# 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

## Snapshots

## Details

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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