Problem def.: Solve the wave problem in a form of a Fourier series
PDE: $u_{tt}=3u_{xx}$
BCs: $u(0,t)=u(\pi,t)=0$
ICs: $u(x,0)=\sin^{3}x,\:u_{t}(x,0)=0,\qquad0<x<\pi$
My solution:
Using separation of variables and BCs, general solution to the PDE is:
$u(x,t)=\sum_{n=1}^{\infty}[A_{n}sin(n\sqrt{3}t)+B_{n}cos(n\sqrt{3}t)]sin(nx)$, where $n=1,2,3,...$
To satisfy ICs:
$u_{t}(x,0)=0=\sum_{n=1}^{\infty}\sqrt{3}nA_{n}sin(nx)\quad\Rightarrow A_{n}=0$
$u(x,0)=\sin^{3}x=\sum_{n=1}^{\infty}B_{n}sin(nx)$
Therefore, the final solution is:
$u(x,t)=\sum_{n=1}^{\infty}[B_{n}cos(n\sqrt{3}t)]sin(nx)$, where $n=1,2,3,...$
$B_{n}=\frac{2}{\pi}\intop_{0}^{\pi}\sin^{3}(x)\sin(nx)dx$
Now my questions are:
- Is this solution correct?
- Is it acceptable to leave $B_n$ in this form (without solving the integral)?
- I have played with Mathematica trying to solve the integral for $B_n$, and using trial and error determined that 'only' n=1 and n=3 produce $B_n$ that is not zero. More specifically: $B_{1}=3/4$ and $B_{3}=-1/4$. Is there an easy way to do this analytically?
Use the fact that $sin^3x=\frac {3sinx-sin3x}{4}$ and try comparing coefficients.