  In this paper, we derive closed form solutions for the quasi-stationary
problems of moving heat sources within the gradient theory of heat transfer.
This theory can be formally deduced from the two-temperature model and it can
be treated as a generalized variant of the Guyer-Krumhansl model with the
fourth order governing equation. We show that considered variant of the
gradient theory allows to obtain a useful singularity-free solutions for the
moving point and line heat sources that can be used for the refined analysis of
the melt pool shape in the laser powder bed fusion processes. Derived solutions
contain single additional length scale parameter that can be related to the
mean particles size of the powder bed. Namely, we show that developed gradient
models allow to describe the decrease of the melt pool depth with the increase
of the powder's particles size that was observed previously in the experiments.
We also derive the dimensionless relations that can be used for the
experimental identification of the model's length scale parameter for different
materials. Semi-analytical solution for the Gaussian laser beam is also derived
and studied based on the Green function method within the considered theory.
