2006APM346Midterm1

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The first mid-term examination for APM 346 will take place over 19-20 October. Problems will be posted on this page on 19 October around 11:00a.m. The exam will be due before 3:00p.m. on 20 October and must be handed in to the receptionist at the main office of the Department of Mathematics, 6th floor, Bahen Center. Students who have typeset their exam may alternatively email it to me before the deadline. Colliand 14:53, 13 October 2006 (EDT)

Although cheating appears to be widespread among Canadian students and elsewhere, it is absolutely unacceptable on this exam and in my course. Students caught cheating will experience unpleasant consequences such as an automatic failing grade in my course. Don't Cheat! Colliand 14:51, 13 October 2006 (EDT)

Please acknowledge resources you have used in preparing your answers to the following problems. Please make an effort to clearly present your ideas. Write your name on your papers.

1 Problem 1
2 Problem 2
3 Problem 3
4 Problem 4
5 Problem 5
6 Problem 6
7 Problem 7
8 Problem 8

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Problem 1

Consider the initial value problem

$\partial_t u = \partial_x^2 u, - \infty < x < + \infty, t >0,$

u (0, x) = φ(x).

Assume that $\phi \in C_0^{100}$ (so $\phi, \partial_x \phi, \dots, \partial_x^{100} \phi$ are all continuous and vanish outside of some big set ${x : | x | < R}$ ).

Prove that

$F(t) = \int_{-\infty}^{+\infty} | \partial_x^3 u (t,x)|^2 dx$

is a non-increasing function of $t$ .

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Problem 2

Find a formula for the surface $z = u (x, y)$ which satisfies

$\sin y~ \partial_x u + \partial_y u = 0$

subject to the initial condition $u (x,0) = f (x).$

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Problem 3

Suppose

x' = a x + b y,

y' = c x + d y

defines a valid change of variables $(x,y) \longmapsto (x',y')$ . Express the Laplacian $\Delta = \partial_x^2 + \partial_y^2$ in terms of the transformed coordinates $(x', y')$ .

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Problem 4

For a solution $u (t, x)$ of the wave equation $\partial_t^2 u - \partial_x^2 u = 0$ , the energy density is defined as $e = \frac{1}{2} (u_t^2 + u_x^2)$ and the momentum density as $p^{ }_{ } = u_t u_x$ .

Show that $\partial_t e = \partial_x p$ and $\partial_t p = \partial_x e$ .
Show that both $e (t, x)$ and $p (t, x)$ also satisfy the wave equation.

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Problem 5

Find the formula for the solution the initial value problem

$\partial_t u = \partial_x^2 u, - \infty < x < + \infty, t >0,$

$u_{ }(0,x) = \pi^{-1/2} e^{-x^2}.$

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Problem 6

Prove the comparison principle for the diffusion equation:
If $u^{ }_{ }$ and $v^{ }_{ }$ are two solutions, and if $u^{ }_{ } \leq v$ for $t = 0$ , for $x = 0$ , and for $x = l$ , then $u \leq v$ for $0 \leq t < \infty, 0 \leq x \leq l$ .

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Problem 7

Suppose $u$ is a smooth solution of the Korteweg-deVries equation $\partial_t u + \partial_x^3 u + u \partial_x u = 0$ for $x \in \mathbb{R}$ and $t > 0$ . Assume that $u$ decays to zero as $x \to \pm \infty$ . Prove that