Working on a PDE that may have solution out there somewhere but I have yet to stumble upon it. I prefer to use the Laplace Transform approach when I can, as I find it more intuitive than some other techniques. The problem is the 1D transient heat equation in it's standard form, with one Neuman and one Robin type boundary.
After separating out the transient and steady state portions, I've gotten to the following form prior to needing to do the final inversion. I'd appreciate any pointers on the inversion. I'd been fairly successful using convolution forms and inversion tables to this point. I'm either missing something regarding how to work with the denominator (whether as hyperbolics or exponentials), or it could be that this approach just dead-ends here for an analytical solution. Links to tutorial material on more complex ways of approaching the inversion would also be appreciated. I have started looking into Residuals but am just getting started now...
$$\Theta(x,s)= -q(\frac{1}{ks^2}+\frac{1}{hs})\frac{\cosh{((L-x)\sqrt{\frac{s}{\alpha}})}}{\sqrt{\frac{s}{\alpha}}\sinh{(L\sqrt{\frac{s}{\alpha}})}+\frac{h}{k}\cosh{(L\sqrt{\frac{s}{\alpha}})}}$$