Question

Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2 (Hint: verify for a general component(Aar(p))j, using trigonometric identities.) (b) Verify that the functions u(p)(x) = sin(pT:r) (p 1, 2, . . .) are eigenfunctions of the continuous differential operator d/dx2 on domain [0, 1] with boundary conditions u(0 0 and (1)0. (Here, u(x) is an eigenfunction of d wth eigenvalue A if u(x)/da2Au(x), and u() satisfies the boundary conditions.) What are the eigenvalues? Compare the eigenvectors and eigenvalues for the discrete and the continuous operators and comment (we would expect the discrete and continuous eigenvalues and eigenvectors to be similar, if the discrete operator is a useful approximation of the continuous operator ...). Are the discrete and continuous eigenvalues similar for sma values of h p?

Answer 1

The given tou .29 henu 0-25 Mo iveny 1% 33 UI "

6-5) λ. 6-5 一1/9 738 ar - 62s) o 35 G. Уо 2 s _@ 一33 니3ナ)buy t l'ur- at 16 一33 1 616 1.1, Au= nqun matrix イTh au