Question

I want 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?
And yes, I have experience with solving PDE's.

Answer 1

complimentary error function/?

Answer 2

yup

Answer 3

@mukushla

Answer 4

method is Combination of Variables