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### Topics - Victor Ivrii

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16
##### Final Exam / FE-P1
« on: April 11, 2018, 02:35:31 PM »
Solve by the method of characteristics the BVP for a wave equation
\begin{align}
&u_{tt}-  16 u_{xx}=0,\qquad 0<x<\infty , \; t>0\label{1-1}\\[2pt]
& u(x,0)=f(x),\label{1-2}\\[2pt]
& u_t(x,0)=g(x),\label{1-3}\\[2pt]
& (u_x  -u )(0,t)= h(t)\label{1-4}
\end{align}
with $f(x)=4e^{-2x}$,   $g(x)=16e^{-2x}$ and  $h(t)=e^{-8t}$. You need to find a continuous solution.

17
##### Final Exam / FE-P2
« on: April 11, 2018, 02:34:19 PM »
$\newcommand{\erf}{\operatorname{erf}}$
Solve  IVP for the heat equation
\begin{align}
&2u_t -   u_{xx}=0,\qquad &&0 <x<\infty,\; t>0,\label{2-1}\\[2pt]
&u|_{x=0}=0,\\
&u|_{t=0}= f(x)\label{2-2}
\end{align}
with $f(x)=e^{-x}$.

Solution should be expressed  through $\displaystyle{\erf(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-z^2}\,dz}$

18
##### APM346--Announcements / Final Exam--Appendix
« on: April 07, 2018, 08:10:24 PM »
On Exam Paper there will be 2 last pages--Appendix

19
##### APM346--Announcements / Sum of all Quizzes
« on: April 05, 2018, 07:08:26 AM »
Quiz 7 on Blackboard

Also Quiz-total =
Code: [Select]
LARGE(Q1:Q7,1)+ LARGE(Q1:Q7,2)+ LARGE(Q1:Q7,3)+ LARGE(Q1:Q7,4)+ LARGE(Q1:Q7,5) (Excel pseudocode), for those who wrote less than 5 quizzes it is undefined and replaced by
Code: [Select]
SUM(Q1:Q7) (nobody submitted more than 1 doctor note for Quizzes, so no adjustment)

Bonus Q will be counted in Bonuses when I count them just before submitting Final Mark

Not calculating for Term Tests since they are pooled with Final Exam

20
##### Web Bonus Problems / Week 13 -- BP3, 4, 5, 6
« on: April 04, 2018, 12:34:15 PM »
$\renewcommand{\Re}{\operatorname{Re}}$
There are several connected bonus problems:
a.
Let $(x\pm i0)^{\nu}= \lim _{\varepsilon\to +0} (x\pm \varepsilon i)^{\nu}$. Prove, it exists for $\Re\nu >-1$.

b. Using equality

[(x\pm \varepsilon i)^{\nu}]'= \nu (x\pm \varepsilon i)^{\nu-1},
\label{eq-11.1.12}

prove that these limits exist (in the sense of distributions) for  $\nu \ne -1,-2,\ldots$.

c. Using

[\ln (x\pm \varepsilon i) ]'=
(x\pm \varepsilon i)^{-1},
\label{eq-11.1.13}

prove that these limits exist (in the sense of distributions) for  $\nu =-1$. Then, using (\ref{eq-11.1.12}), prove that these limits exist (in the sense of distributions) for $\nu=-2,-3,\ldots$.

d. Prove that
\begin{align}
&(x-i0)^{-1} + (x+i0)^{-1}=2 x^{-1},
\label{eq-11.1.14}\\
&(x-i0)^{-1}- (x+i0)^{-1}=2\pi i \delta(x)
\label{eq-11.1.15}
\end{align}
with integral understood in the sense of principal value, use (\ref{eq-11.1.13}) and
\begin{align}
&\ln (x+i0)+\ln (x-i0)= 2\ln |x|,\\
&\ln (x+i0)-\ln (x-i0)= -2\pi i\theta(-x).
\end{align}
You need to justify these formulae.

21
##### Web Bonus Problems / Week 13 -- BP2
« on: April 04, 2018, 12:28:13 PM »
Prove that if $f=\ln |x|$ then
\begin{equation*}
f'(\varphi)= pv \int x^{-1}\varphi (x)\,dx ,
\end{equation*}
where $f'$ is understood in the sense of distributions and the integral is understood as a principal value integral.

22
##### Web Bonus Problems / Week 13 -- BP1
« on: April 04, 2018, 12:27:11 PM »
a. Check that if $\theta(x)$ is a Heaviside function: $\theta(x)=1$ as $x> 0$ and $\theta(x)=0$ as $x\le 0$ then $\theta'(x)=\delta(x)$.

b. Check that if $f(x)$ is a smooth function as $x< a$ and as $x> a$ but has a jump at $a$ then
\begin{equation*}
f'=\overset{\circ}{f}{}'+ \bigl(f(a+0)-f(a-0)\bigr)\delta (x-a),
\end{equation*}
where $f'$ is understood in the sense of distributions and $\overset{\circ}{f}' (x)$ is an ordinary function equal to derivative of $f$ as $x\ne a$. Generalize to piecewise differentiable functions.

23
##### MAT244--Announcements / Review Lecture and TT2 papers
« on: April 04, 2018, 11:27:52 AM »
10-APR-18 13:00--15:00

ES 1050 Earth Sciences Centre

Get your TT2 today MP203 6--9 during lecture breaks -- and after the lecture.

Probably I will not be available between today and Apr 10. I will post if I will come--so follow the forum/blackboard

24
##### APM346--Announcements / Review Lecture
« on: April 04, 2018, 11:23:20 AM »
10-APR-18     10:00--12:00
LM 162      Lash Miller Chemical Labs

25
##### MAT244--Announcements / TT2 papers
« on: April 04, 2018, 09:37:28 AM »
If I recover them from TAs , I will bring all of them (for all 3 sections) to my class 6--9PM, MP203 (but you can get them during the breaks or after the class).

I will post an announcement before class. Otherwise during the survey lecture Apr 10 (time and place TBA)

26
##### Quiz-B / Quiz-B P2
« on: April 02, 2018, 08:26:18 PM »
Simplify (write as the sum of $\delta (x)$, $\delta '(x)$, $\delta''' (x)$, ... with the numerical coefficients)

\sin(x)\delta'''(x).
\tag{1}

27
##### Quiz-B / Quiz-B P1
« on: April 02, 2018, 08:25:31 PM »
If the surface is a surface of revolution $z=u(r)$ with $r^2=x^2+y^2$, then

S=2\pi\int_{D} \sqrt{1+u_r^2}\,rdr .
\tag{1}

Write Euler-Lagrange equation and solve it (find general solution).

28
##### Quiz-7 / Q-T5101
« on: March 30, 2018, 12:25:55 PM »
Problem
a. Determine all critical points of the given system of equations.

b. Find the corresponding linear system near each critical point.

c. Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

d. Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the
nonlinear system.
\left\{\begin{aligned} &\frac{dx}{dt} = (2 + x)( y - x)\\ &\frac{dy}{dt} = (4 - x)( y + x) \end{aligned}\right.

29
##### Quiz-7 / Q7-0901
« on: March 30, 2018, 12:24:45 PM »
a. Determine all critical points of the given system of equations.

b. Find the corresponding linear system near each critical point.

c. Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

d. Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
\left\{\begin{aligned} &\frac{dx}{dt} = (1 + x) \sin (y)\\ &\frac{dy}{dt} = 1 - x - \cos (y) \end{aligned}\right.

30
##### Quiz-7 / Q7-T0801
« on: March 30, 2018, 12:23:34 PM »
a. Determine all critical points of the given system of equations.

b. Find the corresponding linear system near each critical point.

c. Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?

d. Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
\left\{\begin{aligned} &\frac{dx}{dt} = -2x - y -x(x^2 + y^2)\\ &d\frac{dy}{dt} = x - y + y(x^2 + y^2) \end{aligned}\right.

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