We can model circularly symmetric diffusion in a circle with radius $a$ as a function of time and radial position by employing the partial differential equation
$$ \frac{\partial c}{\partial t} = \frac Dr \frac{\partial}{\partial r}\left(r \frac{\partial c}{\partial r}\right) $$
where $c(r,t)$ is the concentration of particles with conditions $c(r,0)=c_0,c(a,t)=1, c(0,t) =$ finite. Solve to find the $c(r,t)$ in terms of bessel functions.
I used separation of variables and found the solution to be
$$ c(r,t)=\frac{J_0(\lambda r)}{J_0(\lambda a)} c_0 e^{-\lambda^2 D t}. $$ I'm not sure if this is correct and I'm not sure how to find what $\lambda$ is. Any guidance would be great!