MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION A
Name:
Work out everything as far as you can before making decimal approximations.
1. Consider d’Alembert’s solution
Z x+ct
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1
u(x, t) = (f (x − ct) + f (x + ct)) +
g(s) ds
2
2c x−ct
of the wave equation for a vibrating string, where f (x) is the odd function with
period 2L which on the interval 0 ≤ x ≤ L gives the initial position of the string,
and similarly g(x) gives the initial velocity on 0 ≤ x ≤ L and is odd and 2L periodic.
Suppose that at initial time we have a flat string with nonzero velocity. At what
later time t will the string be completely flat again? Explain how you obtain your
answer.
Date: July 16, 2001.
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION A
2. Use separation of variables to find the general solution of the heat equation in
a rectangular plate with edges of lengths a and b, and with top and bottom edges
insulated, while right and left edges are kept at 0o .
MATH 3150: PDE FOR ENGINEERS
(continued)
MIDTERM TEST #2 VERSION A
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MATH 3150: PDE FOR ENGINEERS
(continued)
MIDTERM TEST #2 VERSION A
MATH 3150: PDE FOR ENGINEERS
3.
MIDTERM TEST #2 VERSION A
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Consider a Taylor expansion
u(z) =
∞
X
σk z k
k=0
in a complex variable z = x + iy with complex coefficients (constants) σk . Calculate
∇2 u .
(Hint: write z in polar coordinates as
z = reiθ
and use the polar coordinate expression for the Laplacian
∇2 u =
1 ∂2u
∂ 2 u 1 ∂u
+
+ 2 2 .)
2
∂r
r ∂r
r ∂θ
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MATH 3150: PDE FOR ENGINEERS
(continued)
MIDTERM TEST #2 VERSION A
MATH 3150: PDE FOR ENGINEERS
4.
MIDTERM TEST #2 VERSION A
Draw a solution u(x, y) of the Laplace equation
0=
with nonzero second derivatives.
∂2u ∂2u
+ 2
∂x2
∂y
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