Numerical solution of partial differential equation

[Suvadip]

I need to solve the following system of equations for $$n=0,1,2$$ 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.

$$\frac{\partial C_n}{\partial t}-\frac{\partial^2 C_n}{\partial r^2}-\frac{1}{r}\frac{\partial C_n}{\partial r}=\beta n\, f(r,t)C_{n-1}+n(n-1)C_{n-2}$$
$$\frac{\partial \zeta_n}{\partial t}-\frac{\partial^2\zeta_n}{\partial r^2}-\frac{1}{r}\frac{\partial \zeta_n}{\partial r}=\beta n \,g(r,t)\zeta_{n-1}+n(n-1)\zeta_{n-2}$$$$C_n(0,r)=1 \quad\mbox{for}\quad n=0$$
$$=0 \quad\mbox{for}\quad n>0$$$$\zeta_n(0,r)=1 \quad\mbox{for}\quad n=0$$
$$\quad\quad\quad=0 \quad\mbox{for}\quad n>0$$$$\frac{\partial C_n}{\partial r}+\gamma C_n=0 \quad\mbox{at}\quad r=a$$
$$\frac{\partial C_n}{\partial r}=\kappa \frac{\partial \zeta_n}{\partial r} \quad\mbox{at}\quad r=b$$
$$C_n=\lambda\zeta_n \quad\mbox{at}\quad r=b$$
$$\frac{\partial \zeta_n}{\partial r}=0 \quad\mbox{at}\quad r=0$$

 

[Dustinsfl]

You should check out the journal on Numerical Methods for Partial Differential Equations. It comes out in monthly in volumes that are the size of a 300 page textbook. I have volume 29 number 6 Nov 2013 and that may not be much of a help to you but there is bound to be a volume of interest.

You can also view the journal online at wilyonlielibrary.com/journal/num