## Answered: - Directions: answer using complete sentences, and I need pr

Directions: answer using complete sentences, ?and I need process.

and I post ?the comment of the previous homework, I hope that can help you.

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 ? : R2 ? R in ?uid mechanics solves ?? =

? where ? is a measure of the ?uid rotation (the velocity ?eld is perpendicular to the gradient of ?, by the

way). A simple model of a hurricane of radius R has

?=

1

0

|x| &lt; R,

.

|x| ? R.

Use the Green?s function appropriate for two dimensions to express the solution ? 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(?)

where ? is the angle between vectors x and x0 . Finally, evaluate ?(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 ?eld condition limx?? u(x) = 0, where k &gt; 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 ?nd g satis?es the ordinary

differential equation

1 2

(r gr )r ? k 2 g = 0, r &gt; 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

?

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 ?rst quadrant x &gt; 0, y &gt; 0 where the

boundary/far-?eld 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 ?nd 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

Question 7 Consider the problem

?u = f (r, ?) inside a disk of radius a,

with mixed boundary conditions

u(a, ?) = h1 (?) for 0 &lt; ? &lt; ?,

ur (a, ?) = h2 (?) for ? &lt; ? &lt; 2?,

(a) What formal problem (equation and boundary conditions) does the Green?s function satisfy? (actually

?nding such a Green?s function is not trivial!)

(b) Suppose that the Green?s function, written in polar coordinates as G(r, ?; r0 , ?0 ) is known. Express the

solution u(r, ?) in terms of G. Write integrals and functions explicitly in terms of polar coordinates.

Solution details:
STATUS
QUALITY
Approved

This question was answered on: Sep 18, 2020 Solution~0001185456.zip (25.37 KB)

This attachment is locked

We have a ready expert answer for this paper which you can use for in-depth understanding, research editing or paraphrasing. You can buy it or order for a fresh, original and plagiarism-free copy from our tutoring website www.aceyourhomework.com (Deadline assured. Flexible pricing. TurnItIn Report provided)

##### Pay using PayPal (No PayPal account Required) or your credit card . All your purchases are securely protected by .

STATUS

QUALITY

Approved

Sep 18, 2020

EXPERT

Tutor 