Practice Midterm

# Practice Midterm - Partial Diﬀerential...

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Unformatted text preview: Partial Diﬀerential Equations (Math417) March 25, 2010 [Practice] Midterm Exam Your PRINTED name is: Justify your answers as fully as possible. This is an in-class closed book exam. No textbook, no lecture notes and no calculator allowed. Time: 12:05–1:15 PM (70 Minutes) This examination booklet contains 3 problems+1 bonus problem. There are 6 sheets of papers. Indicate below the pages where you do the corresponding problems. Problem 1. Let f (x) = x2 for 0 < x < π . (1) Find the Fourier sine series of f (with [0, π ] as the basic interval). (2) Over the interval [−3π, 3π ], sketch the function to which the series converges. (3) brieﬂy describe how the answers to (2) would change if we studied the Fourier cosine series instead. Pages used: Problem 2. Solve the initial-boundary value problem for the linear wave equation 2 2 u ∂2 =∂u (0 < x < π, t > 0), ∂t ∂x2 ∂u 2 u(x, 0) = 0, ∂t (x, 0) = x (0 < x < π ), u(0, t) = u(π, t) = 0 (t > 0). You may stop when you can say “And now continue as in Problem 1, above.” (assume appropriate continuity) Pages used: Problem 3. Solve the initial-boundary value problem for the linear heat equation (assume appropriate continuity) 2 ∂u = ∂ u + (x2 − 2πx)e−t (0 < x < π, t > 0), ∂t ∂x2 ∂u −t ∂u (0, t) = 2πe , ∂x (π, t) = 0 (t > 0), ∂x u(x, 0) = 4πx − x2 (0 < x < π ). Pages used: 1 2 3 4 5 6 ...
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Practice Midterm - Partial Diﬀerential...

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