I am a novice in PDE, and I met a problem, I reduced it to
$$ \begin{cases} u_{tt}-u_{xx}=f(x,t), -\infty <x<+\infty,t>0 \\ u(x,0)=u_{0}(x), -\infty<x<+\infty \\ u_{t}(x,0)=u_{1}(x),-\infty<x<+\infty \end{cases} $$
Clearly if $f=0$, the solution would be the D'Alembert Formula. So can anyone solve this problem for me?
Also, I am looking for a pde book that is full of solutions for wave equation with different boundary condition, so... any recommendations?
Thank you for your help.
Let $\begin{cases}p=x+t\\q=x-t\end{cases}$ ,
Then $u_x=u_pp_x+u_qq_x=u_p+u_q$
$u_{xx}=(u_p+u_q)_x=(u_p+u_q)_pp_x+(u_p+u_q)_qq_x=u_{pp}+u_{pq}+u_{pq}+u_{qq}=u_{pp}+2u_{pq}+u_{qq}$
$u_t=u_pp_t+u_qq_t=u_p-u_q$
$u_{tt}=(u_p-u_q)_t=(u_p-u_q)_pp_t+(u_p-u_q)_qq_t=u_{pp}-u_{pq}-u_{pq}+u_{qq}=u_{pp}-2u_{pq}+u_{qq}$
$\therefore u_{pp}-2u_{pq}+u_{qq}-(u_{pp}+2u_{pq}+u_{qq})=f\left(\dfrac{p+q}{2},\dfrac{p-q}{2}\right)$
$-4u_{pq}=f\left(\dfrac{p+q}{2},\dfrac{p-q}{2}\right)$
$u_{pq}=-\dfrac{1}{4}f\left(\dfrac{p+q}{2},\dfrac{p-q}{2}\right)$
$u(p,q)=F(p)+G(q)-\dfrac{1}{4}\int_0^q\int_0^pf\left(\dfrac{r+s}{2},\dfrac{r-s}{2}\right)~dr~ds$
$u(x,t)=F(x+t)+G(x-t)-\dfrac{1}{4}\int_0^{x-t}\int_0^{x+t}f\left(\dfrac{r+s}{2},\dfrac{r-s}{2}\right)~dr~ds$
