A system of n masses is connected to a top support by n unit springs of length ∆.x = 1/n, so that the overall length of the mass-spring chain is ℓ = 1. Let K = AT A, where A is the rescaled incidence matrix (11.4). and let G = K-1/∆x.
(a) Explain why the matrix entries gij ≈ G(xi, xj) should approximate the sampled values of the Green's function (11.70) for the continuum limit boundary value problem with c = 1.
(b) Test this expectation for n = 5. 10, 20. How close is the matrix G to the sampled Green's function?