  This article aims to establish a semi-analytical approach based on the
homotopy perturbation method (HPM) to find the closed form or approximated
solutions for the population balance equations such as Smoluchowski's
coagulation, fragmentation, coupled coagulation-fragmentation and bivariate
coagulation equations. An accelerated form of the HPM is combined with the
Elzaki transformation to improve the accuracy and efficiency of the method. One
of the significant advantages of the technique lies over the classic numerical
methods as it allows solving the linear and non-linear differential equations
without discretization. Further, it has benefits over the existing
semi-analytical techniques such as Adomian decomposition method (ADM),
optimized decomposition method (ODM), and homotopy analysis method (HAM) in the
sense that computation of Adomian polynomials and convergence parameters are
not required. The novelty of the scheme is shown by comparing the numerical
findings with the existing results obtained via ADM, HPM, HAM and ODM for
non-linear coagulation equation. This motivates us to extend the scheme for
solving the other models mentioned above. The supremacy of the proposed scheme
is demonstrated by taking several numerical examples for each problem. The
error between exact and series solutions provided in graphs and tables show the
accuracy and applicability of the method. In addition to this, convergence of
the series solution is also the key attraction of the work.
