<p>Population balance equations (PBEs) play a pivotal role in modeling the evolution of various particulate systems, including liquid-liquid dispersions, aerosols, colloids, pharmaceutical granulation, and raindrop fragmentation, as they provide a comprehensive framework to describe particle size distribution dynamics. Among the processes governed by PBEs, coagulation stands out as a key mechanism with significant applications in engineering and environmental fields, such as turbulent flows, gas-phase nanoparticle synthesis, soot formation, and atmospheric dynamics, yet its modeling remains challenging due to the complexity of the coagulation kernels and the diversity of initial conditions. These complexities often hinder the analytical solution of such equations for standard kernel classes. To overcome these challenges, this study presents the Beyond Linear Use of Equation Superposition (BLUES) function method to solve coagulation model. The scheme is applied to a variety of problems and its results are validated against established approaches. The algorithm not only provides accurate estimates for density distribution functions but also determines integral moments with high accuracy. Additionally, a convergence theorem and error analysis are provided, confirming the robustness of the method. This work contributes to advancing computational techniques for modeling particle aggregation phenomena, with wide-reaching applications in industrial and scientific contexts.</p>