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Numerical solution of partial differential equation

suvadip

Member
Feb 21, 2013
69
I need to solve the following system of equations for [TEX]n=0,1,2 [/TEX] 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.




[TEX]\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}[/TEX]
[TEX]\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}[/TEX]


[TEX]C_n(0,r)=1 \quad\mbox{for}\quad n=0[/TEX]
[TEX]=0 \quad\mbox{for}\quad n>0 [/TEX]


[TEX]\zeta_n(0,r)=1 \quad\mbox{for}\quad n=0[/TEX]
[TEX]\quad\quad\quad=0 \quad\mbox{for}\quad n>0[/TEX]


[TEX]\frac{\partial C_n}{\partial r}+\gamma C_n=0 \quad\mbox{at}\quad r=a[/TEX]
[TEX]\frac{\partial C_n}{\partial r}=\kappa \frac{\partial \zeta_n}{\partial r} \quad\mbox{at}\quad r=b[/TEX]
[TEX]C_n=\lambda\zeta_n \quad\mbox{at}\quad r=b[/TEX]
[TEX]\frac{\partial \zeta_n}{\partial r}=0 \quad\mbox{at}\quad r=0[/TEX]
 
Last edited:

dwsmith

Well-known member
Feb 1, 2012
1,673
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 text book. 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