In the classic book "Conduction of Heat in Solids" by Carslaw & Jaeger (1959) the heat diffusion equation
$$ \frac{\partial T}{\partial t}=\alpha\left(\frac{\partial^2T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right) $$
is solved for $T=T(r,t)$ (page 334) for the following case:
- Cylindrical symmetry is assumed.
- No flow of heat in $z$ (vertical) direction.
- The region of interest is $r\geq a$, where $a$ is given.
- Initial temperature is zero everywhere, i.e. $T(r,0)=0$.
- The cylinder $r=a$ is for $t>0$ always at temperature $V$, i.e. $T(a,t>0)=V$.
The solution given is
$$ T(r,t)=V+\frac{2V}{\pi}\int_0^\infty\exp\left[-\alpha u^2t\right]\frac{J_0(ur)Y_0(ua)-Y_0(ur)J_0(ua)}{J_0^2(au)+Y_0^2(au)}\frac{{\rm d}u}{u}, $$ where $J_0(x)$ and $Y_0(x)$ are the Bessel functions of first and second kind (both of order zero).
I find this integral hard to evaluate numerically. Can it be reformulated as e.g. an infinite sum of tabulated functions? Or approximated for the case of case of large time ($\alpha t/a^2\gg 1$) at a large distance ($r\gg a$)?
Any help on this one is appreciated!