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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.

The initial and boundary conditions are:

, for and for ,

, ,

, ,

where is the temperature, is time, and is the position.

This problem has an analytical solution in the form of a Fourier series after separation of variables:

.

This Demonstration plots the solution . The numerical solution obtained using Chebyshev orthogonal collocation is given by the red dots. The analytical solution is given by the blue curve. Excellent agreement between the two solutions is observed. You can vary the value of as well as the number of Chebyshev collocation points, .

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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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