Mathematical Solvability of a Caputo Fractional Polymer Degradation Model Using Further Generalized Functions

Emile Franc Doungmo Goufo,Stella Mugisha

doi:10.1155/2014/392792

Emile Franc Doungmo Goufo, Stella Mugisha

Open Access

https://doi.org/10.1155/2014/392792

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Abstract

The continuous fission equation with derivative of fractional orderα, describing the polymer chain degradation, is solved explicitly. We prove that, whether the breakup rate depends on the size of the chain breaking up or not, the evolution of the polymer sizes distribution is governed by a combination of higher transcendental functions, namely, Mittag-Leffler function, the further generalizedG-function, and the Pochhammer polynomial. In particular, this shows the existence of an eigenproperty; that is, the system describing fractional polymer chain degradation contains replicated and partially replicated fractional poles, whose effects are given by these functions.

Highlights

Polymer degradation is the process where polymers are converted into monomers or mixtures of monomers
Whether the breakup rate depends on the size of the chain breaking up or not, the evolution of the polymer sizes distribution is governed by a combination of higher transcendental functions, namely, Mittag-Leffler function, the further generalized G-function, and the Pochhammer polynomial. This shows the existence of an eigenproperty; that is, the system describing fractional polymer chain degradation contains replicated and partially replicated fractional poles, whose effects are given by these functions
Let us have a look at what is known about the classical kinetics of polymer chain degradation

Summary

Introduction

Polymer degradation is the process where polymers are converted into monomers or mixtures of monomers. In the theory of polymers division, one would expect a conservation of mass, especially when polymers are converted into monomers or mixtures of monomers, but [1, 2] an infinite cascade of division events creating a “dust” of monomers of zero size carrying nonzero mass and leading to nonconservativeness (dishonesty) in the model has been observed Since this remains partially unexplained by classical models of clusters’ fission, extending the analysis to the fractional version and expressing the solutions explicitly may bring more light and a broader outlook about this phenomenon which remains a mystery. The authors in [12] analysed the model in combination with the inverse process, that is, the coagulation process, and provided a similar result for the size distribution

Fractional Fission Differential Problem

Mathematical Analysis

Concluding Remarks