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Topics - Victor Ivrii

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1
MAT244--Announcements / TT1 papers
« on: February 24, 2018, 07:28:30 AM »
TT1 grades are on BlackBoard

Papers will be returned at Lecture sections. We planned to do it on Tutorial sections, but too many students failed to write properly their Tutorial sections (time slot instead of code does not cut).



Term Test 2 papers will be returned at Tutorial sections first, and only one or two weeks later––on Lecture sections.

2
APM346--Announcements / Missed TT or Quiz
« on: February 21, 2018, 05:01:16 AM »
Note to new Fields on Blackboard:

Notes-TT and Notes-Q

For majority these fields are empty, but

* if you missed Term Test 1 or 2, and submitted Doctor's note to me,  the corresponding note appear in Notes-TT .
* if you missed one of Quizzes, and submitted Doctor's note to me,  the corresponding note appear in Notes-Q .

If you submitted Doctor's note and the note did not appear on Blackboard, contact me immediately.

Possible reasons:

* you have not indicated APM346 in the subject and mail spam filter trashed your message.--please resend

* Instead of scan you sent the picture shot by cellphone, and the quality was really poor.

* You did not indicate what Quiz it was, and what Tutorial section you are

_____

Next time any poor quality scans, or not oriented properly will not be considered at all.

3
MAT244--Announcements / Missed TT or Quiz
« on: February 21, 2018, 04:53:01 AM »
Note to new Fields on Blackboard:

Notes-TT and Notes-Q

For majority these fields are empty, but

* if you missed Term Test 1 or 2, and submitted Doctor's note to me,  the corresponding note appear in Notes-TT .
* if you missed one of Quizzes, and submitted Doctor's note to me,  the corresponding note appear in Notes-Q .

If you submitted Doctor's note and the note did not appear on Blackboard, contact me immediately.

Possible reasons:

* you have not indicated MAT244 in the subject and mail spam filter trashed your message.--please resend

* You sent note to other instructor or TA (because you did not follow course outlines)--please send it to me. Other instructors and TAs are asked to ignore such messages

* Instead of scan you sent the picture shot by cellphone, and the quality was really poor.

* You did not indicate what Quiz it was, and what Tutorial section you are
_____
Next time any poor quality scans, or not oriented properly will not be considered at all.

4
Web Bonus Problems / Web Bonus problem -- Reading Week
« on: February 16, 2018, 08:27:43 AM »
$\renewcommand{\Re}{\operatorname{Re}}$
Find Fourier transform of $e^{-\frac{1}{2}a x^2}$, where $a\in \mathbb{C}$ with $\Re a>0$.

To do this

(a) Check that $u=e^{-\frac{1}{2}a x^2}$ satisfies $u'(x) = -ax u(x)$; then using Fourier Transform properties conclude that $\hat{u}$ satisfies $i \omega \hat{u}(\omega)= -i a\hat{u}'(\omega)$ and solving this equation find $\hat{u}$ up to a constant factor $C$.

(b) Then calculate $I:=\hat{u}(0)=\int e^{-\frac{1}{2}a x^2}\,dx$ in the same way as we did it for real $a>0$ but justify the correct "sign" (since $I$ is recovered up to a factor $\pm 1$), comparing with the case of real $a>0$.

5
Term Test 1 / P5 Night
« on: February 15, 2018, 08:27:08 PM »
$\newcommand{\erf}{\operatorname{erf}}$
Find the solution $u(x,t)$ to
\begin{align}
&u_t=u_{xx} && -\infty<x<\infty, \ t>0,\tag{1}\\[2pt]
&u|_{t=0}=\left\{\begin{aligned} &x &&|x|<1,\\ &0 &&|x|\ge 1,\end{aligned}\right.\tag{2}\\
&\max |u|<\infty.\tag{3}
\end{align}
Calculate the integral.

Hint: For $u_t=ku_{xx}$ use
\begin{equation}
G(x,y,t)=\frac{1}{\sqrt{4\pi kt}}\exp (- (x-y)^2/4kt).
\tag{4}
\end{equation}
To calculate integral make change of variables and use $\erf(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-z^2}\,dz$.

6
Term Test 1 / P4 Night
« on: February 15, 2018, 08:23:58 PM »
Consider the PDE  with boundary conditions:
\begin{align}
&u_{tt}+c^2u_{xxxx}  + a u=0,\qquad&&0<x<L, \tag{1}\\
&u|_{x=0} =u_{x}|_{x=0}=0,\tag{2}\\
&u_{xx}|_{x=L} =u_{xxx}|_{x=0}=0|_{x=L}=0, tag{3}
\end{align}
where  $c>0$ and $a$ are constant. Prove that the energy $E(t)$ defined as
\begin{equation}
E(t)= \frac{1}{2}\int_0^L \bigl( u_t^2 + c^2u_{xx}^2 +au^2)\,dx
\tag{4}
\end{equation}
does not depend on $t$.

7
Term Test 1 / P3 Night
« on: February 15, 2018, 08:21:20 PM »
Find   solution to
\begin{align}
&u_{tt}-u_{xx}=0, &&   t>0,\; x>  2-2\sqrt{t+1} \tag{1}\\
&u|_{t=0}=0, && x>0,\tag{2}\\
&u_t|_{t=0}=0, && x>0,\tag{3}\\
&u|_{x=2-2\sqrt{t+1}}= t, &&t>0.\tag{4}
\end{align}

8
Term Test 1 / P2 Night
« on: February 15, 2018, 08:19:03 PM »
$\newcommand{\erf}{\operatorname{erf}}$
Find solution $u(x,t)$ to
\begin{align}
&u_{tt}-9u_{xx}=18 e^{-x^2},\tag{1}\\
&u|_{t=0}=0, \quad u_t|_{t=0}=0.\tag{2}
\end{align}

Hint: Change order of integration over characteristic triangle.
Use $\erf(x)=\frac{2}{\sqrt{\pi}}\int _0^x e^{-z^2}\,dz$.

9
Term Test 1 / P1 Night
« on: February 15, 2018, 08:18:09 PM »
 Consider the first order equation:
\begin{equation}
u_t +  xt  u_x =  x te^{-t^2/2}.
\tag{1} 
\end{equation}

(a)  Find the characteristic curves and sketch them in the $(x,t)$ plane.

(b) Write the general solution.

(c) Solve  equation (1)  with the initial condition $u(x,0)= x$. Explain why the solution is fully  determined by the initial condition.

10
Term Test 1 / P5
« on: February 15, 2018, 07:08:45 PM »
$\newcommand{erf}{\operatorname{erf}}$
Find the solution $u(x,t)$ to
\begin{align}
&u_t=u_{xx} && -\infty<x<\infty, \ t>0,
\tag{1}\\[2pt]
&u|_{t=0}=e^{-|x|}
\tag{2}\\
&\max |u|<\infty.
\tag{3}
\end{align}
Calculate the integral.


Hint: For $u_t=ku_{xx}$ use
\begin{equation}
G(x,y,t)=\frac{1}{\sqrt{4\pi kt}}\exp (- (x-y)^2/4kt).
\tag{4}
\end{equation}
To calculate integral make change of variables and use $\erf(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-z^2}\,dz$.

11
Term Test 1 / P4
« on: February 15, 2018, 07:06:24 PM »
Consider the PDE  with boundary conditions:
\begin{align}
&u_{tt}-c^2u_{xx}  + a u =0,\qquad&&0<x<L,\tag{1}\\
&(u_x -\alpha u_{tt})|_{x=0}=0,\tag{2}\\
&(u_x +\beta u_{tt})|_{x=L}=0\tag{3}
\end{align}
where  $c>0$, $\alpha>0$, $\beta>0$ and $a$ are constant. Prove that the energy $E(t)$ defined as
\begin{equation}
E(t)= \frac{1}{2}\int_0^L \bigl( u_t^2 + c^2u_{x}^2 +au^2)\,dx +\frac{\alpha c^2}{2}u_t(0,t)^2+
\frac{\beta c^2}{2}u_t(L,t)^2
\tag{4}
\end{equation}
does not depend on $t$.

12
Term Test 1 / P3
« on: February 15, 2018, 07:05:01 PM »
Find  continuous solution to
\begin{align}
&u_{tt}-4u_{xx}=0, &&   t>0, x>-t,\tag{1}\\
&u|_{t=0}=0, && x>0,\tag{2}\\
&u_t|_{t=0}=0, && x>0,\tag{3}\\
&u_x|_{x=-t}= \sin(t), &&t>0.\tag{4}
\end{align}

13
Term Test 1 / P2
« on: February 15, 2018, 07:03:41 PM »
Find solution $u(x,t)$ to
\begin{align*}
&u_{tt}-4u_{xx}= \frac{8}{x^2+1},\tag{1}\\
&u|_{t=0}=0, \quad u_t|_{t=0}=0.\tag{2}
\end{align*}

Hint: Change order of integration over characteristic triangle

14
Term Test 1 / P1
« on: February 15, 2018, 06:56:34 PM »
Consider the first order equation:
\begin{equation}
u_t +  3(t^2-1)  u_x =  6t^2.
\tag{1} 
\end{equation}

(a)  Find the characteristic curves and sketch them in the $(x,t)$ plane.

(b)  Write the general solution.

(c) Solve  equation (1)  with the initial condition $u(x,0)= x$. Explain why the solution is fully  determined by the initial condition.


15
Term Test 1 / P4-Morning
« on: February 15, 2018, 05:12:34 PM »
Find the general solution for equation
\begin{equation*}
y''(t)+8y'(t)+25y(t)=9 e^{-4t}+ 104\sin(3t).
\end{equation*}

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