The properties of memristor as the fourth basic circuit element are studied. The mathematical models in integral form for memristors with and without border constraint are developed. The simulation is done for the memristor with border constraint. The influences of the source frequency and model parameters on the memristor’s properties are analyzed and some conclusions are drawn. The model parameters considered include the doping ratio and the initial doping width, and their influnce on the current, voltage-current relation and flux-charge relation of the memrister are investigaled.