scholarly journals A Mathematical Study of a Coronavirus Model with the Caputo Fractional-Order Derivative

2021 ◽  
Vol 5 (3) ◽  
pp. 87
Author(s):  
Youcef Belgaid ◽  
Mohamed Helal ◽  
Abdelkader Lakmeche ◽  
Ezio Venturino
Keyword(s):  
Dynamical System ◽  
Fractional Derivative ◽  
Numerical Simulations ◽  
Minimal Model ◽  
Fractional Operators ◽  
Mathematical Study ◽  
Stability Behavior ◽  
Relevant Role ◽  
Caputo Fractional Order Derivative ◽  
Theoretical Results

In this work, we introduce a minimal model for the current pandemic. It incorporates the basic compartments: exposed, and both symptomatic and asymptomatic infected. The dynamical system is formulated by means of fractional operators. The model equilibria are analyzed. The theoretical results indicate that their stability behavior is the same as for the corresponding system formulated via standard derivatives. This suggests that, at least in this case for the model presented here, the memory effects contained in the fractional operators apparently do not seem to play a relevant role. The numerical simulations instead reveal that the order of the fractional derivative has a definite influence on both the equilibrium population levels and the speed at which they are attained.

MAKING A DISCRETE DYNAMICAL SYSTEM CHAOTIC: THEORETICAL RESULTS AND NUMERICAL SIMULATIONS

2003 ◽  
Vol 13 (11) ◽  
pp. 3437-3442 ◽  
Author(s):  
DEJIAN LAI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical System ◽  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Computer Simulations ◽  
State Feedback ◽  
Controller Design ◽  
Discrete Dynamical System ◽  
Finite Samples ◽  
Dimensional Dynamical System ◽  
Theoretical Results

In this paper, we study state-feedback controller design for controlling the Lyapunov exponents of an n-dimensional dynamical system. We examine some theoretical results and perform numerical simulations for systems with and without noise influence. The controlled Lyapunov exponents are asymptotically normally distributed if the system has noisy inputs. Computer simulations on finite samples are all consistent with the theoretical results.

scholarly journals Fractional Newton–Raphson Method Accelerated with Aitken’s Method

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 47
Author(s):  
A. Torres-Hernandez ◽  
F. Brambila-Paz ◽  
U. Iturrarán-Viveros ◽  
R. Caballero-Cruz
Keyword(s):  
Fractional Derivative ◽  
Iterative Methods ◽  
Fractional Integral ◽  
Order Of Convergence ◽  
Fractional Operators ◽  
Speed Of Convergence ◽  
Newton Raphson ◽  
Raphson Method

In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order α of the derivative is different from one. A simplified way of constructing the Riemann–Liouville (R–L) fractional operators, fractional integral and fractional derivative is presented along with examples of its application on different functions. Furthermore, an introduction to Aitken’s method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken’s method in the F N–R method, where it is shown that F N–R with Aitken’s method converges faster than the simple F N–R.

scholarly journals A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 979
Author(s):  
Sandeep Kumar ◽  
Rajesh K. Pandey ◽  
H. M. Srivastava ◽  
G. N. Singh
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Special Cases ◽  
Collocation Approach ◽  
Linear And Nonlinear ◽  
Simulation Results ◽  
Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

On the kinetics of Hadamard-type fractional differential systems

2020 ◽  
Vol 23 (2) ◽  
pp. 553-570 ◽  
Author(s):  
Li Ma
Keyword(s):  
Differential Operator ◽  
Numerical Simulations ◽  
Type Inequality ◽  
The Other ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Kinetics Of ◽  
Theoretical Results

AbstractThis paper is devoted to the investigation of the kinetics of Hadamard-type fractional differential systems (HTFDSs) in two aspects. On one hand, the nonexistence of non-trivial periodic solutions for general HTFDSs, which are considered in some functional spaces, is proved and the corresponding eigenfunction of Hadamard-type fractional differential operator is also discussed. On the other hand, by the generalized Gronwall-type inequality, we estimate the bound of the Lyapunov exponents for HTFDSs. In addition, numerical simulations are addressed to verify the obtained theoretical results.

scholarly journals A Stochastic SIR Epidemic System with a Nonlinear Relapse

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Ali El Myr ◽  
Abdelaziz Assadouq ◽  
Lahcen Omari ◽  
Adel Settati ◽  
Aadil Lahrouz
Keyword(s):  
Numerical Simulations ◽  
Stationary Distribution ◽  
Stochastic Perturbations ◽  
Sir Epidemic ◽  
Spread Model ◽  
Stochastic Sir ◽  
Theoretical Results

We investigate the conditions that control the extinction and the existence of a unique stationary distribution of a nonlinear mathematical spread model with stochastic perturbations in a population of varying size with relapse. Numerical simulations are carried out to illustrate the theoretical results.

More Dynamical Properties Revealed from a 3D Lorenz-like System

2014 ◽  
Vol 24 (10) ◽  
pp. 1450133 ◽  
Author(s):  
Haijun Wang ◽  
Xianyi Li
Keyword(s):  
Numerical Simulations ◽  
Local Stability ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Mathematical Work ◽  
Chaotic Attractors ◽  
Heteroclinic Cycles ◽  
Homoclinic And Heteroclinic Orbits ◽  
Dynamical Properties ◽  
Theoretical Results

After a 3D Lorenz-like system has been revisited, more rich hidden dynamics that was not found previously is clearly revealed. Some more precise mathematical work, such as for the complete distribution and the local stability and bifurcation of its equilibrium points, the existence of singularly degenerate heteroclinic cycles as well as homoclinic and heteroclinic orbits, and the dynamics at infinity, is carried out in this paper. In particular, another possible new mechanism behind the creation of chaotic attractors is presented. Based on this mechanism, some different structure types of chaotic attractors are numerically found in the case of small b > 0. All theoretical results obtained are further illustrated by numerical simulations. What we formulate in this paper is to not only show those dynamical properties hiding in this system, but also (more mainly) present a kind of way and means — both "locally" and "globally" and both "finitely" and "infinitely" — to comprehensively explore a given system.

scholarly journals Integral transforms of the κ-Hilfer fractional derivative

2021 ◽  
Author(s):  
Felix Costa ◽  
Junior Cesar Alves Soares ◽  
Stefânia Jarosz
Keyword(s):  
Fractional Derivative ◽  
Gamma Function ◽  
Fundamental Theorem ◽  
Beta Function ◽  
Integral Transforms ◽  
Fractional Operators ◽  
Riesz Fractional Derivative ◽  
Hilfer Fractional Derivative ◽  
Fundamental Theorem Of Calculus

In this paper, some important properties concerning the κ-Hilfer fractional derivative are discussed. Integral transforms for these operators are derived as particular cases of the Jafari transform. These integral transforms are used to derive a fractional version of the fundamental theorem of calculus. Keywords: Integral transforms, Jafari transform, κ-gamma function, κ-beta function, κ-Hilfer fractional derivative, κ-Riesz fractional derivative, κ-fractional operators.

DYNAMICS OF A PREY-DEPENDENT DIGESTIVE MODEL WITH STATE-DEPENDENT IMPULSIVE CONTROL

2012 ◽  
Vol 22 (04) ◽  
pp. 1250092 ◽  
Author(s):  
LINNING QIAN ◽  
QISHAO LU ◽  
JIARU BAI ◽  
ZHAOSHENG FENG
Keyword(s):  
Numerical Simulations ◽  
Analytical Method ◽  
Impulsive Control ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
Lambert W Function ◽  
Impulsive Effect ◽  
State Dependent ◽  
Theoretical Results

In this paper, we study the dynamical behavior of a prey-dependent digestive model with a state-dependent impulsive effect. Using the Poincaré map and the Lambert W-function, we find the analytical expression of discrete mapping. Sufficient conditions are established for transcritical bifurcation and period-doubling bifurcation through an analytical method. Exact locations of these bifurcations are explored. Numerical simulations of an example are illustrated which agree well with our theoretical results.

Dynamics of a stochastic three-species competitive model with Lévy jumps

2018 ◽  
Vol 11 (05) ◽  
pp. 1850075
Author(s):  
Yongxin Gao ◽  
Shiquan Tian
Keyword(s):  
Numerical Simulations ◽  
Critical Value ◽  
Persistence In The Mean ◽  
Competitive Model ◽  
Stability In Distribution ◽  
Parameters Analysis ◽  
The Mean ◽  
Lévy Jumps ◽  
White Noises ◽  
Theoretical Results

This paper is concerned with a three-species competitive model with both white noises and Lévy noises. We first carry out the almost complete parameters analysis for the model and establish the critical value between persistence in the mean and extinction for each species. The sufficient criteria for stability in distribution of solutions are obtained. Finally, numerical simulations are carried out to verify the theoretical results.

A study of a modified nonlinear dynamical system with fractal-fractional derivative

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Sunil Kumar ◽  
R.P. Chauhan ◽  
Shaher Momani ◽  
Samir Hadid
Keyword(s):  
Dynamical System ◽  
Fractional Derivative ◽  
Active Control ◽  
Numerical Scheme ◽  
Chaos Control ◽  
Nonlinear Dynamical System ◽  
Control Technique ◽  
Complex Behavior ◽  
Content Type ◽  
Control And Synchronization

Purpose This paper aims to study the complex behavior of a dynamical system using fractional and fractal-fractional (FF) derivative operators. The non-classical derivatives are extremely useful for investigating the hidden behavior of the systems. The Atangana–Baleanu (AB) and Caputo–Fabrizio (CF) derivatives are considered for the fractional structure of the model. Further, to add more complexity, the authors have taken the system with a CF fractal-fractional derivative having an exponential kernel. The active control technique is also considered for chaos control. Design/methodology/approach The systems under consideration are solved numerically. The authors show the Adams-type predictor-corrector scheme for the AB model and the Adams–Bashforth scheme for the CF model. The convergence and stability results are given for the numerical scheme. A numerical scheme for the FF model is also presented. Further, an active control scheme is used for chaos control and synchronization of the systems. Findings Simulations of the obtained solutions are displayed via graphics. The proposed system exhibits a very complex phenomenon known as chaos. The importance of the fractional and fractal order can be seen in the presented graphics. Furthermore, chaos control and synchronization between two identical fractional-order systems are achieved. Originality/value This paper mentioned the complex behavior of a dynamical system with fractional and fractal-fractional operators. Chaos control and synchronization using active control are also described.

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