Solve Burgers' equation $$u_t + uu_x =0,$$ with $u=u(x,t)$ and the side condition $u(x,-1) = x^2$. Find the solution for $u=u(1,2)$
I can't figure out how to use the side condition in order to find the solution.
The method of characteristics seem to lead to a shock formation for $x<0$ is that right?
Let $\begin{cases}p=t+1\\q=x\end{cases}$ ,
Then $u_t=u_pp_t+u_qq_t=u_p$
$u_x=u_pp_x+u_qq_x=u_q$
$\therefore u_p+uu_q=0$
Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dp}{ds}=1$ , letting $p(0)=0$ , we have $p=s$
$\dfrac{du}{ds}=0$ , letting $u(0)=u_0$ , we have $u=u_0$
$\dfrac{dq}{ds}=u=u_0$ , letting $q(0)=f(u_0)$ , we have $q=u_0s+f(u_0)=up+f(u)$ , i.e. $u=F(q-up)=F(x-u(t+1))$
$u(x,-1)=x^2$ :
$F(x)=x^2$
$\therefore u=(x-u(t+1))^2$
$(t+1)^2u^2-(2x(t+1)+1)u+x^2=0$
$u(x,t)=\dfrac{2x(t+1)+1\pm\sqrt{(2x(t+1)+1)^2-4x^2(t+1)^2}}{2(t+1)^2}=\dfrac{2x(t+1)+1\pm\sqrt{4x(t+1)+1}}{2(t+1)^2}$
$\therefore u(1,2)=\dfrac{7\pm\sqrt{13}}{18}$
