Homework 5 Math 456/556 Question 1 Consider the boundary value problem uxx = f (x), u(0) = A, ux (1) = B, (a)

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Homework 5
Math 456/556 Question 1 Consider the boundary value problem
uxx = f (x), u(0) = A, ux (1) = B, (a) Find a solution in terms of the Green’s function (see previous homework!), using the Green’s formula
1 uv − vu dx = [uv − vu ]1 .
0
0 Write your answer in terms of known functions rather than just the generic G(x; x0 ) notation.
(b) Check your result with u(x) = 1 − x so that f ≡ 0. Question 2 Find a solution to
uxx − u = f (x), u (0) = A, lim u(x) = 0, x→∞ in terms of the appropriate Green’s function, using the Green’s formula
∞ u(v − v) − v(u − u) dx = [uv − vu ]∞ .
0 0 Recall that the free-space Green’s function was G∞ = − 1 exp(−|x − x0 |).
2 Question 3 (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
ω= 1
0 |x| < R,
.
|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 it’s a lot more complicated!). Question 4 (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 and far field condition limx→∞ u(x) = 0, where k > 0 is some 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, r > 0.
r2
The trick to solving this is the change of variables g(r) = h(r)/r. The equation for h(r) will have 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
ˆ
lim
x G(x; 0) · n dx = 1,
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.
Question 5 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)
Question 6 Consider Laplace’s equation ∆u = 0 in the upper half-plane.
(a) Find the appropriate Green’s function which is subject to boundary conditions
G(x, 0; x0 , y0 ) = 0, lim G(x, y; x0 , y0 ) = 0 y→∞ (b) Use part (a) to write (in Cartesian coordinates) the solution to Laplace’s equation subject to u(x, 0) =
h(x).
(c) Use part (b) to find an explicit solution (not just an integral) if h(x) = H(x), the step function. (Hint:
you may want to use arctan(u) = 1/(1 + u2 )du where the chosen branch of arctan(u) is such that
arctan(±∞) = ±π/2). Check your answer!
Question 7 Consider the problem
∆u = f (r, θ) inside a disk of radius a,
with mixed boundary conditions
u(a, θ) = h1 (θ) for 0 < θ < π, ur (a, θ) = h2 (θ) for π < θ < 2π, (a) What formal problem (equation and boundary conditions) does the Green’s function satisfy? (actually
finding such a Green’s function is not trivial!)
(b) Suppose that the Green’s function, written in polar coordinates as G(r, θ; r0 , θ0 ) is known. Express the
solution u(r, θ) in terms of G. Write integrals and functions explicitly in terms of polar coordinates.