The GF stated below contains a double summation with eigenfunctions
norms and , eigenvalues and kernel function
, as follows:
The kernel function
The expression for in Eq. (7) is symmetric if and are interchanged and covers several combinations of boundary conditions provided . The special case of is discussed below.
Kernel function with
If the face of the cylinder is of type 2, then the
zero eigenvalue exists. In this case the
kernel function must satisfy
Modified GF for Case R02Z22
A very special condition occurs if the cylinder's entire boundary
has type 2 (Neumann) conditions. In this case the ordinary Green's
function, defined above, does not exist.
However the GF method can be used if we define a modified GF,
following Barton (Elements of Green's Functions and Propagation,
Oxford University Press, 1989), as follows:
Modified Green's function may be used to find temperature with the Green's function solution equation, Eq. (2), with the following additional constraints: the sum of the heat passing through the boundaries must be equal to the total amount of heat introduced by volume energy generation ; and, since the spatial average temperature of the body computed from is zero, the average temperature in the body must be supplied as part of the input data to the problem. Further discussion and numerical examples of steady heat conduction in the cylinder may be found in a paper by Cole (K.D. Cole, "Fast converging series for heat conduction in the circular cylinder," J. of Engineering Mathematics, vol. 49, pp. 217-232, 2004).