The transient two-dimensional thermocapillary convection and molten pool shape in melting or welding with a time-dependent and distributed incident flux are numerically predicted in this study. Determination of the molten pool shapes is crucial, because of its close relationships with the strength, microstructure, and mechanical properties of the fusion zone. In the work, the time-dependent incident flux is assumed to be a function of scanning speed and energy distribution parameter. Transport processes at the time corresponding to the maximum cross section can be identical to those under steady three-dimensional condition. The computed flow patterns and molten pool shapes under the flat free surface exhibits distinct regions for different Marangoni and Prandtl numbers. The effects of Peclet number and beam power on flow and temperature fields and fusion zone shapes are also presented. The computed results are confirmed by comparing the predicted peak speed on the free surface and molten pool width with those obtained from scale analysis provided in the literature.