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
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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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