Transient Heat Conduction Using Chebyshev Collocation

Consider the one-dimensional heat equation given by

, with and .

This equation represents heat conduction in a rod. The boundary conditions are such that the temperature, , is equal to 0 at both ends of the rod:

and for .

Without loss of generality, one can take the thermal diffusivity, , equal to . The initial condition is given by

for

The temperature can be found using either NDSolve (solid colored curve) or the Chebyshev collocation technique (colored dots). Both methods give the same results, which are plotted in the same diagram at various values of time, ranging from 0.01 to 0.1 with a span of 0.01. You can set the number of interior points used by the Chebyshev collocation method.

In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by . These points are the extremums of the Chebyshev polynomial 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 second-order partial spacial derivatives are obtained using .

An affine transformation, , allows shifting from interval to .

Reference

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