MATH
152,
FALL 2004: FINAL
There are five problems. Do all of them. Total score: 160 points.
Problem
#1,
(25
points)
For both of the following functions
f
on [0, 11, state whether the Fourier cosine series
on [0, I] converges in each of the following senses: uniformly, pointwise, in L2.
If
the Fourier series converges pointwise, state what
it
converges to for each x
E
[0,
I ] .
Make sure that you give the reasoning that led you to the conclusions.
1.
f(x)
=
x(sin(nX/L))~,
2.
f
(x)
=
0, for 0
5
x
5
112, and
f
(x)
=
1
for 112
<
x
5
1.
Problem
#2,
(25
points total)
1.
(8
points)
Find the general solution of the PDE
u,
+
2yuy
=
0
on
R,
x
Ry.
2.
(7points)
Now impose in addition that u(0, y)
=
y. Find u explicitly.
3.
(10
points)
Consider the PDE
Urn
+
2yuy
=
x
on
IR,
x
R,.
Find its general solution.
Problem
#3,
(35
points total)
Consider the differential operator
on twice differentiable functions
f
on [0,
I]
which satisfy Dirichlet boundary con
ditions
f
(0)
=
f
(1)
=
0. That is, for these functions
f ,
A
f
=
(x2 fl(x))'. Let
1

(f,
g)
=
So
f
(x)g(x) dx denote the standard inner product on functions.
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 Fall '09
 DeVille
 Fourier Series, Partial differential equation, tempered distribution

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