Toronto Math Forum

$g$ is a degree 3 polynomial, so $P$ will be a degree 1 polynomial. $g$ also has rotational symmetry about the z axis ($x^2+y^2=s^2$ in cylindrical coordinates)

Thank you Vivian for explaining why the symmetry exists!

Thanks Vivian Tan! Great answer, as far as I recall I got the same! 

To prove the energy conservation law, we have the time derivative of $E(t)$. So:

\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{1}{2} \int_0^{\infty} \left[ u_{tt}^*u_t + u_t^*u_{tt} + c^2 \left( u_{xt}^*u_x + u_x^*u_{xt} \right) \right] dx
\end{equation}

We make use of the wave equation to rewrite the equation:

\begin{equation}
\frac{\partial E(t)}{\partial t} =  \frac{1}{2} \int_0^{\infty} \left[ \left( c^2u_{xx} \right)^*u_t + u_t^*\left( c^2 u_{xx} \right) + c^2 \left( u_{xt}^*u_x + u_x^*u_{xt} \right) \right] dx
\end{equation}

We then notice that these terms can be combined as a derivative, since $u_tu_{xx} + u_xu_{xt} = \frac{d}{dx}u_tu_x$ So:

\begin{equation}
\frac{\partial E(t)}{\partial t} =  \frac{c^2}{2} \int_0^{\infty} \left[ \frac{d}{dx} \left( u_x^*u_t \right) + \frac{d}{dx} \left( u_t^*u_x \right) \right] dx
\end{equation}

\begin{equation}
\frac{\partial E(t)}{\partial t} =  \frac{c^2}{2} \left( u_x^*u_t |_0^{\infty} + u_t^*u_x |_0^{\infty} \right)
\end{equation}

We neglect the terms at $\infty$, since we assume the function is fast decaying. We then make use of the boundary condition to rewrite this:

\begin{equation}
\frac{\partial E(t)}{\partial t} =  -\frac{c^2}{2} \left( -i \alpha u_t^*u_t + i \alpha u_t u_t^* \right) = 0
\end{equation}

Thus we have proven the energy conservation law.

Also for solution to P2, should you have $(y+1)^2$ in the denominator of Eq. 2.8, since $(y+1)$ is our $\alpha$?

Quote from: Emily Deibert on November 19, 2015, 01:59:03 AM

Quote from: Fei Fan Wu on November 19, 2015, 01:58:00 AM

I did it a different way. But by Catch's method shouldn't there be the additional constraint that $\lambda=\lambda_1 +\lambda_2$?

Actually I think Catch did mention that, near the top right of the page.

You're right, but then in the end shouldn't the final eigenvalues be $\lambda_n=\left(\frac{n\pi}{a}\right)^2+\left(\frac{n\pi}{b}\right)^2$?

Indeed, good point.

I did it a different way. But by Catch's method shouldn't there be the additional constraint that $\lambda=\lambda_1 +\lambda_2$?

Actually I think Catch did mention that, near the top right of the page.

I got the same answer as Fei Fan Wu, but unfortunately forgot to put the factor of $\frac{1}{2}$ during the test! Oops.

as Xi Yue Wang did in homework 7 and lecture notes here: http://www.math.toronto.edu/courses/apm346h1/20159/PDE-

Rong Wei, which section did you mean? The link does not work for me!