Consider the steady temperature in the cylinder caused either by heating at
the boundaries or by internal energy generation. The temperature satisfies

(1) | |||

Here is the outward normal on each surface of the cylinder (at , , and ). The boundary condition represents one of three types at each surface: type 1 for , , and a specified temperature; type 2 for , and a specified heat flux; and, type 3 for and for convection to surroundings at temperature . Heat transfer coefficient must be uniform on the i boundary.

The temperature can be stated in the form of integrals with the method of
Green's functions. If the Green's function is known, then the
temperature that satisfies equation (1) is given by:

(for volume energy generation)

(for boundary conditions of type 2 and 3)

(2) |

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 , . The two summations represent all possible combinations of boundary conditions, but with only one type of boundary condition on each of three surfaces of the cylinder. Mixed-type boundary conditions are not treated.