Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 332 / 1 / a / i / Solution

Before the river disturbance reaches the divide, use the similarity solutionSubstitution gives the nonlinear boundary-value problemIt can be solved numerically. If $C={\lim }_{\eta \downarrow 0}H{H}^{\prime }>0$, then the riverward flux and near-river profile areThe negative sign means flow toward decreasing $x$; the discharge into the river is $-q(0,t)$.