APPM 4/5350 Worksheet Week 7
Figure 1:
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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
∇
2
u
= 0
inside the domain
r
∈
[0
, R
]
and
θ
∈
[
π
4
,
7
π
4
]
11. Where would you need boundary conditions to solve
δu
δt
=
k
∇
2
u
inside the domain
x
∈
[0
, b
]
and
y
∈
[0
,
∞
)
12. What is
L
(
c
1
u
1
+
c
2
u
2
)
, given that
L
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 Fall '08
 ABLOWITZ
 Fourier Series, Periodic function, Partial differential equation, δu δx

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