In the present study, we have derived the analytical solutions of particle coagulation due to Brownian motion in the free molecule and continuum regime, based on the Taylor-series expansion method of moment (TEMOM). The derivation is rigorous, and the accuracy of the analytical solution is fully dependent on underlying approximations in an expanded Taylor series. The analytical solutions for TEMOM model provide a basic framework to approximate the population balance equation. The analytical results together with the previous asymptotic analysis of TEMOM model will benefit for the understanding of Brownian coagulation mechanism. (C) 2013 Elsevier Ltd. All rights reserved.