Laplace's equation 3: Fourier expansion :: Maths for Physicists and vice versa

We have found the general solutions of Laplace's equation,	$\frac{\partial^2T(x,y)}{\partial x^2}+\frac{\partial^2T(x,y)}{\partial y^2}=0$ .
The general solutions are: $T(x,y)=X(x)Y(y)={e^{kx}\sin{ky};e^{-kx}\sin{ky};e^{kx}\cos{ky};e^{-kx}\cos{ky}}$
Applying the boundary conditions of the hot plate problem leaves	$T(x,y)=e^{-\frac{n\pi}{10}y}\sin{\frac{n\pi}{10}x}$
as physically sensible solutions.
A linear combination of all of these with coefficients b_n is	$T(x,y)=\sum_{n=1}^{\infty}b_ne^{-\frac{n\pi}{10}y}\sin{\frac{n\pi}{10}x}$ .

Compare the linear combination above with the Fourier sine series:	$f(x)=\sum_{n=1}^{\infty}b_n\sin{\frac{n\pi x}{l}}$ .
They are identical for l=10 and y=0 (because then the e-term is 1).
The BC for the heated edge hasn't been used yet:	T(x,0)=100.
Therefore, we can substitute:	$f(x)=T(x,0)=100=\sum_{n=1}^{\infty}b_n\sin{\frac{n\pi x}{10}}$ .
The Fourier coefficients are, generally:	$b_n=\frac{2}{l}\int_0^lf(x)\sin{\frac{n\pi x}{l}}\ur{d}x$ .
So, in this case:	$b_n=\frac{2}{10}\int_0^{10}100\sin{\frac{n\pi x}{10}}\ur{d}x$ .
Integrate:	$=20(-\frac{10}{n\pi})\left[\cos{\frac{n\pi x}{10}}\right]_0^{10}$ ,
insert limits:	$=-\frac{200}{n\pi}(\cos{n\pi}-\cos{0})$ ,
and tidy up:	$={\frac{400}{n\pi} (for n odd); 0 (for n even)}$ .
Finally, insert the b_n into T(x,y):	$T(x,y)=\frac{400}{\pi}\sum_{{\rm odd~} n=1}^{\infty}\frac{1}{n}\ur{e}^{-\frac{n\pi}{10}y}\sin{\frac{n\pi}{10}x}$ .

And __that's it__ !

General solution
1. $\nabla^2T=0$
2. separation of variables: T(x,y)=X(x)·Y(y)
3. leads to 2nd-order ODE: $\frac{1}{X}\frac{d^2X}{dx^2}=-k^2$
4. solved as ODE with constant coefficients: $T(x,y)=e^{\pm kx}[\sin|\cos](ky)$
Specific solution
1. apply boundary conditions to eliminate unphysical sin/cos/exp terms
2. restate solution as linear combination of harmonics: $T(x,y)=\sum_{n=1}^{\infty}b_n\ur{e}^{-\frac{n\pi}{10}y}\sin{\frac{n\pi}{10}x}$ (or the like)
3. find Fourier coefficients: $b_n=\frac{2}{l}\int_0^lf(x)\sin{\frac{n\pi x}{l}}dx$

The next on the list of PDE in physics is the diffusion equation.

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