 Solution

Along a characteristic, $\dot{z}=(1+t{)}^{-2}$ and $\dot{H}=0$. Integration gives $z+1/(1+t)=\xi$, so $\boxed{H=F(\xi )}$. A characteristic entering through $z=z_0$ at time $\tau$ has $\xi =z_0+1/(1+\tau )$ and carries $H=C_0(1+\tau {)}^{-4/3}$. Thus the boundary-fed and initially dry portions giveValues at $\xi \le z_0$ are never sampled in the finite-time physical region and need not be prescribed. The moving coating front is $z=z_0+t/(1+t)$. Explicitly,The incompatible initial corner creates a jump at the leading coating front; a trace convention at the jump does not affect the transported volume. The wall's total stroke is ${\int }_0^{\infty }{v}_{0}dt=1$, so within this leading transport model the final profile isEarlier-deposited fluid is thicker and sits farther up the finite coated interval. Its total scaled volume is $3C_0/7$, agreeing with ${\int }_0^{\infty }{v}_{0}H(z_0,t)dt=C_0{\int }_0^{\infty }(1+t{)}^{-10/3}dt$. This is a finite-stroke coating profile for the leading matched model.