The Frank-Kamenetskii Problem Using Arc-Length Continuation

The Frank–Kamenetskii problem relates to the self-heating of a reactive solid. When the heat generated by a reaction is balanced by conduction in a one-dimensional slab of combustible material, the nonlinear boundary value problem (BVP) for , , and admits up to two solutions. Here, is the dimensionless temperature and is the heat transfer coefficient.

For and , the BVP admits an analytical solution , where is one of the two solutions of the transcendental equation (given by and ).

We use the arc length continuation method and the Chebyshev orthogonal collocation technique (with collocation points) to track the solutions, , in the parameter space.

The plot of the norm of the solution versus clearly indicates that there can be up to two solutions. These two solutions are plotted in blue and magenta for .