The following diffusion problem, in a semi-infinite interval,

u(x,0) = 0, u(0,t) = 1

admits the explicit (similarity) solution:

with erfc(.) = 1−erf(.) the

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 ∫eTurn in on paper.^{−ξ2/4}dξ + C2, with C1, C2 arbitrary constants. 3. This integral is not an elementary function! It can be written in terms of theerror function: erf(z) = 2/√π. This is the area under the bell curve from 0 to z, also known as the_{0}∫^{z}e^{−s2}dsnormal distributionin 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 thatu(x,t) = y(ξ) = 1 − erf( x / 2√(D t) ).