3d heat equation with constant point source

[acme37]

Hi all,

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

$\partial_t u(r,t) = D\Delta u(r,t)$
$u(r,0) = 0$
$u(0,t) = C_o$

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:
$$u(x,t) = C_o \mathrm{erfc}\left(\frac{x^2}{\sqrt{4Dt}}\right)$$
In 3D, I proceed by taking the Laplace transform and solving the resulting Helmholtz equation. After applying the $r\to\infty$ boundary condition,
$$u(r,s)=A\frac{e^{-\sqrt{\frac{s}{D}}r}}{r}$$
But I can't apply the boundary condition at $u(0,s)$ due to the singularity at the origin. So instead I assume we fix the constant temperature condition on a ball of radius $a$, for $a\ll r$. If I do that my solution isn't so bad,
$$u(r,t)=C_o\frac{a}{r}\mathrm{erfc}\left(\frac{(a-r)^2}{4Dt}\right)$$
This matches my simulations rather well, when I add appropriate image sources for my particular geometry. Of course, the solution vanishes as I shrink $a\to0$. 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!

 

[mfb]

An ideal point source does not transmit heat in 3 dimensions, so your solution for the ball with a finite size looks reasonable.

 

[acme37]

Thanks, but I'm a bit confused. What else is the Green's Function if not the response to an ideal point source?

 

[mfb]

There is no response (read: zero temperature change) to an ideal point source.

 

[blue_raver22]

I find this problem very interesting and challenging. The 3D heat equation with a constant point source is a common problem in many fields, including thermal engineering and physics. Your approach of solving it using the Laplace transform and the Helmholtz equation is a valid and well-established method. However, the singularity at the origin in the resulting solution is a known issue and can be difficult to deal with.

One possible solution is to use a different boundary condition at the origin, such as a Neumann boundary condition which specifies the heat flux rather than the temperature. This can help to avoid the singularity and provide a more physically realistic solution. Another approach is to use a regularization technique to smooth out the singularity, such as using a Gaussian distribution instead of a point source.

It is also worth considering the physical implications of the problem. In reality, a truly point source of heat is not possible and there will always be some finite size and distribution of the source. Therefore, using a point source in the mathematical model may not accurately reflect the physical system. In this case, it may be more appropriate to use a distributed source with a finite size and shape.

In conclusion, while the 3D heat equation with a point source can be a challenging problem, there are various techniques and considerations that can help to provide a more accurate and physically meaningful solution. I encourage you to continue exploring different approaches and techniques to further improve your understanding and solution of this problem.