Solution

[Papapetros Vaggelis]

Solve the problem

\(\displaystyle{u_{tt}(x,t)-u_{xx}(x,t)=0\,,(x,t)\in\left(0,\pi\right)\times \left(0,+\infty\right)}\), where

\(\displaystyle{u(x,0)=x\,,0\leq x\leq \pi\,\,,u_{t}(x,0)=\sin\,x\,,0\leq x\leq \pi\,\,,u(0,t)=u(\pi,t)=0\,,t\geq 0\,\,\,,u\in C^{\infty}}\)

Find

\(\displaystyle{M=\max\,\left\{u(x,t)\in\mathbb{R}\,\,,(x,t)\in\left[0,\pi\right]\times \left[0,+\infty\right)\right\}}\).