scholarly journals A Comparative Numerical Study and Stability Analysis for a Fractional-Order SIR Model of Childhood Diseases

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2847
Author(s):  
Mohamed M. Mousa ◽  
Fahad Alsharari
Keyword(s):  
Stability Analysis ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fourth Order ◽  
Numerical Scheme ◽  
Numerical Study ◽  
Sir Model ◽  
Childhood Diseases ◽  
Reliability And Robustness ◽  
Derivatives Of

The objective of this work is to examine the dynamics of a fractional-order susceptible-infectious-recovered (SIR) model that simulate epidemiological diseases such as childhood diseases. An effective numerical scheme based on Grünwald–Letnikov fractional derivative is suggested to solve the considered model. A stability analysis is performed to qualitatively examine the dynamics of the SIR model. The reliability and robustness of the proposed scheme is demonstrated by comparing obtained results with results obtained from a fourth order Runge–Kutta built-in Maple syntax when considering derivatives of integer order. Graphical illustrations of the numerical results are given. The inaccuracy of some results presented in two studies exist in the literature have been clearly explained. Generalizing of the cases examined in another study, by considering a model with fraction-order derivatives, is another objective of this work as well.

Dynamic analysis of a fractional order system

2015 ◽  
Vol 08 (06) ◽  
pp. 1550079
Author(s):  
M. Javidi ◽  
N. Nyamoradi
Keyword(s):  
Stability Analysis ◽  
Fractional Derivative ◽  
Dynamic Analysis ◽  
Fractional Order ◽  
Local Stability ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Order System ◽  
Stability Conditions ◽  
Hurwitz Stability

In this paper, we investigate the dynamical behavior of a fractional order phytoplankton–zooplankton system. In this paper, stability analysis of the phytoplankton–zooplankton model (PZM) is studied by using the fractional Routh–Hurwitz stability conditions. We have studied the local stability of the equilibrium points of PZM. We applied an efficient numerical method based on converting the fractional derivative to integer derivative to solve the PZM.

scholarly journals A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdon Atangana ◽  
Aydin Secer
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Advantages And Disadvantages ◽  
Derivatives Of

The purpose of this note is to present the different fractional order derivatives definition that are commonly used in the literature on one hand and to present a table of fractional order derivatives of some functions in Riemann-Liouville sense On other the hand. We present some advantages and disadvantages of these fractional derivatives. And finally we propose alternative fractional derivative definition.

scholarly journals Application of local meshless method for the solution of two term time fractional-order multi-dimensional PDE arising in heat and mass transfer

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 95-105 ◽  
Author(s):  
Imtiaz Ahmad ◽  
Hijaz Ahmad ◽  
Mustafa Inc ◽  
Shao-Wen Yao ◽  
Bandar Almohsen
Keyword(s):  
Mass Transfer ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Caputo Fractional Derivative ◽  
Meshless Method ◽  
Fractional Part ◽  
Higher Dimensions ◽  
Numerical Treatment ◽  
Dimensional Diffusion ◽  
Derivatives Of

In this article, we presented an efficient local meshless method for the numerical treatment of two term time fractional-order multi-dimensional diffusion PDE. The demand of meshless techniques increment because of its meshless nature and simplicity of usage in higher dimensions. This technique approximates the solu?tion on set of uniform and scattered nodes. The space derivatives of the models are discretized by the proposed meshless procedure though the time fractional part is discretized by Liouville-Caputo fractional derivative. The numerical re?sults are obtained for 1-, 2- and 3-D cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

scholarly journals A Numerical Scheme to Solve Fractional Optimal Control Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Shakoor Pooseh ◽  
Ricardo Almeida ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Numerical Scheme ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Terminal Time ◽  
Dynamic Constraint ◽  
Fractional Optimal Control Problems ◽  
Free Terminal Time

We review recent results obtained to solve fractional order optimal control problems with free terminal time and a dynamic constraint involving integer and fractional order derivatives. Some particular cases are studied in detail. A numerical scheme is given, based on expansion formulas for the fractional derivative. The efficiency of the method is illustrated through examples.

scholarly journals Study of a Fractional-Order Chaotic System Represented by the Caputo Operator

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Ndolane Sene
Keyword(s):  
Fractional Derivative ◽  
Lyapunov Exponents ◽  
Fractional Order ◽  
Chaotic System ◽  
Caputo Fractional Derivative ◽  
Numerical Scheme ◽  
Chaotic Behavior ◽  
Bifurcation Diagrams ◽  
Numerical Discretization ◽  
Good Agreement

This paper is presented on the theory and applications of the fractional-order chaotic system described by the Caputo fractional derivative. Considering the new fractional model, it is important to establish the presence or absence of chaotic behaviors. The Lyapunov exponents in the fractional context will be our fundamental tool to arrive at our conclusions. The variations of the model’s parameters will generate chaotic behavior, in general, which will be established using the Lyapunov exponents and bifurcation diagrams. For the system’s phase portrait, we will present and apply an interesting fractional numerical discretization. For confirmation of the results provided in this paper, the circuit schematic is drawn and simulated. As it will be observed, the results obtained after the simulation of the numerical scheme and with the Multisim are in good agreement.

scholarly journals Dynamics of the helmholtz oscillator with fractional order damping

2013 ◽  
Vol 23 ◽  
pp. 12-15
Author(s):  
Adolfo Ortiz ◽  
Jesús Seoane ◽  
J. Yang ◽  
Miguel Sanjuán
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Numerical Scheme ◽  
Initial Conditions ◽  
Critical Value ◽  
Periodic Motions ◽  
Damping Term ◽  
Fractional Damping ◽  
Runge Kutta Method ◽  
History Of

The dynamics of the nonlinear Helmholtz Oscillator with fractional order damping are studied in detail. The discretization of differential equations according to the Grünwald-Letnikov fractional derivative definition in order to get numerical simulations is reported. Comparison between solutions obtained through a fourth-order Runge-Kutta method and the fractional damping system are comparable when the fractional derivative of the damping term a is fixed at 1. That proves the good performance of the numerical scheme. The effect of taking the fractional derivative on the system dynamics is investigated using phase diagrams varying a from 0.5 to 1.75 with zero initial conditions. Periodic motions of the system are obtained at certain ranges of the damping term. On the other hand, escape of the trajectories from a potential well result at a certain critical value of the fractional derivative. The history of the displacement as a function of time is shown also for every a selected.

scholarly journals Theory Analysis of Left-Handed Grünwald-Letnikov Formula with0<α<1to Detect and Locate Singularities

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Shaoxiang Hu ◽  
Ping Liang
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Theory Analysis ◽  
Left Handed ◽  
Local Extrema ◽  
Derivative Formula ◽  
Derivatives Of

We study fractional-order derivatives of left-handed Grünwald-Letnikov formula with0<α<1to detect and locate singularities in theory. The widely used four types of ideal singularities are analyzed by deducing their fractional derivative formula. The local extrema of fractional derivatives are used to locate the singularities. Theory analysis indicates that fractional-order derivatives of left-handed Grünwald-Letnikov formula with0<α<1can detect and locate four types of ideal singularities correctly, which shows better performance than classical 1-order derivatives in theory.

On the Appearance of the Fractional Derivative in the Behavior of Real Materials

1984 ◽  
Vol 51 (2) ◽  
pp. 294-298 ◽  
Author(s):  
P. J. Torvik ◽  
R. L. Bagley
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Newtonian Fluid ◽  
Transient Response ◽  
Ad Hoc ◽  
Constitutive Relationship ◽  
Viscoelastic Materials ◽  
Frequency Dependent ◽  
Integer Order ◽  
Derivatives Of

Generalized constitutive relationships for viscoelastic materials are suggested in which the customary time derivatives of integer order are replaced by derivatives of fractional order. To this point, the justification for such models has resided in the fact that they are effective in describing the behavior of real materials. In this work, the fractional derivative is shown to arise naturally in the description of certain motions of a Newtonian fluid. We claim this provides some justification for the use of ad hoc relationships which include the fractional derivative. An application of such a constitutive relationship to the prediction of the transient response of a frequency-dependent material is included.

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