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2d problem with Neumann boundary conditions
Let us redo the above calculation, replacing the Dirichlet boundary conditions (149)
with the following simple Neumann boundary conditions:

(161) 
In this case, we can express in the form

(162) 
which automatically satisfies the boundary conditions in the direction. Likewise,
we can write the source term as

(163) 
where

(164) 
since

(165) 
Finally, the boundary conditions in the direction become

(166) 
at , and

(167) 
at , where

(168) 
etc. Note, however, that the factor in front of the integrals in Eqs. (164)
and (168) takes the special value for the harmonic.
As before, we truncate the Fourier expansion in the direction, and discretize in the
direction, to obtain the set of tridiagonal matrix equations specified in Eqs. (158), (159), and (160). We can solve these equations to obtain the , and then reconstruct
the from Eq. (162). Hence, we have solved the problem.
Next: The fast Fourier transform
Up: Poisson's equation
Previous: 2d problem with Dirichlet
Richard Fitzpatrick
20060329