# The Arnold Problem

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Consider the unsteady-state evaporation of a liquid, the Arnold problem. The governing equation is:

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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] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, *Transport Phenomena*, 2nd ed., New York: John Wiley & Sons, 2002.

[4] J. H. Arnold, *Transactions of American Institute of Chemical Engineers*, 40, 1944 pp. 361–378.

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