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### Topics - Victor Ivrii

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1
##### APM346--Announcements / FE and FM
« on: December 20, 2016, 06:14:23 AM »
Final Exam:
Count   53
Minimum Value   17.33
Maximum Value   40.00
Range   22.67
Average   27.37
Median   28.00
Standard Deviation   6.15

This is what I submitted

2
##### FE / FE7
« on: December 13, 2016, 08:01:10 PM »
Solve using (partial) Fourier transform with respect to $y$
\begin{align}
&\Delta u:=u_{xx}+u_{yy}=0, &&x>0,\label{7-1}\\
&u|_{x=0}= g(y),\label{7-2}\\
&\max |u|<\infty\label{7-3}
\end{align}
with $g(y)=\frac{2}{y^2+1}$.

Hint. Fourier transform of $g(y)$ is $\hat{g}=e^{-|\eta|}$.

3
##### FE / FE6
« on: December 13, 2016, 08:00:15 PM »
Solve as $t>0$
\begin{align}
&u_{tt}-\Delta u  =0, \label{6-1}
\end{align}
with initial conditions
\begin{align}
&0 &&r\ge 1,\end{aligned}\right.\qquad u_t(x,y,z,0)=0\label{6-2}
\end{align}
and solve by a separation of variables.

Hint. Use spherical coordinates, observe that solution must be spherically symmetric: $u=u(r,t)$ (explain why).
Also, use equality

r  u_{rr}+2 u_r= (r u)_{rr}.
\label{6-3}

4
##### FE / FE5
« on: December 13, 2016, 07:58:11 PM »
Consider Laplace equation in the half-strip
\begin{align}
&u_{xx} +u_{yy}=0 \qquad  y>0, \ 0 < x< \pi \label{5-1}
\end{align}
with the boundary conditions
\begin{align}
&u  (0,y)=u(\pi, y)=0,\label{5-2}\\
&u_y(x,0)=g(x)\label{5-3}
\end{align}
with $g(x)=\cos(x)$    and condition $\max |u|<\infty$.

• Write the associated eigenvalue problem.
• Find all  eigenvalues and corresponding eigenfunctions.
• Write the solution in the form of  a series expansion.

5
##### FE / FE4
« on: December 13, 2016, 07:55:55 PM »
Consider the Laplace equation in the sector
\begin{align}
&u_{xx} +  u_{yy} =0\qquad &&\text{in  } x^2+y^2 < 16, x> -\sqrt{3}|y|  ,
\label{4-1}
\end{align}
with the boundary conditions
\begin{align}
& u =1\qquad &&\text{for  }  x^2+y^2=16,\label{4-2}\\
& u=0 &&\text{for  }  x=-\sqrt{3}|y| .\label{4-3}
\end{align}

• Look for solutions $u$ in the form of  $u(r,\theta)= R(r) P(\theta)$  (in polar coordinates) and derive a set of  ordinary differential equations for $R$ and $P$. Write the correct  boundary conditions for $P$.
• Solve the eigenvalue problem for $P$ and find all eigenvalues.
• Solve the differential equation  for $R$.
• Find the solution $u$ of (\ref{4-1})--(\ref{4-3}).

6
##### FE / FE3
« on: December 13, 2016, 07:51:04 PM »
Solve by the method of separation of variables
\begin{align}
&4 u_{tt}-  u_{xx}=0,\qquad 0<x<2, \; t>0,\label{3-1}\\[2pt]
& u (0,t)= u (2,t)=0,\label{3-2}\\[2pt]
& u(x,0)=f(x),\label{3-3}\\[2pt]
& u_t(x,0)=g(x)\label{3-4}\end{align}
with f(x)=\left\{\begin{aligned} &x &&0<x<1,\\ &2-x &&1<x<2,\end{aligned}\right.    and  $g(x)=0$.  Write the answer in terms of  Fourier series.

7
##### FE / FE2
« on: December 13, 2016, 07:50:03 PM »
$\newcommand{\erf}{\operatorname{erf}}$
Solve  IVP for the heat equation
\begin{align}
&4u_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 $f(x)=xe^{-x^2}$.

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

8
##### FE / FE1
« on: December 13, 2016, 07:49:01 PM »
Solve by the method of characteristics the BVP for a wave equation
\begin{align}
&u_{tt}-  9 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 (0,t)= h(t)\label{1-4}
\end{align}
with $f(x)=4\cos(x)$,    $g(x)=6\sin (x)$ and  $h(t)=\sin (3t)$. You need to find a continuous solution.

9
##### APM346--Announcements / Term Mark Calculated
« on: December 01, 2016, 04:43:29 AM »
Term=Q+Test1+Test2 Remind: possible points = 60

STATISTICS
 Count 56 Minimum Value 8 Maximum Value 63 Range 55 Average 44.09 Median 45 Standard Deviation 11.82
====================

 Greater than 100 4 90 - 100 10 80 - 89 10 70 - 79 10 60 - 69 7 50 - 59 8 40 - 49 5 30 - 39 1 20 - 29 0 10 - 19 1

10
##### APM346--Announcements / Term Test 2 graded
« on: November 28, 2016, 08:36:58 AM »
UPDATED

 Count 53 Minimum Value 7 Maximum Value 21 Range 14 Average 15.14 Median 16 Standard Deviation 3.95

 Greater than 100 2 90 - 100 14 80 - 89 12 70 - 79 4 60 - 69 8 50 - 59 7 40 - 49 4 30 - 39 2

=================

Cumulative (in % to 60 )

STATISTICS
 Count 56 Minimum Value 20 Maximum Value 105 Range 85 Average 74.17 Median 77.5 Standard Deviation 19.22

-----------
 Greater than 100 4 90 - 100 8 80 - 89 14 70 - 79 9 60 - 69 7 50 - 59 8 40 - 49 3 30 - 39 2 20 - 29 1

11
##### TT2 / TT2-P5
« on: November 17, 2016, 03:28:26 AM »
Find Fourier transforms of the  functions

f_\pm (x)= e^{-\varepsilon |x|}\theta(\pm x)

and write these function as a Fourier integrals, where $\theta$ is a Heaviside function: $\theta(t)=1$ for $t>0$ and $\theta(t)=0$ for $t<0$.

Bonus (1pt).
Write Fourier transforms of the  functions $g(x)=f_+(x)+ f_-(x)$ and $h(x)= f_+(x)- f_-(x)$.

12
##### TT2 / TT2-P4
« on: November 17, 2016, 03:27:07 AM »
Consider Laplace equation in the sector

u_{rr} +\frac{1}{r}u_r +\frac{1}{r^2}u_{\theta\theta}=0 \qquad  r<8,\,0<\theta<\frac{3}{2}\pi \label{4-1}

with the Dirichlet boundary conditions as $\theta=0$ and $\theta=\frac{3}{2}\pi$

u|_{\theta=0}=u|_{\theta=\frac{3}{2}\pi}=0\label{4-2}\\

and the Dirichlet boundary condition as $r=8$

u|_{r=8}=1.\label{4-3}

Using separation of variables find solution as a series.

13
##### TT2 / TT2-P3
« on: November 17, 2016, 03:25:28 AM »
Using Fourier method find eigenvalues and eigenfunctions of Laplacian in the rectangle $\{0<x<a,\, 0<y<b\}$ with Neumann 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|_{y=0}=u_y|_{y=b}=0.\label{3-2}
\end{align}

14
##### TT2 / TT2-P2
« on: November 17, 2016, 03:24:00 AM »
Solve
\begin{align}
&u_{xx}+u_{yy}=0\qquad &-\infty<x<\infty, \ 0<y<\infty,\label{2-1}\\
&u|_{y=0}=g(x)=\left\{\begin{aligned} &1 &&|x|<1,\\ &0 &&|x|>1,\end{aligned}\right.\label{2-2}\\
&\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.

15
##### TT2 / TT2-P1
« on: November 17, 2016, 03:22:16 AM »
Solve by Fourier method
\begin{align}
Hint: $\lambda_n\ge 0$. Also remember how solution looks like in the case of double eigenvalues.