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# Transient Two-Dimensional Heat Conduction Using Chebyshev Collocation

Consider the two-dimensional heat equation given by
, with and ,
which represents heat conduction in a two-dimensional domain. The boundary conditions are such that the temperature, , is equal to 0 on all the edges of the domain:
and for
and for
Without loss of generality, one can take the thermal diffusivity, , equal to . The initial condition is given by
for .
The dimensionless temperature, , can be found using either NDSolve or the Chebyshev collocation technique. As shown in the table of data given at , the two methods give the same results. Also, three contours of the dimensionless temperature, , (i.e., 0.25, 0.5, and 0.75) at are shown using the red curves and dashed black curves for the solution obtained with NDSolve and Chebyshev collocation, respectively. Again, perfect agreement is observed. Finally, you can set the value of the dimensionless time, and the contour plot tab lets you display the contour plot of the solution obtained using the Chebyshev collocation method.

### 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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