1.3  Operator Notation

Textbooks on the BEM still tend to leave the integral equation in the form (1) and work from there. However, it is much easier to work from the operator notation, this makes the equations more compact and clear to read and it simplifies the transition from boundary integral equation to boundary element method.

An integral equation contains integral operators. For example if z is a function defined on S then applying the following operation to z for all points p on S

ó
õ

S 
 G(p,q)  z(q) dSq = n(p)       (p Î S)
gives a function n. This can be viewed as the application of an operator to the function z to return the function n. More simply we may write

{L z}S(p) = n(p) .
(2)
In (2) the L represents the integral operator

{L z}G(p) º ó
õ

G 
 G(p,q) z(q) dSq
and the subscript (G) refers to the domain of integration. Here G is used as a variable, representing either a whole surface or a patch of surface.

In operator notation the integral equation (1) can be written in the alternative shorthand notation

{M f}S(p) + 1
2
 f(p) = {L v}S(p)   or   {(M + 1
2
 I) f}S(p) = {L v}S(p)
(3)
where v(q) = f/ nq, the M represents the other integral operator;

{M z}G(p) º ó
õ

G 
 G
nq
 (p,q)  z(q) dSq ,
(4)
and I the identity operator.
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