Author Topic: Web bonus problem : Week 4 (#2)  (Read 2039 times)

Victor Ivrii

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Zaihao Zhou

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Re: Web bonus problem : Week 4 (#2)
« Reply #1 on: October 21, 2015, 09:38:54 PM »
To prove energy conservation law, we need to show $\partial E(t)/ \partial t = 0$
\begin{equation}     \frac{\partial E(t)}{\partial t} = \frac{1}{2} \int_0^\infty (2u_tu_{tt} +2c^2u_xu_{xt}+f(u)u_t) dx    \end{equation}
\begin{equation}     \frac{\partial E(t)}{\partial t} = \frac{1}{2} \int_0^\infty (2u_t(u_{tt}+f(u)) +2c^2u_xu_{xt}) dx    \end{equation}
\begin{equation}     \frac{\partial E(t)}{\partial t} = \frac{1}{2} \int_0^\infty (2c^2u_tu_{xx} +2c^2u_xu_{xt}) dx    \end{equation}
\begin{equation}     \frac{\partial E(t)}{\partial t} = c^2 \int_0^\infty \partial_x(u_tu_x) dx    \end{equation}
\begin{equation}     \frac{\partial E(t)}{\partial t} = c^2 ( u_tu_x|_{x=\infty} - u_tu_x|_{x=0} )   \end{equation}

For Dirichlet condition, $ u|_{x=0} = 0   \Rightarrow $  u_x|_{x=0} = 0 $u_t|_{x=0} = 0$Incorrect! You meant not $u_x$ but ?

We also know $u$ vanishes at $\infty$, thus $\partial E(t) / \partial t = 0 $
For Newmann condition, $ u_x|_{x=0} = 0  $. We also know $u$ vanishes at $\infty$, thus $\partial E(t) / \partial t = 0 $

Sorry don't know how to strike through an equation.
« Last Edit: October 22, 2015, 12:39:27 PM by Zaihao Zhou »

Emily Deibert

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Re: Web bonus problem : Week 4 (#2)
« Reply #2 on: October 22, 2015, 11:44:33 AM »
I believe it should perhaps be $u_t|_{x=0}=0$ for the last part.
« Last Edit: October 22, 2015, 12:42:57 PM by Emily Deibert »