Your program must perform the following:$$

$1)$ Use the explicit method to calculate the temperature disturbution with M $=51,101,201.$

$2)$ Use Equation $$t $=($t$\_$max$)/($N$-1)[$s$].$ to determine the least number of time steps, N$,$ that satisfies the stability condition in Equation $$t $<$x$^2/2\backslash$alpha $.$ Round N$,$ to the largest integer and recalculate the time step $$t from Equation $$t $=($t$\_$max$)/($N$-1)[$s$].$

$3)$ Save the variables corresponding to L$,$ k$,\backslash$alpha $,,$ T$\_\backslash$infty $,$ h$,$ and the three array storing the temperature disturbutions for M $=51,101,$ and $201$ to file named Case$1$Explicit.mat

$4)$ Figure: Plot the temperature profile T vs x at approximately $1$ min, $2\mathrm{.}5$ min $5$ min, $10$ min $($estimate based on your $$t$).$ Your figure must contain $3\backslash$times $1$ subplots, one for each set of nodes $($M$)$ where each plot has four lines for the four times on the same set of axes. Annotate your plots with title, axis labels, and a legend.

$5)$ Use the implicit method to calculate the temperature disturbution with M $=101$ and N $=101,201401.$

$6)$ Save the variables corresponding to L$,$ k$,\backslash$alpha $,,$ T$\_\backslash$infty $,$ h$,$ and the three arrays storing the temperature disturbutions for N $=101,201,$ and $401$ to a file named Case$1$Implicit.mat

$7)$ Figure: Plot the temperature distribution T vs x at approximately $1$ min, $2\mathrm{.}5$ min, $5$ min, $10$ min $($estimate based on your $$t$).$ Your figure must contain $1\backslash$times $3$ subplots, one for each set of time nodes $($N$)$ where each plot has four lines for the fours times on the same set of axes. Annotate you plots with a title, axis labels, and a legend.