The Arnold Problem

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



where is the diffusion coefficient.

The initial and boundary conditions are:

, ,

, ,

, ,

where is the interfacial gas-phase concentration and is the position.

This equation has an analytical solution:

, where and is a solution of the nonlinear equation: .

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 both solutions is observed. You can vary the values of , , and as well as the number of Chebyshev collocation points, .


Contributed by: Housam Binousand Brian G. Higgins (June 2013)
Open content licensed under CC BY-NC-SA



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 .


[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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