ps5 - Problem Set 5 February 16, 2007 ACM 95b/100b Niles A....

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Problem Set 5 February 16, 2007 Due February 23, 2007 ACM 95b/100b 3pm in Firestone 303 Niles A. Pierce (2 pts) Include grading section number 1. (3 × 8 pts) Solve each of the following boundary value problems a) y 00 + y = 0 , y (0) = 0 , y 0 ( π ) = 1 b) y 00 + y = 0 , y 0 (0) = 1 , y ( L ) = 0 c) y 00 + 4 y = sin x, y (0) = 0 , y ( π ) = 0 2. (15 pts) Consider the eigenvalue problem y 00 + λy = 0 , y (0) = 0 , y ( L ) = 0 . Show that there are no complex eigenvalues. 3. (15 pts) Find the Fourier series for the function f ( x ) = H ( x ) on the interval x [ - π,π ], where H ( x ) is the Heavyside step function. Plot a sequence of partial sums and observe that the maximum overshoot does not tend to zero as the number of terms is increased. This behavior is known as Gibb’s phenomenon. Use your plots to estimate the approximate magnitude of the maximum overshoot as the number of terms increases to infinity. What happens to the location of the maximum overshoot as the number of terms increases? 4. (3
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ps5 - Problem Set 5 February 16, 2007 ACM 95b/100b Niles A....

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