Can anybody help me with this time-dependent PDE in polar coordinate:
$$u_t=k\left(u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta \theta}\right)$$
with Boundary condition: $$u(r,\theta, 0)=\frac{1}{\pi R}\delta(r-R),\ \ \ a<r<b$$ $$u(a,\theta,t)=0, \ \ \ t>=0$$ $$u_r(b,\theta,t)=0, \ \ \ t>=0$$ $$u_\theta(r,0,t)=0, \ \ \ t>=0$$ $$u_\theta(r,\pi,t)=0, \ \ \ t>=0$$
I eliminated the time dependence using Laplace transform, then I couldn't find a general solution to separate $\theta$ and $r$.
After Laplace transformation: $$\hat{u}_{rr}+\frac{1}{r}\hat{u}_r+\frac{1}{r^2}\hat{u}_{\theta \theta}-ks\hat{u}=-\frac{k}{\pi R}\delta(r-R)$$ where $\hat{u}{(r,s, \theta)}=\int^{\infty}_0u(r,\theta,t)e^{-st}dt$