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2-d 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: 2-d problem with Dirichlet
Richard Fitzpatrick
2006-03-29