PDE14

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domolopa

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

$u(0,y) = 0\,$
$u(a,y) = 0\,$
$u(x,0) = 0\,$
$u(x,b) = f(x)\,$

$0<x<a,\,\,\,\,\, 0<y<b\,$

Seperate variables and plug back into the DE. The equation is now seperable. Separate it and set it equal to a constant. Put a negative sign or power of two on the constant if it makes the later math easier.

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

$X''Y + XY'' = 0\,$

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

The ODE for $X(x)\,$ is

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

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

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

$X(a) = c_2\sin(\lambda a) = 0 \Rightarrow \lambda = \frac{n\pi}{a}, n=1,2,...\,$

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

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

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

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

$u(x,b) = \sum_{n=1}^\infty A_n \sinh \frac{n\pi b}{a}\sin \frac{n\pi x}{a} = f(x)\,$

Through Fourier analysis the equation is solved for the coefficients in the integrand.

$A_n\sinh\frac{n\pi b}{a} = \frac{2}{a}\int_0^a f(x)\sin\frac{n\pi x}{a}\,dx\,$

The final solution is:

$u(x,y) = \sum_{n=1}^\infty \frac{ \frac{2}{a}\int_0^a f(x)\sin\frac{n\pi x}{a}\,dx}{\sinh\frac{n\pi b}{a}}\sin\frac{n\pi x}{a}\sinh\frac{n\pi y}{a}\,$

Partial Differential Equations

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