Analytical solution for a transient, three-dimensional temperature distribution due to a moving laser beam

This paper presents an analytical solution of the transient temperature distribution in a finite solid when heated by a moving heat source. The analytical solution is obtained by solving the transient three-dimensional heat conduction equation in a finite domain by the method of separation of variables (SOV). Meanwhile previous studies focus on analytical solutions for semi-infinite domains, here an analytical solution is provided for a finite domain. The non-homogeneous equation is solved by using the Laplace transform for a unit impulse and then convoluted with the actual heat source. Two different distributions are used: a Gaussian distribution and a spatially uniform plane heat source.