WOLFRAM NOTEBOOK

WOLFRAM|DEMONSTRATIONS PROJECT

The Frank-Kamenetskii Problem

number of collocation points

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+1

for

0<x<1

, and

u(x=0)=u(x=1)=0

admits two steady solutions. Here,

is the dimensionless temperature. The BVP admits an analytical solution given by

u(x)=lncoshx-

cosh



, where

is one of the two solutions of the nonlinear equation

θ=

cosh(θ/4)

(i.e.,

θ≈3.0362

and

θ≈7.1350

). 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.

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.

I understand and wish to continue anyway »