General solution for the heat equation of a 1-D circle

[GwtBc]

Homework Statement

Modify the initial conditions (for the diffusion equation of a circle) to have the initial conditions $g(\theta)= \sum_{n=-\infty}^{\infty}d_{n}e^{2\pi in\theta}$
Using the method of Green's functions, and $S(\theta,t)= \frac{1}{\sqrt{4\pi Dt}}\sum_{n=-\infty}^{n=\infty}e^{-(2\pi n)^2Dt}e^{2\pi in\theta}$ derive the identity

$S_{g}(\theta,t)= \frac{1}{\sqrt{4\pi Dt}}\sum_{n=-\infty}^{n=\infty}d_{n}e^{-(2\pi n)^2Dt}e^{2\pi in\theta}$

Homework Equations

The Attempt at a Solution

We tried many times to use convolution of S with g but this did not produce the desired result (We also tried convolution with the second derivative of g after messing around for a bit with separation of variables in the heat equation).

We've only been recently introduced to the concept of Green's functions and convolution and there does not seem to be consensus amongst sources about what this actually means. Some sources have definite integrals with arbitrary terminals, some have the terminals as theta and 0 and others state indefinite sums.

To be clear '1-D circle' simply means that the shape was constructed by joining together the ends of a 1-dimensional line.

 

[Orodruin]

Hint: Solving the heat equation on a circle is equivalent to solving it on the entire real line with a periodic source.

 

[GwtBc]

I believe that we have actually solved it, in principle, since S up there is the Green's function for the case of a circle.