Solve Diffusion Equation with Laplace Transform

Solve the diffusion equation from this differential equation (Fick's Second Law). Note that there are two different independent variables t (time) and x (distance).

$\displaystyle \frac {\partial c}{\partial t} = D \frac {\partial ^2 c}{\partial x^2} $

With these boundary conditions:

$\displaystyle c(x, t) = \begin{cases} 0, & \mbox{for } 0 < x < \infty; t = 0 \\ 0, & \mbox{for } x \to -\infty \\ 0, & \mbox{for } x \to +\infty \\ c_0, & \mbox{for } x = 0 ; t > 0 \\ \end{cases}$

The professor said to use Laplace Transform. I'm familiar with Laplace transforms and initial value problems, but I'm not sure how to use them here since there are two independent variables and there seems to be a missing limit condition:

$\displaystyle \frac {\partial c}{\partial x}(x=0)$

Any help is appreciated. Thanks!

Re: Solve Diffusion Equation with Laplace Transform

Quote:

Originally Posted by

**Ackbeet** $\displaystyle \mathcal{L}\left\{\frac{\partial c(x,t)}{\partial t} =D\frac{\partial^{2}c(x,t) }{\partial x^{2}} \right\}\to s C(x,s)-c(x,0)=D\frac{\mathrm{d}^{2} C(x,s)}{\mathrm{d} x^{2}}. $

My capital letters and lowercase letters are very intentional. Lowercase c represents the original concentration function. Capital C represents the LT in time of the concentration function.

How is the Laplace transform for two-dimensional diffusion as $\displaystyle \frac{\partial c}{\partial t} =D\left\{\frac{\partial^{2}c(x,t) }{\partial x^{2}} + \frac{\partial^{2}c(y,t) }{\partial y^{2}}\right\} $?