Practice Final, APMA 4200, 2011.
PROBLEM 1.
Consider the function
g
(
x
) =
x
for
x
∈
(0
, π
)
,

x
for
x
∈
(

π,
0)
.
1.1.
Show that the function is even.
1.2.
Calculate its Fourier coefficients. We recall that a 2
π
periodic function can be represented as
f
(
x
) =
a
0
+
∞
X
n
=1
a
n
cos(
nx
) +
b
n
sin(
nx
)
,
b
n
=
1
π
Z
π

π
f
(
y
) sin(
ny
)
dy,
a
0
=
1
2
π
Z
π

π
f
(
y
)
dy,
a
n
=
1
π
Z
π

π
f
(
y
) cos(
ny
)
dy.
1.3.
Is the function
g
(
x
) extended by periodicity a continuous function on the whole line
x
∈
(
∞
,
∞
)?
1.4.
Justify that you can differentiate term by term in the Fourier series expansion of
g
(
x
) and deduce the
Fourier series expansion of the function
h
(
x
) =
1
for
x
∈
(0
, π
)
,

1
for
x
∈
(

π,
0)
.
PROBLEM 2.
Consider the heat equation
∂u
∂t
=
k
∂
2
u
∂x
2
,
0
< x < π,
t >
0
∂u
∂x
(0
, t
) = 0
,
∂u
∂x
(
π, t
) = 0
,
t >
0
,
u
(
x,
0) =
x
for 0
< x < π.
(1)
2.1.
Using the method of separation of variables
u
(
x, t
) =
G
(
t
)
φ
(
x
), write the ordinary differential equations
that
G
and
φ
must satisfy (including boundary conditions for
φ
).
2.2.
Solve for
G
(
t
) knowing
G
(0).
2.3.
Solve the eigenvalue problem for
φ
(we assume to save time that
φ
00
+
λφ
= 0 with the proper boundary
conditions has no nontrivial solution when
λ <
0).
2.4.
Write down the most general solution
u
(
x, t
) that can be constructed from the above method of separation
of variables. Justify the formula.
2.5.
Solve for
u
(
x, t
) in (1) as a superposition of sines and cosines in the
x
variable.
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 Fall '11
 R
 Fourier Series, Partial differential equation, Boundary conditions

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