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 Incoming links: Drift-diffusion equation with a potential

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 1 / a / Solution 2026-10-18

Substitution into the diffusive flux gives

J (x, t) = - e^{- λ t} D (x) (u^{'} (x) + β V^{'} (x) u (x)) .

The continuity equation for this drift-diffusion equation with a potential therefore reduces, after cancelling the nonzero exponential, to

[D (u^{'} + β V^{'} u)]^{'} = - λ u .

The reflecting boundary conditions become

D (u^{'} + β V^{'} u) = 0

at both endpoints. Thus each solution of this eigenvalue problem supplies a separated solution of the original diffusion equation.

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