The continuous fission equation with derivative of fractional order
Polymer degradation is the process where polymers are converted into monomers or mixtures of monomers. Polymers range from familiar synthetic plastics such as polystyrene (also called styrofoam) to natural biopolymers such as DNA and proteins that are fundamental to biological structure and function. Historically, products arising from the linkage of repeating units by covalent chemical bonds have been the primary focus of polymer science; emerging important areas of the science now focus on noncovalent links. Polyisoprene of latex rubber and the polystyrene of styrofoam are examples of polymeric natural/biological and synthetic polymers, respectively. In biological contexts, essentially, all biological macromolecules, that is, proteins (polyamides), nucleic acids (polynucleotides), and polysaccharides, are purely polymeric and are composed in large part of polymeric components, for instance, isoprenylated/lipid-modified glycoproteins, where small lipidic molecule and oligosaccharide modifications occur on the polyamide backbone of the protein. 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 [
The binary fission integrodifferential equation,
We aim to investigate the evolution of the number density of particles described by the fractional fission integrodifferential equation
Our analysis consists of two distinct cases: the case where the breakup rate depends on the size of the chain breaking up and the case where it does not depend. This will help us compare and analyse the two scenarios.
Firstly we assume that the rate of breakup is independent of the length of polymer. Model (
We note that by the differential expression (
Hence we put
This case represents a process where the rate of fission increases with size. Such a process can occur when the polymers are under tenseness or in a destructive force field such as ultrasound. Model (
If we take
If we compare this distribution to the previous case where the breakup rate is independent of the length of polymer, we see that the second model shows a much slower production of daughter particles due to fission. This is an expected outcome given the relative behaviour of the two breakup speeds.
We have used the model of fractional
It is obvious to see that taking
The authors declare that there is no conflict of interests regarding the publication of this paper.