# Tracking the Frank-Kamenetskii Problem

Tracking 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) +α=0 for , , and +hu(x=1)=0 admits up to two solutions. Here, is the dimensionless temperature and is the heat transfer coefficient.

u

xx

u

e

0<x<1

u(x=0)=0

u

x

x=1

u

h

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

h=∞

α=e

u(x)=lncoshx-cosh

1

2

θ

2

θ

4

θ

θ=

2e

cosh(θ/4)θ≈3.0362

θ≈7.1350

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

N+1=13

u(x)

α

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 .

u=

N+1

∑

i=1

2

(u())

x

i

α

α=2.5