Solve Fick's second law of diffusion

[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?

 

[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).

 

[hunt_mat]

Possibly try a similarity solution?

 

[Chestermiller]

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.

 

[blue_raver22]

The solution to Fick's second law of diffusion can be found using separation of variables and the method of eigenfunction expansion. This involves assuming a solution of the form c(x,t) = X(x)T(t) and substituting it into the equation. This leads to separating the equation into two ordinary differential equations, one for X(x) and one for T(t). From there, boundary conditions can be applied to solve for the coefficients and obtain the final solution. In this case, the boundary conditions at x=0 and x=\infty were used to solve for the constant A and the error function, erfc, was used to represent the concentration profile over time.