This article aims to provide approximate solutions for the non-linear collision-induced breakage equation using two different semi-analytical schemes, i.e., variational iteration method (VIM) and optimized decomposition method (ODM). The study also includes the detailed convergence analysis and error estimation for ODM in the case of product collisional ($K(\epsilon,\rho)=\epsilon\rho$) and breakage ($b(\epsilon,\rho,\sigma)=\frac{2}{\rho}$) kernels with an exponential decay initial condition. By contrasting estimated concentration function and moments with exact solutions, the novelty of the suggested approaches is presented considering three numerical examples. Interestingly, in one case, VIM provides a closed-form solution, however, finite term series solutions obtained via both schemes supply a great approximation for the concentration function and moments.

翻译：本文旨在利用两种半解析方法，即变分迭代法(VIM)和优化分解法(ODM)，为非线性碰撞诱导破碎方程提供近似解。研究还详细分析了在乘积碰撞核($K(\epsilon,\rho)=\epsilon\rho$)与破碎核($b(\epsilon,\rho,\sigma)=\frac{2}{\rho}$)条件下，采用指数衰减初始条件时ODM的收敛性及误差估计。通过将预估的浓度函数和矩与精确解进行对比，结合三个数值算例展示了所提方法的新颖性。值得注意的是，在一种情形下VIM给出了闭式解，但两种方法得到的有限项级数解均能为浓度函数和矩提供高度逼近。