1.6 Direct and Indirect Boundary Integral Equations

There are two fundamental approaches to the derivation of an integral equation formulation of a partial differential equation. The first is often termed the direct method and the integral equations are derived through the application of Green's second theorem and an example of this has already been given in Section 1.2.

The other technique is termed the indirect method. This is based on the assumption that the solution can be expressed in terms of a source density function defined on the boundary. For example it is assumed that the solution of the Helmholtz equation can be written in the form

f(p) =

ó
õ

G(p,q) s(q)dS_q ,

where s is the source density function defined on S only. The following integral equation can be derived from the above

¶f

¶n_p

(p) =

¶n_p

ó
õ

G(p,q) s(q)dS_q +

s(p) =

ó
õ

¶G

¶n_p

(p,q) s(q)dS_q +

s(p) .

In operator notation the above integral equations are written

f(p) = { L s}(p) and v(p) = {(M^t +

I) s}(p).

Note the relationship between the operator M^t and the operator M introduced earlier; M^t is known at the transpose of M and is arrived at simply by swapping the arguments p and q in the definition.

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