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Messages - Victor Ivrii

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Final Exam / FE-P7
« on: April 11, 2018, 02:47:10 PM »
Solve using (partial) Fourier transform with respect to $y$
&\Delta u:=u_{xx}+u_{yy}=0, &&x>0,\label{7-1}\\
&u_x|_{x=0}= h(y),\label{7-2}\\
&u\to 0 &&\text{as   }x\to +\infty\label{7-3}
with $h(y)=\frac{4y}{(y^2+1)^2}$.

Hint. Fourier transform of $g(y)=\frac{2}{y^2+1}$ is $\hat{g}=e^{-|\eta|}$ and $h(y)=-g'(y)$.

Final Exam / FE-P6
« on: April 11, 2018, 02:46:28 PM »
Solve as $t>0$
&u_{tt}-\Delta u  =0, \label{6-1}\\
\left\{\begin{aligned} &r^{-1}\sin(r) &&r:=\sqrt{x^2+y^2+z^2}<\pi,\\
&0 &&r\ge \pi,\end{aligned}\right.\qquad \label{6-2}
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}.

Final Exam / FE-P5
« on: April 11, 2018, 02:43:36 PM »
 Consider Laplace equation in the half-strip
&u_{xx} +u_{yy}=0 \qquad  y>0, \ 0 < x< \frac{\pi}{2} \label{5-1}\\
&u_x  (0,y)=u_x(\frac{\pi}{2}, y)=0,\label{5-2}\\
with $g(x)=1$  and condition $\max |u|<\infty$.

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

Final Exam / FE-P4
« on: April 11, 2018, 02:41:32 PM »
Consider the Laplace equation in the sector
&u_{xx} +  u_{yy} =0\qquad &&\text{in    } \frac{1}{4} \le x^2+y^2 < 4, y>0  ,
\label{4-1} \\
& u =1\qquad &&\text{for  }  x^2+y^2=4,\label{4-2}\\
& u =-1\qquad &&\text{for  }  x^2+y^2=\frac{1}{4},\label{4-3}\\
&  u=0 &&\text{for\ \ }  y=0 ,\label{4-4}
 where $\theta$ is a polar angle.
a  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 $P$.

b  Solve the eigenvalue problem for $P$ and find all eigenvalues.

c  Solve the differential equation  for $R$.

d  Find the solution $u$ of (\ref{4-1})--(\ref{4-4}).

Final Exam / FE-P3
« on: April 11, 2018, 02:36:13 PM »
Solve by the method of separation of variables
& u_{tt}-   u_{xx}+ 4u =0,\qquad 0<x<\pi , \; t>0,\label{3-1}\\[2pt]
& u (0,t)= u (\pi ,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 $f(x)=0$    and  $g(x)=x^2-\pi x$.  Write the answer in terms of  Fourier series.

Final Exam / FE-P1
« on: April 11, 2018, 02:35:31 PM »
Solve by the method of characteristics the BVP for a wave equation
&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}
with $f(x)=4e^{-2x}$,   $g(x)=16e^{-2x}$ and  $h(t)=e^{-8t}$. You need to find a continuous solution.

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

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

APM346––Home Assignments / Re: Question 9 from 4.2P
« on: April 11, 2018, 03:54:10 AM »
MathJax produced this output because of unbalanced { . I fixed them.

You are trying your best to make things as complicated as possible. We considered it during Review lecture. After we found $X'(l(/X(l)$ we can find $T(t)$ from the boundary condition, then plug it into equation $T''=-\lambda T$ and get extremely simple equation to $\omega =\sqrt{\lambda}$.

MAT244--Misc / Re: Periodic Solutions on final?
« on: April 10, 2018, 01:04:28 AM »
Will periodic Solutions and Limit cycles be  there on the final exam?
The scope of final exam was published in the announcements. Read and search before posting. This topic is locked for a good.

MAT244--Announcements / Re: Review Lecture and TT2 papers
« on: April 09, 2018, 12:52:37 PM »
Who graded whart

Q1 - Francis
Q2 - Malors
Q3 - Li
Q4 - Christopher

Web Bonus Problems / Re: Week 13 -- BP3, 4, 5, 6
« on: April 09, 2018, 10:54:40 AM »
I added tags to comment

(A): multiple errors: it should be $=\exp \bigl[\nu \bigr( \log|x+iy|+i\arg (x+iy)\bigr)\bigr]$. First factor $\nu$ is applied to everything second, we do not allow $(x+iy)$ to cross $(-\infty,0]$; so angle is defined uniquely and it rans from $-\pi$ to $\pi$.
Anyway: $[(x\pm i \varepsilon)^\nu]' =\nu (x\pm i \varepsilon)^{\nu-1}$ from complex variables and here $\nu$ could be even complex.

As $\varepsilon\to +0$, there is a limit in $\mathscr{D}'$ (even in $L^1_{loc}$) of $(x\pm i \varepsilon)^\nu$ provided $\Re\nu >-1$; but then there exists a limit of its derivative, i.e. $\nu (x\pm i \varepsilon)^{nu-1}$ and we can divide by $\nu\ne 0$. So we defined $(x\pm i \varepsilon)^{\nu}$ as long as $\Re \nu >-2$ and $\nu \ne -1$.

Repeating, we define $(x\pm i \varepsilon)^{\nu}$ as long as $\Re \nu >-3$ and $\nu \ne -1,-2$. ... and so on... as $\nu\ne -1,-2,\ldots$.

Remark. To mitigate the latter restriction, $f_\nu ^\pm :=\frac{(x\pm i0)^\nu}{\Gamma(\nu+1)}$ could be considered where $\Gamma$ is Euler's $\Gamma$-function; it has simple poles at $0,-1,-2,\ldots$ and $\Gamma(\nu+1)=\nu\Gamma(\nu)$.

(B), (C) are out of the window: they do not follow; also what is $x^{-\nu}$ for $x<0$ and $\nu\notin\mathbb{Z}$? What is $\delta^{\nu-1}$ for $\nu\ne 1,2,\ldots$? We can define those but a posteriori.

I suggested a simple way: look at $\log (x\pm i0)$ as $x>0$ and $x<0$; obviously $\log (x\pm i0)=\log|x|$ as $x>0$ and $\log (x\pm i0)=\log|x|\pm i\pi$ as $x<0$. In other words $\log (x\pm i0)=\log |x| \pm i\pi \theta(-x)$. Differentiating in $\mathscr{D}'$ we get
(x\pm i0)^{-1}= (\log |x|)' +\pm i \pi (\theta (-x))'=x^{-1} \mp i\pi \delta(x)
where (see other of bonus problems, $(\log |x|)' =x^{-1}$ in vp sence, and
$(\theta (-x))'=-\delta(x)$.

PS Gelfand--Shilov 1--6 (+coauthors in higher volumes) is a truly remarkable book, but IMHO, sometimes they go too far. F.e. considering F.T. of distributions not in $\mathscr{S}'$ they get distributions over some classes of the entire analytic functions , in particular they get $\delta (x-c)$ with any $c\in \mathbb{C}$ (which is definitely a perversion). Unfortunately, G.Shilov died too young (at 58) and I never met him.  I.Gelfand was a great mathematician.

Quiz-B / Re: Quiz-B P2
« on: April 08, 2018, 07:44:48 PM »
, everything is much simpler : There is a very standard formula (basically binomial for derivatives)
(uv)^{(n)}= \sum_{j=0}^n \frac{n!}{k!((n-j)!} u^{(j)}v^{(n-j)}.
\langle x^k\delta^{(n)} ,\varphi\rangle = \langle \delta^{(n)} ,x^k\varphi\rangle=
(-1)^n(x^k \varphi)^{(n)} (0) = (-1)^n\Bigl[\sum_{j} (x^k)^{(j)} \varphi ^{(n-j)}\Bigr](0).
However,  $j>k\implies (x^k)^{(j)}=0$, $j\le k\implies  (x^k)^{(j)}=\frac{k!}{(k-j)!}x^{k-j}$ and therefore in (**) only term with $j=k$ is not $0$. So we get
(-1)^n k! \varphi^{(n-k)}(0) = (-1)^{k}k!\langle \delta^{(n-k)},\varphi\rangle.


APM346––Home Assignments / Re: Question 9 from 4.2P
« on: April 08, 2018, 07:29:23 PM »
Look at place I marked OK. Plug it into boundary condition as $x=l$ and find $T(t)$ from here, and plug it into the second order equation *

Basically you already did everything

Quiz-B / Re: Quiz-B P2
« on: April 08, 2018, 02:22:58 PM »
In (*) there is no $f$ (or, rather, very specific $f$)

Web Bonus Problems / Re: Week 13 -- BP3, 4, 5, 6
« on: April 08, 2018, 01:54:42 PM »
Andrew, you right with powers but losing numerical factor.
Zhongnan  is right (copy-paste-not-clening-up error). I corrected

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