Homework 5A: Green`s functions Math 456/556

Homework 5A: Green’s functions
Math 456/556
1. (The vortex patch) The so-called streamfunction ψ : R2 → R in fluid
mechanics solves ∆ψ = ω where ω is a measure of the fluid rotation (the
velocity field is perpendicular to the gradient of ψ, by the way). A simple
model of a hurricane of radius R has
(
1
|x| < R,
ω=
.
0
|x| ≥ R.
Use the Green’s function appropriate for two dimensions to express the
solution ψ as an integral in polar coordinates. You will want to use the law
of cosines
|x − x0 | = |x0 |2 + |x|2 − 2|x||x0 | cos(ϕ)
where ϕ is the angle between vectors x and x0 . Finally, evaluate ψ(0) (you
can actually evaluate the integral in general, but its a lot more complicated!).
2. (The Helmholtz equation in 3D) We want the Green’s function G(x; x0 )
for the equation ∆u − k 2 u = f (x) in R3 , where k is a constant.
(a) Like the Laplace equation, we suppose that G only depends on the distance from x to x0 , that is G(x; x0 ) = g(r) where r = |x − x0 |. Going to
spherical coordinates, we find g satisfies the ordinary differential equation
1 2
(r gr )r − k 2 g = 0,
r2
r > 0.
The trick to solving this is the change of variables g(r) = h(r)/r. The equation for h(r) has simple exponential solutions.
(b) The constant of integration in part (a) is found just like the Laplacian
Green’s function in R3 , using the normalization condition
Z
∇x G(x; 0) · n
ˆ dx = 1,
lim
r→0 Sr (0)
where Sr (0) is a spherical surface of radius r centered at the origin. (Recall
this comes from integrating the equation for G on the interior of Sr (0) and
applying the divergence theorem).
(c) Finally, write down the solution to ∆u − k 2 u = f (x) assuming u → 0
as |x| → ∞. Write the answer explicitly as iterated integrals in Cartesian
coordinates.
3. Using the method of images, the Green’s function for the Laplacian on a
sphere |x| < a for problems with Dirichlet boundary conditions is
G(x, x0 ) = −
1
(a/r0 )
,
+
4π|x − x0 | 4π|x − a2 x0 /r02 |
r0 = |x0 |.
In the case x0 = 0, the correct formula is G(x, 0) = −1/(4π|x|) + 1/(4πa)
(a) Check that G is indeed correct: show formally that ∆G = δ(x − x0 ), at
least restricted to the domain |x| < a. Hint: recall
1
∆
= −δ(x − x0 ).
4π|x − x0 |
(b) show G(x; x0 ) = 0 if |x| = a, i.e. the Green’s function satisfies the
Dirichlet boundary condition. Hint: rewrite each norm using the law of
cosines, and then set |x| = a.
(c) It turns out that the solution to ∆u = 0 can be written in terms of the
boundary values of u by the formula
Z
u(x)∇x G(x; x0 ) · n
ˆ dx.
u(x0 ) =
|x|=a
Use this to show that that u(0) is just the average of the values of u on the
spherical surface |x| = a. This just generalizes the so-called mean value
theorem for the two-dimensional Laplace equation.
4. Find the Green’s function for ∆u = f (x, y) in the first quadrant x >
0, y > 0 where the boundary/far-field conditions are
u(x, 0) = 0,
u(0, y) = 0,
lim |∇u| = 0.
|x|→∞
(Hint: locate image sources in each quadrant, choosing their signs appropriately)