Cylindrical Coordinates. Transient 1-D.

The Heat Equation for transient 1-D Green's Functions in cylindrical-radial coordinates is

$${\frac{1}{r}} {\frac{\partial }{\partial r}} \left(\vphantom{ r\frac{\partial G}{\partial r}}\right. {\frac{\partial G}{\partial r}} \left.\vphantom{ r\frac{\partial G}{\partial r}}\right) {\frac{1}{\alpha }} \delta \prime \delta \tau {\frac{1}{\alpha }} {\frac{\partial G}{\partial t}}$$

Note that the cylindrical-radial Dirac delta function $\delta$(r - r^$\prime$) has vector arguments and has units of [ meters^-2].