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Messages - Victor Ivrii

Pages: 1 2 [3] 4 5 ... 89
31
FE / Re: Final exam coverage
« on: November 27, 2016, 01:16:05 AM »
No. No.Nothing from "calculus of variations" or "distributions" on FE, but Q8 can contain former and some other bonuses the latter

32
Chapter 8 / Re: HA 10, problem 3c
« on: November 24, 2016, 06:01:44 AM »
Shentao is correct.

33
Chapter 8 / Re: Typo
« on: November 24, 2016, 06:00:41 AM »
Thanks. There are also more serious omissions. Corrected.

34
Chapter 8 / Re: Typo
« on: November 23, 2016, 10:25:09 AM »
Indeed.

35
Chapter 8 / Re: HA 10, problem 3a
« on: November 22, 2016, 09:37:28 PM »
Now it is correct

36
Chapter 8 / Re: HA 10, problem 3a
« on: November 22, 2016, 05:51:03 PM »
Laplacian is not $0$, and what $g$ you are looking for?

37
Chapter 8 / Re: HA 10, problem 3a
« on: November 22, 2016, 09:52:48 AM »
Solution is a harmonic polynomial but not necessarily homogeneous. On the other hand it must have prescribed boundary value (as $x^2+y^2+z^2=R^2$).

38
Chapter 8 / Re: HA 10, trouble with question 2
« on: November 22, 2016, 09:50:15 AM »
For some $l,m$ you need multiply by $\sin(\phi)$

39
Chapter 9 / Re: Are these typos?
« on: November 20, 2016, 10:35:52 AM »
1) Yes, it is typo. Corrected

2) No, since the wave equation is in the form
$$
\Delta u - c^{-2}u_{tt}= fe^{i\omega t}.
$$

40
TT2 / Re: TT2-P5
« on: November 18, 2016, 07:31:06 AM »
If we are talking about functions we do not care about their values in a few particular points (distribution will be another matter but for them "value at some particular point" is not defined). Heaviside function could be defined at $0$ as 0 (to make it semi-continuous from the left), $1$ (to make it semi-continuous from the right), or $\frac{1}{2}$ (as a half-sum of the limits).

41
TT2 / TT2-P5
« on: November 17, 2016, 03:28:26 AM »
Find Fourier transforms of the  functions
\begin{equation}
f_\pm (x)= e^{-\varepsilon |x|}\theta(\pm x)
\end{equation}
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 $h(x)= f_+(x)- f_-(x)$.

42
TT2 / TT2-P4
« on: November 17, 2016, 03:27:07 AM »
Consider Laplace equation in the sector
\begin{equation}
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}
\end{equation}
with the Dirichlet boundary conditions as $\theta=0$ and $\theta=\frac{3}{2}\pi$
\begin{equation}
u|_{\theta=0}=u|_{\theta=\frac{3}{2}\pi}=0\label{4-2}\\
\end{equation}
and the Dirichlet boundary condition as $r=8$
\begin{equation}
u|_{r=8}=1.\label{4-3}
\end{equation}
Using separation of variables find solution as a series.

43
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}

44
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.

45
TT2 / TT2-P1
« on: November 17, 2016, 03:22:16 AM »
Solve by Fourier method
\begin{align}
& u_{tt}-u_{xx}=0\qquad 0<x<\pi,\label{1-1}\\
& u|_{x=0}=-u|_{x=\pi},\qquad u_x|_{x=0}=-u_x|_{x=\pi},\label{1-2}\\
&u| _{t=0}=1,\qquad u_t|_{t=0}=0.\label{1-3}
\end{align}

Hint: $\lambda_n\ge 0$. Also remember how solution looks like in the case of double eigenvalues.

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