# Finding Green's function

I have this problem. I am studying a physical problem and I came to this equation:

$$\frac{\partial}{\partial t} (R^{1/2} \Sigma) = \frac{12\nu}{s^2} \frac{{\partial}^2}{{\partial s}^2} (R^{1/2} \Sigma)$$

Which, looking for a sparable solution $R^{1/2} \Sigma = T(t) S(s)$ gives

$$\frac{T'}{T} = \frac{12 \nu} {s^2} \frac{S''}{S}= const= -\lambda^2$$

The t depending part is an exponential and the s depending part is a Bessel function. And I'm ok so far (thanks to the help here). Then the book says: "It is interesting to find the Green's function which is by definition the solution fo $\Sigma(R,t)$, taking the initial mass distribution as a delta function: (We're talking about an annulus of radius ($R_0$)

$$\Sigma(R,0)= \frac{m}{2 \pi R_0} \delta(R-R_0)$$

using the dimensionless variables $x=R/R_0$ and $\tau= 12 \nu \, t \, R_0^{-2}$, the result is

$$\Sigma(x,t) = \frac{m}{\pi R_0^2\, \tau \, x^{1/4}} exp \Biggl(-\frac{1-x^2}{\tau} \Biggr) I_{1/4}(2x/\tau)$$

Where $I_{1/4}$ is a modified Bessel function.

Now my problem is that i had a very poor knowledge of the green function method. I found some green function in special and easy cases but I never did it with this kind of operator. I wouldn't know how to do it in this case. Can somebody enlighten me? Any help would be appreciated.

I'm studying the general theory of the Green's function method but I don't have the time to master it all before using it in this problem. So I need to work out this problem before I learn all there is to learn about Green's function