An inverse problem: recovering the fragmentation kernel from the short-time behaviour of the fragmentation equation

TL;DR

This paper introduces a new method to recover the fragmentation kernel from short-time measurements of particle size distributions, using a power series representation and stability analysis in the fragmentation equation.

Contribution

It provides a novel power series solution representation and a stability result that enables robust kernel recovery from experimental data.

Findings

Successful kernel reconstruction from short-time data

New stability results using Wasserstein-type norm

Robustness of the method against noise and initial data variations

Abstract

The present paper provides a new representation of the solution to the fragmentation equation as a power series in the Banach space of Radon measures endowed with the total variation norm. This representation is used to justify how the fragmentation kernel, which is one of the two key parameters of the fragmentation equation, can be recovered from short-time experimental measurements of the particle size distributions when the initial condition is a delta function. A new stability result for this equation is also provided using a Wasserstein-type norm. We exploit this stability to prove the robustness of our reconstruction formula with respect to noise and initial data.