Part B $-$ Finite difference method

Figure $2$: Finite difference discretization of noninsulated uniform rod between two bodies of constant temperature.

$-T_{i+1}+(2+h^{\text{'}}{x}^2)T_i-T_{i-1}=h^{\text{'}}{T}_a{x}^2,i=1,$dots,$n-1$

To solve $(1)$ using the finite difference method, we discretize the bar into $n$ segments of length, $x,$ as shown in Figure $2$ and approximate the second derivative of temperature using finite divided differences. Then, after rearranging, the BVP in $(1)$ can be expressed as a set of simultaneous algebraic equations,

Goals

Write a Matlab script, Script$\_$NonInsulatedRod$\_$FiniteDifference.m$,$ that solves the BVP in $(1)$ using the finite difference method and considering $x=2m$ and $x=0\mathrm{.}1m.$ To do so$,$ please:

$($a$)$ Develop the system of equations associated with $(5)$ in a general form such that the same equations can be used for any specified segment length, $x,$ i$.$e$.,$ DO NOT hard$-$code the specific finite difference equations for a given $x.$ No te: Matlab's backslash operator will come in handy to solve the system of equations in $(5).$

$($b$)$ Plot the analytical solution in $(2)$ alongside the two finite difference solutions. Please consider axis limits$,x\underset{?}{lim}([6\mathrm{.}5,7\mathrm{.}5])$ and ylim$([130,160]),$ so that the difference in the two numerical solutions is visible $-$ see example in Figure $7.$