# PDEMOC2

$\displaystyle u_t + u u_x = 0, u(x,0)=x\,$ This is Burgers' equation.

Let $\displaystyle x=x(t)\,$ so $\displaystyle u=u(x(t),t)\,$ .

$\displaystyle u_t = u_x x'(t) + u_t = u_t + u u_x = 0\,$ so $\displaystyle u\,$ is constant with respect to $\displaystyle t\,$ and

$\displaystyle u(x(t),t) = u(x(0),0) = x_0\,$

$\displaystyle x'(t) = u\,$

$\displaystyle x(t) = ut+x_0\,$

Since $\displaystyle u(x,t) = x-ut\,$ , the solution is $\displaystyle u(x,t) = \frac{x}{1+t}, t>-1\,$ .