Partial Differential Equations: Exercise Set 8
1. Find the Fourier series of the function f (x) = | sin x| (x ∈ R), and
plot some partial sums of the series.
2. Solve the heat equation ut = kuxx in (0, l) with boundary conditions
u(0, t) = u(l, t) = 0 and initial condition
3x
0 < xl < 23
2l
u(x, 0) =
3 − 3 xl 23 < xl < 1
3. Solve the wave equation utt = c2 uxx in (0, l) with boundary conditions
u(0, t) = u(l, t) = 0 and initial conditions
3x
0 < xl < 23
2l
u(x, 0) =
3 − 3 xl 23 < xl < 1
and ut (x, 0) = 0.
4. Solve the heat equation ut = kuxx in x ∈ (0, l) with boundary conditions u(0, t) = 0, ux (l, t) = 0 and initial condition u(x, 0) = x/l.
5. What is the steady-state solution of the heat equation problem in the
example of section 8.5 of the notes? What is its Fourier sine series?
6. Solve utt = c2 uxx + g(x) sin ωt in x ∈ (0, l) with boundary conditions
u(0, t) = u(l, t) = 0 and initial conditions u(x, 0) = 0, ut (x, 0) = 0.
You may assume that ω is not an integer
P multiple of πc/l. Hint: assume a solution of the form u(x, t) = m um (t)Xm (x) and obtain the
decoupled ODEs
1
(f (t), Xn ) = u00n (t) + λn un (t),
kXn k2
|
{z
}
fn (t)
each of which (with initial conditions un (0) = u0n (0) = 0) has the
solution
Z t
p
1
un (t) = √
fn (τ ) sin( λn (t − τ )) dτ.
λn 0
Answers
1.
3.
2
π
+
P
4
π
P
2 2
2
cos 2nx 2. n≥1 n29π2 sin 2nπ
sin nπx
e−n π kt/l
3
l
nπ(x+ct)
nπ(x−ct)
2nπ
sin 3 sin
+ sin
l
l
1
n≥1 1−4n2
P
9
n≥1 2n2 π 2
2(−1)n
n≥0 (n+ 1 )2 π 2
2
1 2 2
2
sin((n + 21 )πx/l)e−(n+ 2 ) π kt/l
P
2(1−(−1)n )
x
5. u(x, ∞) = 2k
(l − x) = ∞
n=1 kn3 π 3 /l2 sin(nπx/l)
R
P
(nπc/l) sin(ωt)−ω sin(nπct/l)
l
2 l
6. n≥1 nπc
g(ξ)
sin(nπξ/l)
dξ
sin(nπx/l)
l 0
(nπc/l)2 −ω 2
4.
P