I need to solve the following system of equations for $\displaystyle n=0,1,2 $ subject to the given initial and boundary conditions. Is it possible to solve the system numerically. If yes, please give me some idea which scheme I should use for better accuracy and how should I proceed. The coupled boundary conditions are challenging for me. Please help.

$\displaystyle \frac{\partial C_n}{\partial t}-\frac{\partial^2 C_n}{\partial r^2}-\frac{1}{r}\frac{\partial C_n}{\partial r}=\beta n\, f(y,t)C_{n-1}+n(n-1)C_{n-2}$

$\displaystyle \frac{\partial \zeta_n}{\partial t}-\frac{\partial^2\zeta_n}{\partial r^2}-\frac{1}{r}\frac{\partial \zeta_n}{\partial r}=\beta n \,g(y,t)\zeta_{n-1}+n(n-1)\zeta_{n-2}$

$\displaystyle C_n(0,y)=1 \quad\mbox{for}\quad n=0$

$\displaystyle =0 \quad\mbox{for}\quad n>0 $

$\displaystyle \zeta_n(0,y)=1 \quad\mbox{for}\quad n=0$

$\displaystyle \quad\quad\quad=0 \quad\mbox{for}\quad n>0$

$\displaystyle \frac{\partial C_n}{\partial r}+\gamma C_n=0 \quad\mbox{at}\quad r=a$

$\displaystyle \frac{\partial C_n}{\partial r}=\kappa \frac{\partial \zeta_n}{\partial r} \quad\mbox{at}\quad r=b$

$\displaystyle C_n=\lambda\zeta_n \quad\mbox{at}\quad r=b$

$\displaystyle \frac{\partial \zeta_n}{\partial r}=0 \quad\mbox{at}\quad r=0$