MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION A)
Name:
This test has 10 pages and 10 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. Draw the graph of
y = tan x
labelling all asymptotes and zeros. Include at least 3 periods in your graph.
What is the period of tan x?
Date: December 11, 2001.
1
2
MATH 3150: PDE FOR ENGINEERS
2.
(
x2
f (x) =
−x2
FINAL EXAM (VERSION A)
0≤x<π
−π ≤ x < 0
and f (x) has period 2π.
(a) Draw the graph of f (x).Include at least two periods in your graph.
(b) Calculate the real Fourier amplitudes am and bm of f (x). Hint:
Z
x2 sin x dx = −x2 cos x + 2 cos x + 2x 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 about 60% of the energy is stored among the amplitudes
a0 , a1 , b1 , a2 , b2 . Hint:
π 2 ∼ 10
π 4 ∼ 100
π 6 ∼ 1000.
So collect all terms over a common denominator of π 6 .
MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION A)
3
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
3. Suppose that u(x, y, t) satisfies
2 3 !
2 4 !
∂u
∂2u
∂ u
∂u
∂u
+1
= −u
+
+
− exp
2
2
∂t
∂x
∂y
∂x
∂y
(which is so horrible an equation that no one can solve it). Suppose that
the function u at time t = 0 looks like a round peak as in figure 1. Does
the top of the peak move up or down? Explain.
4
MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION A)
x
Figure 2. A circular ring, with x axis wound around it
4. Solve the heat equation
∂u
∂2u
= c2 2
∂t
∂x
in a circular ring of radius r (see figure 2) by finding a kernel K(x, t) so
that
Z
2πr
u(s, 0)K(x − s, t) ds .
u(x, t) =
0
Note that u(x, t) has period 2πr in x, so you can use complex Fourier series
(not Fourier transforms, since it isn’t infinitely long).
MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION A)
5. Find the Fourier transform of the function
(
1 if − 1 ≤ x ≤ 1
f (x) =
0 otherwise
5
6
MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION A)
6. Show that according to the heat equation, heat propagates with infinite
speed. To do this, take initial temperature u(x, 0) which is positive near
x = 0, and not negative anywhere, and zero everywhere except in a little
region near x = 0. Now apply the heat kernel
2
G(x, t) =
2
e−x /4c t
√
2c πt
in the equation
Z
∞
u(s, 0)G(x − s, t) ds
u(x, t) =
−∞
giving final temperature at time t. How does this show you that a small
amount of heat spreads out to all points of space at as small a time as you
like?
MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION A)
7
7. Take a disk of unit radius and heat the top half of the edge of the disk
to 1o and the bottom 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. Try to estimate (without making
any decimal approximations—just use fractions) the temperature at the
point
π
(x, y) = 0,
.
10
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 2π
1
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
u(r, θ) = a0 +
rm (am cos (mθ) + bm sin (mθ))
m=1
8
MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION A)
8. Suppose that I have a function
y = f (x)
which is periodic with period T . Suppose that it has energy E(f ) = kf k2 .
If I rescale the x variable by an amount α, making a new variable X = αx
and a new function F (X) = f (x),
(a) what is the period of F (X)? and
(b) by what amount do I have to rescale the y variable to keep the same
energy as f ?
MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION A)
9
9. Suppose that f (x) has period T and average value 0.
(a) Using complex Fourier series, show that the energy in the derivative
df /dx satisfies
2 2
df 2
≥ 2π
kf k .
dx T
(b) Which functions f (x) satisfy
2 2
df 2
= 2π
kf k ?
dx T
10
MATH 3150: PDE FOR ENGINEERS
FINAL EXAM (VERSION A)
10. Write down the general solution of the heat equation for temperature u(x, t)
in a wire of length L with left end at 0o and insulated right end. Explain
how you got your answer. (You don’t have to explain how to dig out the
amplitudes at time t = 0.)