APM346-2015F > Test 2

TT2-P3

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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 λn for U = (pi*n/a)^2 + (pi*n/b)^2
     

Emily Deibert:

--- Quote from: Catch Cheng on November 19, 2015, 12:43:05 AM ---Please correct if something is wrong, thank you.

--- End quote ---

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 $\lambda=\lambda_1 +\lambda_2$?

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