Suppose we have a laterally insulated metal rod of length 3, parametrized by length, starting with 0 at the left hand endpoint. Suppose the left hand endpoint is kept at temperature 9, the right hand endpoint is kept at temperature 1, and the initial temperature at time $t=0$ is $\sin(x)$.
Set up the initial-boundary-value-problem (IBVP).

The equation is:
$u_{tt}=\alpha^2 u_{xx}$
$u_t=\alpha^2 u_{xx}$
$u_t=\alpha^2 u_{xx}+u$
$u_{tt}+u_{xx}=0$

Range of idependent variables:
$0 < x < {}$ , $t>0$,

Boundary and initial conditions:
$u(0,t) = {}$ for $t>0$,
$u($ $,t) = {}$ for $t>0$,
$u(x,0) = {}$

help (formulas)