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Messages - Victor Ivrii

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16
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 $\hat{g}=e^{-|\eta|}$.

17
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}
&u(x,y,z,0)=\left\{\begin{aligned} &1\quad &&r:=\sqrt{x^2+y^2+z^2}<1,\\
&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
\begin{equation}
r  u_{rr}+2 u_r= (r u)_{rr}.
\label{6-3}
\end{equation}

18
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.


19
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}).



20
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.

21
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 $\displaystyle{\erf(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-z^2}\,dz}$.

22
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.

23
Q-bonus / Re: Q-bonus
« on: December 13, 2016, 08:11:21 AM »
The solution suggested is a perfect one. If we do not use the fact that $L$ does not depend explicitly on $x$ we end up with the second order equation which after transformations becomes ... I am too lazy to do it directly, so I do it from the fact that $u(1+u'^2)=C$: Differentiating we $u'(1+u'^2)+2 u u'u''=0$ or $2uu''+u'^2+1=0$. This is done only to show the correct 2-nd order equation.

24
Chapter 4 / Re: problem involving fourier series
« on: December 13, 2016, 07:11:19 AM »
You make your conclusion basing on the equation and the boundary conditions. The easiest way is to change variables $x'=x+\frac{\pi}{2}$; then $\lambda_n=n^2$ and $X_n(x')=\cos (nx')$ from the standard problem; returning to old coordinates 
\begin{equation}
X_n(x)=\cos (nx+\frac{\pi 2}{2})=\left\{\begin{aligned}
&(-1)^m \cos (2mx) && n=2m,\\
&(-1)^{m+1}\sin ((2m+1)x) && n=2m+1.
\end{aligned}\right.
\end{equation}
We can drop sign at our wish. So, you got correctly only half of the eigenfunctions, but in this particular problem it would lead to a correct solution. Indeed, since "initial" functions $x^2$ and $0$ are even only $\cos(2mx)$ would be in the end. However, it will cost you points!

If we observe that the functions $x^2$ and $0$ are even, and thus solution must be even, we can reduce interval to $(0,\frac{\pi}{2})$ and set conditions $u_{x=0}=u_{x=\pi/2}=0$ which would lead to  $\lambda_m=4m^2$ and $X_m=\cos (2mx)$ from the beginning.

25
Chapter 2 / Re: energy integrals
« on: December 11, 2016, 08:31:26 AM »
Take values at infinity equal 0 (we assume that solution decays there)

26
Q-bonus / Re: Q-bonus
« on: December 02, 2016, 03:17:16 AM »
While I did not ask too calculate an integral on the Quiz, can you do this?

You can either write
$$x=  \int \sqrt{\color{brown}{\frac{c^2 u}{1-c^2 u}}}\,du$$
and make a standard substitution (colored expression $=t^2$) or simply use Mathematica, Maple or just WolframAlpha:
http://www.wolframalpha.com/input/?i=integrate+sqrt(c%5E2+u%2F(1-c%5E2+u))+du
Not that the formula is of any good,  but taking $c=1$ f.e. (you can achieve it by $u:=c^2u$) you can
http://www.wolframalpha.com/input/?i=plot+(integrate+sqrt(1+u%2F(1-+u))+du)
which results in the plot, $x$ is vertical (up), $u$ horizontal (to the right)  and you are looking for the red line

27
APM346--Announcements / Term Mark Calculated
« on: December 01, 2016, 04:43:29 AM »
Term=Q+Test1+Test2 Remind: possible points = 60

STATISTICS
Count56
Minimum Value8.00
Maximum Value63.00
Range55.00
Average44.09
Median45.00
Standard Deviation11.82
====================
 GRADE DISTRIBUTION
 
Greater than 1004
90 - 10010
80 - 8910
70 - 7910
60 - 697
50 - 598
40 - 495
30 - 391
20 - 290
10 - 191

28
APM346--Announcements / Re: Q1--7 graded
« on: November 30, 2016, 02:48:26 PM »
Q7 has been graded. Marks on Blackboard.

Q7
STATISTICS
Count34
Minimum Value1.00
Maximum Value4.00
Range3.00
Average2.97
Median3.00
Standard Deviation0.87

GRADE DISTRIBUTION
Greater than 1000
90 - 1007
80 - 897
70 - 7910
60 - 693
50 - 594
40 - 490
30 - 390
20 - 293

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

Q= Q1+... +Q7 with two worst dropped

STATISTICS
Count56
Minimum Value2.00
Maximum Value22.00
Range20.00
Average15.46
Median16.75
Standard Deviation4.82
GRADE DISTRIBUTION
 
Greater than 1006
90 - 10015
80 - 8911
70 - 798
60 - 693
50 - 596
40 - 492
30 - 390
20 - 294
10 - 191

29
Chapter 10 / Re: General extremisation of functionals of arc length
« on: November 28, 2016, 06:11:26 PM »
Correct. As it was mentioned both in the lectures and in textbook, if $L=L(u,u')$, so does not depend explicitly of $x$ then one does not need to write Euler-Lagrange equation, but a simpler one $u'L_{u'}=L=C$.It is equivalent to "multiply by $u'$ and integrate".

If $f(u)=u$ it is an area of the surface of the revolution of $x=f(u)$ around ``$x$''


30
APM346--Announcements / Term Test 2 graded
« on: November 28, 2016, 08:36:58 AM »
UPDATED

Count53
Minimum Value7.00
Maximum Value21.00
Range14.00
Average15.14
Median16.00
Standard Deviation3.95

GRADE DISTRIBUTION
Greater than 1002
90 - 10014
80 - 8912
70 - 794
60 - 698
50 - 597
40 - 494
30 - 392

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

Cumulative (in % to 60 )

STATISTICS
Count56
Minimum Value20.00
Maximum Value105.00
Range85.00
Average74.17
Median77.50
Standard Deviation19.22

-----------
GRADE DISTRIBUTION
Greater than 1004
90 - 1008
80 - 8914
70 - 799
60 - 697
50 - 598
40 - 493
30 - 392
20 - 291

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