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### Messages - Victor Ivrii

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1
##### Web Bonus Problems / Re: Web bonus problem -- Week 4
« on: Today at 05:34:09 AM »
Well, $r(0)=g(0)$ is obvious. And this is the only condition needed for the first problem, (a); no for the second (a).

You do not need to find solution to get compatibility conditions

2
##### MAT244--Misc / Re: Lecture Notes
« on: Today at 05:24:14 AM »
Are we allow to post our own lecture notes on this forum? (in case some students missed their own lecture)
Sure. But type them. Not crappy mugshots of bad handwriting

3
##### Web Bonus Problems / Re: Web bonus problem--Week 3
« on: Today at 05:22:22 AM »
Usually web bonus is given to the first poster (or may be for improvements if the first poster did not get everything right). This time I was more generous. But not infinitely. Bonus got first 5 or so

4
##### MAT244--Tests / MOVED: Week 2 Quiz?
« on: Today at 05:13:48 AM »

5
##### MAT244--Lectures / MOVED: lecture notes for week 3
« on: Today at 05:12:15 AM »
This topic has been moved to MAT244--Misc.

http://forum.math.toronto.edu/index.php?topic=907.0 Post in the proper board!

6
##### Web Bonus Problems / Re: Web bonus problem--Week 3
« on: Today at 05:10:32 AM »
When I wrote "type" I meant type on forum. Like I do:
$$y'+2\cot(t)y=\cos(t)$$,
then
$$I=\exp\Bigl(2\cot(t)\,dt\Bigr)=\exp \bigl(2\ln \sin(t)\bigr)=\sin^2(t)$$
and so on, not to put image (even typed) on forum.

Don't cut lines too short... Also $\exp$, $\cos$ etc should be upright (in $\LaTeX$ it is \exp, \cos).

No external images! Post attachments

7
##### Web Bonus Problems / Re: Web bonus problem--Week 3
« on: January 22, 2018, 01:53:50 PM »
Looks OK. Could you type it?

8
##### Web Bonus Problems / Web bonus problem--Week 3
« on: January 20, 2018, 09:49:36 AM »
a. Find the general solution of
$$y'+2\cot(t)y=\cos(t).$$
b. Find solution, which is defined on $(-\infty,\infty)$;
c. Find solution, which is defined on $(-\infty,\infty)$ and periodic.

9
##### Web Bonus Problems / Re: Web bonus problem--Week 3
« on: January 20, 2018, 06:08:43 AM »
Right. What is $G$? Note, how I reformatted the first half of your post

10
##### MAT244--Announcements / Enrolment in Lecture and Tutorial sections
« on: January 20, 2018, 01:27:19 AM »
I have received plenty of emails complaining "cannot enrol into ...", "can you help me?". The answer is "no": instructors are not involved in the enrolment. You need to talk to your College or Faculty Registrar. Also try Undergraduate Administrator of Math Department.

Some Tutorial sections take place in the rooms, which have extra seats (in comparison with the declared maximal size of the section). However some Tutorial sections don't have extra seats. (*)

Some students ask "Can you be enrolled into one section and attend another?" I cannot answer without knowing exactly which sections you are talking about. However it is possible as an exception to a very limited number of students because of (*).

Inevitably, some students (hopefully not many) decide to drop the course. In this case, please, drop from both Lecture and Tutorial sections. First, it will help other students, those who cannot get into desired section. Second, it will help instructors and TAs: we need to know, how many copies of the quiz papers should we make for a particular tutorial section (they will be different). Finally, stopping to attend the course without official dropping the Lecture section brings the financial and academic penalty. Probably, it is not the so  for the Tutorial section, but do you want to take the risk?

11
##### Web Bonus Problems / Re: Web bonus problem--Week 3
« on: January 19, 2018, 03:17:38 PM »
OK. Part 1 done

12
##### Web Bonus Problems / Web bonus problem -- Week 4
« on: January 19, 2018, 05:53:52 AM »
1. Consider problem:
\begin{align}
&u_{tt}-c^2u_{xx}=f(x,t),&&x>0, t>0,\\
&u|_{t=0}=g(x),\\
&u_t|_{t=0}=h(x),\\
&u|_{x=0}=r(t).
\end{align}
Assuming that all functions $g,h,r$ are very smooth, find conditions necessary for solution $u$ be
a. $C$ in $\{x>0,t>0\}$ (was on the lecture);
b. $C^1$ in $\{x>0,t>0\}$;
c. $C^2$ in $\{x>0,t>0\}$;
d. $C^3$ in $\{x>0,t>0\}$;
where $C^n$ is the class on $n$-times continuously differentiable functions.

2. Consider problem:
\begin{align}
&u_{tt}-c^2u_{xx}=f(x,t),&&x>0, t>0,\\
&u|_{t=0}=g(x),\\
&u_t|_{t=0}=h(x),\\
&u_x|_{x=0}=s(t).
\end{align}
Assuming that all functions $g,h,s$ are very smooth, find conditions necessary for solution $u$ be
a. $C$ in $\{x>0,t>0\}$ (automatically);
b. $C^1$ in $\{x>0,t>0\}$;
c. $C^2$ in $\{x>0,t>0\}$;
d. $C^3$ in $\{x>0,t>0\}$.

Hint: These compatibility conditions are on $f,g,h,s$ and may be their derivatives as $x=t=0$. Increasing smoothness by $1$ ads one condition. You do not need to solve problem, just plug and may be differentiate.

13
##### Web Bonus Problems / Re: Web bonus problem--Week 3
« on: January 19, 2018, 12:31:35 AM »
In (5) I may agree that left-hand expression is equal to the right-hand expression, but the middle is a mystery

In (6) what is $L$ is also a mystery.

14
##### Web Bonus Problems / Web bonus problem--Week 3
« on: January 18, 2018, 04:02:11 PM »
From formula
$$u(x,t)=\frac{1}{2}\bigl( g(x+ct) +g(x-ct)\bigr) \tag{1}$$
for the problem
\left\{\begin{aligned} &u_{tt}-c^2u_{xx}=0&&-\infty <x <\infty,\\ &u|_{t=0}=g(x),\\ &u_t|_{t=0}=0 \end{aligned}\right. \tag{2}
derive
$$v(x,t)=\frac{1}{2c}\int_{x-ct}^{x+ct}g(x')\,dx' \tag{3}$$
for the problem
\left\{\begin{aligned} &v_{tt}-c^2v_{xx}=0&&-\infty <x <\infty,\\ &v|_{t=0}=0,\\ &v_t|_{t=0}=g(x). \end{aligned}\right. \tag{4}
Also. from (4) for (3) derive (2) for (1).

Hint: prove that $v=\int _0^t u(x,t')\,dt'$.

15
##### APM346--Misc / Re: Grading Scheme
« on: January 17, 2018, 12:55:05 AM »
Would anyone be willing to explain the grading scheme that is posted in the course outline? I am unfamiliar with the notation. How do we receive bonus marks for lecture? For the bonus forum question, do we receive bonus marks for making any meaningful contribution or just the right answer? Is it then a bit of a race to finish it first?