Find steady state solution

Question

The governing equation: ∂Ω/∂t = (∂²Ω^/∂x²)-D_H(∂Ω/∂x)

Find the steady-state solution Ω(x) from the boundary conditions: Ω(0) =1,Ω(1)=0

Answer 1

In the steady state condition the left hand side is zero.  This leaves a fairly simple second order differential equation for Ω(x).   There are two constants of integration which can be evaluated from the two boundary conditions.

 

The answer is  Ω(x) =  ( e^{DH x}  - e^DH) /(1 -e^DH)