In this paper, the growth kinetics of liquid phase electroepitaxy of III-V binary and ternary compound semiconductors have been studied. Concentration profiles of solute atoms such as As, P, Sb in front of a crystal (GaAs, InP, InAs, GaSb) interface growing in a In- or Ga-rich solution for the case of growth of binary compounds and for the case of ternary compounds (InGaAs, AlGaSb, InAsP) have been constructed using a computer simulation technique. The effect due to Peltier heating or cooling and electromigration during the growth have been incorporated to simulate the concentration profiles. The growth velocity in the absence and presence of convection, due to the Peltier effect and electromigration are calculated under different conditions. The results are discussed in detail.