This linear system is $AT=b$ where $T=(T_{-}0,T_{-}1,$dots,$T_{-}N)$ is the vector of $N+1$ unknown temperatures, $A$ is the $(N+1)(N+1)$ tridiagonal matrix

$A=\left[\begin{array}{ccccc}1& ?& ?& ?& ?\\ \frac{1}{h^2}& -\frac{2}{h^2}& \frac{1}{h^2}& ?& ?\\ ?& \frac{1}{h^2}& -\frac{2}{h^2}& \frac{1}{h^2}& ?\\ ?& \text{d}\text{d}\text{o}\text{t}\text{s}& \text{d}\text{d}\text{o}\text{t}\text{s}& \text{d}\text{d}\text{o}\text{t}\text{s}& ?\\ ?& \frac{1}{h^2}& -\frac{2}{h^2}& \frac{1}{h^2}& ?\\ ?& 1& ?& ?& ?\end{array}\right]$

and $b$ is the column vector of $N+1$ right$-$hand side terms:

$b=(T_H,0,0,0,$dots,$0,T_C{)}^T$

For the rest of this lab, use the parameter values: $L=10m,T_H=200C,T_C=40C,.$

Task $1$:

Duplicate and rename your programme from Exercise $1.$ Revise the construction of the vector b in your programme to match equation. You can now solve the linear system $AT=b$

Ensure that you change the comments in your program to reflect what it does. Solve the system to find the temperature profile $T(x)$ with a suitable number of steps $N.$

Draw a graph of $T(x)$ against $x.$

In $[]$: # create Matrix A

In $[]$: # Solve system and plot

Does the graph agree with what you would expect the temperature profile for an insulated rod to look like? Is there any need for $T_H>T_C?$

In $]$ :$\%$config InlineBackend.figure$\_$format$=$'retina'

Figure below shows a thin rod, of length $L,$ whose ends are kept at fixed temperatures $T_H$ and $T_C$ where $T_H>T_C$ and whose sides are insulated so there is no heat loss through the

sides. The arrows represent heat flow.

In the steady$-$state situation the temperature $T$ is a function of $x,$ the distance along the rod, but not a function of time. $T(x)$ is determined by Laplace's equation in $1$D:

$\frac{d^{2}T}{d{x}^2}=0,T(L)=T_C$

As in the lectures, we divide the rod into $N$ intervals. In lectures the grid used was separated by $x=\frac{L}{N}$; hereweuse $\frac{\frac{?}{?}}{}x=h$$$.$ We replace the derivative term by a central difference

approximation, and evaluate the resulting equation at each grid point. This gives a set of linear equations

$\frac{T_{k-1}-2T_k+T_{k+1}}{h^2}=0,k=1,$dotsN$-1$

From the boundary conditions, we also have

$T_0=T_H,$ and $,T_N=T_C$

Taking the above two equations gives a total of $N+1$ equations in the unknowns $T_0,$dots,$T_N.$

This linear system is $AT=b$ where $T=(T_{-}0,T_{-}1,$dots,$T_{-}N)$ is the vector of $N+1$ unknown temperatures, $A$ is the $(N+1)(N+1)$ tridiagonal matrix