Industrial Mathematics - Alexiades

      Explicit similarity solution for diffusion in semi-infinite interval

The following diffusion problem, in a semi-infinite interval,
      ut = D uxx ,   0 < x < Inf ,   t > 0
      u(x,0) = 0,   u(0,t) = 1

admits the explicit (similarity) solution:
      u(x,t) = erfc( 0.5 x / √(D t) ).
with erfc(.) = 1−erf(.) the complementary error function.
This solution can be found as follows:
1. Seek a similarity solution:  Set  ξ = x / √(D t)  and u(x,t) = y(ξ).
   Show that the diffusion equation transforms to an ODE for y(ξ):
	y'' + (ξ/2) y' = 0.

2. This is linear first order in y', so it can be solved explicitly.  Show that:
	y(ξ) = C1 ∫e−ξ2/4dξ + C2,   with C1, C2 arbitrary constants.

3. This integral is not an elementary function! It can be written in terms of the 
	 error function:  erf(z) = 2/√π 0z e−s2 ds . 
   This is the area under the bell curve from 0 to z, also known as the 
   normal distribution in statistics. Note that erf(0)=0 and erf(∞)=1.

   Show that y(ξ) = A erf(ξ/2) + B,  A,B arbitrary constants.

4. Apply the initial condition and the boundary condition to show that
	u(x,t) = y(ξ) = 1 − erf( x / 2√(D t) ).
Turn in on paper.