of the GF Method.
The GF method can be viewed as a restatement of a boundary
value problem into integral form. The GF method is useful if the GF is
known (or can be found), and if the integral expressions can be evaluated.
If these two limitations can be overcome, the GF method offers several
advantages for the solution of linear heat conduction problems. The advantages
of the GF method are the following:
1. The GF method is flexible and powerful.
2. The solution procedure is systematic.
3. The GF method gives analytical solutions in the form of integrals.
4. The GF for 2-D and 3-D transient cases can be found by multplication
of 1-D cases.
5. Alternative form of the solution can improve series convergence.
6. Time partitioning can improve series convergence.
1. The GF method is flexible and powerful. The same GF for a given geometry
(including type of boundary conditions) can be used as a building block
to the temperature resulting from: space-variable initial conditions; time-
and space-variable boundary conditions; and, time- and space-variable energy
2. The solution procedure is systematic. Many GF are given in this Library,
so the derivation of the GF can be omitted in these cases, and the solution
for the temperature can be written immediately in the form of integrals.
The systematic procedure saves time and reduces the possibility of error,
which is particularly important for two- and three- dimensional geometries.
For complicated problems in which the heat conduction is caused by several
non-homogeneous terms, and the effect of each term can be considered separately.
3. The GF method gives analytical solutions in the form of integrals. The
solution takes the form of a sum (superposition) of several integrals,
one for each non-homogeneous term in the problem. The analytical expressions
for temperature can be: evaluated with high accuracy; evaluated only where
needed for great computer-use efficiency; differentiated to find heat flux
or sensitivity coefficients; or, integrated to find average temperature.
The integrals can always be evaluated numerically (quadratures) if they
cannot be found in closed form.
4. Multiplication of GF for 2-D and 3-D transient cases. Two- and three-dimensional
transient GF can be found by simple multiplication of one-dimensional transient
GF for the rectangular coordinate system for most boundary conditions (type
0,1,2 and 3), provided that the body is homogeneous and the body is orthogonal
(for which each boundary is defined by a fixed value of one coordinate,
such as x=a). This multiplicative property can result in
great simplification in the derivation of the temperature, as well as providing
a very compact means to catalog the GF in these cases. For cylindrical
coordinates, the multiplicative property of the GF applies to certain 2-D
5. Alternative form of the solution can improve series convergence. For
heat conduction in finite bodies, infinite series solutions for heat conduction
problems driven by non-homogeneous boundary conditions sometimes exhibit
slow convergence, requiring a very large number of terms to obtain accurate
numerical values. For some of these problems an alternative formulation
of the Green's Function Solution Equation reduces the number of required
6. Time partitioning can improve series convergence. Time partitioning
arises naturally from the GF method by splitting the time integrals in
the solution into small-time and large-time partitions and using rapidly-converging
forms of the GF in each partition. The time-partitioning method can give
accurate values of the temperature using only a few terms of the infinite