A review on semi-analytical methods for aggregation and coupled aggregation–breakage equations

Abstract

This work reviews the semi-analytical technique (SAT) and proposes a unique SAT based on the homotopy analysis method (HAM), called accelerated HAM (AHAM) (recently proposed in Hussain et al . (Hussain S, Arora G, Kumar R. 2024 Semi-analytical methods for solving non-linear differential equations: a review. J. Math. Anal. Appl. 531 , 127821 (doi: 10.1016/j.jmaa.2023.127821 ))) to solve pure aggregation and coupled aggregation–fragmentation population balance equations (PBEs), which are nonlinear integro-partial differential equations. The novelty of this approach is demonstrated through a comparative analysis of numerical results against existing findings from the Adomian decomposition (Singh R, Saha J, Kumar J. 2015 Adomian decomposition method for solving fragmentation and aggregation population balance equations. J. Appl. Math. Comput. 48 , 265–292 (doi: 10.1007/s12190-014-0802-5 )), homotopy analysis (Kaur G, Singh R, Briesen H. 2022 Approximate solutions of aggregation and breakage population balance equations. J. Math. Anal. Appl. 512 , 126166 (doi: 10.1016/j.jmaa.2022.126166 )), homotopy perturbation (Kaur G, Singh R, Singh M, Kumar J, Matsoukas T. 2019 Analytical approach for solving population balances: a homotopy perturbation method. J. Phys. A Math. Theor. 52 , 385201 (doi: 10.1088/1751-8121/ab2cf5 )) and optimized decomposition (Kaushik S, Kumar R. 2023 A novel optimized decomposition method for Smoluchowski’s aggregation equation. J. Comput. Appl. Math. 419 , 114710 (doi: 10.1016/j.cam.2022.114710 )) methods for these models. In addition, the theoretical convergence analysis and error estimation of the proposed method are also investigated. To validate the scheme, we analyse several numerical test cases and the results illustrate that the suggested technique offers the most accurate estimates for the two models under consideration.