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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 inclass 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 initialboundary 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 initialboundary 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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 Spring '08
 GOLDBERG
 Equations

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