I'm trying solve a single dimension diffusion problem, where
$\frac{\partial c(x,t)}{\partial t} = D \frac{\partial^2 c(x,t)}{\partial x^2}$ over $0 \leq x\leq L$.
and $c(0,t)=c_0$, $\frac{\partial c}{\partial x}(L,t)=0$, and $c(x,0)=0$.
Using the Laplace transform of my relation,
I get the ODE $C(x,s)=\frac{D}{s}\frac{\partial^2 C(x,s)}{\partial x^2}$ with the following conditions:
$C(L,s)=0$ (I think that's right?) and $C(0,s)=\frac{c_0}{s}$
Accordingly, I think I've found the following as a potential solution in the Laplace domain, where I assume $c_0=1$:
$C(x,s)=\frac{\text{csch}\left(L \sqrt{\frac{s}{D}}\right) \sinh \left(\sqrt{\frac{s}{D}} (L-x)\right)}{s}$
Now my question is, how can this be applied with the inverse Laplace transform to find the solution in the time domain?
Ultimately, I'm looking for a solution that I can integrate with respect to x from 0 to L. I've looked through Crank's book on "The Mathematics of Diffusion" but those solutions for "diffusion in a plane" are infinite sums Fourier series.