Use the Gaussian elimination algorithm on p$.24$ to calculate

a finite$-$difference approximation to Worked Example $2\mathrm{.}1$ for one

time$-$step taking $x=0\mathrm{.}1$ and $r=1$ using either

$($i$)$ the fully implicit backward time$-$difference equation

$u_{i,j+1}-u_{i,j}=r(u_{i-1,j+1}-2u_{i,j+1}+u_{i+1,j+1}),$