Let $L,m,h$ be known constants, consider a quite complicated difference equation system,
$\left\{ {\begin{array}{*{20}{l}} {4u(x,y) - u(x - h,y) - u(x + h,y) - u(x,y - h) - u(x,y + h) = 0} \\ {u(2h,y) = \sqrt {\frac{{m + 1}}{2}} \sin (\frac{{\pi ky}}{L})} \\ {u(x,0) = u(x,L) = 0} \end{array}} \right.$
where $\left\{ {\begin{array}{*{20}{l}} {x = - \infty ,..., - h,0,h} \\ {y = h,2h,...,mh} \end{array}} \right.$.
I have the solutions are $u(x,y) = {e^{ \pm w(k)x}}\sin (\frac{{\pi ky}}{L})$ where $w(k)$ is the unique solution of $\cosh (w(k)x) = 2 - \cos (\frac{{\pi kh}}{L})$, but I don't know how they derive this.