Transient Two-Dimensional Heat Conduction Using Chebyshev Collocation

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Consider the two-dimensional heat equation given by
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Contributed by: Housam Binous, Brian G. Higgins, and Ahmed Bellagi (April 2013)
Open content licensed under CC BY-NC-SA
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In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by . These points are the extremums of the Chebyshev polynomial of the first kind
.
The Chebyshev derivative matrix at the quadrature points,
, is an
matrix given by
,
,
, for
, and
for
and
,
where for
and
.
The discrete Laplacian is given by , where
is the
identity matrix,
is the Kronecker product operator,
, and
is
without the first row and first column.
An affine transformation, , allows shifting from the interval
to
.
Reference
[1] L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.
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