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APPM 4/5350 Worksheet Week 7
Figure 1:
www.xkcd.com
1. What are the requirements to differentiate a Fourier sine series on
[0
,L
]
term by term?
2. What are the requirements to differentiate a Fourier series on
[

L,L
]
term by term?
3. Draw a function that satisﬁes
∇
2
u
= 0
on
[0
,a
]
when
u
(0) = 0
and
u
(
a
) = 8
.
4. Draw a function that does not satisfy
∇
2
u
= 0
on
[0
,a
]
but does satisfy
u
(0) = 0
and
u
(
a
) = 8
.
5. What is the solution to
∇
2
u
= 0
in a circle of radius
R
with
f
(
R,θ
) = 0
?
6. Draw the Fourier cosine and sine series, and their periodic extensions, of
f
(
x
) =
x
2
on
[0
,L
]
7. Prove that the solution to Laplace’s equation with
f
(
r,θ
) =
g
(
θ
)
on the boundary is unique.
8. Explain why
f
(
x
) =
g
(
y
)
∀
x,y
implies
f
(
x
) =
c
and
g
(
y
) =
c
where
c
is a constant.
9. Write down the coefﬁcients that solve
f
(
x
) =
∑
∞
n
=0
a
n
φ
n
(
x
)
if the set
{
φ
n
(
x
)
}
∞
n
=0
form an orthogonal
basis for the space that contains
f
(
x
)
.
10. Where would you need boundary conditions to solve
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This note was uploaded on 10/24/2010 for the course APPM 4350 taught by Professor Ablowitz during the Fall '08 term at Colorado.
 Fall '08
 ABLOWITZ

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