PDE15

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\Delta u = u_{xx}+u_{yy}=0\,

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

u(x,1) = A 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 = A x(1-x)\,

From Fourier series coefficients,

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

Which can be integrated by parts and results in

= \begin{cases} 0 & n=2,4,...\\ \frac{8A}{n^3\pi^3} & \frac{n-1}{2}\isin\mathbb{Z} \end{cases}, n=1,2,...\,

u(x,y) = \sum_{n=1}^\infty c_n \sinh n\pi y \sin n \pi x, \,\,\, c_n = \begin{cases}\frac{8A}{n^3\pi^3\sinh n\pi} & n=1,3,5,... \\ 0 & n=2,4,... \end{cases}\,

Making a change of index variable to only odd numbers, the final solution is

u(x,y) = \frac{8A}{\pi^3}\sum_{n=1}^\infty \frac{\sinh(2n-1)\pi y \sin(2n-1)\pi x}{(2n-1)^3\sinh(2n-1)\pi}\,


Main Page : Partial Differential Equations

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