### Show Posts

This section allows you to view all posts made by this member. Note that you can only see posts made in areas you currently have access to.

### Messages - Djirar

Pages: [1] 2
1
##### Final Exam / problem 4
« on: December 20, 2012, 01:33:07 PM »
Let $u$ solve the initial value problem for the wave equation in one dimension
\begin{equation*}
\left\{\begin{aligned}
& u_{tt}-  u_{xx}= 0 ,\qquad&& ~{\mbox{in}} ~\mathbb{R} \times (0,\infty),\\[3pt]
&u (0,x) = f(x), \qquad&& ~{\mbox{on}}~ \mathbb{R} \times \{t=0\} ,\\[3pt]
&u_t(0,x)= g(x),  \qquad&& ~{\mbox{on}}~ \mathbb{R} \times \{t=0\} .
\end{aligned}\right.
\end{equation*}
Suppose $f(x)=g(x)=0$ for all $|x|>1000.$ The  kinetic energy is
$$k(t)= \frac{1}{2}\int_{-\infty}^{+\infty} u_t^2 (t,x) dx$$
and the potential energy is
$$p(t)= \frac{1}{2}\int_{-\infty}^{+\infty} u_x^2 (t,x) dx.$$
Prove
• $k(t)+ p(t)$ is constant with $t$ (so does not change as $t$ changes),
• $k(t)=p(t)$ for all large enough times $t$.

problem 4 part (a)

2
##### Final Exam / problem 5
« on: December 20, 2012, 01:30:49 PM »
Suppose $\Delta u = 0$ and satisfies $|u| < 1000$ everywhere on $\mathbb{R}^2$.

Prove that $u$ is a constant function. In other words, show that there exists a constant $C$ so that $u (x) = C$ everywhere on $\mathbb{R}^2$.

Hint: Mean Value Theorem

3
##### Final Exam / problem 2
« on: December 20, 2012, 01:30:26 PM »
Consider a $2\pi$-periodic function $f$ with full Fourier series
$$\sum_{n \in \mathbb{Z}} c_n e^{i n x}.$$
Suppose that the Fourier coefficients decay fast enough to satisfy
$$\sum_{n \in \mathbb{Z}} |n| \cdot |c_n| < 17.$$
Prove that $f'$ is bounded.

4
##### Final Exam / Problem 1
« on: December 20, 2012, 01:30:05 PM »
Solve the first-order equation $2 u_t + 3 u_x =0$ with the auxiliary condition $u = \sin x$ when $t=0$.

5
##### Home Assignment 5 / Re: Problem 4
« on: December 15, 2012, 11:06:40 PM »
includes solution to part (d) and what I think is the graph to part (e).

6
##### Home Assignment 5 / Re: Problem 6
« on: December 15, 2012, 10:44:26 PM »
Solutions to Problem 6.

7
##### Home Assignment 5 / Re: Problem 5
« on: December 15, 2012, 10:40:57 PM »
Solutions to problem 5.

8
##### Misc Math / Re: Lec 26 #3
« on: November 26, 2012, 04:17:35 PM »
I think that for part (3) we are supposed to take
$\mathbf{U}=\ w \nabla u$

then you take (permutting)

$\mathbf{U}=\ u \nabla w$

9
##### Home Assignment 5 / Re: Problem 1
« on: October 30, 2012, 07:42:37 PM »
Do you want the solution in the complex form or real form of the Fourier series?

10
##### Term Test 1 / Re: TT1 = Problem 1
« on: October 16, 2012, 08:54:19 PM »
My solution. Please check there might be mistakes. Sorry for quality of print and hand writing.

Djirar, I posted my solution 10 seconds after you

I was scribbling my solutions as fast as I could

Edit: I forgot to write down the axes to part a.  The vertical is x-axis and the horizontal is y-axis. I hope I didn't forget this on the test

Do you mean the $t$ axis?

Yes $t$ not $y$. Thank you.

11
##### Term Test 1 / Re: TT1 = Problem 1
« on: October 16, 2012, 08:46:51 PM »
I think Qitan Cui is correct. In my haste I must have messed up the integration of the bonus part.

12
##### Term Test 1 / Re: TT1 = Problem 1
« on: October 16, 2012, 08:32:04 PM »
My solution. Please check there might be mistakes. Sorry for quality of print and hand writing.

Djirar, I posted my solution 10 seconds after you

I was scribbling my solutions as fast as I could

Edit: I forgot to write down the axes to part a.  The vertical is x-axis and the horizontal is y-axis. I hope I didn't forget this on the test

13
##### Term Test 1 / Re: TT1 = Problem 1
« on: October 16, 2012, 08:25:30 PM »
My solution. Please check there might be mistakes. Sorry for quality of scan and handwriting.

14
##### Term Test 1 / Re: TT1 = Problem 4
« on: October 16, 2012, 08:15:40 PM »
part b.

15
##### Term Test 1 / Re: TT1 = Problem 5
« on: October 16, 2012, 08:07:48 PM »
I solved the question in the same way as Aida Razi did. Of course for part b, C(x) > 0  for all x.

Pages: [1] 2