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

$u_t(x,t)-u_{}(x,t)=4,0x1,t0$

subject to the conditions

$u(0,t)=0,u(l,t)=0,t>0,$ and $u(x,0)=x(1-x),0x1.$

With $n=10$ for $x$ and step size $k=0\mathrm{.}01$ for $t,$ apply backward$-$difference method to find its numerical

solution at $t=k$ : Use Jupyter notebook to write Python code to form $A,w^{(0)}$ and $b$ of the linear

equations $A{w}^{(1)}=w^{(0)}+b.$ Then use numpy.linalg.solve to solve $A{w}^{(1)}=w^{(0)}+b$ for $w^{(1)}.$