Method of reflections for the wave equation. For the Wave Equation:

$ \left\{\begin{matrix}
u_{tt}-c^2u_{xx}=0 & x>0, t>0\\ 
u(x,0)=f(x) & x>0 &  \hspace{0.5cm} (1) \\ 
u_{t}(x,0)=g(x) & x>0 &  \hspace{0.5cm} (1)  \\
u(0,t)=h(t) & t>0  \\ 
\end{matrix}\right. $

The general solution is:

$ u(x,t) = \frac{f(x+ct)-f(ct-x)}{2}+ \frac{1}{2c}\int_{x+ct}^{ct-x}g(s)ds + h(t-\frac{x}{c})  $

My notes explain how to obtain it. But then they say that when $h\equiv0$, we can use the method of reflections to solve the problem for $ x\in{\mathbb{R}} $ with the restrictions (1).


*

*Can someone post an example using method of reflections? (*)

*Also I don't get why we need to solve it for $x\in{\mathbb{R}}$... Isn't the problem defined for $x>0$ ?


(*) I have been searching for it but I found nothing. Maybe it is a problem with the name, I have translated the name method of reflections from spanish método de reflexiones.
 A: *

*Here is an example of an application of the method of reflections to a PDE problem:


Suppose we have the following half-line interval problem 
\begin{align*}
\left\{\begin{matrix}
v_t-kv_{xx}, & ~\text{in}~0<x<\infty,~0<t<\infty, \\ 
v(x,0)=\phi(x), & ~\text{for}~t=0, \\ 
v(0,t)=0, & ~\text{for}~x=0,
\end{matrix}\right.
\end{align*}
with Dirichlet boundary conditions. 
Introduce an odd extension 
\begin{align*}
\left\{\begin{matrix} 
\phi(x) & \text{for}~x>0, \\ 
-\phi(-x) & \text{for}~x<0, \\ 
0 & \text{for}~x=0,
\end{matrix}\right.
\end{align*}
to the whole line that is unique. Suppose $u(x,t)$ is a solution to the problem
\begin{equation*}
u_t-ku_{xx}=0,~u(x,0)=\phi_{\text{odd}}(x),~x\in (-\infty,\infty),~t\in (0,\infty)
\end{equation*}
of the form
\begin{equation*}
u(x,t)=\int^{\infty}_{-\infty}S(x-y,t)\phi_{\text{odd}}(y)dy.
\end{equation*}
Write the above as
\begin{equation*}
u(x,t)=\int^{\infty}_{0}S(x-y,t)\phi(y)dy-\int^{0}_{-\infty}S(x-y,t)\phi(-y)dy.
\end{equation*}
Doing a change of variables (i.e. transforming $y\rightarrow -y$ in the second integral )
\begin{equation*}
u(x,t)=\int^{\infty}_{0}[S(x-y,t)-S(x+y,t)]\phi(y)dy
\end{equation*}
gives the solution
\begin{equation*}
v(x,t)=\frac{1}{\sqrt{4\pi kt}}\int^{\infty}_{0}[e^{-(x-y)^2/4kt}-e^{-(x+y)^2/4kt}]\phi(y)dy
\end{equation*}
where we have used
\begin{equation*}
u(x,t)=\frac{1}{\sqrt{4\pi kt}}\int^{\infty}_{-\infty}e^{-(x-y)^2/4kt}\phi(y)dy.
\end{equation*}


*

*With the second point you raised, the solution we are searching for means that we can find a solution for any choice of $x$ or $t$ (as long as it is valid for the given conditions). In the question, we have results for $u$ for any $x$ as long as $t=0$. What about $t\neq 0$?


See "PDEs: An Introduction" by Walter Strauss.
A: Here is your complete answer. Please look at the link pdf. It deals both Dirichlet and Neumann boundary conditions
http://web.math.ucsb.edu/~grigoryan/124A/lecs/lec13.pdf
