Consider the steady temperature in the cylinder caused either by heating at
the boundaries or by internal energy generation. The temperature satisfies
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(1) |
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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:
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(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.