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Top1-D Mathematical Modeling
Consider a laser beam hitting a 1-D rod with finite length 
. The beam moves along the rod which is insulated at the two endpoints. Mathematically, the temperature distribution of the rod can be modeled by the nonhomogeneous heat equation:
(1) where
denotes the temperature rise of the rod at position
and time
,
is the thermal diffusivity of the rod,
is the thermal conductivity, and
is the energy distribution of the moving laser beam. In the case of a dithering laser beam shown in Figure 1 (1),
can be expressed as
(2)Figure 1. (1) A dithering laser beam on a 1-D rod. (2) 1-D temperature distribution along the rod from various numerical methods. (3) 1D maximum temperature rise of steel AISI 4340 versus frequency of the dithering laser beam. (4) The curve in (3) is well approximated by the function 
.
where
is the position of the dithering Gaussian beam,
is the initial position of the laser beam,
is the intensity of the laser beam,
is the effective radius of the laser beam, and
is a constant used for the Gaussian model. The initial condition for
is zero which assumes that the rod has the same temperature as the ambient initially. The boundary conditions impose that the rod is insulated at the two ends: