MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #3
Name:
This test has 7 pages and 5 problems.
Work out everything as far as you can before making decimal approximations. No
calculators, computers or slide rules are allowed.
1. A rectangular sheet is tied on all four sides. The top and bottom sides
have length a = 1 and the left and right sides have length b = 1. The
wave velocity is c = 1/π. Initially the sheet is flat, with initial velocity
g(x, y) = x(1 − y). Find the position u(x, y, t) of the sheet at time t.
Date: April 22, 2002.
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #3
2. The steady state of heat in a rectangular plate of width a = 1 and length
b = 1 with left bottom corner at the origin of coordinates and right top
corner at (1, 1), with temperatures f1 (x) = x at bottom, f2 (x) = x + 1 at
top, g1 (y) = y at left side and g2 (y) = y + 1 at right side is:
(a)
x2 + y 2
(b)
∞
X
(−1)m+n
sin (πmx) sin (πny)
mn
n=0
(c)
r cos θ + ir sin θ
(d)
sin(πx) sin(πy)
(e)
x+y
(f )
∞
X
n=1
rn
1
(−1)m+n
cos(nθ) +
sin(nθ)
mn
mn
(g)
1−x−y
(h)
0
(i) y
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #3
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3. Suppose that the same rectangular plate as in the previous problem has
initial temperature f (x, y) = 1o and diffusivity constant c = 1/π. Find the
temperature U (x, y, t) as a function of time. Remember that U = s + u
where s is the steady state, and u = f − s at time t = 0.
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MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #3
Figure 1. The heated disk
1
0.8
0.6
0.4
0.2
0
1
0.5
0
–1
–0.5
–0.5
0
–1 1
0.5
Figure 2. The steady state inside the disk
4. Take a disk of radius 1 and place thermostats on its edge so that the temperature is 1o on a quarter of the disk, and 0o on the other three quarters.
Align the disk so that the quarter heated to 1o is half above and half below
the x axis, with the center of the disk at the origin of the coordinates, as
in figure 1.
(a) Find the steady state temperature s(r, θ) at all points of the disk.
(b) Write down the first four terms of the expansion into modes of the
steady state.
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #3
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(c) The steady state temperature at (x, y) = (0, 1/10) is : (you can use
the approximation π ∼ 3)
(a)
0.2466
(b)
0.2792
(c)
0.1900
(d)
0.25
(e)
1
(f ) 0
(g)
− 0.001
(h)
3.14159
Hint: recall that if we write the temperature of the edge of the disk as
f (θ), and write the Fourier amplitudes of f (θ) as
Z 2π
1
a0 =
f (θ) dθ
2π 0
Z
1 2π
am =
f (θ) cos(mθ) dθ
π 0
Z
1 2π
bm =
f (θ) sin(mθ) dθ
π 0
then the steady state temperature inside the plate, in polar coordinates, is
∞
X
s(r, θ) = a0 +
rn (an cos (nθ) + bn sin (nθ)) .
n=0
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MATH 3150: PDE FOR ENGINEERS
(continued)
MIDTERM TEST #3
MATH 3150: PDE FOR ENGINEERS
MIDTERM TEST #3
5. Calculate ∇2 u where
1
u(x, y) = p
x2
+ y2
.
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