= Heat-kernel solution
For initial data $u_0$ on the real line, Fourier transformation or convolution with the fundamental solution gives
$$
u(x,t)=\int_{-\infty}^{\infty}
\frac{e^{-(x-\xi)^2/(4Dt)}}{\sqrt{4\pi Dt}}
u_0(\xi)\,d\xi.
$$
For a <bounded function> $u_0$, the <convolution> is well defined at every positive time because the <heat kernel> has integral one. The <error-function diffusion front> uses a <Heaviside step function> <initial condition> and gives one smoothed interface. The <heat equation with interval-indicator initial data> instead has two interfaces; subtracting their two <error functions> produces the finite-interval profile.
Back to article page
