Analysis and Simulations of the Discrete Fragmentation Equation with Decay

TL;DR

This paper studies the discrete decay--fragmentation equation, proving existence, uniqueness, and properties of solutions, supported by numerical simulations to illustrate the theoretical findings.

Contribution

It establishes conditions for the existence, uniqueness, and specific properties of solutions to the discrete decay--fragmentation equation using semigroup theory.

Findings

Solutions exist and are unique under certain conditions.

The solution semigroup can be analytic and compact.

Numerical simulations support the theoretical results.

Abstract

Fragmentation--coagulation processes, in which aggregates can break up or get together, often occur together with decay processes in which the components can be removed from the aggregates by a chemical reaction, evaporation, dissolution, or death. In this paper we consider the discrete decay--fragmentation equation and prove the existence and uniqueness of physically meaningful solutions to this equation using the theory of semigroups of operators. In particular, we find conditions under which the solution semigroup is analytic, compact and has the asynchronous exponential growth property. The theoretical analysis is illustrated by a number of numerical simulations.