I have difficulties solving the following exercise: Consider the IBVP on the half line $(0,\infty)$ (with $T \in (0,\infty)$:
$u_{xx}-u_{tt}=0$, on $(0,T)\times(0,\infty)$
$u(0,x)=u_{0}(x), x>0$,
$u_t(0,x)=v_{0}(x), x>0$ and finally the boundary condition:
$u(t,0)=h(t), t\in(0,T)$, where $u_0 \in C^2([0,\infty))$,$v_0 \in C^1([0,\infty))$,
$h \in C^2([0,T))$.
Question: Find a formula for solutions of the preceding IBVP, and show uniqueness of its solutions $u \in C^2([0,T)×[0,\infty))$. In addition, provide compatibility conditions for $u_0, v_0, h$ which ensure the existence of such a solution u, and determine the largest open superset of $[0,T)×[0,\infty)$ to which u can be extended as the unique $C^2$ solution of the 1-dimensional wave equation.
The answer is supposed to be quite similar to D'Alamberts formula. I tried to derive a formula in the same fashion but it didn't really work. Compatibility conditions: Probably $u_0(0)=h(0)$ and $v_0(0)=h'(0)$. Uniqueness at least is quite clear, but that is it.
Thanks in advance
Follow the method in this similar question. First note that the initial conditions and the boundary conditions are separated by the characteristic line $x=t$, so we find the solution in 2 different regions
$$ u(x,t) = \begin{cases} u_1(x,t), & 0 \le x < t \\ u_2(x,t), & x > t \end{cases} $$
For $x > t$, the solution is given by d'Alembert's formula
$$ u_2(x,t) = \frac{u_0(x+t)+u_0(x-t)}{2} + \frac{1}{2}\int_{x-t}^{x+t} v_0(s)\ ds $$
For $0 < x < t$, we find a solution of the form
$$ u_1(x,t) = F(t+x) + G(t-x) $$
The given B.C. gives
$$ u_1(0,t) = F(t) + G(t) = h(t) $$
And the continuity condition at $x=t$ gives
$$ u_1(t,t) = F(2t) + G(0) = u_2(t,t) = \frac{u_0(2t)+u_0(0)}{2} + \frac{1}{2}\int_0^{2t}v_0(s)\ ds $$
Solving for $F$ and $G$ gives
\begin{align} F(z) &= -G(0) + \frac{u_0(z)+u_0(0)}{2} + \frac{1}{2}\int_0^z v_0(s)\ ds \\ G(z) &= G(0) + h(z) - \frac{u_0(z)+u_0(0)}{2} - \frac{1}{2}\int_0^z v_0(s)\ ds \end{align}
So finally
$$ \implies u_1(x,t) = h(t-x) + \frac{u_0(t+x)-u_0(t-x)}{2} + \frac{1}{2}\int_{t-x}^{t+x} v_0(s)\ ds $$
If the boundary function $h(t)$ has compact support, then the solution in $0<x<t$ is additionally bounded above by the characteristic line $x = t - T$, so the domain of $u_1$ is $\{x > 0, x < t < x + T\}$