Math 602
Homework 8
1. Review questions.
(i) u1 , u2 , u3 , . . . form an othonormal basis for the Hilbert space H
if....(define!)
(ii) y1 , y2 , y3 , . . . form an othogonal basis for the Hilbert space H if....(define!)
(iii) If u1 , u2 , u3 , . . . form an othonormal basis for the Hilbert space H
then any f ∈ H can be written as
f = f1 u1 + f2 u2 + . . . =
∞
X
fn un
n=1
where the numbers fn are calculated from f using the formula....? and kf k2
can be found using fn using the formula....?
(iv) If y1 , y2 , y3 , . . . form an othogonal basis for the Hilbert space H
then any f ∈ H can be written as
f = c1 y1 + c2 y2 + . . . =
∞
X
cn yn
n=1
where the numbers cn are calculated
from f using the formula....?
P∞
(v) Define in which sense a series n=1 fn un converges to, and equals, f .
2. A solution u(x, t) of the heat equation ut = α uxx (or of any evolution
equation) is called stationary if it does not depend on t: u(x, t) ≡ u(x) (the
system remains in the same state for all time).
(i) Show that the stationary solutions of the heat equation are linear
functions: u(x) = mx + n.
(ii) Consider u(x, t) a solution of the heat equation for x ∈ [0, L] with the
boundary conditions u(0, t) = A, u(L, t) = B (where A, B are constants).
Find a stationary solution us (x) so that v(x, t) = u(x, t) − us (x) satisfies the
heat equation with zero boundary conditions on [0, L].
(iii) Use separation of variables (and (ii)) to find the solution of the heat
equation for x ∈ [0, L] with the boundary conditions u(0, t) = A, u(L, t) = B
and initial condition u(x, 0) = p(x).
3. Find the eigenvalues and eigenfunctions of the Sturm-Liouville problem on [−π, π] with periodic boundary conditions:
y 00 + λy = 0,
y(−π) = y(π), y 0 (−π) = y 0 (π)
4. (i) Find the Fourier series of f (x) = x on the interval x ∈ [−1, 1].
(ii) Find the sin-series of f (x) = x on the interval x ∈ [0, 1].
(iii) For which x the Fourier series converges to f (x)? What is the limit of
the Fourier series at the other points x in [−1, 1]?
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