I am trying to simulate transitional processes in RF filters. Mathematical model of the circuit is a system of $N$ equations. Each looks like this: $$a\frac{d^2i_j(t)}{dt^2}=bi_j+c\frac{di_j(t)}{dt}+d\frac{d^2i_{j-1}(t)}{dt^2}+e\frac{d^2i_{j+1}(t)}{dt^2}$$ , where $j$ is the index of the equation in the system. It doesn't look like any equation system I've seen because each equation depends on the solutions of it's neighbors. What kind of equation are those? Can I split them into ODEs, and, if not, how can I solve this system?
Edit
Forgot to add that one of the equations would have an additional summand $g\sin{(\omega t)}$ corresponding to the driving force.
Essentially you have a coupled system of the form $$ A\ddot x=B\dot x+Cx $$ where $A,B,C$ are tridiagonal resp. diagonal matrices. For a numerical solution you need a first order system. Introduce $y=\dot x$ then this system is \begin{align} \dot x&=y\\ \dot y&=A^{-1}(By+Cx) \end{align}