1. No category

Homework 4: Green’s functions and distributions
Math 456/556
1. Show that
Z
b
d[φ] =
φ0 (x)dx
a
is a linear mapping from D → R. What is this distribution in terms of delta
functions?
2. Recall that distributions can themselves have derivatives, simply by defining
d0 by the linear functional
Z ∞
0
d(x)φ0 (x) dx = −d[φ0 (x)].
d [φ] = −
−∞
What rule represents the derivative of the delta function δ(x − x0 )?
3. Find the distributional Laplacian (in R2 ) of f (r, θ) = ln(r) in terms of δ
functions. You need to use the definition, regarding ∆f as a mapping of some
smooth φ
Z
Z
∞
2π
f (r, θ)∆φ rdrdθ,
∆f [φ] =
0
0
and integrate by parts to see how this distribution acts on φ.
4. Consider the 1D problem
d2 u
= f (x),
dx2
u(0) = 0,
du
(L) = 0.
dx
A. Find the Green’s function G(x; x0 ) which is continuous at x = x0 , satisfies
the jump condition
lim Gx (x; x0 ) − lim− Gx (x; x0 ) = 1.
x→x+
0
x→x0
and boundary conditions
Gxx (x; x0 ) = 0 when x 6= x0 ,
G(0; x0 ) = 0, Gx (L; x0 ) = 0.
B. Write the solution u in terms of the f and the Green’s function you found.
Verify your formula for the source term f (x) = x2 .
5. Find the Green’s function for the following problem:
d2 u
2
− 2 u = f (x),
dx2
x
u(0) = 0,
lim u(x) = 0.
x→∞
(Hint: the homogeneous equation is of Euler type, so solutions are powers of
x) Write down the solution u(x) in terms of the Green’s function.