New Generalised Semi-Analytical Approach for the Multidimensional Nonlinear Collisional Fragmentation Equations

The nonlinear collisional fragmentation equation is valuable for studying particle collisions and can model the evolution of raindrops, liquid-liquid dispersion, bubble columns, astrophysical planetary phenomena, and granulation processes in the pharmaceutical industry. Solving this equation analytically is highly tedious due to the complex structures of rate kernels, along with the nonlinear nature of the equation. This work aims to develop a generalised and efficient semi-analytical method that combines the Laplace decomposition method with Padé approximation to solve multidimensional nonlinear integro-partial differential equations. The Laplace decomposition method yields a series solution that represents the collisional fragmentation process over short time periods, while the combined Laplace decomposition with Padé approximation approach effectively captures the long-term dynamics. The mathematical formulation is validated through a detailed convergence analysis in a Banach space. Several examples including binary, Austin's and a gelling kernel are examined to demonstrate its accuracy and robustness. Most of the cases analysed in one-dimension, explicit (closed-form) solutions for the number density functions are derived for the first time. For the remaining multidimensional cases, accuracy is evaluated by comparison with the finite volume scheme [Das et al. (2020), SIAM J. Sci. Comput. 42, B1570-B159], the homotopy perturbation method [Yadav et al. (2023), Proceedings of the Royal Society A, 479(2279), 20230567] and the blues function method [Hussain et al. (2024), Physics of Fluids, 36, 103359]. The new approach captures both number density functions and their integral moments with high precision. Errors in number density functions are computed using various series solutions.