Math 6720
Appl. Complex Var. & Asymptc Mthds
Alexander Balk
NAME: .....................................................................
HW11
due Wednesday, 4/13/2016
1. Orszag’s 2/3 Rule. Suppose we represent unknown functions by Fourier expansions
M
X
u(x) =
uk ei k
2π
x
T
,
v(x) =
k=−M
M
X
vk e i k
2π
x
T
k=−M
with 2M + 1 Fourier coefficients. When our equation is nonlinear, we need to compute convolutions
X
wk =
up vq ,
p+q=k
where the sum is over all integers p, q ∈ [−M, M ], such that p + q = k. We can do this computationally fast,
via the FFT:
(i) Choose sufficiently big N being a power of 2 (later we will see how big it should be)
and pad Fourier coefficients vectors with zeros, so that the resulting vectors have length N

0
..
.




 0

 u−M


F =  ...

 u
 M

 0
 .
 .
 .





















0
..
.








 0 


 v−M 




G =  ... 


 v

 M 


 0 
 . 
 . 
 . 


 0,
− N2 ≤ k ≤ −M − 1,
uk ,
−M ≤ k ≤ M,
Fk =

 0, M + 1 ≤ k ≤ N − 1;
2
0


 0,
− N2 ≤ k ≤ −M − 1,
vk ,
−M ≤ k ≤ M,
Gk =

 0, M + 1 ≤ k ≤ N − 1;
2
0
then make two IFFTs
N
−1
2π
1 2X
Fk ei k N n ,
fn =
N
N
k=−
N
−1
2π
1 2X
G k ei k N
gn =
N
N
k=−
2
(ii) Calculate the point-wise product
(iii) Make the FFT
n
,
n = 0, 1, . . . , N − 1.
2
hn = fn gn .
Hk =
N
−1
X
hn e−i k
2π
n
N
.
k=0
QUESTIONS:
(a) Show that the calculation (i)-(ii)-(iii) results in
1
Hk =
N
X
Fp Gq .
p+q≡k(modN )
so that the sum is not just (linear) convolution, but includes all aliases. The idea is to choose N sufficiently
large so that all aliases vanish.
(b) Suppose we want to find all wk , k = −2M, −2M + 1. . . . , 2M . How big should we choose N ?
What is the relation between wk and Hk ?
Suggestion: We want to elliminate all aliases p + q − k ≡ 0(modN ), besides the main term, with p + q − k = 0.
p, q ∈ [−M, M ], k ∈ [−2M, 2M ].
(c) Suppose we want to find only Fourier coefficients wk of the same order: k = −M, −M + 1, . . . , M ,
as in the expansions for u and v. How big should we choose N ?
Suggestion: Now all three p, q, k ∈ [−M, M ].
(Aliasing would still occur for modes with k > |M |, but they are usually absent from our approximation anyway.)
2. Solve Laplace’s equation
φxx + φyy = 0
in the domain
D = {z : |z − 1| > 1 & |z − 2| < 2}
subject to the boundary condition
φ(x, y) = a when |z − 1| = 1
and
φ(x, y) = b when |z − 2| = 2
(a and b are real parameters).
Suggestion: Conformally transform the domain between the circles onto a domain between two parallel straight lines and use new
symmetry to reduce the PDE to an ODE.
3. Solve Laplace’s equation
φxx + φyy = 0
in the domain
D = {z : |z| < 1 & |z − i| <
√
2}
subject to the boundary condition
φ(x, y) = a on the ark of the circle |z| = 1
(a and b are real parameters).
Suggestion: Explore the maps w =
z−1
z+1
and ζ = ln w.
and
φ(x, y) = b on the ark of the circle |z − i| =
√
2
4. (a) A function f (x) has two asymptotic expansions as x → x0 over the same asymptotic sequence {φn (x)}
(n = 0, 1, 2, . . .)
f (x) ∼
∞
X
an φn (x),
f (x) ∼
n=0
∞
X
bn φn (x)
n=0
(x is a real variable, an and bn are real constant coefficients). Is it true that an = bn ?
(b) Two functions f (x) and g(x) have the same asymptotic expansion as x → x0 over the same asymptotic
sequence {φn (x)} (n = 0, 1, 2, . . .)
f (x) ∼
∞
X
n=0
an φn (x),
g(x) ∼
∞
X
an φn (x)
n=0
(x is a real variable, an are real constant coefficients). Is it true that f (x) = g(x) in some neighborhood of x0 ?
5. Consider the exponential integral
Ei(x) =
Z ∞ −t
e
x
t
dt.
(a) Find its asymptotic expansion as x → ∞, using integration by parts.
Suggestion: Using a few integrations by parts guess the formula
e−x
Ei(x) =
x
(
(−1)N −1 (N − 1)!
1
2
1 − + 2 − ... +
x x
xN −1
)
+ RN (x).
with the reminder
N
RN (x) = (−1) N !
Z ∞ −t
e
x
tN +1
dt.
Prove this formula integrating by parts one more time to find the same formula with N replaced by N + 1.
Then prove that your representation is an asymptotic expansion.
(b) What is the asymptotic sequence in this case?
(c) Is the expansion convergent?
(d) Use table 10.1 in Keener’s textbook (page 441) to find the optimal number of terms N̄ (x) and the corresponding
accuracies ¯(x) for x = 2, 3, 5.