$1.$ Modify the fCrankNicolson function and CNscript codes to accommodate a Type $3$ boundary condition at x $=0$ and type $2$ at x $=$ L $[$m$].$ Let the first boundary condition be characterized by BiL $=$ hL$/$k and environment temperature T$\backslash$omicron $\backslash$omicron $[$deg C$].$ Pass these parameters to the new function, such as in the following function header line:

fCrankNicolson$\_2($nx$,$ dt$,$ Told, BiL, T$\_$inf, q$\_$j$,$ q$\_$jp$1)$

Use these parameters to call your function:

L $=2$;

nx $=40$;

dt $=0\mathrm{.}01$;

T$\_$inf $=5$;

x $=[0$:L$/$nx:L$]$; tfinal $=1\mathrm{.}8$;

and let q$($t$)=1\mathrm{.}0$ W$/$m$2$ constant beginning at time t $=0$ s$.$ Make a plot showing T$($x$)$ at different times, all on the same graph: t $=0\mathrm{.}1,0\mathrm{.}45,0\mathrm{.}90,1\mathrm{.}35,$ and $1\mathrm{.}8$ s

$2.$ Test the code with the script letting BiL $=0.$ How do you expect the solution to behave? What do you see in the results that confirms or informs your expectation?

$3.$ Test the code with the script letting BiL $=100.$ How does this affect the solution at x $=0?$ What$$s going on in the results at $$small$$ times at x $=0?($you can investigate letting dt be smaller, say dt $=0\mathrm{.}001,$ if you wish$).$

$4.$ Test the code with a moderate value of BiL $=20.$ What do you see in the results? Approximately how long does it take for steady state?