A biomathematical view on the fractional dynamics of cellulose degradation

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Emile Franc Doungmo Goufo
Keyword(s):  
Fractional Order ◽  
Cellulose Degradation ◽  
Fractional Dynamics ◽  
Cellulose Chain ◽  
Classical Models ◽  
Transcendental Functions ◽  
Chain Breaking

AbstractWe perform a biomathematical analysis of a model of cellulose degradation with derivative of fractional order γ. In the theory of biopolymers division, the phenomenon of shattering remains partially unexplained by classical models of clusters’ fragmentation. Thus, we first examine the case where the breakup rate H is independent of the size of the cellulose chain breaking up, following by the case where H is proportional to the size of the cellulose chain. Both cases show that the evolution of the biopolymer sizes distribution is governed by a combination of higher transcendental functions, namely the Mittag-Leffler function, the further generalized G-function and the Pochhammer polynomial. In particular, this shows existence of an eigen-property, that is, the system describing fractional cellulose degradation contains replicated and partially replicated fractional poles, whose effects are given by these functions.

scholarly journals Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Emile Franc Doungmo Goufo ◽  
Stella Mugisha
Keyword(s):  
Fractional Order ◽  
Evolution Equations ◽  
Multiplication Operator ◽  
Fractional Dynamics ◽  
Perturbation Theorem ◽  
Bounded Perturbation ◽  
The Cauchy Problem ◽  
Fractional Models ◽  
Classical Models ◽  
Sudden Appearance

AbstractClassical models of clusters’ fission have failed to fully explain strange phenomenons like the phenomenon of shattering (Ziff et al., 1987) and the sudden appearance of infinitely many particles in some systems with initial finite particles number. Furthermore, the bounded perturbation theorem presented in (Pazy, 1983) is not in general true in solution operators theory for models of fractional order γ (with 0 < γ ≤ 1). In this article, we introduce and study a model that can be understood as the fractional generalization of the clusters’ fission process.We make use of the theory of strongly continuous solution operators for fractional models (analogues of C0-semigroups for classical models) and the subordination principle for fractional evolution equations (Bazhlekova, 2000, Prüss, 1993) to analyze and show existence results for clusters’ splitting model with derivative of fractional order. In the process, we exploit some properties of Mittag-Leffler relaxation function (Berberan-Santos, 2005), the He’s homotopy perturbation (He, 1999) and Kato’s type perturbation (Banasiak, 2006) methods. The Cauchy problem for multiplication operator in the fractional dynamics is first considered, before we perturb it. Some additional concepts like Laplace transform, Hille-Yosida theorem and the dominated convergence theorem are use to finally show that there is a solution operator to the full fractional model that is positive and contractive.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Emile Franc Doungmo Goufo ◽  
Stella Mugisha
Keyword(s):  
Fractional Order ◽  
Polymer Chain ◽  
Polymer Degradation ◽  
Generalized Functions ◽  
Degradation Model ◽  
Transcendental Functions ◽  
Chain Breaking

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.

Formulation of a State Equation Including Fractional-Order State-Vectors

2007 ◽  
Author(s):  
Masaharu Kuroda
Keyword(s):  
Fractional Calculus ◽  
Control Theory ◽  
Fractional Order ◽  
State Equation ◽  
The State ◽  
Space Representation ◽  
Fractional Dynamics ◽  
Modern Control Theory ◽  
Order State ◽  
Modern Control

In recent years, applications of fractional calculus have flourished in various science and engineering fields. Particularly in engineering, control engineering appears to be expanding aggressively in its applications. Exemplary are the CRONE controller and the PIλDμ controller, which is categorizable into applications of fractional calculus in classical control theory. A state equation can be called the foundation of modern control theory. However, the relationship between fractional derivatives and the state equation has not been examined sufficiently. Consequently, a systematic procedure referred to by every researcher on the fractional-calculus side or control-theory side has not yet been established. For this study, therefore, involvement of fractional-order derivatives into a state equation is demonstrated here for ready comprehension by researchers. First, the procedures are explained generally; then the technique to incorporate the fractional-order state-vector into a conventional state equation is given as an example of the applications. The state-space representation in this study is useful not only for modeling a controlled system with fractional dynamics, but also for design and implementation of a controller to control fractional-order states. After we complete installation of the basic parts, we can apply the benefits of modern control theory, including robust control theories such as H-infinity and μ-analysis and synthesis in their integrities, to this fractional-order state-equation.

Dynamic Behaviors of a Fractional Order Viscoelastic Two-Member Truss

2011 ◽  
Vol 90-93 ◽  
pp. 951-957
Author(s):  
Yuan Ping Li ◽  
Wei Zhang
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Chaotic Motion ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Fractional Dynamics ◽  
Dynamic Behaviors ◽  
Period Doubling Bifurcations

The fractional dynamics equation of a viscoelastic two-member truss system, in which fractional derivative model introduced to simulate the materials’ characteristics, is proposed. The simplified single DOF differential equation is developed combined with boundary conditions and symmetry. Dynamic behaviors of the fractional single DOF system with harmonic loads are discussed by numerical calculations. The results show that: the system may lead to chaotic motion via period-doubling bifurcations or intermittent routes; the dynamical character is greatly inflected by the varying of excitation amplitude or damping coefficient or fractional order.

scholarly journals Fractional Dynamics of Computer Virus Propagation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Carla M. A. Pinto ◽  
J. A. Tenreiro Machado
Keyword(s):  
Steady State ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Initial Conditions ◽  
Computer Virus ◽  
Fractional Dynamics ◽  
Fractional Order Systems ◽  
Virus Propagation ◽  
Integer Order ◽  
Fast Transients

We propose a fractional model for computer virus propagation. The model includes the interaction between computers and removable devices. We simulate numerically the model for distinct values of the order of the fractional derivative and for two sets of initial conditions adopted in the literature. We conclude that fractional order systems reveal richer dynamics than the classical integer order counterpart. Therefore, fractional dynamics leads to time responses with super-fast transients and super-slow evolutions towards the steady-state, effects not easily captured by the integer order models.

scholarly journals Why Do Big Data and Machine Learning Entail the Fractional Dynamics?

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 297
Author(s):  
Haoyu Niu ◽  
YangQuan Chen ◽  
Bruce J. West
Keyword(s):  
Machine Learning ◽  
Big Data ◽  
Complex Systems ◽  
Fractional Order ◽  
Big Data Analytics ◽  
Future Research ◽  
Fractional Dynamics ◽  
Complex Signals ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed

Fractional-order calculus is about the differentiation and integration of non-integer orders. Fractional calculus (FC) is based on fractional-order thinking (FOT) and has been shown to help us to understand complex systems better, improve the processing of complex signals, enhance the control of complex systems, increase the performance of optimization, and even extend the enabling of the potential for creativity. In this article, the authors discuss the fractional dynamics, FOT and rich fractional stochastic models. First, the use of fractional dynamics in big data analytics for quantifying big data variability stemming from the generation of complex systems is justified. Second, we show why fractional dynamics is needed in machine learning and optimal randomness when asking: “is there a more optimal way to optimize?”. Third, an optimal randomness case study for a stochastic configuration network (SCN) machine-learning method with heavy-tailed distributions is discussed. Finally, views on big data and (physics-informed) machine learning with fractional dynamics for future research are presented with concluding remarks.

scholarly journals A Fractional Measles Model Having Monotonic Real Statistical Data for Constant Transmission Rate of the Disease

2019 ◽  
Vol 3 (4) ◽  
pp. 53 ◽  
Author(s):  
Ricardo Almeida ◽  
Sania Qureshi
Keyword(s):  
Fractional Order ◽  
Transmission Rate ◽  
Classical Model ◽  
Real Life ◽  
Nonlinear Least Squares ◽  
Vital Role ◽  
Type Model ◽  
Stable Equilibrium Point ◽  
Classical Models ◽  
Local Operators

Non-Markovian effects have a vital role in modeling the processes related with natural phenomena such as epidemiology. Various infectious diseases have long-range memory characteristics and, thus, non-local operators are one of the best choices to be used to understand the transmission dynamics of such diseases and epidemics. In this paper, we study a fractional order epidemiological model of measles. Some relevant features, such as well-posedness and stability of the underlying Cauchy problem, are considered accompanying the proofs for a locally asymptotically stable equilibrium point for basic reproduction number R 0 < 1 , which is most sensitive to the fractional order parameter and to the percentage of vaccination. We show the efficiency of the model through a real life application of the spread of the epidemic in Pakistan, comparing the fractional and classical models, while assuming constant transmission rate of the epidemic with monotonically increasing and decreasing behavior of the infected population. Secondly, the fractional Caputo type model, based upon nonlinear least squares curve fitting technique, is found to have smaller residuals when compared with the classical model.

Formulation of A State Equation Including Fractional-Order State Vectors

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Masaharu Kuroda
Keyword(s):  
Fractional Calculus ◽  
Control Theory ◽  
Fractional Order ◽  
State Equation ◽  
The State ◽  
Space Representation ◽  
Fractional Dynamics ◽  
Modern Control Theory ◽  
Order State ◽  
Modern Control

In recent years, applications of fractional calculus have flourished in various science and engineering fields. Particularly in engineering, control engineering appears to be expanding aggressively in their applications. Exemplary are the CRONE controller and the PIλDμ controller, which is categorizable into applications of fractional calculus in classical control theory. A state equation can be called as the foundation of modern control theory. However, the relationship between fractional derivatives and the state equation has not been examined sufficiently. Consequently, a systematic procedure referred to every researcher on the fractional-calculus side or control-theory side has not yet been established. For this study, therefore, involvement of fractional-order derivatives into a state equation is demonstrated here for ready comprehension by researchers. First, the procedures are explained generally; then the technique to incorporate a fractional-order state vector into a conventional state equation is given as an example of the applications. The state-space representation in this study is useful not only to model a controlled system with fractional dynamics but also for design and implementation of a controller to control fractional-order states. After introducing the basic parts, the benefits of modern control theory including robust control theories, such as H∞ and μ-analysis and synthesis in their integrities, can be applied to this fractional-order state equation.

scholarly journals Existence Results for a Michaud Fractional, Nonlocal, and Randomly Position Structured Fragmentation Model

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Emile Franc Doungmo Goufo ◽  
Riëtte Maritz ◽  
Stella Mugisha
Keyword(s):  
Fractional Order ◽  
Continuous Solution ◽  
Semigroup Theory ◽  
Time Derivative ◽  
Existence Results ◽  
Fragmentation Model ◽  
Classical Models ◽  
Sudden Appearance ◽  
Many Particles ◽  
Number Of Particles

Until now, classical models of clusters’ fission remain unable to fully explain strange phenomena like the phenomenon of shattering (Ziff and McGrady, 1987) and the sudden appearance of infinitely many particles in some systems having initial finite number of particles. That is why there is a need to extend classical models to models with fractional derivative order and use new and various techniques to analyze them. In this paper, we prove the existence of strongly continuous solution operators for nonlocal fragmentation models with Michaud time derivative of fractional order (Samko et al., 1993). We focus on the case where the splitting rate is dependent on size and position and where new particles generating from fragmentation are distributed in space randomly according to some probability density. In the analysis, we make use of the substochastic semigroup theory, the subordination principle for differential equations of fractional order (Prüss, 1993, Bazhlekova, 2000), the analogy of Hille-Yosida theorem for fractional model (Prüss, 1993), and useful properties of Mittag-Leffler relaxation function (Berberan-Santos, 2005). We are then able to show that the solution operator to the full model is positive and contractive.

scholarly journals Fractional Dynamics of Stuxnet Virus Propagation in Industrial Control Systems

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2160
Author(s):  
Zaheer Masood ◽  
Muhammad Asif Zahoor Raja ◽  
Naveed Ishtiaq Chaudhary ◽  
Khalid Mehmood Cheema ◽  
Ahmad H. Milyani
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Order System ◽  
Fractional Dynamics ◽  
Virus Propagation ◽  
Industrial Control ◽  
State Behavior ◽  
Industrial Control Systems ◽  
Derivative System ◽  
Slow Evolution

The designed fractional order Stuxnet, the virus model, is analyzed to investigate the spread of the virus in the regime of isolated industrial networks environment by bridging the air-gap between the traditional and the critical control network infrastructures. Removable storage devices are commonly used to exploit the vulnerability of individual nodes, as well as the associated networks, by transferring data and viruses in the isolated industrial control system. A mathematical model of an arbitrary order system is constructed and analyzed numerically to depict the control mechanism. A local and global stability analysis of the system is performed on the equilibrium points derived for the value of α = 1. To understand the depth of fractional model behavior, numerical simulations are carried out for the distinct order of the fractional derivative system, and the results show that fractional order models provide rich dynamics by means of fast transient and super-slow evolution of the model’s steady-state behavior, which are seldom perceived in integer-order counterparts.

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