Solved: [M] The band matrix A shown below can be used to
Chapter 2, Problem 32E(choose chapter or problem)
Problem 32E
[M] The band matrix A shown below can be used to estimate the unsteady conduction of heat in a rod when the temperatures at points p1,…, p2 on the rod change with time.2
The constant C in the matrix depends on the physical nature of the rod, the distance Δx between the points on the rod, and the length of time Δt between successive temperature measurements. Suppose that for k = 0, 1, 2,…, a vector tk in ℝ4 lists the temperatures at time k Δ t . If the two ends of the rod re maintained at 0° then the temperature vectors satisfy the equation Atk + 1 = tk(k = 0, 1,,…), where
a. Find the LU factorization of A when C = 1. A matrix such as A with three nonzero diagonals is called a tridiagonal matrix. The L and U factors are bidiagonal matrices.
b. Suppose
Use the LU factorization of A to find the temperature distributions t1, t2, t3, and t4.
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