Toronto Math Forum
APM3462015F => APM346Lectures => Web Bonus = Oct => Topic started by: Victor Ivrii on October 04, 2015, 05:51:09 AM

Problem 1 http://www.math.toronto.edu/courses/apm346h1/20159/PDEtextbook/Chapter2/S2.7.P.html#problem2.7.P.1 (http://www.math.toronto.edu/courses/apm346h1/20159/PDEtextbook/Chapter2/S2.7.P.html#problem2.7.P.1)

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.

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