I'm trying apply the Laplace transformation to solve the non-dimensional heat conduction PDE for a hollow cylinder with convection boundary conditions and a non-homogenous initial condition.
$$\frac{\partial^2 \theta(r,t)}{\partial r^2}+\frac{1}{r}\frac{\partial \theta}{\partial r}=\frac{\partial \theta}{\partial t}$$
Initial condition:
$$\theta(r,t=0)=1$$
Boundary conditions:
$$\frac{\partial \theta}{\partial r}\big\rvert_{r=\mathrm{a/R}}=\mathrm{Bi_1}\theta\big\rvert_{r=\mathrm{a/R}}$$
$$\frac{\partial \theta}{\partial r}\big\rvert_{r=\mathrm{b/R}}=\mathrm{Bi_2}\theta\big\rvert_{r=\mathrm{b/R}}$$
where $\mathrm{a, b, R, Bi_1, Bi_2}$ are physical constants, and $\theta, r, t$ are dimensionless temperature, space and time, respectively.
My attempt:
I first applied the laplace transformation,
$$\frac{\partial^2 \overline{\theta}}{\partial r^2}+\frac{1}{r}\frac{\partial \overline{\theta}}{\partial r}=s\overline{\theta}-1$$
IC:
$$\overline{\theta}(r,s=\infty)=1/s$$
BCs:
$$\frac{\partial \overline{\theta}}{\partial r}\big\rvert_{r=\mathrm{a/R}}=\mathrm{Bi_1}\overline{\theta}\big\rvert_{r=\mathrm{a/R}}$$
$$\frac{\partial \overline{\theta}}{\partial r}\big\rvert_{r=\mathrm{b/R}}=\mathrm{Bi_2}\overline{\theta}\big\rvert_{r=\mathrm{b/R}}$$
I solved for the homogeneous and particular solutions of the now ODE to get,
$$\overline{\theta}(r,s)=A I_0(s r)+B K_0(s r) + 1/s$$
Which doesn't look like it has enough constants to apply the initial and boundary conditions. What could I be doing incorrectly?
Edit- I just found this: https://math.stackexchange.com/a/243003/642640. It is a similar problem, and it looks like the solution starts off looking similar to mine, however the arguments of the modified bessel functions look different, and I'm not sure how to do that correctly. I'm also still unclear on how to treat the boundary conditions.
