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

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

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

$\displaystyle C_n(0,r)=1 $

$\displaystyle \zeta_n(0,r)=1 $

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

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

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

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

By using Crank Nicholson method and Thomas algorithm I can solve the following, but the above one gives trouble for me.

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

$\displaystyle C_n(0,r)=1 $

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

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