Fourier transform of Laplace Equation

• Jul 6th 2010, 07:29 AM
punkstart
Fourier transform of Laplace Equation
if given :

$\displaystyle u_{xx} + u_{yy} = 0\ \ \ \ \ \ \ \ \ -\infty\ <\ x\ <\infty ,\ \ \ \ y>0$

$\displaystyle u(x,0) = f(x)$

has unique bounded solution:

$\displaystyle u(x,y)= \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{y}{y^2+(x-\alpha)^2}f(\alpha)d\alpha$

Use the above information to solve:

$\displaystyle u_{xx} + u_{yy} = 0\ \ \ \ \ \ \ \ \ -\infty\ <\ x\ <\infty ,\ \ \ \ y>0$

$\displaystyle u(x^w,0)=\left[\item\begin{array}{clr}{\,1&if&x>0\\-1&if&x<0\end{array}\right$

Ok, now it's the first time i've seen the use of an exponent in a boundary condition. This question is worth 5 marks, which makes me think that my lecturer feels that this is easy,and not a lot of work. But i'm not sure of anything here.

Where should i start ? If the condition was simply $\displaystyle u(x,0) = \left[\item\begin{array}{clr}{\,1&if&x>0\\-1&if&x<0\end{array}\right$ then i could easily solve for $\displaystyle f(\alpha)$ in the given form of the solution and work from there, but now with that pesky exponent w, i'm not sure.

Any ideas welcome !