Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 356 / 2 / c

Solution

The adsorption mechanism is uniform along the two vertical sides, and

X

evolves independently of

Y

. Consequently the mean first-passage time depends only on the initial

x

a_{1} = 1

τ^{''} (x) + τ^{'} (x) = - 1, - 1 < x < 1.

J \cdot n = κ p

. The boundary term in the adjoint relation is

\int_{\partial Ω} (τ J \cdot n + p \partial_{n} τ) d s,

because the

X

diffusion coefficient is one. It vanishes for every admissible

p

precisely when

\partial_{n} τ = - κ τ .

For

κ = 1

τ^{'} (- 1) = τ (- 1), τ^{'} (1) = - τ (1) .

The general solution of the ordinary differential equation is

τ (x) = - x + C + D e^{- x} .

The right condition gives

C = 2

, and the left gives

D = - 2/ e

. At the prescribed initial position,

τ = τ (0) = 2 - \frac{2}{e} .

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