APPM 4350/5350: Fourier Series and Boundary Value Problems
Homework #3 Monday, September 20, 2010
Due: Friday, October 8, 2010 — Note date due!
1.
(a) Given the equation for
y
=
y
(
x
):
x
2
d
2
y
dx
2
+ (2
b
+ 1)
x
dy
dx
+
cy
= 0
where
b, c
are real constants and
b
2
> c
.
(b) Transform the equation from independent variable
x
to
z
where
x
=
e
z
.
Hint: the
equation should have constant coefficients.
(c) Solve for the general solution
y
=
y
(
z
) in terms of real functions.
(d) Transform the solution back to
y
=
y
(
x
)
.
(e) Show how to get the same solution
y
=
y
(
x
) by finding special solutions of the
form:
y
s
(
x
) =
x
p
and then finding
p
.
(f) Put the solution in real form if
b
2
< c
; find the solution if
b
2
=
c
.
2. In this problem you will solve for the exterior ideal flow around a circle; this problem
was discussed in class.
However here you will use the velocity potential:
u
=
∇
φ
where
u
is the velocity of the flow. The problem is posed as follows. Find the most
general solution
φ
=
φ
(
r, θ
) where
φ
satisfies
∇
2
φ
= 0
,
∂φ
∂r
(
r
=
R, θ
) = 0
,
φ
→
Ux
=
Ur
cos
θ
as
r
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 Fall '08
 ABLOWITZ
 Fourier Series

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