 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/2/b/solution

By orthogonality and the sampling property of the Dirac delta distribution,

a_{n} = ρ_{0} \int_{0}^{L} δ (x - x_{0}) ψ_{n} (x) d x = ρ_{0} \frac{2}{L} sin \frac{nπ x _{0}}{L} .

Consequently the reaction-diffusion equation solution is

ρ (x, t) = \frac{2 ρ _{0}}{L} n = 1 \sum \infty sin \frac{nπ x _{0}}{L} sin \frac{nπ x}{L} exp [(γ - \frac{D n ^{2} π ^{2}}{L ^{2}}) t] .

For every

t > 0

this eigenfunction expansion is smoothed by the Gaussian decay in

n

t ↓ 0

, it recovers the specified initial condition in the sense of distributions.

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