Separation of variables in Fourier Series

**tttcomrader** · September 15th 2007, 04:25 PM

Problem: Find the Fourier Series of $f(x) = 0.05sin \pi x$ , with the initial velocity $g(x)=0$ , $c= \frac{1}{n}$ , $L = 1$ .

My solution: I use the separation of variables method, then I have:

$u(x,t)= \sum^{\infty}_{n=1}sin \frac{n \pi x}{L}[b_{n}cos( \frac {cn \pi}{L})t+ \hat{b_{n}} sin( \frac {cn \pi}{L})t]$

where

$b_{n}=\frac{2}{L} \int^{L}_{0}f(x)sin( \frac {cn \pi}{L})dx$
$\hat{b_{n}} = \frac{2}{cnx} \int^{L}_{0}g(x)sin( \frac {cn \pi}{L})dx$

Now I know that [tex]\hat{b_{n}}=0[tex] since $g(x)=0$ , but when I solve for $b_{n}$ , I have:

$b_{n}=2 \int^{1}_{0}0.05sin( \pi x)sin(n \pi x)dx$

I use the table of integrals, and I get bn = 0, is there something I'm doing wrong here? (Most likely)

Please check, thanks.

K

**ThePerfectHacker** · September 15th 2007, 04:50 PM

What are you trying to do?

**tttcomrader** · September 15th 2007, 06:37 PM

I'm trying to find the Fourier Series representation of this f(x), and I'm using the separation method to do it.

**ThePerfectHacker** · September 15th 2007, 06:59 PM

So you are trying to solve the wave equation.

Use the following facts,
$\int_0^L \sin \frac{\pi n x}{L}\sin \frac{\pi m x}{L} dx= L\delta_{nm}$

$\int_0^L \sin \frac{\pi nx}{L}\cos \frac{\pi mx}{L}dx=0$

$\int_0^L \cos \frac{\pi nx}{L}\cos \frac{\pi mx}{L} dx = L\delta_{nm}$

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