Given PDE: $\nabla^2 u(x,y)=(α^2-β^2 ) u(x,y)$
The domain is the square $[0,1] \times [0,1]$, which the boundary nodes are $(0,0),(1,0),(1,1),(0,1)$.
It is given that $u(0.5,0.5)=1$.
We would like to know how to started with using linear approximation to get the equation which connected those nodes?
Updates: If we would like to solve the PDE before linear approximation, what should we do?