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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 4

Take the <Fourier transform> in $x$. Since $\widetilde{\theta_{xx}}=-k^2\widetilde\theta$, the transformed <heat equation> is
$$
\frac{\partial\widetilde\theta}{\partial t}
=-Dk^2\widetilde\theta,
\qquad
\widetilde\theta(k,0)=\widetilde\Theta(k).
$$
Therefore
$$
\widetilde\theta(k,t)=e^{-Dk^2t}\widetilde\Theta(k).
$$
The given transform pair and the <convolution theorem> yield the <heat-kernel solution>
$$
\boxed{
\theta(x,t)=\frac1{\sqrt{4\pi Dt}}
\int_{-\infty}^{\infty}
\exp\left[-\frac{(x-\xi)^2}{4Dt}\right]
\Theta(\xi)\,d\xi}.
$$

Solution

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

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

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

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