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Home Assignment 6 / Bonus Web Problem--4

« on: October 27, 2012, 06:21:01 AM »

Don't post until November 7, 21:30

Prove the following properties of convolution:

• (a) $f*g=g*f$
• (b) $(f*g)'=f'*g = f*g'$, where $'$ denotes the derivative in one variable
• (c) $f*(g*h)=(f*g)*h$
• (d) Let $x_+^\lambda := x^\lambda$ as $x>0$ and $0$ as $x<0$. Prove that for $f$ which fast decays as $x\to -\infty$ and $n=1,2,\ldots$
  \begin{equation*}
  \frac{x_+^{n-1}}{(n-1)!} * f =\underbrace{\int_{-\infty}^x \int_{-\infty}^{x_1} \ldots \int_{-\infty}^{x_{n-1}}}_{\text{$n$ integrals}} f(x_n)\, dx_n \cdots dx_1
  \end{equation*}

677

Home Assignment 4 / Web Bonus Problem--3

« on: October 26, 2012, 09:14:14 AM »

Oscillations of the beam  (with left end clamped and right end free) are described by an equation
\begin{equation*}
u_{tt} + K u_{xxxx}=0, \qquad 0<x<l
\end{equation*}
with $K>0$ and the boundary conditions
\begin{equation*}
u(0,t)=u_{x}(0,t)=u_{xx}(l,t)=u_{xxx}(l,t)=0.
\end{equation*}

• (a) Find  equation describing frequencies and corresponding  eigenfunctions
  (You may assume that all eigenvalues are real and positive).
• (b) Solve  this equation graphically.
• (c) Prove  that eigenfunctions corresponding to different eigenvalues are orthogonal.
• (d) Bonus  Prove that eigenvalues are simple, i.e. all eigenfunctions corresponding to the same eigenvalue are proportional.

Compare with eigenvalues of Problem 2 of HA2

678

Home Assignment 4 / Bonus Web Problem--2

« on: October 20, 2012, 06:28:59 AM »

Oscillations of the beam  (with both its ends  having fixed positions but not directions, imagine beam lying on supports) are described by an equation
\begin{equation*}
u_{tt} + K u_{xxxx}=0, \qquad 0<x<l
\end{equation*}
with $K>0$ and the boundary conditions
\begin{equation*}
u(0,t)=u_{xx}(0,t)=u(l,t)=u_{xx}(l,t)=0.
\end{equation*}

• (a) Find  equation describing frequencies and corresponding  eigenfunctions
  (You may assume that all eigenvalues are real and positive).
• (b) Solve  this equation graphically.
• (c) Prove  that eigenfunctions corresponding to different eigenvalues are orthogonal.
• (d) Bonus  Prove that eigenvalues are simple, i.e. all eigenfunctions corresponding to the same eigenvalue are proportional.

Compare with eigenvalues of Problem 2 of HA2

679

Term Test 1 / TT1

« on: October 16, 2012, 07:02:16 PM »

tex and pdf files attached

680

Term Test 1 / TT1 = Problem 5

« on: October 16, 2012, 06:30:11 PM »

The functions $H$ and $Q$ are defined as follows:
\begin{equation*}
H(x) = \left\{\begin{aligned}
&1, \qquad&&x>0,\\
&0, \qquad&&x \leq 0,
\end{aligned}\right.
\end{equation*}   
\begin{equation*}
Q(x) = \left\{\begin{aligned}
&1, \qquad&&|x| \leq 1,\\
&0, \qquad&&|x| > 1.
\end{aligned}\right.
\end{equation*}
Consider the function
\begin{equation*}
C(x) = \int_{-\infty}^{+\infty} H(x-y) Q(y) dy.
\end{equation*}

• (a) Graph the function $C(x)$.
• (b) Identify the set of points $x$ where $C(x) >0.$

681

Term Test 1 / TT1 = Problem 4

« on: October 16, 2012, 06:29:34 PM »

Consider the initial value problem for the diffusion equation on the line:
\begin{equation*}
\left\{\begin{aligned}
&u_{t} = k  u_{xx},\qquad&&x \in \mathbb{R},\\
&u (x,0) = f(x), \qquad&&x \in \mathbb{R}.
\end{aligned}\right.
\end{equation*}

• (a) Assuming $f$ is smooth and vanishes when $|x|>10$, give a formula for the solution $u(x,t)$.
• (b) Using the formula from part (a), prove that
  \begin{equation*}
  \lim_{t \searrow 0} u(x,t) = f(x).
  \end{equation*}
  

682

Term Test 1 / TT1 = Problem 3

« on: October 16, 2012, 06:28:21 PM »

Consider the PDE  with boundary conditions:
\begin{align*}
&u_{tt}+K u_{xxxx} + \omega^2 u =0,\qquad&&0<x<L,\\[3pt]
&u(0,t)=u_x(0,t)=0,\\[3pt]
&u(L,t)=u_x(L,t)=0,
\end{align*}
where  $K>0$ is constant. Prove that the energy $E(t)$ defined as
\begin{equation*}
E(t)= \frac{1}{2}\int_0^L \bigl( u_t^2 + K u_{xx}^2 + \omega^2 u^2)\,dx
\end{equation*}
does not depend on $t$.

683

Term Test 1 / TT1 = Problem 2

« on: October 16, 2012, 06:27:29 PM »

Consider the initial value problem for the wave equation posed on the left half-line:
\begin{equation*}
\left\{\begin{aligned}
&u_{tt}-  u_{xx}= 0 ,\qquad&&-\infty <x< 0\\
&u (x,0) = f(x), \qquad&&-\infty < x < 0 ,\\
&u_t(x,0)= g(x), \qquad&&-\infty < x < 0.
\end{aligned}\right.
\end{equation*}
Do the initial conditions uniquely determine the solution in the region $\{ (t,x): t \in \mathbb{R}, -\infty < x < 0 \}$? Explain your answer with convincing arguments.

684

Term Test 1 / TT1 = Problem 1

« on: October 16, 2012, 06:26:37 PM »

Consider the first order equation:
\begin{equation}
u_t + x u_x = 0.
\label{eq-1} 
\end{equation}

• (a) Find the characteristic curves and sketch them in the $(x,t)$ plane.
• (b) Write the general solution.
• (c) Solve  equation (\ref{eq-1})  with the initial condition $u(x,0)= \cos(2x)$.

Explain why the solution is fully  determined by the initial condition.

• (d) bonus Describe domain in which solution of
  \begin{equation}
  u_t + x^2 u_x = 0, \qquad x>0
  \label{eq-2} 
  \end{equation}
  is fully determined by the initial condition $u(x,0)=g(x)$ ($x>0$)?

685

General Discussion / Please, don't

« on: October 14, 2012, 10:24:45 AM »

Every year before, during, and after Term Test I am getting emails

This morning I went to seafood on Spadina. After this ...

where ... replace a graphical description of a certain medical condition.

I love seafood and I love certain places on Spadina but I also know that attending unknown eateries there is not dissimilar to a Russian roulette. Therefore, if you really want to beta-test an unfamiliar eatery there, please do it after term test, not before it.

BTW, it covers also a Final Exam--and especially Final Exam.

686

Misc Math / Bonus Web Problem 1

« on: October 09, 2012, 03:49:08 PM »

This is not very difficult problem but it contains one tricky point

Consider heat equation with thermo-conductivity depending on the temperature:
\begin{equation}
u_t= (u^m u_x)_x
\label{eq-1}
\end{equation}
with $m>0$ and find solution(s) $u$ which are self-similar
\begin{equation}
u_\lambda:= \lambda u(\lambda x, \lambda^\gamma t )=u(x,t)\qquad \forall \lambda>0
\label{eq-2}
\end{equation}
and
\begin{equation}
u(x,t)\to 0 \qquad\text{as  } x\to \pm \infty.
\label{eq-3}
\end{equation}
Hint: first find $\gamma$ and then plugging $\lambda =t^{1/\gamma}$ reduce $u$ to a function of one variable and PDE (\ref{eq-1}) to ODE (Follow lecture 9 with the necessary modifications)

687

General Discussion / Registration has been tightened

« on: October 03, 2012, 01:52:18 PM »

Due to spammers--actually wannabe spammers--the were erased by admins before causing any harm--registration has been tightened: ReCaptcha has been upgraded, security questions changed and registration from Immediate became Requires email validation (next step--Requires approval by admins).

688

Home Assignment 2 / Problem 1--not done yet!!!

« on: October 02, 2012, 12:17:36 PM »

http://forum.math.toronto.edu/index.php?topic=30.0

689

Home Assignment 1 / WTH?

« on: September 25, 2012, 02:08:17 AM »

Aida, WTH? In MAT244 your scan was a golden standard of scanning
http://weyl.math.toronto.edu:8888/MAT244-2012S-Forum/index.php?topic=38.msg111#msg111 (see, I remember, so good it was), and now this? Such posts defy the whole purpose of this forum, it is not to submit your papers for grading (you submit it to TA) but even this is difficult as quality of scan is poor (actually you used cellphone without taking care of settings), but to share the solution with your classmates who can comment, find errors or correct them.

From this point the typed solution using MathJax is far the best as I can edit it, just marking the place where I see an error, and anyone can copy-paste code from it. But apart of this your former clean black-white scan with perfect position of the paper was the very best thing.

Here you use colour (not even grayscale) scan and some papers are horizontal and some diagonal ... everyone has a really hard time to read them. I admit, some of your submissions a better than othersbut  from the point of view of this forum I must consider all of them non-existent . It looks like you just decided to capture the space preventing anyone else from posting the solutions.

So, everyone should feel to post solutions.

------

Also:
some people used paper clips despite our request to staple and some even tried to use "poor man paper clips" just folding several times the corner of the paper and adding a bit of saliva. Sorry, no of these constructions is robust enough to prevent separation of pages. I hope I caught every such attempt and stapled, but I am not sure. I am also not sure if Prof Colliander managed with this. So, if you have not stapled and your pages are separated and lost, you know whom to blame.

Not everyone indicated the section where to bring your papers back (so get such papers from TA who graded it).

Please use the standard paper (letter size). Someone used A4 (a bit more narrow and a longer overseas size). It looks like a little thing but also causes problems when dealing with tens of papers.

And finally, again: please, do not come to submit papers during the class -- only before lecture, during the break or after. Everyone in the night class feels tired (instructor in the least degree because of adrenalin rush) and such distractions are rather disruptive.

690

Home Assignment 1 / HA1-pdf

« on: September 22, 2012, 01:40:44 PM »

Here is HA1.pdf - Home assignment 1 printed to pdf (updated Mon 24 Sep 2012 05:03:22 EDT)