I have heat equation 1D in cylindrical coordinates:
$$u_{\rho\rho}+\frac{1}{\rho}u_{\rho}=u_{t},\;0<\rho<1,t>0 $$
with boundary conditions $u(1,t)=0,\;t>0$ and initial condition $u(\rho,0)=T$. I approximate $u_{\rho\rho}$ with central differences of second order and $u_{\rho}$, $u_{t}$ with central and backward-scheme. So my domain is:
for $\rho\neq 0.$ If $\rho=0$ then heat equation is $$2u_{\rho\rho}=u_{t}.$$ My question is. Why in the last case $\rho=0$ is not a boundary but a line of symmetry.