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I'd like to solve the following problem in 3 dimensions:

[itex]\partial_t u(r,t) = D\Delta u(r,t)[/itex]

[itex]u(r,0) = 0[/itex]

[itex]u(0,t) = C_o[/itex]

In words, I am looking at a point 'source' that is turned on at t=0 and held at constant temperature. The ultimate goal is to then convolve this solution with constant sources distributed arbitrarily in space.

In 1D, I can find the solution:

[tex]u(x,t) = C_o \mathrm{erfc}\left(\frac{x^2}{\sqrt{4Dt}}\right)[/tex]

In 3D, I proceed by taking the Laplace transform and solving the resulting Helmholtz equation. After applying the [itex]r\to\infty[/itex] boundary condition,

[tex]u(r,s)=A\frac{e^{-\sqrt{\frac{s}{D}}r}}{r}[/tex]

But I can't apply the boundary condition at [itex]u(0,s)[/itex] due to the singularity at the origin. So instead I assume we fix the constant temperature condition on a ball of radius [itex]a[/itex], for [itex]a\ll r[/itex]. If I do that my solution isn't so bad,

[tex]u(r,t)=C_o\frac{a}{r}\mathrm{erfc}\left(\frac{(a-r)^2}{4Dt}\right)[/tex]

This matches my simulations rather well, when I add appropriate image sources for my particular geometry. Of course, the solution vanishes as I shrink [itex]a\to0[/itex]. So my question is, is there a way to solve the original problem of a point source? Otherwise using this as a sort of Green's Function for a distributed constant temperature boundary seems suspect.

I feel like this issue comes up a bunch in EM but I'm blanking on how to deal with it here. By the way, a second approach I've tried is to integrate the normal Green's Function over time (i.e. convolving with a step source). There again I get a singularity at r=0.

Thanks!