 Solution

ID: 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/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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