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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 ? : 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| < 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 > 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 > 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 > 0, y > 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

 

arctan(??) = ??/2). Check your answer!

 

Question 7 Consider the problem

 

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

 

with mixed boundary conditions

 

u(a, ?) = h1 (?) for 0 < ? < ?,

 


 

ur (a, ?) = h2 (?) for ? < ? < 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.

 


 

 


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