This paper presents accurate solutions for nonlinear transcendental equations of the selective harmonic elimination technique used in three-phase PWM inverters feeding the induction motor by particle swarm optimization (PSO). With the proposed approach, the required switching angles are computed efficiently to eliminate low order harmonics up to the $23^{rd}$ from the inverter voltage waveform, whereas the magnitude of the fundamental component is controlled to the desired value. A set of solutions and the evaluation of the proposed method are presented. The obtained results prove that the algorithm converges to a precise solution after several iterations. The salient contribution of the paper is the application of the particle swarm algorithm to attenuate successfully any undesired loworder harmonics from the inverter output voltage. The current paper demonstrates that the PSO is a promising approach to control the operation of a three-phase voltage source inverter with a selective harmonic elimination strategy to be applied in induction motor drives.