 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-332/3/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The depth-integrated ice flux is

q = \int_{0}^{h} u d z = u_{b} h - \frac{ρ g}{3 μ} h^{3} h_{x} .

Local mass conservation with accumulation rate

a

says

h_{t} + q_{x} = a .

Substituting the flux gives

\frac{\partial h}{\partial t} + \frac{\partial}{\partial x} (u_{b} h) = \frac{ρ g}{3 μ} \frac{\partial}{\partial x} (h^{3} h_{x}) + a,

as required. Equivalently, after using

u_{b} = - β h_{x}

h_{t} = \frac{\partial}{\partial x} [(β h + \frac{ρ g}{3 μ} h^{3}) h_{x}] + a .

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