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Time Fractional Fisher–KPP and Fitzhugh–Nagumo Equations
- Source :
- Entropy, Vol 22, Iss 1035, p 1035 (2020), Entropy, Volume 22, Issue 9
- Publication Year :
- 2020
- Publisher :
- MDPI AG, 2020.
-
Abstract
- A standard reaction–diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction–subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction–diffusion equations. In this paper, we formulate clear examples of reaction–subdiffusion systems, based on<br />equal birth and death rate dynamics, Fisher–Kolmogorov, Petrovsky and Piskunov (Fisher–KPP) equation dynamics, and Fitzhugh–Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction–diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.
- Subjects :
- General Physics and Astronomy
lcsh:Astrophysics
010103 numerical & computational mathematics
01 natural sciences
Article
Law of mass action
Reaction rate
Chemical kinetics
reaction–diffusion equations
Reaction–diffusion system
lcsh:QB460-466
Applied mathematics
0101 mathematics
Diffusion (business)
Nuclear Experiment
lcsh:Science
Mathematics
fractional diffusion
Rate equation
Birth–death process
lcsh:QC1-999
Term (time)
continuous time random walks
010101 applied mathematics
reaction kinetics
lcsh:Q
lcsh:Physics
Subjects
Details
- Language :
- English
- ISSN :
- 10994300
- Volume :
- 22
- Issue :
- 1035
- Database :
- OpenAIRE
- Journal :
- Entropy
- Accession number :
- edsair.doi.dedup.....11c3e11b0cdce33dd25698cab8a01ea9