Name.
3150 Sample Exam2S2Ol3
Partial Differential Equations 3150
Sample Midterm Exam 2
Exam Date: Monday, 22 April 2013
Instructions: This exam is timed for 50 minutes. You will be given double time to complete the
exam. No calculators, notes, tables or books. Problems use only chapters 1 to 4 of the textbook. No
answer check is expected. Details count 3/4, answers count 1/4.
la. (CH3. Finite String: Fourier Series Solution)
(a) [75%] Display the series formula without derivation details for the finite string problem
I
j
u(x, t)
u(0,t)
u(L,t)
u(x,0)
ut(z,0)
c
u
2
(x, 1),
0,
0,
0,
g(x),
=
=
=
=
t > 0,
t>0,
t>0,
< L,
0 <
0<z<L.
(b) [25%] Display an explicit formula for the Fourier coefficients which contains the symbols L,
g(x).
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3150 Sample Exam 2 S2013
lb. (CH3. d’Alembert’s Solution: Finite String)
Let
=
f
0.3z
0.3(1—x)
0<x0/5,
0.5<x< 1.
and define g(x)
=
0 on 0< x
1. Assumed is
dAlemberts solution to the vibrating string problem on 0 < x < 1, t> 0, which is the formula
u(x,t)
Assume L
=
1 and c
(a) [25%] Define
f,
=
(f(z
-
ct) + f(z + ct)) +
1
x+ct
L
gs)ds.
1/7r.
g as appropriate periodic extensions of
f
and g, respectively.
(b) [75%] Display a piecewise-defined formula for u(z, ir/3) on 0 <x < 1.
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-
—peM
i,” os ‘
C
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rr P..
If
-,
(1÷2)
( (
,,
-
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z
-c—c
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-
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c
2
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± ((
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2=
)
t
‘
P
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-
-x
i
-
U..
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3150 Sample Exam 2 S2013
Name.
2. (CR3. Heat Conduction in a Bar)
Consider the heat conduction problem in a laterally insulated bar of length 1 with one end at
zero Celsius and the other end at 100 Celsius. The initial temperature along the bar is given by
function f(x).
u(0,t)
u(l,t)
()
o<<i, t>o,
=
Ut
U(X,0)
=
0,
100,
=
f(x),
=
t>0,
O<x<1.
(a) [25%] Find the steady-state temperature ui(z).
(b) [50%] Solve the bar problem with zero Celsius temperatures at both ends, but f(x) replacej
(x,t). The answer has Fourier coefficients in integral form,
2
Qr). Call the answer u
1
by f(x) —u
unevaluated, to save time.
(c) [25%] Explain why
(ta-)
u(x)
+ u2(x,t).
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0
‘
A
U(x,t) =
QQ4t4.t
-it,
çjL-.?
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C
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n
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2
c
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+
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L
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1A
qt)-
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