= Exponential drainage transform for porous-medium diffusion
{title2=$h=e^{-\Gamma t}H(x,\tau),\quad\tau=(1-e^{-\Gamma t})/\Gamma$}
The <porous medium equation> $h_t=\lambda(hh_x)_x-\Gamma h$ with constant $\Gamma>0$ becomes $H_\tau=\lambda(HH_x)_x$ under the displayed transformation. Substitution gives $h_t+\Gamma h=e^{-\Gamma t}\tau'H_\tau$ and $\lambda(hh_x)_x=\lambda e^{-2\Gamma t}(HH_x)_x$, so $\tau'=e^{-\Gamma t}$. This transformation applies to every solution with appropriately transformed initial and boundary conditions, not just a <similarity solution>. A <one-sided constant-volume porous gravity current> therefore has remaining physical volume $Ve^{-\Gamma t}$ and a limiting front $[9\lambda V/(\phi\Gamma)]^{1/3}$. The finite transformed time $\tau\to1/\Gamma$ means that arbitrary initial profiles need not reach the universal large-time source shape before drainage removes them.
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