 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-v/6/a/solution

Let

A

be the constant cross-sectional area, and suppose that the temperature is uniform on each cross-section. Consider a slice

[a, b]

. Its stored thermal energy is

ρ c A \int_{a}^{b} T (x, t) d x

. Since the lateral surface is insulated and there is no internal heat generation, conservation of energy gives

ρ c A \frac{d}{d t} \int_{a}^{b} T (x, t) d x = A q (a, t) - A q (b, t),

where

q

is the heat flux in the positive

x

direction. By Fourier's law,

q = - k T_{x}

, so

ρ c \int_{a}^{b} T_{t} d x = k T_{x} (b, t) - k T_{x} (a, t) = k \int_{a}^{b} T_{xx} d x .

The slice is arbitrary; therefore the local heat equation is

T_{t} = κ T_{xx}, κ = \frac{k}{ρ c} > 0.

The constant

κ

is the thermal diffusivity. The cancellation of

A

uses a rod of uniform cross-section; the one-dimensional model assumes no transverse temperature variation.

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