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### Messages - Thomas Nutz

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##### Home Assignment Y / Re: Problem 1
« on: December 19, 2012, 02:19:38 PM »
The question says we should do it with a Fourier transform, but why don't I just take
$$u(x,y)=e^{-|y|-ix}$$? That does give me a zero Laplacian and satisfies the boundary condition, or am I wrong here?

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##### Term Test 2 / Re: TT2--Problem 1
« on: December 16, 2012, 01:30:08 PM »
Is eq. (3)
$$(f \star g)'=f\star g'$$ sufficient to prove that $(f \star g)'$ is continuous? I.e. can we assume that the convolution of two continuous functions is continuous, or do we have to take difference quotients as Ian did in his solution?

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##### Misc Math / Re: Mean-value theorem
« on: November 26, 2012, 06:34:06 PM »
I see, thanks.

Are we considering closed domains, i.e. $\Sigma \in \Omega$? I think we have to, otherwise the $max_{\Omega}u \geq max_{\Sigma}u$ does not pull, correct?

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##### Misc Math / Re: Mean-value theorem
« on: November 26, 2012, 01:51:41 PM »
or rather $\nabla G \cdot \frac{\partial}{\partial x}(\vec{v})$?

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##### Misc Math / Re: Mean-value theorem
« on: November 26, 2012, 01:48:06 PM »
Hi,

what is meant by $\frac{\partial G}{\partial v_x}$ (eq. 7 in Lec. 26)? is $\frac{\partial G}{\partial v_x}=\nabla G \cdot (\nabla \cdot \vec{v})$?

Thanks!

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##### Home Assignment 7 / Re: Problem 3
« on: November 17, 2012, 03:21:05 PM »
Are we supposed to derive eq. 19 in lecture notes 24 (laplacian outside the disk) for problem 3b) or can we start from this formula (it given in the context of an exercise)?

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##### Home Assignment Y / Re: Problem 2
« on: November 13, 2012, 05:21:52 PM »
Ok I did
$$u(x,y)=\int_0^{\infty}\hat{u}(\omega,y)\sin(\omega x)d\omega$$
and took the resulting eigenvalue problem as
$$\hat{u}_{yy}-\omega^2\hat{u}=0$$
which yields eigenvalues $\omega=i\pi n$
and eigenfunctions
$$\hat{u}(\omega,y)=A(\omega) e^{i \pi n y}+B(\omega)e^{-i \pi n y}$$
Is that wrong?

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##### Misc Math / Re: Lecture 20 Notes
« on: November 13, 2012, 12:24:36 PM »
In the section \textbf{Laplace equation in half-plane} it says

"The problem $u_{yy}+u_{xx}=0$, $y>0$,  $u(x,0)=g(x)$ [...] is not uniquely solvable". As an example the function $u(x,y)=y$ is given, which satisfies the Laplace equation. But it does obviously not satisfy $u(x,0)=g(x)$, so I don't see how we can conclude immediately that only bounded solutions are unique...

Can anybode help me with this?
Thanks!

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##### Home Assignment 7 / Problem 3
« on: November 12, 2012, 04:40:07 PM »
Dear Professor,

do we need to find a closed expression for the solution in problem 3, or can we express it as a sum over n?

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##### Home Assignment 7 / Re: Problem 2
« on: November 12, 2012, 02:09:39 PM »
Sorry I don't understand. I did find an ODE for $u(r)$, which is solved by Bessel functions. When you ask "try to find solutions", what do you expect me to do?

Thanks again!

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##### Home Assignment 7 / Problem 2
« on: November 12, 2012, 01:13:34 PM »
Hey,

I get an ODE for problem 2, but it is not of the Euler type and I don't know how to solve it. Could anyone give me a hint?

Thanks!

12
##### Home Assignment 5 / plotting instead of sketching
« on: October 25, 2012, 05:17:54 PM »
Dear Professor,

can we plot the sum of some series elements or do you want us to draw it by hand?

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##### Home Assignment X / Re: Problem2
« on: October 15, 2012, 12:04:21 PM »
I obtain the solution
$$u(x,t)=\frac{1}{2}((x-ct)e^{x+ct}+(x-ct)e^{x-ct})+ \int_{x-ct}^{x+ct}e^{-x}dx+\int_0^{t}\int_{x-ct}^{x+ct}e^{-x-t}dxdt$$
which becomes
$$\frac{2ce^{-x}}{1-c^2}+\frac{e^{-t}(e^{-ct}+e^{ct})}{1+c}$$
(plugging in $c=5$ doesn't really simplify it)

Did anyone get the same?

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##### Home Assignment X / Re: problem 3
« on: October 13, 2012, 07:46:50 PM »
Thanks for your very quick response!
Is the 3 in your first point supposed to be a 2? I obtain
$$(u_t^*u_t)_t=3Re(u_{tt}u_t^*)$$
as I obtain (with u_t=f(t)+ig(t))
$$(u_t^*u_t)_t=u^*_tu_{tt}+u_tu_{tt}^*=(f-ig)(f'+ig')+(f+ig)(f'-ig')=2(ff'+gg')=2Re(u_{tt}u_t^*)$$

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##### Home Assignment X / problem 3
« on: October 13, 2012, 06:11:36 PM »
Dear all,

I don't know what to do with problem 3. We are asked to find conditions on the three parameters $\alpha$, $\beta$ and $\gamma$ s.t. the integral
$$E(t)=\frac{1}{2}\int_0^L (|u_t|^2+c^2|u_x|^2+\gamma |u|^2)dx$$

is time-independent, where u satisfies b.c. and $u_{tt}-c^2u_{xx}+\gamma u=0$.

The time independence of the integral means that
$$\frac{\partial}{\partial t}u_tu^*_t+c^2\frac{\partial}{\partial t}u_xu^*_x+\gamma \frac{\partial}{\partial t}u u ^* =0$$

but I can`t find $u$, as there is this $u$ term in the wave equation, and the boundary conditions do not help me with this equation neither. Any hints? Thanks a lot!

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