I want to find the explicit solution of the following difference equation
$e_{i,j+1}=re_{i-1,j}+(1-2r)e_{i,j}+re_{i+1,j}+km_{i,j}$
where $r>0$, $k>0$ and $m_{i,j}$ are known and $e_{i,0}=0$. My approach is that as in the solution of PDE by seperation of variables, i.e,
$e_{i,j}=X_{i}T_{j}$
but I couldn't go further. Is there any other approach to solve this?