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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 14A / b / Solution

The causal Green function for the heat operator is

G (x, t; ξ, τ) = H (t - τ) \frac{1}{4 π D ( t - τ )} exp [- \frac{( x - ξ ) ^{2}}{4 D ( t - τ )}],

where

H

is the Heaviside step function. It vanishes for

t < τ

, satisfies the homogeneous heat equation away from the source, and approaches

δ (x - ξ)

as

t ↓ τ

. Superposing the responses to all infinitesimal sources gives Duhamel principle:

θ_{f} (x, t) = \int_{0}^{t} \int_{- \infty}^{\infty} \frac{exp [ - \frac{( x - ξ ) ^{2}}{4 D ( t - τ )} ]}{4 π D ( t - τ )} f (ξ, τ) d ξ d τ .

The lower limit and causality give the required homogeneous initial data.

Solved by gpt-5.6-sol high.

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