The Frank-Kamenetskii Problem
The Frank–Kamenetskii problem relates to the self-heating of a reactive solid. When the heat generated by reaction is balanced by conduction in a one-dimensional slab of combustible material, the nonlinear boundary value problem (BVP) for , and admits two steady solutions. Here, is the dimensionless temperature. The BVP admits an analytical solution given by , where is one of the two solutions of the nonlinear equation (i.e., and ). The two analytical solutions are indicated by the blue and magenta curves. The dots represent the numerical solutions obtained using the Chebyshev collocation method. You can change the number of collocation points. You can clearly see that the analytical and numerical solutions are in agreement.
In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by , at the collocation points. These points are extremums of the Chebyshev polynomial of the first kind .
The Chebyshev derivative matrix at the quadrature points , , is 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 .
 P. Moin, Fundamentals of Engineering Numerical Analysis, Cambridge, UK: Cambridge University Press, 2001.