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APM346-2015F => APM346--Lectures => Web Bonus = Oct => Topic started by: Victor Ivrii on October 04, 2015, 05:51:09 AM

Title: Web bonus problem : Week 4 (#2)
Post by: Victor Ivrii on October 04, 2015, 05:51:09 AM

Problem 1 http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter2/S2.7.P.html#problem-2.7.P.1 (http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter2/S2.7.P.html#problem-2.7.P.1)

Title: Re: Web bonus problem : Week 4 (#2)
Post by: Zaihao Zhou 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.

Title: Re: Web bonus problem : Week 4 (#2)
Post by: Emily Deibert on October 22, 2015, 11:44:33 AM

I believe it should perhaps be $u_t|_{x=0}=0$ for the last part.