# Solving the equation for thickness evolution

$\frac{\partial \varphi }{\partial t}=-\nabla \cdot Q+S_{\varphi }$
On the left-hand side we have the change in φ per unit time, which is given by the negative of the divergence (the $\nabla \cdot$ operation) of the flux, Q, plus any contribution from sources, $S_{\varphi }$.