APM346-2012 > Term Test 1

TT1 = Problem 3

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Victor Ivrii:
Consider the PDE  with boundary conditions:
\begin{align*}
&u_{tt}+K u_{xxxx} + \omega^2 u =0,\qquad&&0<x<L,\\[3pt]
&u(0,t)=u_x(0,t)=0,\\[3pt]
&u(L,t)=u_x(L,t)=0,
\end{align*}
where  $K>0$ is constant. Prove that the energy $E(t)$ defined as
\begin{equation*}
E(t)= \frac{1}{2}\int_0^L \bigl( u_t^2 + K u_{xx}^2 + \omega^2 u^2)\,dx
\end{equation*}
does not depend on $t$.

Ian Kivlichan:
Solution to Question 3!

Re-wrote solution more nicely at Prof. Ivrii's request. Original at http://i.imgur.com/l4Pw2.jpg

Aida Razi:
Solution is attached,

Ian Kivlichan:

--- Quote from: Aida Razi on October 16, 2012, 07:39:46 PM ---Solution is attached,

--- End quote ---

Aida: I'm not sure your solution is correct: u(0,t)=0 and u_x(0,t)=0 don't necessarily imply that u_xx(0,t) = 0. Consider for example u(x, t) = x^2. There, u_xx(0, t) = 2 despite u(0,t)=0 and u_x(0,t)=0.

Up until crossing out u_xx on the last line, though, I think your solution is still right, and your final answer is definitely right. ;P

Victor Ivrii:
Aida, your conclusion that out of integral terms vanish is correct but not a justification. Ian is correct both in the solution and finding flaws in Aida' solution. Sure these terms disappear as bot $u_t$ and $u_{tx}$ vanish as $x=0$ and $x=L$ due to boundary condition.

Remark. With $\gamma=0$ this equation
\begin{equation}
u_{tt}+Ku_{xxxx}+\gamma u=0
\end{equation}
 describes oscillations of the beam (beam resists to bending rather than stretching; in fact resistance to bending is due to the fact that while bend the outer part of the beam (which is on its convex side) stretches and inner part of the beam (which is on its concave side) squeezes.

Picture attached taken from

http://www.flickr.com/photos/mitopencourseware/3364718263/



With $\gamma<0$ this equation describes oscillation of the rotating shaft. We have $\gamma=\omega^2$ which is less interesting but makes $E$ positively definite.

PS. Ian, your posts are virtually useless for a class: too poor handwriting makes it almost impossible to read for anyone who does not know solution. Could you repost?
PPS. No need--Zarak did it nicely, and you did it too :D

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