Chapter 7.4, Problem 36E

A machine contains the grid of wires shown in the accompanying sketch. At the seven indicated points, the temperature is kept fixed at the given values (in °C). Consider the temperatures $T_1\left(t\right)=T_2\left(t\right)$ , and $T_3\left(t\right)$ at the other three mesh points. Because of heat flow along the wires, the temperatures $T_i\left(t\right)$ changes according to the formula
$T_i\left(t+1\right)=T_i\left(t\right)-\frac{1}{10}{\displaystyle \sum \left(T_i\left(t\right)-T_{\text{adj}}\left(t\right)\right)}$ ,
where the sum is taken over the four adjacent points in the grid and time is measured in minutes. For example,
$T_2\left(t+1\right)=T_2\left(t\right)-\frac{1}{10}\left(T_2\left(t\right)\right)-\frac{1}{10}\left(T_2\left(t\right)-200\right)-\frac{1}{10}\left(T_2\left(t\right)-0\right)-\frac{1}{10}\left(T_2\left(t\right)-T_3\left(t\right)\right)$ .
Note that each of the four terms we subtract represents the cooling caused by heat flowing along one of the wires. Let
$\stackrel{\to }{x}\left(t\right)=\left[\begin{array}{c}{T}_1\left(t\right)\\ T_2\left(t\right)\\ T_3\left(t\right)\end{array}\right]$ .

1. Find a $3\times 3$ matrix A and a vector $\stackrel{\to }{b}$ in ${ℝ}^3$ such that $\stackrel{\to }{x}\left(t+1\right)=A\stackrel{\to }{x}\left(t\right)+\stackrel{\to }{b}$ .

Introduce the state vector $\stackrel{\to }{y}\left(t\right)=\left[\begin{array}{c}{T}_1\left(t\right)\\ T_2\left(t\right)\\ T_3\left(t\right)\\ 1\end{array}\right]$,

with a dummy” 1 as the last component. Find a $4\times 4$ matrix B such that $\stackrel{\to }{y}\left(t+1\right)=B\stackrel{\to }{y}\left(t\right)$ . (This technique for convening an affine system into a linear system is introduced in Exercise 35; see also Exercise 32.)

Suppose the initial temperatures are $T_1\left(0\right)=T_2\left(0\right)=T_3\left(0\right)=0$ . Using technology, find the temperatures at the three points at $t=10$ and $t=30$ . What long-term behavior do you expect?

Using technology, find numerical approximations for the eigenvalues of the matrix B. Find an eigenvector for the largest eigenvalue. Use the results to confirm your conjecture in part (c).