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Partial Differential Equations 3150
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Midterm Exam 2
Exam Date: Monday, 22 April 2013
I
Instructions: This exam is timed for 50 minutes. You will be given extra 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.
1. (CH3. Finite String: Fourier Series Solution)
(a) [75%] Display the series formula without derivation details for the finite string problem
Utt(X,t)
u(0,t)
u(L,t)
u(x,0)
u(x,0)
=
=
=
=
(x,t),
c
u
2
0,
0,
f(x),
g(x),
0&lt;x&lt;L,
t&gt;0,
t&gt;0,
t&gt;0.
0&lt;x&lt;L,
0&lt;x&lt;L.
Symbols f arid g should not appear explicitly in the series for u(x, t). Expected in the
formula for u(x, t) are product solutions times constants.
(b) [25%] Display an explicit formula for the Fourier coefficients which contains the symbols
L, f(x), g(x).
Mryk
ijj?
4
(r.)(/ CflTTCt/L
yrrx/L)
-
11
Vc. pfor
4
Y)oyD4
Le,
(ri1rx/L- 4t (‘WrTCt/L_-
(flY))
Z
-t
?rc&plusmn;-JL-)
‘TiTC
r
r.Lrr1L)3
Cfl1Y1L’j
ALLc
-
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Name.
3150 Exam 2 S2013
2. (CH3. Heat Conduction in a Bar)
Consider the heat conduction problem in a laterally insulated bar of length 2 with one end
at 50 Celsius and the other end at zero Celsius. The initial temperature along the bar is
given by function f(z), which is a symbol used throughout the problem, devoid of a specific
formula.
zz 0&lt;x&lt;2, t&gt;0,
2
c
,
1
=
t &gt;0,
u(0,t) = 50,
0,
t&gt;0,
n(2,t)
u(x,0) = f(x), 0&lt;x&lt;2.
(x).
1
u
c
(a) [25%] Find the steady-state temperature
A
(b) [50%] Solve the bar problem with zero Celsius temperatures at both ends, but f(x)
replaced by f(x) ui (x). Call the answer u2(x, t). The answer has Fourier coefficients in
integral form, unevaluated to save time.
—
(c)
[25%] Display an answer check for the solution u(x, t)
=
ui(x) + u2(X, t).
2(
-
‘1)
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3150 Exam 2 S2013
3. (CH4. Rectangular Membrane)
Consider the general membrane problem
utt(x, y, t)
u(x, y, t)
u(x,y,0)
u(x,y,0)
=
=
=
=
c (n(x, y. t) + u(r, y, t)),
2
0
f(x,y),
g(x.y),
0 &lt; x &lt;a, 0 &lt;p &lt; b, t
on the boundary,
0&lt;x&lt;a, 0&lt;y&lt;b,
0&lt; x &lt;a, 0&lt; p &lt; b.
&gt;
f o c.)
0,
Solve the problem for a = b = c = 1, f(x,y) = 1, g(x,y) = 0. Expected are displays for the
normal mqcles, a superposition formula for n(x, y, t), and explicit numerical values
EEe generalized rF.cients.
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3150 Exam 2 S2013
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4. (CH4. Steady-State Heat Conduction on a Disk)
Consider the heat conduction problem on a disk
urr(r,0) + ur(r,6) + u&uuml;(r,6)
z(a,8) =f(O),
?‘
0
0,
0&lt; r &lt;a,
pulse(6,0,7r/2), that is, f(8)
‘
=
1 on 0
&lt;
0
&lt;
v/2,
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iW,c
[)1
l”-
0&lt;0 &lt;2,
0&lt;8 &lt;2.
Solve for u(r,9) when a 1 and f(0)
f(9) 0 on ir/2 &lt; 8 &lt; 2ir.
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