TT2-P3

[Victor Ivrii]

Using Fourier method find eigenvalues and eigenfunctions of Laplacian in the rectangle $\{0<x<a, <y<b\}$ with Dirichlet boundary conditions:
\begin{align}
&u_{xx}+u_{yy}=-\lambda u\qquad 0<x<a,\ 0<y<b,\label{3-1}\\[3pt]
&u_{x=0}=u_{x=a}=u_{y=0}=u_{y=b}=0.\label{3-2}
\end{align}

[Catch Cheng]

Please correct if something is wrong, thank you.

[Rong Wei]

furthermore， we will have λ_n for U = (pi*n/a)^2 + (pi*n/b)^2
     

[Emily Deibert]

Please correct if something is wrong, thank you.

Catch, I am very confused---why is $Y$ a function of $x$ in your last step?!

[Bruce Wu]

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

[Rong Wei]

Please correct if something is wrong, thank you.

Catch, I am very confused---why is $Y$ a function of $x$ in your last step?!

I guess she means y, but clerical error

[Emily Deibert]

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.

[Emily Deibert]

Please correct if something is wrong, thank you.

Catch, I am very confused---why is $Y$ a function of $x$ in your last step?!

I guess she means y, but clerical error

Yes, you must be right! For a second I was worried that I had done everything wrong!

[Bruce Wu]

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 $\lambda_n=(\frac{n\pi}{a})^{2}+(\frac{n\pi}{b})^{2}$?

[Emily Deibert]

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.

[Rong Wei]

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 $\lambda_n=(\frac{n\pi}{a})^{2}+(\frac{n\pi}{b})^{2}$?

I think so, maybe she lost that step