I am not an expert in Mathematica. I want to keep off from tedious calculation
I want to solve (in symbolic sens) this system: $\quad AU^{j+1}+BU^{j}=F^{j}$
where:
$*$ ${U}^{j}$ a $(N;1)$ vector $\quad\mathbf{U}^{j}=\left[\begin{array}{c}u_{1}^{j} \\ \vdots \\ u_{N}^{j}\end{array}\right]$ and $\quad\mathbf{U}^{1}=\left[\begin{array}{c}\phi(x_{2}) \\ \phi(x_{2}) \\\phi(x_{3}) \\ \vdots \\ \phi(x_{N-1}) \\ \phi(x_{N-1})\end{array}\right] $
$*$ ${F}^{j}$ a $(N-2;1)$ vector$\quad\mathbf{F}^{j}=\left[\begin{array}{c}kf_{2}^{j} \\ \vdots \\ kf_{N-1}^{j}\end{array}\right] $
$*$ $A$ a $(N-2;N)$ matrix $A=\left(\begin{array}{rrrrrr} -\lambda &-1& -\lambda&0&0&\cdots&0\\ 0&-\lambda &-1& -\lambda&0&\cdots&0\\ 0&0&-\lambda &-1& -\lambda&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&0&\cdots&-1&-\lambda\\ \end{array}\right)$
$*$ $B$ a $(N-2;N)$ matrix $B=\left(\begin{array}{rrrrrr} 0&1&0&0&\cdots&0&0\\ 0&0&1&0&\cdots&0&0\\ 0&0&0&1&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\ddots&\vdots&\vdots&\\ 0&0&0&\cdots&0&1&0\\ \end{array}\right)$
Thanks a lot for your time, you can ask me to clarify anything
n
,U1
, and some closed for expression for theFj
you might be able to useRSolve
. $\endgroup$