Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 2 / 20 / c

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- Solution c

Solution

Insert the linear initial profile into the transformed integral from (b):

u (x, t) = \frac{a}{π} \int_{R} (x + 2 t η) e^{- η^{2}} d η .

The constant part integrates to

a x

. The other part is an absolutely integrable odd function, so its integral vanishes. Hence

u (x, t) = a x (t \geq 0) .

As a direct check,

u_{t} = u_{xx} = 0

and

u (x, 0) = a x

. Thus a linear temperature profile is stationary on the whole line. More generally, every affine function is preserved by this heat-kernel solution.

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