Consider two-dimensional heat conduction in the finite cylinder.
The temperature is given by

(12) | |||

The temperature may be stated in the form of integrals with the method of Green's functions. If the Green's function, , is known, the temperature is given by

(for volume energy generation)

(for boundary conditions of type 2 and 3)

(13) |

The same Green's function appears in each integral but is evaluated at locations appropriate for each integral. Here position is located on surface . Surface differential is associated with appropriate surfaces of the cylinder: on surface , ; and, at or , .

**GF, 2D Cylinder**

The steady Green's function represents the response at point
caused
by a point source of heat located at
.
The GF for the finite cylinder satisfies the following equations:

(14) | |||

Note that the boundary conditions are homogeneous and of the same type as the temperature problem.

The Green's function for the cylinder with axisymmetric heat conduction
is given by

(15) | |||

where are Bessel functions of order . The norms, eigenconditions, and kernel functions are identical to those used for the three-dimensional GF. The above single-sum GF may be derived either by a direct solution of the defining equation for , or, by integrating the 3D GF over . A physical interpretation of this approach, called the method of descent, is to distribute 3D point sources to form a ring-shaped source appropriate for axisymmetric 2D heating. A double-sum form of the GF may also be found from the transient GF by the limit method; see Beck et al. (1992, p. 249) for a discussion of case R01Z11.