Analytical and computational perspectives on classical and non-classical multidimensional aggregation population balance models

Semi-analytical methods offer an efficient way to solve population balance equations (PBEs) by combining analytical formulations with numerical computation to capture complex particle dynamics. While classical PBEs are computationally demanding owing to nonlinear integral terms, rewriting them in a nonlinear divergence (conservative) form significantly reduces this burden. This study introduces a hybrid semi-analytical method for non-classical divergence-based multidimensional aggregation PBEs. The approach integrates the Laplace transform (LT) with the projected differential transform to construct a recursive series solution without requiring Adomian polynomials, unlike the Adomian decomposition method (ADP). To improve long-term accuracy, Padé approximations are employed to address the limitations of standard series solutions. The method is validated through number density functions (NDFs) and their moments across various aggregation kernels. Results show that it is about four times more efficient than existing semi-analytical techniques. For one-dimensional (1D) models, it reduces CPU time by 40–50% and achieves roughly 50% lower errors, closely matching exact solutions. Extension to two-dimensional (2D) models also yields accurate results, though at a higher computational cost owing to nonlinear triple integrals. The approach also demonstrates strong numerical stability and robustness across varying initial conditions (ICs) and parameter ranges.