The exact solution of the wave equation on $[-2,2]$ with initial data $u_0(x)=\max(1-|x|,0), u_t(x)=0 $ and boundary conditions

Question

I am trying to find the exact solution for the computation of the following wave equation.

\begin{equation*} u_{tt} = u_{xx}~~~on~~x\in(-2,2)~~,t\in(0,3.8) \end{equation*} Initial conditions are $u_t(0,x)=0$ and

The value of $u$ at $t=0$ is\begin{cases} 1-|x| &\text{if } |x| \leq 1,\\ 0&\text{otherwise}. \end{cases}

The boundary conditions are $u(t,-2)=0$ and $u_x(t,2)=0$

According to the book, I got the solution $u(x,t)=(1/2)[u_0(x-t)+u_0(x+t)]$. This solution does not satisfies the boundary values. I need this exact solution to compute the error.

Can anyone please let me know the exact solution of this problem?

Answer 1

The boundary conditions are (kind of) satisfied with the solution formula if you extend the initial condition function $u_0$ from $[-2,2]$ to $\Bbb R$ as a function with period $16$ by instituting an odd symmetry at $x=-2+8k$ and even symmetry at $x=2+8k$, $k\in\Bbb Z$.

The "kind-of" is because the wave function is not differentiable everywhere, so that any conditions on derivatives are only valid in some weak sense.