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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.