 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-4/14a/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

t

gives

U_{xx} - s U = - f (x) .

The bounded resolvent solution supplied in the hint can be written

U (x, s) = \int_{- \infty}^{\infty} \frac{e ^{- s ∣ x - ξ ∣}}{2 s} f (ξ) d ξ .

Part (a) inverts the kernel, yielding the heat kernel

K (∣ x - ξ ∣, t) = \frac{1}{4 π t} exp (- \frac{( x - ξ ) ^{2}}{4 t}) .

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