Exact solutions of population balance equation

•Group analysis of the population balance equation.•Invariant solutions of integro-differential equations.•Invariant solutions of moment system of equations.

Population balance equations have been used to model a wide range of processes including polymerization, crystallization, cloud formation, and cell dynamics, but the lack of analytical solutions necessitates the use of numerical techniques. The one-dimensional homogeneous population balance equation with time dependent but size independent growth rate and time dependent nucleation rate is investigated. The corresponding system of equations is solved analytically in this paper.