Past exam of the mathematics course of the University of Cambridge / 2014 / ib / Paper 2 / 16D (Methods)

Take the Fourier transform in $x$. The transform of the source is $\delta (t-t^{\prime })e^{-ik{x}^{\prime }}$, so the Green function equation becomesChoose the causal Green function, zero for $t<t^{\prime }$. Away from $t^{\prime }$ the solution is a decaying exponential, and integrating across $t^{\prime }$ gives the jump $\tilde{G}(t^{\prime }+)-\tilde{G}(t^{\prime }-)=e^{-ik{x}^{\prime }}$. ThereforeThe phase shifts the kernel's centre to $x^{\prime }$. Applying the inverse Fourier transform and the Fourier transform of a Gaussian with $a=D(t-t^{\prime })$ givesFor positive elapsed time this heat kernel solves the homogeneous diffusion equation and has unit integral; as elapsed time tends to zero it approaches the spatial delta distribution, supplying the required impulse.

The Duhamel principle superposes the response to source impulses:This expression uses the source at the origin, is causal, and has zero initial data in the distributional sense. Its integral over the plane is ${\int }_0^{t}p(s)ds$, consistent with the injected total amount.

For the finite-duration source and $t>t_0$, set $b=(x^2+y^2)/(4D)$. The causal diffusion from a finite-duration planar point source becomesWith $\eta =b/\tau$, or by differentiating the supplied exponential integral, this isIf $b/(t-t_0)\ll 1$, both arguments are small. Their logarithmic constants and $\ln b$ cancel, leavingAt the origin this logarithmic expression is exact for $t>t_0$, obtained as the limit $b\to 0$. The leading spatial correction is $-p_{0}b{t}_0/[4\pi Dt(t-t_0)]$. If also $t\gg t_0$, the leading value is $p_0{t}_0/(4\pi Dt)$.