Consider the two-dimensional Poisson equation (8.6) in a rectangular domain \(0 \leq x \leq L_{x}, 0 \leq y \leq L_{y}\). The boundary conditions are

\[\frac{\partial u}{\partial x}(0, y)=g_{1}(y), u\left(L_{x}, y\right)=g_{2}(y), \frac{\partial u}{\partial y}(x, 0)=g_{3}(x), u\left(x, L_{y}\right)=g_{4}(x)\]

The computational grid is uniform with steps \(\Delta x=L_{x} / N_{x}\) and \(\Delta y=\) \(L_{y} / N_{y}\). Develop the system of finite difference equations approximating the entire problem (PDE and boundary conditions) with the second order of accuracy. Write the scheme in the matrix form similar to (8.9) and (8.10). Take into account that the equations and expressions for coefficients are different for internal and boundary grid points.