Heat Equation, Neumann, Fourier

[andrewcheong]

Hi, it's been a while since I touched mathematics and I'm a little rusty... I'm looking at a problem right now that I find difficult to understand, conceptually. Any insight would be greatly appreciated. (A direct solution would help immensely as well, not only because that's what I need to submit, but also because I can usually figure out the concept from looking at the solution.) LaTeX output doesn't seem to be working for me; I've tried FF and IE. I'll place LaTeX between $$.

Homework Statement

Use Fourier Transform in $(x_1,x_2)$ to find $u(x_1,x_2,x_3)$ when $\Delta u=0$; in $(x_1,x_2)\in \Re^2$; $x_3<0$. $\frac{\partial u}{\partial x_3}=f(x_1,x_2)$ at $x_3=0$.

(Note that $\Delta$ is the Laplace operator.)

Homework Equations

N/A

The Attempt at a Solution

First of all, note that this problem is not dependent on time. I believe it's because $\Delta u=0$, right? This is the first "concept" that confuses me; what kind of problem does not depend on time, and what exactly are we looking for? Am I mistaken? Does it indeed depend on time implicitly?

And the solution to be found here is $u(x_1,x_2,x_3)$, which gives heat at any point in space, regardless of time?

On the original problem (handwritten and photocopied by the professor) the $x_3<0$ was emphasized. Why?

Is it perhaps that $\Delta u=0$ only applies to $x_1$ and $x_2$, but not $x_3$?

So confused. Any advice and direction would be greatly appreciated. Thanks.

 

[Mapes]

It looks like a steady-state diffusion problem on the half-space, or semi-infinite medium (this is why $x_3<0$ is emphasized). The heat flux at the surface is a function of location. $\Delta u=0$ applies to all three variables; it's the temperature field within the half-space.