, 
as the corresponding dummy integration variables. Considering the problem
(2) as an example, requiring that G satisfy
over 
, the last term in (1.12) reduces to u(x,y).
Replacing 
in the left-hand side by the known function 
, and subjecting G to boundary conditions on 
which result in the elimination of any normal derivative terms in the boundary
integral. We can solve for the desired u(x,y) in terms
of quantities which are known, and eventually, the Green's function can
be determined. As we mentioned before, the domain
that we considered in this report is a unit square, lets say 
, and according to equation (3.18) imposed
the boundary condition on G, therefore, the corresponding boundary
value problem on the Green's function will become
and the eigenfunction method can be used to determine G.
Now, the eigenvalue problem corresponding to (3.19) is
Solving this by separation of variables, we seek
Inserting this into (3.20), and dividing through by XY we have
Since the left-hand side is a function of
alone, and the right-hand side is a function
alone, we observe that (3.22) can hold for
all
's and
's in
only if both sides are constant, say 
. As for the boundary conditions on X and Y, the conditions
imply that X(0)=X(1)=Y(0)=Y(1)=0. Thus
For the first ordinary differential equation,
Either A and/or 
must be zero; A=0 is unacceptable since it leads to the trivial
solution 
and hence 
, whereas eigenfunctions are, by definition, nontrivial solutions of (3.20).
To have a nontrivial solution, the constant k must be chosen such
that k coincides with a zero of the sine function, that is, 
, where 
. The constant A remains arbitrary, and
Inserting
into the second equation of (3.23), and
proceeding as above, we have
so that, for nontrivial solutions, we need
or, the eigenvalues of (3.23) are given by
and the corresponding eigenfunctions are
Next, we expand the quantities in (3.19) in terms of these eigenfunctions
Unknowns 
are the Fourier coefficient and 
are given by
where we define the two-dimensional inner product by
Accordingly,
so that
Now, inserting (3.29) and (3.30)
into (3.19), and noting from (3.20)
that 
, we have
so that, equating coefficients of 
,
and
With 
given by (3.28), 
by (3.27), and
we have, finally,
