MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION D)
Name:
This test has 9 pages and 8 problems.
No calculators are allowed. You are permitted one sheet of notes with writing
on both sides. Work out everything as far as you can before making decimal
approximations.
1. Consider the heat equation in a wire whose diffusivity varies over time:
∂u
∂2u
= k(t) 2
∂t
∂x
where k(t) is some positive function of time. Assume the wire is infinitely
long in both directions.
(a) Find the general solution to the heat equation in the form of convolution
u(x, t) = G(x, t) ∗ u(x, 0)
and
Rt
(b) write out explicitly the function G(x, t) in terms of 0 k(t) dt and x.
Hint: the ordinary differential equation
dZ
= A(t)Z(t)
dt
has solution
Rt
Z(t) = e 0 A(t) dt Z(0).
(c) Calculate G(x, t) for k(t) = e−t an exponentially decaying diffusivity.
Date: May 9, 2002.
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MATH 3150: PDE FOR ENGINEERS
(continued)
FINAL EXAM (VERSION D)
MATH 3150: PDE FOR ENGINEERS
2.
FINAL EXAM (VERSION D)
3
(
x
0≤x<π
f (x) =
−x −π ≤ x < 0
and f (x) has period 2π.
(a) Draw the graph of f (x), including at least two periods in your graph.
(b) Calculate the real Fourier amplitudes am and bm of f (x). Hint:
Z
x cos x dx = cos x + x sin x .
(c) Find the energy of f (x).
(d) In the form of a fraction, find the total percentage of energy contained
among the amplitudes a0 , a1 , b1 , a2 , b2 . Do not use any decimal approximations.
(e) Show that over 95% of the energy is stored among the amplitudes
a0 , a1 , b1 , a2 , b2 . Hint: π 4 ∼ 97.
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MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION D)
3. True or false: the Fourier transform of the function
(
e−|x| if − 1 ≤ x ≤ 1
f (x) =
0
otherwise
is
fˆ(ω) =
1
.
1 + ω2
MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION D)
5
4. Calculate the kernel K(x, t) for the heat equation with convection
∂u
∂2u
∂u
=k
+ c2 2
∂t
∂x
∂x
where c and k are constants, so that the solution of the heat equation with
convection will be
Z ∞
1
u(s, 0)K(x − s, t) ds.
u(x, t) = √
2π −∞
It might help to know the integral:
Z ∞
2
2
1
1
√
e−ap eipx dp = √ e−x /4a
2a
2π −∞
where a is any constant.
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MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION D)
5. Expand the function
f (x) = arctan x
in a Taylor expansion to four terms about x = 0. Hint:
1
f 0 (x) =
= 1 − x2 + x4 − x6 + . . .
1 + x2
(a geometric series), and the Taylor expansion is
f (x) = f (0) +
f 0 (0)
f 00 (0) 2
x+
x + ···
1!
2!
MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION D)
7
6. Take a disk of unit radius and heat the right half of the edge of the disk to 1o
and the left half of the edge to 0o . Let the disk sit with these temperatures
on its edges for a long time, until it reaches a steady state. Center the disk
at origin of coordinates. Draw a graph of u as a function of radius r at
θ = 0, from r = 0 to r = 1, clearly labelling the maximum and minimum
values on the vertical axis. 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 π
1
a0 =
f (θ) dθ
2π −π
Z
1 π
am =
f (θ) cos(mθ) dθ
π −π
Z
1 π
bm =
f (θ) sin(mθ) dθ
π −π
then the steady state temperature inside the plate, in polar coordinates, is
∞
X
u(r, θ) = a0 +
rm (am cos (mθ) + bm sin (mθ))
m=1
Hint: now compare with the last problem.
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MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION D)
7. Take a function f (x) with period T and form the function g(x) = f (ax) for
some constant a. The period of g(x) is
(a) aT
(b) T /a
(c) 2π
(d) 2πa/T
(e) none of the above.
The energy of g(x) is related to the energy of f (x) by
(a) E(g) = E(f )
(b) E(g) = a2 E(f )
(c) E(g) = aE(f )
(d) E(g) = a1 E(f )
(e) E(g) = a12 E(f )
(f ) none of the above.
MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION D)
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8. (a) Suppose that u(x, t) satisfies the wave equation
2
∂2u
2∂ u
=
c
∂t2
∂x2
with u = 0 at x = 0 and x = L at all times t. Why is it true that
∂ ∂u ∂u
∂ 2 u ∂u ∂u ∂ 2 u
=
+
?
∂x ∂x ∂t
∂x2 ∂t
∂x ∂x∂t
(b) Show that u(x, t) has constant (in time) energy, where the energy of
u(x, t) is defined to be
2
2 !
Z
1 L
∂u
∂u
+ c2
dx.
E=
2 0
∂t
∂x
2
Hint: differentiate in t, use the wave equation to get rid of ∂∂t2u . Do
not use any Fourier series or transforms. Do not separate variables.
Warning: this notion of energy is different from the one we studied for
Fourier series.