We consider the approximations to the eigenvalues and eigenfunctions of the one-

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We consider the approximations to the eigenvalues and eigenfunctions of the one-dimensional Laplace Operator L(u) = -d62 /dx^2 on the unit interval [0, 1] with boundary u(0) = u(1) = 0. A scalar lambda is an eigenvalue of L if there exists a twice -differentiable function u s. t. -u"(x) = lambda u(x) on [0, 1] with u(0) = u(1) = 0. In this case u is said to be an eigenfunction of L corresponding to the eigenvalue lambda. This continuous problem can be approximated by the discrete eigenvalue problem h^-2 T_N u = lambda u Where we have set T = [2 -1 0 -1 -1 0 -1 2], u = [u_1 u_2 u_n], with u_i = u(x_i). It can be shown that the N times N matrix T_N has eigenvalues v_j = 2(1 - cos xj/N + 1) for j = 1: N, corresponding to the eigenvectors u_j (k) = squareroot 2/N + 1 sin (jk pi/N + 1) is the kth entry in u_j. Notice that the eigenvectors are normalized w.r.t. the 2-norm. Also notice that eigenvalues of T_n lie in the interval (0, 4). Hence, the eigenvalues of h^-2 T_N lie in the interval (0, 4(N + 1)^2). (a) Express the spectral condition number kappa = lambda_N/lambda_1 of T_N (which is the same as the condition number of h^-2 T_N) as a simple function of N for N rightarrow infinity. Describe the conditioning of the T_N. (b) Use Matlab command eig to find the eigenvalues of T_N for N = 5. (c) Use Matlab to plot the eigenvalues of T_N for N = 21 (from smallest to largest). (d) Again using N = 21, use Matlab to plot the eigenvectors u_j of T_N for j = 1, 2, 3, 5, 11, 21. The plot should be of the form (k, u_j (k)) where k = 1: 21. Note that as j increases, the behavior of the eigenvectors (approximate eigenfunctions) becomes increasingly oscillatory: these eigenfunctions are often called "high energy modes".

Explanation / Answer

c0 = 2.4*1000; c1 = 67*1000; c2 = 58*1000; c3 = 57*1000; c4 = 50*1000; c5 = 38*1000; k0 = 1200*1000; k1 = 33732*1000; k2 = 29093*1000; k3 = 28621*1000; k4 = 24954*1000; k5 = 19059*1000; m1 = 6800; m2 = 5897; m3 = 5897; m4 = 5897; m5 = 5897; m6 = 5897; Mmat = [m1 0 0 0 0 0; 0 m2 0 0 0 0; 0 0 m3 0 0 0; 0 0 0 m4 0 0; 0 0 0 0 m5 0; 0 0 0 0 0 m6]; Kmat = [(k0+k1) -k1 0 0 0 0 -k1 (k1+k2) -k2 0 0 0 0 -k2 (k2+k3) -k3 0 0 0 0 -k3 (k3+k4) -k4 0 0 0 0 -k4 (k4+k5) -k5 0 0 0 0 -k5 k5]; A = Kmat/Mmat; %Calculate eigenvalues and eigenvectors of A [V,D] = eig(A); %Check using det det(A-D(1,1)*eye(6)) det(A-D(2,2)*eye(6)) det(A-D(3,3)*eye(6)) det(A-D(4,4)*eye(6)) det(A-D(5,5)*eye(6)) det(A-D(6,6)*eye(6))

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