Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 356 / 2 / c / Solution

A centered nearest-neighbour discretization of drift--diffusion is obtained from

k_{i}^{+} = \frac{D}{h ^{2}} + \frac{v ( x _{i} )}{2 h}, k_{i}^{-} = \frac{D}{h ^{2}} - \frac{v ( x _{i} )}{2 h} .

For sufficiently small

h

these rates are nonnegative. Taylor expansion of the mean equations gives

\partial_{t} c = D \partial_{x}^{2} c - \partial_{x} (v c) .

Absorption into the target to the left of the first compartment gives

c (0, t) = 0,

while the absence of outward jumps at the right endpoint gives the zero-flux boundary condition

v (L) c (L, t) - D \partial_{x} c (L, t) = 0.

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