Solve the wave equation with fixed endpoints and the given initial displacement and velocity.

$\displaystyle a^2\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2}, 0< x < L, t > 0$

$\displaystyle u(0,t) = 0, u(L,t) = 0, t > 0$

$\displaystyle u(x,0) = \begin{cases} 0, &0 < x < \frac{L}{2}\\ 1, &\frac{L}{2}<x<L\\ \end{cases}$

$\displaystyle \hskip 10pt \frac{\partial u}{\partial t}\Bigg\vert_{t=0}=x\!\left(L-x\right),0<x<L$

Now we can solve the PDE using the series solution $u(x,t) = \displaystyle\sum\limits_{n=1}^\infty \left( A_n \cos\left(\frac{n\pi a}{L}t\right) + B_n \sin\left(\frac{n\pi a}{L}t\right) \right) \sin\left( \frac{n\pi}{L}x\right)$. The coefficients $A_n$ and $B_n$ are Fourier coefficients we have computed many times:

An example:

$A_n$

$=$

-------------

$\displaystyle\int$

$dx$

$\hskip 12pt$		$=$				$\Bigg\vert$
$\hskip 12pt$		$=$

$B_n=$

--------------

$\displaystyle\int$

$dx$

$\ =\ $