I'm trying to solve telegraph equation (transmission line) with no losses. I got this equation (wave eq) $$LC\frac{\delta^2u}{\delta t^2}=\frac{\delta^2u}{\delta x^2}$$ that I wan't to solve using Crank-Nicolson in MATLAB, but I'm stuck with Crank-Nicolson difference scheme. So can i write it like this
$$ \frac{\partial u}{\partial t} = v => \frac{u_{i+1}^n - u_{i}^n}{\Delta t} = v_{i}^n\ (1) $$
$$ \frac{\delta v}{\delta t}=\frac{1}{LC} \frac{\partial^2 u}{\partial x^2} => \frac{v_{i+1}^{n} - v_i^n}{\Delta t} = \frac{1}{2LC}\frac{u_{i+1}^{n+1}- 2u_{i+1}^n+u_{i+1}^{n-1}+u_{i}^{n+1}-2u_{i}^n+u_{i}^{n-1}}{\Delta x^2} (2) $$ Can I substitute $v_{i}^n$ in (2) using (1) and how can I write $v_{i+1}^{n}$ ?
If I use this eq $LC\frac{\delta^2u}{\delta t^2}=\frac{\delta^2u}{\delta x^2}$, and if I write $\frac{\delta ^2 u}{\delta x^2}=\frac{u_{i+1}^{n+1}- 2u_{i+1}^n+u_{i+1}^{n-1}+u_{i}^{n+1}-2u_{i}^n+u_{i}^{n-1}}{\Delta x^2}$ what is correct way to write $\frac{\delta^2u}{\delta t^2}$ ? Thanks
\implies($\implies$) by $=>$ ...? – CiaPan Sep 01 '23 at 10:00