PROJECT 10: HEAT EQUATION In this project we will model heat in a metal rod. We recall Newton's law of cooling: the rate of change in temperature is proportional to the difference in temperature dT dt = k(A - T), where A is the ambient temperature. For simplicity we will assume k = 1. We will model a rod with a finite number of equally spaced particles. We denote the temperature of each particle at time t as x:( ). Each particle is influenced by the heat of the neighbor on the right as well as on the left. Consequently, we have dx (21+1 - ) + (-:-1-2.) dt =4141 + 21-1-20 We will assume that the temperature on the left endpoint is fixed at one temperature To, and the right endpoint of the rod is fixed at another temperature Ty. If we have N+1 points, then the temperatures at each point are {=0, 11,...,IN-1,{n}. Furthermore, ro(t) = T, and xn = Ty. Then (+) is a system of N - 1 linear equations. Problem 1 Assume N = 5, so that (*) is a system of 4 equations. If T, = Ty = 0, and if (*) is written as x' = Ar, then write the 4 x 4 matrix A. By finding the eigenvalues and eigenvectors of A, the solution of (+) in Problem 1 can be found. To find the behavior of each r(t) as t+00, it is sufficient to find the eigenvalues of A