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Heat kernel

Heat-kernel solution

For $u_t=Du_{xx}$ and $u(x,0)=\mathbf1_{[-a,a]}(x)$, convolution with the <heat kernel> gives
$$
u(x,t)=\frac12\left[
\operatorname{erf}\left(\frac{x+a}{2\sqrt{Dt}}\right)
-\operatorname{erf}\left(\frac{x-a}{2\sqrt{Dt}}\right)
\right].
$$

Heat equation with interval-indicator initial data

Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 14C / Solution

Heat equation with interval-indicator initial data

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Heat equation with interval-indicator initial data

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