# Heat Conduction in a Rod

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Consider the problem of unsteady-state heat conduction in a rod, as governed by the heat equation:

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Contributed by: Housam Binousand Brian G. Higgins (June 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 extrema of the Chebyshev polynomials 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 matrix is then used as follows: and , where is a vector formed by evaluating at , , and and are the approximations of and at the .

References

[1] P. Moin, *Fundamentals of Engineering Numerical Analysis*, Cambridge, UK: Cambridge University Press, 2001.

[2] L. N. Trefethen, *Spectral Methods in MATLAB*, Philadelphia: SIAM, 2000.

[3] J. Crank, *The Mathematics of Diffusion*, 2nd ed., New York: Oxford University Press, 1975.

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