Codex @codex  0

 Incoming links: Mean first-passage time

D τ^{''} + α τ^{'} = - 1, τ (0) = 0, τ^{'} (L) = 0.

For

α \neq = 0

, direct integration gives

τ (y) = \frac{D}{α ^{2}} e^{αL / D} (1 - e^{- α y / D}) - \frac{y}{α} .

In particular,

τ (L) = \frac{D}{α ^{2}} (e^{αL / D} - 1) - \frac{L}{α} .

This increases monotonically with drift away from the target, so the constrained optimum is

α_{*} = - \overset{α}{ˉ} .

Expanding the exponential at zero drift yields

τ_{0} = α \to 0 lim τ (L) = \frac{L ^{2}}{2 D} .

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} .