first step is to approximate the boundary S by a set
of n panels S* = åj = 1n DS*j,
as described in Chapter 2.
The integral operators are approximated by the technique
outlined in Chapter 3.
For example the application of collocation to indirect
integral equations gives an eigenvalue problem of the
form (6.7) with Ak = Lk
for the Dirichlet boundary condition and Ak = Mkt +1/2 I
for the Neumann boundary condition. For the more general Robin
condition (6.2) the eigenvalue problem is of the form (6.4)
with
Ak = Da Lk + Db (Mkt +
1
2
I)
where the matrices Lk and Mk are
as defined by the techniques employed in Section 1.1 and Chapter 3,
Da,
Db
are diagonal matrices with
[Da]ii = a(pi),
[Db]ii = b(pi) and
the pi are the collocation points.
For the Dirichlet boundary condition the discrete equivalent
is identical to that of the indirect method stated earlier;
Ak = Lk. For the Neumann problem
and Ak = Mk +1/2 I.
For the more general Robin
condition (6.2) the eigenvalue problem
cannot be straightforwardly put in the form (6.4).
It can be written in the form