Question $3$

The Finite Difference $($FD$)$ grid for a metal rod is given below. The temperatures at the end are constant, while the temperature at $=0\mathrm{.}424-T_{i-1}^{m+1}+2(1+)T_i^{m+1}-T_{i+1}^{m+1}=T_{i-1}^m+2(1-)T_i^m+T_{i+1}^{m}0$ changes with time and can $be$ described $by$ the heat conduction Partial Differential Equation. Assume $=0\mathrm{.}424$

$3\mathrm{.}1$ Use Crank$-$Nicolson's method $to$ determine the temperature distribution $in$ the rod after $3$ and $6$ seconds. Crank$-$Nicolson's formula:

$-T_{i-1}^{m+1}+2(1+)T_i^{m+1}-T_{i+1}^{m+1}=T_{i-1}^m+2(1-)T_i^m+T_{i+1}^m$