PDEFT2

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$u_{xx}+u_{yy}=0\,$

$u(0,y) = 0\,$
$u(1,y) = 0\,$
$u(x,0) = 0\,$

$u(x,1) = B x(1-x)\,$
$t>0,\,\,0<x<\infty\,$

$u(x,y) = X(x)Y(y)\,$

$\frac{X''}{X} = -\frac{Y''}{Y} = -\lambda^2\,$

$X'' + \lambda^2 X = 0\,$

$X(x) = c_1 \cos \lambda x + c_2 \sin \lambda x\,$

$X(0) = c_1 = 0\,$

$X(1) = c_2\sin\lambda = 0,\lambda = n\pi, n=1,2,...\,$

$X(x) = c_2 \sin n\pi x,n=1,2,...\,$

$Y''-\lambda^2 Y = 0\,$

$Y(y) = c_3 \cosh \lambda y + c_4 \sinh \lambda y\,$

$Y(0) = c_3 = 0\,$

$Y(y) = c_4 \sinh n\pi y, n=1,2,...\,$

$u(x,y) = \sum_{n=1}^\infty A_n \sin n\pi x \sinh n\pi y\,$

$u(x,1) = \sum_{n=1}^\infty A_n \sin n\pi x \sinh n\pi = B x(1-x)\,$

From Fourier series coefficients,

$A_n\sinh n\pi = 2 B \int_0^1 x(1-x)\sin n\pi x\,dx\,$

Which can be integrated by parts and results in

$u(x,y) = \begin{cases} 0 & \frac{n}{2}\isin\mathbb{Z}\\ \frac{8B}{n^3\pi^3} & \frac{n-1}{2}\isin\mathbb{Z} \end{cases}, n=1,2,...\,$

Partial Differential Equations

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