PDE24

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


u(x,0) = f(x)\,
\lim_{x^2+y^2\rightarrow\infty} u(x,y) = 0\,

-\infty<x<\infty,\,\,\,\,\, 0<y<\infty\,



This is called the Dirichlet problem for the half-plane. Let u = XY\, so that

X''+\lambda^2X = 0,\,\,-\infty<x<\infty\,

Y''-\lambda^2Y = 0,\,\,0<y<\infty\,

X(x) = \begin{cases}\frac{1}{2}A_0 & \lambda=0 \\ c_1\cos \lambda x + c_2 \sin \lambda x & \lambda>0\end{cases}\,

Y(y) = \begin{cases}\frac{1}{2}B_0 & \lambda=0 \\ c_3 e^{\lambda y} + c_4 e^{-\lambda y} & \lambda>0\end{cases}\,

The exponentials were chosen over the hyperbolic trig functions because one of the exponetials tends to zero as its argument tends to infinity, while both h-trig functions go to infinity.

To satisfy the limit condition, we must set c_3 = A_0 = B_0 = 0\,

Since we are dealing with \lambda^2\, above, \lambda\, only needs to take on nonnegative values. This is reflected in the integrals in the solution below:

  • u(x,y) = \int_0^\infty\left[A_\lambda\cos\lambda x + B_\lambda\sin\lambda x\right] e^{-\lambda y}\,d\lambda\,

u(x,0) = \int_0^\infty A_\lambda\cos \lambda x + B_\lambda \sin \lambda x\,d\lambda = f(x)\,

The Fourier coefficients are:

A_\lambda = \frac{1}{\pi} \int_{-\infty}^\infty f(x) \cos \lambda x\,dx\,

B_\lambda = \frac{1}{\pi} \int_{-\infty}^\infty f(x) \sin \lambda x\,dx\,

Partial Differential Equations

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