MATH 519 Homework
Fall 2013
41. For f, g ∈ L2 (0, π) find a solution of the PDE problem
ut = uxx + g(x) 0 < x < π
u(0, t) = u(π, t) = 0
u(x, 0) = f (x)
t>0
t>0
0<x<π
in the form of a time dependent Fourier sine series
u(x, t) =
∞
X
bn (t) sin (nx)
n=1
The coefficients bn (t) should be expressed explicitly in terms of the corresponding Fourier
sine coefficients of f and g.
P
42. If f ∈ H 1 (T) and cn is the n’th Fourier coefficient of f , show that ∞
n=−∞ |cn | is
convergent and so the Fourier series of f is uniformly convergent.
43. Construct a test function φ ∈ C0∞ (R) with the following properties: 0 ≤ φ(x) ≤ 1
for all x ∈ R, φ(x) ≡ 1 for |x| < 1 and φ(x) ≡ 0 for |x| > 2. (Suggestion: think about
what φ0 would have to look like.)
44. Show that
T (φ) =
∞
X
φ(n) (n)
n=1
defines a distribution, T ∈ D0 (R). (As usual, φ(n) means the n’th derivative of φ.)
45. Let λn > 0, λn → +∞ and set
fn (x) = sin λn x
gn (x) =
sin λn x
πx
a) Show that fn → 0 in D0 (R) as n → ∞.
b) Show that gn → δ in D0 (R) as n → ∞.
R∞
(You may use without proof the fact that the value of the improper integral −∞
π.)
sin x
x
dx =