I have the following problem:
( $u_t$ means the first derivative with respect to time )
$ u_t - 4 u_{xx} = 0 \quad x \in ]0, \pi[, \quad t>0 $
with initial condition given by
$ u(0,t) = 0 = u(\pi,t) \quad t>0 $
and
$u(x,0) = 1 \quad x\in ]0,\pi[$.
I have to solve this exercise using the separation of variables technique. The problem is that, once I found a general solution, I'm not able to find the coefficient of the fourier series because they're all zero. Any suggestion?
The solution will be $$ u(x,t)=\sum_{n=1}^\infty a_n\,e^{-4n^2}\sin(n\,x),\quad 0<x<\pi,\quad t>0, $$ where the coefficients $a_n$ are such that $$ \sum_{n=1}^\infty a_n\sin(n\,x)=1,\quad 0<x<\pi. $$ You have to develop the constant function $1$ in a series of sines. Extend it to $[-\pi,\pi]$ by impurity and get $$ a_n=\frac{2}{\pi}\int_0^\pi\sin(n\,x)\,dx,\quad n\ge1. $$