Modified variational iteration method and its convergence analysis for solving nonlinear aggregation population balance equation

Abstract

This study proposes a novel approach based on the variational iteration method to solve the nonlinear aggregation population balance equation. The approach provides great flexibility by allowing the selection of appropriate linear operators and efficiently determining the Lagrange multiplier in the nonlinear aggregation population balance equation. The mathematical derivation is supported by conducting a detailed convergence analysis using the contraction mapping principle in the Banach space. Furthermore, error estimates for the approximate solutions are derived, thereby improving our understanding of the accuracy and reliability of the proposed method. To validate the new approach, the obtained solutions are compared with the exact solutions for analytically tractable kernels. However, for more complex physically relevant kernels including polymerization, Ruckenstein/Pulvermacher, and bilinear kernels, due to lack exact solutions, the obtained series solutions corresponding to different initial conditions are verified against the finite volume scheme (kumar et al., 2016). The outcomes illustrate that the proposed approach offers superior approximations of number density functions with fewer terms and demonstrates higher accuracy over extended time domains than the traditional variational iterative method. The new approach also has the tendency to capture the zeroth and first order moments of the number density function with high precision.