# Solve Fick's second law of diffusion

1. Aug 5, 2012

### abstracted6

I'm curious how to solve Fick's second law of diffusion $$\frac{∂c}{∂t}=D \frac{∂^2c}{∂x^2}$$For conditions:$$c(x,0)=0$$$$c(0,t)=A$$$$c(\infty,t)=0$$Physically this means:
-c(x,t) is the concentration at point x at time t.
-Initially there is no concentration of diffusing species.
-At x=0 for all t the is a constant concentration "a".
-As x goes to infinity for all time, the concentration is 0.
-D is the diffusivity, assume it is a constant.

The solution is:$$c(x,t)=A erfc(\frac{x}{2\sqrt{Dt}})$$
What method was used to arrive at that solution?

2. Aug 5, 2012

### JJacquelin

Hi !

May be this formula was obtained thanks to the Laplace method for PDE resolution.
Normally we would have to use the double Laplace transform (relatively to x AND t), which would be rather arduous.
But the PDE and boundary conditions are simple enough to use the usual single Laplace transform (relatively to t only).

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3. Aug 7, 2012

### hunt_mat

Possibly try a similarity solution?

4. Aug 10, 2012

### Staff: Mentor

Transport Phenomena by Bird, Stewart, and Lightfoot show how to solve this (singular perturbation boundary layer problem) using similarity solutions. Look for the analogous viscous flow startup problem.