# Fick's Second Law: Laplace Transform to solve PDE in Spherical Coords

1. Dec 9, 2008

### johndoe3344

Fick's second law in general form:
$$\frac{\partial C}{\partial t} = D\nabla^2 C$$

In spherical form:
$$\frac{\partial C}{\partial t} = D\frac{1}{r^2}\frac{\partial}{\partial r}\left( r^2\frac{\partial C}{\partial r} \right)$$

(Assume all changes in phi and theta to be zero, so we are only concerned with the r component here.)

Let's say that C(t=0) = 0

If we laplace transform:
LHS becomes: $$p\bar{C}}$$

Where C bar is the laplace transform of C, and C(t=0) = 0.

I'm stuck on the right hand side. The textbook just skips the math and gives the solution. Any help would be appreciated.

Thanks.

Last edited: Dec 9, 2008