Problem 16.5
Find the Fourier cosine series of the function
f
(
x
) =
x,
0
≤
x
≤
π
2
π

x,
π
2
≤
x
≤
π
Problem 16.6
Find the Fourier cosine series of
f
(
x
) =
x
on the interval [0
, π
]
.
Problem 16.7
Find the Fourier sine series of
f
(
x
) = 1 on the interval [0
, π
]
.
Problem 16.8
Find the Fourier sine series of
f
(
x
) = cos
x
on the interval [0
, π
]
.
Problem 16.9
Find the Fourier cosine series of
f
(
x
) =
e
2
x
on the interval [0
,
1]
.
Problem 16.10
For the following functions on the interval [0
, L
], find the coefficients
b
n
of
the Fourier sine expansion.
(a)
f
(
x
) = sin
(
2
π
L
x
)
.
(b)
f
(
x
) = 1
(c)
f
(
x
) = cos
(
π
L
x
)
.
126
SECOND ORDER LINEAR PARTIAL DIFFERENTIAL EQUATIONS
Problem 16.11
For the following functions on the interval [0
, L
], find the coefficients
a
n
of
the Fourier cosine expansion.
(a)
f
(
x
) = 5 + cos
(
π
L
x
)
.
(b)
f
(
x
) =
x
(c)
f
(
x
) =
1
0
< x
≤
L
2
0
L
2
< x
≤
L
Problem 16.12
Consider a function
f
(
x
)
,
defined on 0
≤
x
≤
L,
which is even (symmetric)
around
x
=
L
2
.
Show that the even coefficients (
n
even) of the Fourier sine
series are zero.
Problem 16.13
Consider a function
f
(
x
)
,
defined on 0
≤
x
≤
L,
which is odd around
x
=
L
2
.
Show that the even coefficients (
n
even) of the Fourier cosine series are zero.
Problem 16.14
The Fourier sine series of
f
(
x
) = cos
(
πx
L
)
for 0
≤
x
≤
L
is given by
cos
πx
L
=
∞
X
n
=1
b
n
sin
nπx
L
,
n
∈
N
where
b
1
= 0
,
b
n
=
2
n
(
n
2

1)
π
[1 + (

1)
n
]
.
Using termbyterm integration, find the Fourier cosine series of sin
(
πx
L
)
.
Problem 16.15
Consider the function
f
(
x
) =
1
0
≤
x <
1
2
1
≤
x <
2
(a) Sketch the even extension of
f.
(b) Find
a
0
in the Fourier series for the even extension of
f.
(c) Find
a
n
(
n
= 1
,
2
,
· · ·
) in the Fourier series for the even extension of
f.
(d) Find
b
n
in the Fourier series for the even extension of
f.
(e) Write the Fourier series for the even extension of
f.
17 SEPARATION OF VARIABLES FOR PDES
127
17 Separation of Variables for PDEs
Finding analytic solutions to PDEs is essentially impossible.
Most of the
PDE techniques involve a mixture of analytic, qualitative and numeric ap
proaches. Of course, there are some easy PDEs too. If you are lucky your
PDE has a solution with separable variables. In this chapter we discuss the
application of the method of separation of variables in the solution of PDEs.
17.1 Second Order Linear Homogenous ODE with Con
stant Coefficients
In this section, we review the basics of finding the general solution to the
ODE
ay
00
+
by
0
+
cy
= 0
(17.1)
where
ab,
and
c
are constants. The process starts by solving the
character
istic equation
ar
2
+
br
+
c
= 0
which is a quadratic equation with roots
r
1
,
2
=

b
±
√
b
2

4
ac
2
a
.
We consider the following three cases:
•
If
b
2

4
ac >
0 then the general solution to (
17.1
) is given by
y
(
t
) =
Ae

b

√
b
2

4
ac
2
a
t
+
Be

b
+
√
b
2

4
ac
2
a
t
.
•
If
b
2

4
ac
= 0 then the general solution to (
17.1
) is given by
y
(
t
) =
Ae

b
2
a
t
+
Bte

b
2
a
t
.
•
If
b
2

4
ac <
0 then
r
1
,
2
=

b
2
a
±
i
√
4
ac

b
2
2
a
and the general solution to (
17.1
) is given by
y
(
t
) =
Ae

b
2
a
t
cos
√
4
ac

b
2
2
a
t
+
Ae

b
2
a
t
sin
√
4
ac

b
2
2
a
t.
128
SECOND ORDER LINEAR PARTIAL DIFFERENTIAL EQUATIONS
17.2 The Method of Separation of Variables for PDEs
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 Fall '14
 FundaAkleman
 Differential Equations, Equations, Partial Differential Equations, Partial differential equation, PDEs, Theory of Partial Diﬀerential Equation