MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION B
Name:
Work out everything as far as you can before making decimal approximations.
1. Consider the general solution of the wave equation for a vibrating string with
fixed ends:
∞
πnx X
u(x, t) =
sin
(bn cos (λn t) + b∗n sin (λn t))
L
n=1
where
πcn
.
L
Suppose that at initial time we have initial position f (x) = 0 (a flat string). At
what later time t will the string be completely flat again?
λn =
Date: July 16, 2001.
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION B
2.
Use separation of variables to find the general solution of the heat equation
2
∂u
∂ u ∂2u
= c2
+
∂t
∂x2
∂y 2
in a rectangular plate with edges of lengths a and b, and with top, bottom and
right edges insulated, while the left edge is kept at 0o . Hint: recall the solutions of
the harmonic oscillator type equation
1 d2 h
=λ
h dt2
√ √ 

A
cos
λt
+
B
sin
λt


h(t) = A + Bt


A cosh p|λ|t + B sinh p|λ|t
−
are
if λ > 0
if λ = 0
if λ < 0
where A and B can be chosen to be any constants. Also recall the identities about
cosh and sinh:
d
cosh x = sinh x
dx
d
sinh x = cosh x
dx
sinh 0 = 0
cosh 0 = 1
and the fact that cosh x is positive for every value of x.
MATH 3150: PDE FOR ENGINEERS
(continued)
MIDTERM TEST #2 VERSION B
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MATH 3150: PDE FOR ENGINEERS
(continued)
MIDTERM TEST #2 VERSION B
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION B
5
3.
Recall that the steady state of the heat equation in a disk of radius R with
edge temperature f (θ) is
∞ n
X
r
u(r, θ) = a0 +
(an cos (nθ) + bn sin (nθ)) .
R
n=1
Find the total heat
ZZ
Q=
ZZ
u(x, y) dx dy =
u(r, θ) r dr dθ
where the integral is carried out over the whole disk.
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #2 VERSION B
1
0.8
0.6
0.4
0.2
–1
–0.5
y0
0.5
1
1
0.5
0
x
–0.5
–1
Figure 1. A round peak
4.
Looking at the heat equation in polar coordinates
2
∂u
∂ u 1 ∂u
1 ∂2u
= c2
+
+
∂t
∂r2
r ∂r
r2 ∂θ2
and recalling first and second derivatives from calculus, explain mathematically
why a round peak in the graph of u (as in figure 1) representing a blob of heat will
be pushed downward (will cool).