A Novel Optimized Decomposition Method for Smoluchowski's Aggregation Equation

TL;DR

This paper introduces a new semi-analytical method called the optimized decomposition method (ODM) for solving Smoluchowski's aggregation equation, demonstrating improved accuracy and efficiency over existing methods through theoretical and numerical validation.

Contribution

The paper presents a novel ODM technique that converges to the exact solution and outperforms the Adomian decomposition method for Smoluchowski's equation.

Findings

ODM converges to the exact solution for tested kernels

ODM outperforms ADM in accuracy and efficiency

Numerical validation confirms theoretical advantages

Abstract

The Smoluchowski's aggregation equation has applications in the field of bio-pharmaceuticals \cite{zidar2018characterisation}, financial sector \cite{PUSHKIN2004571}, aerosol science \cite{shen2020efficient} and many others. Several analytical, numerical and semi-analytical approaches have been devised to calculate the solutions of this equation. Semi-analytical methods are commonly employed since they do not require discretization of the space variable. The article deals with the introduction of a novel semi-analytical technique called the optimized decomposition method (ODM) (see \cite{odibat2020optimized}) to compute solutions of this relevant integro-partial differential equation. The series solution computed using ODM is shown to converge to the exact solution. The theoretical results are validated using numerical examples for scientifically relevant aggregation kernels for which the exact solutions are available. Additionally, the ODM approximated results are compared with the solutions obtained using the Adomian decomposition method (ADM) in \cite{singh2015adomian}. The novel method is shown to be superior to ADM for the examples considered and thus establishes as an improved and efficient method for solving the Smoluchowski's equation.