18.085 Computational Science and Engineering
Answers to Problem Set 10
April 26, 2010
TAs: Dorian Croitoru, Nan Li
4.2.1.a) We have
2 ˆ π ˆ π
1
F (x, y)e−imx e−iny dxdy
2π
−π −π
ˆ πˆ π
ˆ π
ˆ π
1
1
−imx −iny
−imx
e
e
dxdy
=
e
dx
e−iny dy.
4π 2 0 0
4π 2 0
0
cmn
=
=
´π
−imπ
otherwise, we
c00 =
e−imx dxequals π if m = 0, and 1−eim
get
−1
1/(2πim) if m odd
if m, n odd
1/4, cm0 = c0m =
for m 6= 0, and cmn = π2 mn
0
otherwise
0
otherwise
for m, n 6= 0.
´ ´
´0 ´0
π π
b) cmn = 4π1 2 0 0 e−imx e−iny dxdy + −π −π e−imx e−iny dxdy . Simi-
Since
0
larly
to a), we obtain c00 = 1/2, cm0 = c0m = 0 for m 6= 0 and cmn =
−2
π 2 mn
if m, n odd
for m, n 6= 0.
otherwise
4.2.2. Since sin mx andPsin
Pny are odd in x and y respectively, S(x, y) will
have a double sine series
bmn sin mx sin ny if S(x, y) is odd in x and y :
−S(x, y) = S(−x, y) = S(x, −y). The double sine functions are orthogonal:
ˆ π ˆ π
1
1 if k = m and l = n
(sin
kx
sin
ly)(sin
mx
sin
ny)dxdy
=
.
0
otherwise
π 2 −π −π
0
4.2.3. Using the orthogonality from the previous problem,
bmn
1
= 2
π
ˆ
π
ˆ
π
S(x, y) sin mx sin nydxdy.
−π
−π
4.2.7. FromPthe (1-dimensional) Fourier
series of the delta function we obtain
P
1
1
−im(x+y)
−imx −imy
δ(x + y) = 2π
e
=
e
.
m∈Z
m∈Z e
2π
1
1
4
4
2
2
4.2.8. x = 8 (8x − 8x
+ 1) + 2 (2x − 1) + 38 = 18 T4 + 21 T2 + 38 T0 .


x −1 ...
 −1 2x −1


4.2.10. Let Bn (x) = 
 (n by n matrix). We compute

.
.

.
−1 
−1
−1 2x
its determinant by expanding along the last row:
det Bn
=
2x det Bn−1 − (−1) det
Bn−2
0 ... − 1
0
−1
= 2x det Bn−1 − det Bn−2 .
Thus det Bn is a polynomial in x satisfying the same recurrence as Tn . Since
det B1 = x = T1 , det B2 = 2x2 − 1 = T2 , it follows that det Bn = Tn for all n.
1
4.2.20. We have
(pu0m )0 + (q + λm w)um
=
0
(pu0n )0 + (q + λn w)un
=
0
Multiplying the rst equation by un ,the second by um and then subtracting,
´b
we obtain un (pu0m )0 − um (pu0n )0 + (λm − λn )wum un = 0. Integrating, a (λm −
´b
b
λn )wum un dx = a un (pu0m )0 − um (pu0n )0 dx = [un (pu0m ) − um (pu0n )]a = 0, by
´b
the boundary conditions. Since λm 6= λn , we get a wum un dx = 0.
4.2.21. Bessel's equation: r2 B 00 + rB 0 + λr2 B = n2 B ⇒ rB” + B 0 + (λr −
2
n /r)B = 0 ⇔ (rB 0 )0 + (λr − n2 /r)B = 0 which is a Sturm-Liouville equation
with u = B, p = r, q = −n2 /r, w = r. Alternatively, one can try u = B(r/2) and
p = r2 . Legendre's equation: (1−x2 )P ”−2xP 0 +λP = 0 ⇔ ((1−x2 )P 0 )0 +λP =
0 is a Sturm-Liouville equation with u = P, p = 1 − x2 , w = 1, q = 0.
2