Laplace transform-based approximation methods for solving pure aggregation and breakage equations

The varied applications of the aggregation and breakage equations in several fields of science have attracted many researchers to explore accurate novel methods to calculate their solutions. Due to the complexity of these models, the exact solutions are computable only for a few cases of aggregation and breakage kernel parameters. So, to obtain solutions for physically relevant kernels, various numerical and semi-analytical approaches have been explored. It is observed in the literature that the numerical methods are accurate, but they require some unrealistic assumptions. This has led to the development of semi-analytical methods that need fewer parameters and are bereft of discretization of the variables. The researchers explore accurate and less time-consuming methods to solve such equations. So, the objective of this article is the introduction of novel and accurate semi-analytical techniques to solve the pure aggregation and breakage equations. We have used the Laplace optimized decomposition method (LODM) to calculate the series solution for the aggregation equation and the Laplace Adomian decomposition method (LADM) to solve pure breakage equation. The novelty of this work is that it deals with the theoretical convergence of the LADM and LODM solutions toward the exact solutions. In addition to this, several numerical test cases are presented to validate our theoretical findings. For the aggregation equation, LODM results are compared with the solutions obtained via well-developed finite volume technique. The methods are found to be highly accurate to solve these partial integro-differential equations.

Automated summary

The aggregation and breakage equations have many applications in different fields of science, prompting researchers to find accurate methods to solve them. Due to their complexity, exact solutions are only possible for certain parameters. Therefore, numerical and semi-analytical approaches have been explored to obtain solutions for physically relevant kernels. However, numerical methods often make unrealistic assumptions, leading to the development of semi-analytical methods. In this article, the authors introduce novel and accurate semi-analytical techniques for solving the pure aggregation and breakage equations.