I scaled my coordinates like:
$x-x_0=(t_0-t)^{1/2}*m$
$u(x,t)=(t_0-t)^{-1/2}*U(m)$
$u$ was a function of $x$ and $t$ $u(x,t)$. So how can I find $u_t$ and $U_m$ "the derivation" with respect to new time and new m coordinates?
I need to change $u_t$ with new $u_t$ and $u_x$ with new $U_m$ in the Euler x-momentum equation:
$u_t + u u_x =-p_x$
which needs to be transfered to the new coordinates.
The answer should be
$\frac{1}{2} U + \frac{m}{2}U_m + U*U_m = -P(m)_m$
with P is from $p(x,t)=(t_0-t)^{-1}P(m)$.