APM346-2012 > Home Assignment X

problem 3

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Thomas Nutz:
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!

Victor Ivrii:
$\renewcommand{\Re}{\operatorname{Re}}$

--- Quote from: Thomas Nutz 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 $u_{tt}-c^2u_{xx}+\gamma u=0$.

--- End quote ---
Yes

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

--- End quote ---
No-we are looking for time independence of integral in the spacial limits, not of the integrand (the latter would be much stronger requirement).


* You can differentiate products. Actually you do not need double each term: f.e. $(u_t u^*_t)_t = 2\Re(u_{tt}u_{t}^* )$.
* In the second term integrate by parts by $x$ to transform $u_x u^*_{tx}$ into $\pm u_{xx}u^*_t$ plus boundary terms.
Equivalently
\begin{equation}
\mathcal{E} _t + \mathcal{P}_x =  2\Re u^*_t f
\end{equation}
where $\mathcal{E}=|u_t|^2+c^2 |u_x|^2 + \gamma |u|^2$ and you need to find expression for $\mathcal{P}$, and $f=0$ if equation is fulfilled.


* You need to assemble integral terms and prove that they together are $0$.
* Finally, you need to consider boundary terms and using boundary conditions find when they are $0$ (to prove (a)). Similarly for (b) but derivative must be -- what?[/list]

Thomas Nutz:
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))
$$
(u_t^*u_t)_t=u^*_tu_{tt}+u_tu_{tt}^*=(f-ig)(f'+ig')+(f+ig)(f'-ig')=2(ff'+gg')=2Re(u_{tt}u_t^*)
$$

Victor Ivrii:

--- Quote from: Thomas Nutz 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?

--- End quote ---
Yes--corrected

--- Quote --- I obtain
$$
(u_t^*u_t)_t=3Re(u_{tt}u_t^*)
$$
as I obtain (with u_t=f(t)+ig(t))
$$
(u_t^*u_t)_t=u^*_tu_{tt}+u_tu_{tt}^*=(f-ig)(f'+ig')+(f+ig)(f'-ig')=2(ff'+gg')=2Re(u_{tt}u_t^*)
$$

--- End quote ---

Too complicated: just use that $2Re (v)=v+v^*$ and then $(u_tu^*_t)_t=u_{tt}u^*_t +u^*_{tt}u_t =2\Ree(u_{tt}u^*_t )$ as the second term is complex-conjugate to the first one.

Bowei Xiao:
Are we supposed to deal with the complex variables in Term test?or just the real valued like last year term test1?

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