### Author Topic: TT2--P3  (Read 2829 times)

#### Victor Ivrii

• Elder Member
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##### TT2--P3
« on: March 23, 2018, 06:10:04 AM »
Using Fourier method find eigenvalues and eigenfunctions of Laplacian in the rectangle $\{0<x<a,\, 0<y<b\}$ with the boundary conditions:
\begin{align}
&u_{xx}+u_{yy}=-\lambda u\qquad 0<x<a,\ 0<y<b,\label{3-1}\\[3pt]
&u_x|_{x=0}=u_x|_{x=a}=u|_{y=0}=u|_{y=b}=0.\label{3-2}
\end{align}

#### Jilong Bi

• Jr. Member
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##### Re: TT2--P3
« Reply #1 on: March 23, 2018, 09:42:24 AM »
First separate of variable $$u(x,y) =X(x)Y(y),$$
take derivative
$$X''Y + XY'' = -\lambda XY$$
$$\implies \frac{X''}{X}+\frac{Y''}{Y}= -\lambda$$
By the given condition
$$X''+\lambda_1X = 0 , X'(0)+X'(a) = 0$$
This is Neumann boundary condition
As n = 0,$$\lambda_1 = 0 ,X_0 = \frac{1}{2}$$
As n = 1,2,...,$$\lambda_1 = \frac{n^2\pi ^2}{a^2},X_1 = \cos\frac{n\pi x}{a}$$
For Y, this is Dirichlet boundary condition  $$Y''+\lambda_2Y= 0 , Y(0)+Y(b) = 0$$
As m = 1,2,...,$$\lambda_2 = \frac{m^2\pi ^2}{b^2},Y_1 = \sin\frac{m\pi y}{b}$$
$\lambda =\lambda_1 + \lambda_2$, for n,m = 1,2,....
$$\implies \lambda = \pi^2(\frac{m^2}{b^2}+\frac{n^2}{a^2})$$
for n,m = 1,2,....
$$u(x,y) =\cos\frac{n\pi x}{a} \sin\frac{m\pi y}{b}$$
For n = 0, $\lambda$  = 0 and u(x,y) = 0