Tracking the Frank-Kamenetskii Problem

heat coefficient h

10

solution u versus α

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)

u

xx

+α

u

e

=0

for

0<x<1

,

u(x=0)=0

, and

u

x

x=1

+hu(x=1)=0

admits up to two solutions. Here,

u

is the dimensionless temperature and

h

is the heat transfer coefficient.

For

h=∞

and

α=e

, the BVP admits an analytical solution given by

u(x)=lncoshx-

1

2

θ

2

cosh

θ

4

, where

θ

is one of the two solutions of the transcendental equation

θ=

2e

cosh(θ/4)

(i.e.,

θ≈3.0362

and

θ≈7.1350

).

We use the homotopy continuation method and the Chebyshev orthogonal collocation technique (with

N+1=13

collocation points) to track the solutions,

u(x)

, in the

α

parameter space.

The plot of the norm of the solution

u=

N+1

∑

i=1

2

(u(

x

i

))

versus

α

clearly indicates that there can be up to two solutions. These two solutions are plotted in blue and magenta for

α=2.5

.