### Show Posts

This section allows you to view all posts made by this member. Note that you can only see posts made in areas you currently have access to.

### Messages - Victor Ivrii

Pages: 1 [2] 3 4 ... 89
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

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

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

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.

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

 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

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

Q7
STATISTICS
 Count 34 Minimum Value 1 Maximum Value 4 Range 3 Average 2.97 Median 3 Standard Deviation 0.87

GRADE DISTRIBUTION
 Greater than 100 0 90 - 100 7 80 - 89 7 70 - 79 10 60 - 69 3 50 - 59 4 40 - 49 0 30 - 39 0 20 - 29 3

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

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

STATISTICS
 Count 56 Minimum Value 2 Maximum Value 22 Range 20 Average 15.46 Median 16.75 Standard Deviation 4.82
GRADE DISTRIBUTION

 Greater than 100 6 90 - 100 15 80 - 89 11 70 - 79 8 60 - 69 3 50 - 59 6 40 - 49 2 30 - 39 0 20 - 29 4 10 - 19 1

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

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

GRADE DISTRIBUTION
 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

-----------
GRADE DISTRIBUTION
 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

Pages: 1 [2] 3 4 ... 89