$(15$ points$)$ Consider the Poisson equation

$u_{}(x,y)+u_{yy}(x,y)=4,onR=[0,1][0,1]$

subject to boundary condition

$u(x,0)=0,u(x,1)=100x,u(0,y)=0,u(1,y)=100y$

With $m=n=4,$ apply the finite$-$difference method in section $12\mathrm{.}1$ to find its numerical solution:

Use Jupyter notebook to write Python code to form A and $b$ of the linear equations $Aw=b.$ Then

use numpy.linalg.solve to solve $Aw=b$ for $w.$