Finite Lenght Wave equation With Only Initial Conditions.

Question

Let $u(x,t)$ be a solution of $$u_{tt}=u_{xx}; 0<x<1, u(x,0)=x(1-x), u_t(x,0)=0$$ Then $u(1/2,1/4)$ is

$1$. $3/16$.

$2$. $1/4$.

$3$. $3/4$.

$4$. $1/16$.

If i apply D’Alembert formula for Wave equation with $f(x)=x(1-x)$ and $g(x)=0$, I got $u(x,t)=\frac{f(x-t)+f(x+t)}{2}$ so $u(1/2,1/4)= 3/16$. But my question is that can I use D’Alembert for finite string problem ? Or I did mistake? Actually I use D’Alembert because it’s initial value problem. Boundary conditions are not given. Please suggest. Thank you.

Answer 1

You can use D'Alembert's formula for a finite string. It works over any domain. It just might be easier to use the typical separation solution for the finite string. After all, it is THE solution to this equation. ALL solutions of the wave equation are of the form $u(x,t) = f(x+t) + g(x-t)$.