Theoretical analysis of a discrete population balance model for sum kernel

TL;DR

This paper provides a theoretical analysis of a discrete population balance model based on the Oort-Hulst-Safronov equation, focusing on existence, conservation, differentiability, and uniqueness of solutions for specific coagulation kernels.

Contribution

It establishes existence, density conservation, differentiability, and uniqueness results for the discrete OHS model with various coagulation kernels.

Findings

Existence and density conservation for $V_{i,j} \\leqs (i+j)$.

Differentiability of solutions for $V_{i,j} \\leqs i^{\\alpha}+j^{\\alpha}$ with $0 \\leqs \\alpha \\leqs 1$.

Uniqueness under bounded second moment.

Abstract

The Oort-Hulst-Safronov equation, shorterned as OHS is a relevant population balance model. Its discrete form, developed by Dubovski is the main focus of our analysis. The existence and density conservation are established for the coagulation rate $V_{i,j} \leqs (i+j),$ $\forall i,j \in \mathbb{N}$. Differentiability of the solutions is investigated for the kernel $V_{i,j} \leqs i^{\alpha}+j^{\alpha}$ where $0 \leqs \alpha \leqs 1$. The article finally deals with the uniqueness result that requires the boundedness of the second moment.