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 0∫z 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) ) = erfc( x / 2√(D t) ).
May be typeset or hand-written (but clear and neat).