# Laplace equation w/ dirichlet boundary conditions - Partial Diff Eq.

1. Nov 8, 2012

### bossman007

1. The problem statement, all variables and given/known data

The steady state temperature distribution $$T(x,y)$$ in a flat metal sheet obeys the partial differential equation:

$$\displaystyle \frac{\partial^2 T}{\partial x^2}+ \frac{\partial^2 T}{\partial y^2} = 0$$

Seperate the variables in this equation just like in the one-dimensional wave equation and find T everywhere on a square flat plate of sides S with the boundary conditions:

$$T(0,y)=T(S,y)=T(x,0) , T(x,S)=T_0$$

2. Relevant equations

3. The attempt at a solution

this is what I have so far..not sure what to do next???

Last edited: Nov 8, 2012
2. Nov 8, 2012

### bossman007

I'm not bumping my post, sorry! I forgot to mention that classmates mentioned $$sinh$$ shows up in the fourier series and I don't know why also. Any help getting started would be greatly appreciated

many thanks

3. Nov 9, 2012

### vela

Staff Emeritus
The differential equation for Y(y) has a sign mistake in it, though the solutions you wrote down are correct. Instead of exponentials, you can write down the solution in terms of the hyperbolic functions
\begin{align*}
\cosh ky &= \frac{e^{ky}+e^{-ky}}{2} \\
\sinh ky &= \frac{e^{ky}-e^{-ky}}{2}
\end{align*} It's analogous to how you can write down the solutions for X(x) in terms of sin kx and cos kx or in terms of eikx and e-ikx. You're just choosing a different basis.

Now you have to apply the boundary conditions.

4. Nov 9, 2012

### bossman007

Thanks vela !