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Thread: Numerical solution of partial differential equation

  1. #1
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    Numerical solution of partial differential equation

    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 $
    $\displaystyle \frac{\partial C_n}{\partial r}+\gamma C_n=0 \quad\mbox{at}\quad r=a$
    $\displaystyle \frac{\partial C_n}{\partial r}=0 \quad\mbox{at}\quad r=0$
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  2. #2
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    Re: Numerical solution of partial differential equation

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