Consider the following quasi-linear PDE : $u_t + uu_x = -2u$, with the boundary condition $u(0,t) = e^{-t}$.
Show, using the method of characteristics, that the solution to this boundary value problem is $u(x,t) = e^{-2t} / ( x + \sqrt{x^2 + e^{-2t}} )$.
So far I've followed the usual method when solving this and I've obtained that $t = r + s$,
$u = e^{-s-2r}$
$x = e^{-s}(1-e^{-2r})$ / 2,
but I'm not sure where to go from here to find my solution $u(x,t)$. Any help?
Best Answer
What you need to finish the solution is to eliminate $s$ and $r$, writing them in terms of $t$ and $x$.
Thus \begin{align*}x^2+e^{-2t}&=\frac{1}{4}e^{-2s}(1+e^{-4r}-2e^{-2r})+e^{-2t}\\ &=\frac{1}{4}(e^{-2s}+e^{-2s-4r}-2e^{-2s-2r})+e^{-2t}\\ &=\frac{1}{4}(e^{-2s}+u^2+2e^{-2t}\\ &=\frac{1}{4}e^{-2s}(1+e^{-2r})^2\end{align*}
Hence $\sqrt{x^2+e^{-2t}}=\frac{1}{2}(e^{-s}+u)$. Also $2x=e^{-s}-u$, so eliminating the $s$ variable gives $$u=\sqrt{x^2+e^{-2t}}-x$$ (This is equal to the given solution.)