How do I solve this PDE (diffusion equation) using the sepration of variables method? $$\frac{\partial u}{\partial t} =\nu\left(\frac{\partial^2u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} \right), 0 < r < a, t >0.$$
Subject to the conditions $$\frac{\partial u(0, t)}{\partial r}=0, t > 0$$
$$u(a, t) = 0, t > 0$$ $$u(r, 0) = \frac{\rho}{4\mu}(a^2 - r^2), 0 < r < a.$$
$\rho , \mu,$ and $\nu$ are constants.
Usually with these type of problems we would set $u(r, t) = R(r) T(t)$ and then find some series solution, but all the problems I have experience of solving were homogenous heat equations $u_t = ku_{rr}$. This is different.

The method of separation of variables leads to a general solution in forme of infinite series, involving Bessel functions (in attachment) :

The boundary conditions allow to find the explicit form of the coefficients, thanks to the Fourier-Bessel series : http://mathworld.wolfram.com/Fourier-BesselSeries.html


Am sorry to post the result on attached images. Because a lack of time it was this or nothing. Sorry again.
