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Topics - Victor Ivrii

Pages: 1 2 [3] 4 5 ... 42
31
Quiz-7 / Q7-T0501
« on: March 30, 2018, 12:22:31 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.
$$
\left\{\begin{aligned}
&\frac{dx}{dt} = 1 - y\\
&\frac{dy}{dt} = x^2 - y^2
\end{aligned}\right.$$

32
Quiz-7 / Q7-T0401
« on: March 30, 2018, 12:19:48 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.
$$
\left\{\begin{aligned}
&\frac{dx}{dt} = 1 - xy\\
&\frac{dy}{dt} = x - y^3
\end{aligned}\right.$$

33
Quiz-7 / Q7-T0301
« on: March 30, 2018, 12:18:39 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.
$$
\left\{\begin{aligned}
&\frac{dx}{dt} = y +x(1-x^2 - y^2)\\
&\frac{dy}{dt} = -x + y(1-x^2 - y^2)
\end{aligned}\right.$$

34
Quiz-7 / Q7-T0201
« on: March 30, 2018, 12:17:11 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.
$$
\left\{\begin{aligned}
&\frac{dx}{dt} = x + x^2 + y^2\\
&\frac{dy}{dt} = y - xy
\end{aligned}\right.$$


35
Quiz-7 / Q7-T0101
« on: March 30, 2018, 12:14:55 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.
$$
\left\{\begin{aligned}
&\frac{dx}{dt} = x - x^2 - xy\\
&\frac{dy}{dt} = 3y - xy - 2y^2
\end{aligned}\right.$$


36
APM346--Announcements / Bonus Quiz
« on: March 30, 2018, 07:14:20 AM »
During Lecture coming Monday (April 2) I will offer a bonus quiz--which means that the mark for it will be simply added to the final mark)
There will be two questions:

1. one is problems from
http://www.math.toronto.edu/ivrii/PDE-textbook/Chapter10/S10.P.html

(based on the material of http://www.math.toronto.edu/ivrii/PDE-textbook/Chapter10/S10.1.html)


2. one based on definitions from
http://www.math.toronto.edu/ivrii/PDE-textbook/Chapter11/S11.1.html


37
MAT244--Announcements / Final Exam coverage
« on: March 29, 2018, 05:08:26 AM »
Chapters 2,3,4, 7--what was covered by Quizzes and Term Tests

Chapter 9: Sections 9.1--9.5. In Nonlinear system pay attention that there are two cases: general and integrable. See my explanations at http://forum.math.toronto.edu/index.php?topic=1100.0

38
Due to holiday coming Friday, March 30:
You may attend any Tutorial Tuesday--Thursday and write Quiz 7 with them
Consult http://www.math.toronto.edu/courses/mat244h1/20181/timetable.html to see times and locations

Each of these rooms can accommodate at least 15 students, and MP134 and GB221 twice as much. Note that Students from the "host" tutorial sections have priority.
For those of you who could not use this arrangement, there will be tutorial and quiz April 5, but it will  be a completely different Quiz. Sorry for this inconvenience.

39
Term Test 2 / TT2--P5
« on: March 23, 2018, 06:18:55 AM »
Find Fourier transforms of the  function
\begin{equation}
f(x)=\cos^2(x)e^{-|x|}
\end{equation}
and write this function as a Fourier integral.

40
Term Test 2 / TT2--P4N
« on: March 23, 2018, 06:18:18 AM »
Consider Laplace equation in the disc with a cut
\begin{equation}
u_{rr} +\frac{1}{r}u_r +\frac{1}{r^2}u_{\theta\theta}=0 \qquad  r<9,\,0<\theta<2\pi
\label{4-1}\end{equation}
with the Neumann boundary conditions as $\theta=0$ and $\theta=2\pi$}
\begin{equation}
u_\theta|_{\theta=0}=u_\theta|_{\theta=2\pi}=0\label{4-2}
\end{equation}
and the Dirichlet boundary condition as $r=9$
\begin{equation}
u|_{r=9}=\pi-\theta.\label{4-3}
\end{equation}
Using separation of variables find solution as a series.

41
Term Test 2 / TT2--P3N
« on: March 23, 2018, 06:15:52 AM »
Using Fourier method find eigenvalues and eigenfunctions of Laplacian in the rectangle $\{0<x<a,\, 0<y<b\}$ with the boundary conditions:
\begin{align}
&u_{xx}+u_{yy}=-\lambda u\qquad 0<x<a,\ 0<y<b,\label{3-1}\\[3pt]
&u|_{x=0}=u_x|_{x=a}=u|_{y=0}=u_y|_{y=b}=0.\label{3-2}
\end{align}

42
Term Test 2 / TT2--P2N
« on: March 23, 2018, 06:15:25 AM »
Solve
\begin{align}
&u_{xx}+u_{yy}=0\qquad &-\infty<x<\infty, \ 0<y<1,\label{2-1}\\
&u|_{y=0}=0,\label{2-2a} \\
&(u_y+\alpha u)|_{y=1}=g(x)=\left\{\begin{aligned} &1 &&|x|<1,\\ &0 &&|x|>1,\end{aligned}\right.
\label{2-2b}\\
&\max|u|<\infty. \label{2-3}\end{align}

Hint: Use partial Fourier transform with respect to $x$. Write solution as a Fourier integral without calculating it.

Find restriction to $\alpha$ , so that there will be no singularities.

43
Term Test 2 / TT2--P1N
« on: March 23, 2018, 06:14:32 AM »
Solve by Fourier method
\begin{align}
& u_{tt}-u_{xx}=0\qquad 0<x<\pi,\label{1-1}\\
& u_x|_{x=0}= 0,\qquad (u_x+\alpha u)|_{x=\pi}=0\label{1-2}\\
&u| _{t=0}=\cos (x),\qquad u_t|_{t=0}=0\label{1-3}
\end{align}
with $\alpha\in \mathbb{R}$.

Hint: We know that $\lambda_n$ are real but since we do not know the sign of $\alpha$ we do not know if it all $\lambda_n\ge 0$; so you must consider the case of some of $\lambda_n<0$.

Note: Only find equations for eigenvalues.

44
Term Test 2 / TT2--P4
« on: March 23, 2018, 06:12:46 AM »
Consider Laplace equation in the disc with a cut
\begin{equation}
u_{rr} +\frac{1}{r}u_r +\frac{1}{r^2}u_{\theta\theta}=0 \qquad  r<9,\,0<\theta<2\pi
\tag{1}
\end{equation}
with the Dirichlet boundary conditions as $\theta=0$ and $\theta=2\pi$}
\begin{equation}
u|_{\theta=0}=u|_{\theta=2\pi}=0
\tag{2}
\end{equation}
and the Neumann boundary condition as $r=9$
\begin{equation}
u_r|_{r=9}=1.
\tag{3}
\end{equation}
Using separation of variables find solution as a series.

45
Term Test 2 / TT2--P3
« on: March 23, 2018, 06:10:04 AM »
Using Fourier method find eigenvalues and eigenfunctions of Laplacian in the rectangle $\{0<x<a,\, 0<y<b\}$ with the boundary conditions:
\begin{align}
&u_{xx}+u_{yy}=-\lambda u\qquad 0<x<a,\ 0<y<b,\label{3-1}\\[3pt]
&u_x|_{x=0}=u_x|_{x=a}=u|_{y=0}=u|_{y=b}=0.\label{3-2}
\end{align}

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