\[u_t = \nu u_{xx},\quad 0 < x < h(t),\quad t > 0\]

\[h(0) = 0 \]

\[u(0,t) = f(t),\quad t > 0\]

\[u(h(t),t) = 0,\quad t > 0 \]

\[\frac{dh(t)}{dt}=-\alpha\,u_x(h(t),t),\quad t>0 \]

In this equation both \(u(x,t)\) are \(h(t)\) unknowns. Find appropriate boundary conditions and parameter values \(\nu, \alpha, f(t) \) from physical considerations.

Google: Stefan problem, free boundary, moving boundary

Challenges: free/moving boundary.