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 Solving the Oscillation Equation With Fractional Order Damping Term Using a New Fourier Transform Method

2017 ◽  
Vol 13 (5) ◽  
pp. 7393-7397
Author(s):  
OZLEM OZTURK MIZRAK
Keyword(s):  
Fourier Transform ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fourier Transformation ◽  
Cosmic Time ◽  
Damping Term ◽  
Fractional Damping ◽  
Transform Method ◽  
Fourier Transform Method ◽  
Oscillation Equation

We propose an adapted Fourier transform method that gives the solution of an oscillation equation with a fractional damping term in ordinary domain. After we mention a transformation of cosmic time to individual time (CTIT), we explain how it can reduce the problem from fractional form to ordinary form when it is used with Fourier transformation, via an example for 1 < alpha < 2; where alpha is the order of fractional derivative. Then, we give an application of the results.

scholarly journals Fractional-Derivative Approximation of Relaxation in Complex Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Kin M. Li ◽  
Mihir Sen ◽  
Arturo Pacheco-Vega
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
System Identification ◽  
Complex Systems ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Initial Conditions ◽  
Time Dependent ◽  
Fractional Order Differential Equation

In this paper, we present a system identification (SI) procedure that enables building linear time-dependent fractional-order differential equation (FDE) models able to accurately describe time-dependent behavior of complex systems. The parameters in the models are the order of the equation, the coefficients in it, and, when necessary, the initial conditions. The Caputo definition of the fractional derivative, and the Mittag-Leffler function, is used to obtain the corresponding solutions. Since the set of parameters for the model and its initial conditions are nonunique, and there are small but significant differences in the predictions from the possible models thus obtained, the SI operation is carried out via global regression of an error-cost function by a simulated annealing optimization algorithm. The SI approach is assessed by considering previously published experimental data from a shell-and-tube heat exchanger and a recently constructed multiroom building test bed. The results show that the proposed model is reliable within the interpolation domain but cannot be used with confidence for predictions outside this region. However, the proposed system identification methodology is robust and can be used to derive accurate and compact models from experimental data. In addition, given a functional form of a fractional-order differential equation model, as new data become available, the SI technique can be used to expand the region of reliability of the resulting model.

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 Qualitative Analysis of Class of Fractional-Order Chaotic System via Bifurcation and Lyapunov Exponents Notions

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Ndolane Sene
Keyword(s):  
Lyapunov Exponents ◽  
Fractional Order ◽  
Chaotic System ◽  
Numerical Scheme ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Fractional Chaotic System ◽  
The Stability ◽  
Predictor Corrector

This paper presents a modified chaotic system under the fractional operator with singularity. The aim of the present subject will be to focus on the influence of the new model’s parameters and its fractional order using the bifurcation diagrams and the Lyapunov exponents. The new fractional model will generate chaotic behaviors. The Lyapunov exponents’ theories in fractional context will be used for the characterization of the chaotic behaviors. In a fractional context, the phase portraits will be obtained with a predictor-corrector numerical scheme method. The details of the numerical scheme will be presented in this paper. The numerical scheme will be used to analyze all the properties addressed in this present paper. The Matignon criterion will also play a fundamental role in the local stability of the presented model’s equilibrium points. We will find a threshold under which the stability will be removed and the chaotic and hyperchaotic behaviors will be generated. An adaptative control will be proposed to correct the instability of the equilibrium points of the model. Sensitive to the initial conditions, we will analyze the influence of the initial conditions on our fractional chaotic system. The coexisting attractors will also be provided for illustrations of the influence of the initial conditions.

An approach for approximate solution of fractional-order smoking model with relapse class

2018 ◽  
Vol 11 (06) ◽  
pp. 1850077 ◽  
Author(s):  
Anwar Zeb ◽  
Vedat Suat Erturk ◽  
Umar Khan ◽  
Gul Zaman ◽  
Shaher Momani
Keyword(s):  
Approximate Solution ◽  
Comparative Study ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Approximate Solutions ◽  
Kutta Method ◽  
Unique Positive Solution ◽  
Runge Kutta Method ◽  
Proposed Model ◽  
Definition Of

In this paper, we develop a fractional-order smoking model by considering relapse class. First, we formulate the model and find the unique positive solution for the proposed model. Then we apply the Grünwald–Letnikov approximation in the place of maintaining a general quadrature formula approach to the Riemann–Liouville integral definition of the fractional derivative. Building on this foundation avoids the need for domain transformations, contour integration or involved theory to compute accurate approximate solutions of fractional-order giving up smoking model. A comparative study between Grünwald–Letnikov method and Runge–Kutta method is presented in the case of integer-order derivative. Finally, we present the obtained results graphically.

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 The fractional nonlinear $${\mathcal{PT}}$$ dimer

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Mario I. Molina
Keyword(s):  
Fractional Derivative ◽  
Laplace Transformation ◽  
Initial Conditions ◽  
Critical Value ◽  
Order Derivative ◽  
Marked Contrast ◽  
Transformation Technique ◽  
First Order ◽  
Main Effect ◽  
Gain Loss

AbstractWe examine a fractional discrete nonlinear Schrodinger dimer, where the usual first-order derivative in the time evolution is replaced by a non integer-order derivative. The dimer is nonlinear (Kerr) and $${\mathcal{{PT}}}$$ PT -symmetric, and for localized initial conditions we examine the exchange dynamics between both sites. By means of the Laplace transformation technique, the linear $${{\mathcal{{PT}}}}$$ PT dimer is solved in closed form in terms of Mittag–Leffler functions, while for the nonlinear regime, we resort to numerical computations using the direct explicit Grunwald algorithm. In general, we see that the main effect of the fractional derivative is to produce a monotonically decreasing time envelope for the amplitude of the oscillatory exchange. In the presence of $${\mathcal{{PT}}}$$ PT symmetry, the oscillations experience some amplification for gain/loss values below some threshold, while beyond threshold, the amplitudes of both sites grow unbounded. The presence of nonlinearity can arrest the unbounded growth and lead to a selftrapped state. The trapped fraction decreases as the nonlinearity is increased past a critical value, in marked contrast with the standard (non-fractional) case.

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.

scholarly journals Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ndolane Sene ◽  
Babacar Sène ◽  
Seydou Nourou Ndiaye ◽  
Awa Traoré
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Fractional Integration ◽  
Order Derivative ◽  
Graphical Representations ◽  
Fractional Order Derivative ◽  
Black Scholes

The value of an option plays an important role in finance. In this paper, we use the Black–Scholes equation, which is described by the nonsingular fractional-order derivative, to determine the value of an option. We propose both a numerical scheme and an analytical solution. Recent studies in fractional calculus have included new fractional derivatives with exponential kernels and Mittag-Leffler kernels. These derivatives have been found to be applicable in many real-world problems. As fractional derivatives without nonsingular kernels, we use a Caputo–Fabrizio fractional derivative and a Mittag-Leffler fractional derivative. Furthermore, we use the Adams–Bashforth numerical scheme and fractional integration to obtain the numerical scheme and the analytical solution, and we provide graphical representations to illustrate these methods. The graphical representations prove that the Adams–Bashforth approach is helpful in getting the approximate solution for the fractional Black–Scholes equation. Finally, we investigate the volatility of the proposed model and discuss the use of the model in finance. We mainly notice in our results that the fractional-order derivative plays a regulator role in the diffusion process of the Black–Scholes equation.

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