Homework 7, due October 29 Math 4041, Fall 2009
In the following problems we let a,b > 0and µk=kπ/a,k= 1,2, . . . .
1. Let
qk(y) = −eµk(y−b)−e−µk(y−b)
eµkb−e−µkb, k ∈N.
Fix some such kand define
u(x, y) = qk(y) sin πk
ax.
Verify that ∆u= 0 and that
u(x, 0) = sin πk
ax, u(x, b) = 0 for 0 < x < a,
u(0, y) = u(a, y) = 0 for 0 < y < b.
2. For qkas in the previous problem show:
a) For all δwith 0 < δ < b/2 there are C0and C1such that
for all k:|q0
k(y)| ≤ C0e−C1kif δ < y < b −δ
b) Same kind of estimate, but for q00
k(y).
3. Let a > 0, let
u0(x) = (xif 0 ≤x≤a/2
0 if a/2< x ≤a.
Compute the integrals
ck=2
aZa
0
u0(s) sin πk
asds, k ∈N
and verify:
a) P∞
k=1 |ck|does not converge;
b) P∞
k=1 |ck|2converges.
You may resort to quoting results from your calculus book.
4. Let a,b > 0 and let u0be the function of the previous problem. Find the solution
of
∆u= 0 in 0 < x < a, 0< y < b,
u(x, 0) = u0(x, 0), u(x, b) = 0 for 0 < x < a,
u(0, y) = u(a, y) = 0 for 0 < y < b.
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