Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 13A / b

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- Solution b

Solution

The boundary conditions are now

C (0, t) = 0

and

C_{x} (L, t) = 0

. Their wave numbers are

k_{n} = \frac{( n + 1/2 ) π}{L}, n = 0, 1, \dots .

The principal growth rate is

μ - D π^{2} / (4 L^{2})

, so long-term growth occurs when

μ > \frac{D π ^{2}}{4 L ^{2}} .

Closing one end reduces the critical growth rate by a factor of four because there is only one absorbing escape boundary.

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