I need to solve:
$\displaystyle \nabla ^2 T = 0$ with T=T(r) and r=a/T=T1 and r=b/T=T2
Can anyone offer advice as to the solution?
I'm assuming that you're only in 2D. Laplace's equation in polar coords is
$\displaystyle T_{rr} + \frac{1}{r} T_r + \frac{1}{r^2} T_{\theta \theta} = 0. $
Asumming that $\displaystyle T = T(r)$ reduces this PDE to the ODE
$\displaystyle T_{rr} + \frac{1}{r} T_r = 0 $
which you can solve. Then impose your boundary conditions.