Fall2006APM346Homework

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This page contains a list of homework problems for the Fall 2006 APM 346.

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1 Homework Assignment 1 (due September 18)
2 Homework Assignment 2 (due September 25)
3 Homework Assignment 3 (due October 2)
4 Homework Assignment 4 (due October 11)
5 Homework Assignment 5 (due October 16)
6 Homework Assignment 6 (due October 30)
7 Homework Assignment 7 (due November 8)
8 Homework Assignment 8 (due November 13)
9 Homework Assignment 9 (due Wednesday November 22 EXTENSION!)
- 9.1 Green's Formulas Exercise
- 9.2 Mean Value Property for Harmonic Functions Exercise
10 Homework Assignment 10 (due Wednesday December 6 EXTENSION!)

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Homework Assignment 1 (due September 18)

From p.5 of Strauss' PDE text:

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(5:10)

Show that the solutions of the differential equation $u''' - 3 u'' + 4 u = 0$ form a vector space. Find a basis for it.

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(5:11)

Verify that $u (x, y) = f (x) g (y)$ is a solution of the PDE

u u x y = u x u y

for all pairs of (differentiable) functions $f$ and $g$ of one variable.

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(5:12)

Verify by direct substitution that

$u_n(x,y)=\sin nx\,\sinh ny$

is a solution of $u x x + u y y = 0$ for every $n > 0$ .

The first homework assignment can be handed in at the beginning of class on Wednesday 20 September. Future homework assignments which arrive late will not be marked. Colliand 10:23, 18 September 2006 (EDT)

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Homework Assignment 2 (due September 25)

From p.9 of Strauss' PDE text:

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(9:1)

Solve the first-order equation $2 u t + 3 u x = 0$ with the auxiliary condition $u = sin x$ when $t = 0$ .

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(9:7)

Solve $a u x + b u y + c u = 0$ .

(Hint. $\frac{d}{ds} p = - c p$ .)

From p.19 of Strauss' PDE text:

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(9:5)

Derive the equation of one-dimensional diffusion in a medium that is moving along the $x$ axis to the right at constant speed $V$ .

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(9:9)

This is an exercise on the divergence theorem

$\iiint_D \nabla\cdot\mathbf{F}\,d\mathbf{x} = \iint_{\partial D}\mathbf{F}\cdot\mathbf{n}\,dS,$

valid for any boundary domain

D

in space with boundary surface $\partial D$ and unit outward normal vector $\mathbf{n}$ . Verify it in the following case by calculating both sides separately:

$\mathbf{F}=r^2\mathbf{x}$ ,

where $\mathbf{x}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$ ,

r 2 = x 2 + y 2 + z 2

, and

D =

the ball of radius

a

and center at origin.

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(9:10)

If $\mathbf{f}(\mathbf{x})$ is continuous and $|\mathbf{f}(\mathbf{x})| \leq 1/(|\mathbf{x}|^3+1)$ for all $\mathbf{x}$ , show that

$\iiint_{\mathrm{all\ space}} \nabla\cdot\mathbf{f}\,d\mathbf{x}=0.$

(Hint: Take

D

to be a large ball, apply the divergence theorem, and let its radius tend to infinity.)

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Homework Assignment 3 (due October 2)

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(27:1)

Consider the problem

$\frac{d^2u}{dx^2}+u=0$

u (0) = 0

and

u (L) = 0

consisting of an ODE and a pair of boundary conditions. Clearly, the function $u (x) = 0$ is a solution. Is the solution unique, or not? Does the answer depend on $L$ ?

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(27:4)

Consider the Neumann problem

Δ u = f (x, y, z)

D

$\frac{\partial u}{\partial n}=0$ on $\partial D$ .

(a) What can we surely add to any solution to get another solution? So we don't have uniqueness.

(b) Use the divergence theorem and the PDE to show that

$\iiint_D f(x,y,z)\,dx\,dy\,dz=0$

is the necessary condition for the Neumann problem to have a solution.

(c) Can you get physical interpretation of part (a) and/or part (b) for either heat flow or diffusion?

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(31:3)

Among all the equations of the elliptic form, show that the only ones that are unchanged under all rotations (rotational invariant) have the form $a (u x x + u y y) + b u = 0$ .

(Hint. Introduce rotated coordinates

x', y'

. Represent $\partial_x , \partial_y$ in terms of $\partial_{x'}, \partial_{y'}$ . Reexpress equation in rotated coordinates.)

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(31:6)

Consider the equation $3 u y + u x y = 0$ .

(a) What is its type?

(b) Find the general solution. (Hint: Substitute $v = u y$ .)

(c) With auxiliary conditions $u (x,0) = e - 3 x$ and $u y (x,0) = 0$ , does a solution exist? Is it unique?

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Homework Assignment 4 (due October 11)

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(36:1)

Solve $u t t = c 2 u x x$ , $u (x,0) = e x$ , $u t (x,0) = sin x$ .

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(36:2)

Solve $u t t = c 2 u x x$ , $u (x,0) = log(1 + x 2)$ , $u t (x,0) = 4 + x$ .

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Homework Assignment 5 (due October 16)

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(68:2)

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(81:1)

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(81:2)

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Homework Assignment 6 (due October 30)

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(87:1)

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(87:2)

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(87:3)

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(108:2)

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(108:3)

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(108:5)

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Homework Assignment 7 (due November 8)

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(119:10)

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(120:12)

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(120:13)

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Homework Assignment 8 (due November 13)

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(139:2)

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(139:5)

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(140:12)

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Homework Assignment 9 (due Wednesday November 22 EXTENSION!)

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Green's Formulas Exercise

Let $U$ be an open bounded connected region in $\mathbb{R}^n$ with smooth boundary $\partial U$ . Let $\overline{U}$ denote the closure of $U$ . Let $ν$ denote the outward pointing unit normal vector field along $\partial U$ . Let $d x$ denote the volume element on $\mathbb{R}^n$ and let $d S$ denote the surface area element on $\partial U$ . Let $u, v \in C^2 (\overline{U}).$

Theorem (Green's Formulas):
- $\int_U \Delta u dx = \int_{\partial U} \frac{\partial u}{\partial \nu} dS.$
- $\int_U Du \cdot Dv dx = - \int_U u \Delta v dx + \int_{\partial U} u \frac{\partial v}{\partial \nu} dS.$
- $\int_U u \Delta v - v \Delta u dx = \int_{\partial U} u \frac{\partial v}{\partial \nu}- v \frac{\partial u}{\partial \nu} dS.$

In class on 15 November, we proved the first of Green's Formulas. Prove the other two formulas.

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Mean Value Property for Harmonic Functions Exercise

Let $B (x, r)$ denote the open ball of radius $r$ centered at $x \in {\mathbb{R}^d}.$ The boundary of this ball, denoted $\partial B (x,r)$ , is the sphere or radius $r$ with center $x$ . Let $|\partial B^{} (x,r) |$ denote the surface area of this sphere and $| B (x, r) |$ denote the volume of the ball. In class on 16 November, we proved: