An accurate approach and its convergence analysis for the multidimensional nonlinear collisional breakage equations

Particulate processes such as bubble breakage, granule preparation, and ore extraction are often modeled using population balance equations. While most studies rely on linear fragmentation models, these fail to capture realistic dynamics where breakage occurs due to particle collisions, leading to nonlinear collisional breakage equations. The mathematical complexity of nonlinear kernels and integrals has limited progress in this direction, with very few analytical solutions available. In this work, we develop an accurate approach based on a modified variational iteration method for solving multidimensional collisional breakage equations. The method allows flexible operator selection and efficient determination of the Lagrange multiplier. A rigorous convergence and error analysis is carried out using the fixed-point theorem. The approach is further extended to two- and three-dimensional models, including collisional fragmentation cases. Numerical experiments confirm the accuracy and efficiency of the method compared with existing approaches.