Partial differential equations (ACM30220) Assignment 3

Partial differential equations (ACM30220)
Assignment 3
Issue Date: 4 November 2014
Due Date: 14 November 2014
1. Using d’Alembert’s formula, or any other means, solve the wave equation
utt = uxx ,
−∞ < x < ∞,
subject to the initial conditions
u(x, t = 0) = f (x) = 0,
(
1 |x| ≤ L,
ut (x, t = 0) = g(x) =
.
0 otherwise
[5 marks]
2. Solve the wave equation
1 ∂ 2u
∂ 2u ∂ 2u
=
+
c2 ∂t2
∂x2 ∂y 2
in two space dimensions (x, y) ∈ (0, L) × (0, L), subject to the following boundary
conditions:
u(x, y = 0) = u(x, y = L) = 0,
u(x = 0, y) = u(x = L, y) = 0,
and the following initial conditions:
u(x, y, t = 0) = f (x, y),
ut (x, y, t = 0) = g(x, y).
Hints: Attempt a separable solution u(x, y, t) = X(x)Y (y)T (t) and show that
u(x, y, t) =
=
∞ X
∞
X
n=1 m=1
∞ X
∞
X
n=1 m=1
where
Xn (x)Ym (y)Tnm (t),
sin
nπx L
sin
mπy L
[Anm cos (knm ct) + Bnm sin (knm t)] ,
π√ 2
n + m2 .
L
and Bnm by double integration.
knm =
Determine the constants Anm
1
[10 marks]
Partial differential equations
The wave equation
3. Consider Bessel’s equation of integer order,
r2
d2 y
dy
+ r + (k 2 r2 − m2 )y = 0,
2
dr
dr
m ∈ Z.
(1)
The general solution to this equation is
ym (r) = αJm (kr) + βKm (kr),
(2)
where Jm (·) and Km (·) are known power series. The function Jm (·) is well behaved
for all values of its argument, while Km (·) has a singularity at zero.
(a) Consider solutions ym (r) to Bessel’s equation in an interval R1 < r < R2 ,
such that
ym (R1 ) = ym (R2 ) = 0.
(3)
Argue that the condition (3) implies that
αm Jm (kR1 ) + βm K(kR1 ) = 0,
αm Jm (kR2 ) + βm K(kR2 ) = 0
and hence Jm (kR1 )K(kR2 ) − K(kR1 )Jm (kR2 ) = 0. Argue further that this
gives rise to a (possibly infinite) set of eigenvalues
kmn : Jm (kmn R1 )K(kmn R2 ) − Km (kmn R1 )Jm (kmn R2 ) = 0,
p
2 + β 2 = 1.
with associated eigenvectors (αmn , βmn ), with αmn
mn
(b) Let
Imn (r) = αmn Jm (kmn r) + βmn Km (kmn r).
Argue that the general solution to Bessel’s equation (1) with boundary conditions (3) is
X
ym (r) =
Cn Imn (r),
n
where the Cn ’s are arbitrary constants.
(c) Demonstrate that the Imn functions are orthogonal, in the sense that
Z
R2
rImn (r)Imn′ (r) dr = 0,
R1
n 6= n.
2
Partial differential equations
The wave equation
Hints for (c):
• Consider
dImn
d2 Imn
2
+r
+ (kmn
r2 − m2 )Imn = 0,
2
dr
dr
2
dImn′
2 d Imn′
2
2
2
r
+
r
+ (kmn
′ r − m )Imn′ = 0.
2
dr
dr
r2
(4a)
(4b)
Take [Eq. (4a)]Imn′ -[Eq. (4b)]Imn and show that it can be re-written as
dImn
dImn′
d
Imn′
− Imn
r
dr
dr
dr
dImn
dImn′
2
2
+ r Imn′
− Imn
+ r2 (kmn
− kmn
′ )Imn Imn′ = 0.
dr
dr
2
• Show further that this can be re-written as
dImn′
dImn
d
2
2
− Imn
r Imn′
+ r(kmn
− kmn
′ )Imn Imn′ = 0.
dr
dr
dr
• Integrate the result from r = R1 to r = R2 and use the fact that Imn (R1 ) =
Imn (R2 ) = 0.
[10 marks]
4. Solve the wave equation
1 ∂ 2u
= ∇2 u
c2 ∂t2
in two dimensions, in polar coordinates, with
∂
1 ∂2
1 ∂
2
r
+ 2 2,
∇ =
r ∂r
∂r
r ∂ϕ
in the annulus R1 < r < R2 , and subject to the following boundary and initial
conditions:
u(R1 , ϕ, t) = u(R2 , ϕ, t) = 0,
u(r, ϕ, 0) = f (r, ϕ),
ut (r, ϕ, 0) = g(r, ϕ).
[5 marks]
3

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