scholarly journals Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1099
Author(s):  
Jocelyn Sabatier
Keyword(s):  
Impulse Response ◽  
Infinite Domain ◽  
Time Constants ◽  
Fractional Model ◽  
Infinite Memory ◽  
Infinite Dimensional ◽  
Fractional Models ◽  
Fractional Differential ◽  
Dimensional Models ◽  
Diffusive Representation

Using a small number of mathematical transformations, this article examines the nature of fractional models described by fractional differential equations or pseudo state space descriptions. Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit infinitely small and high time constants. This impulse response can be rewritten as a diffusive representation whose Fourier transform permits a representation of a fractional model by a diffusion equation in an infinite space domain. Fractional models can thus be viewed as doubly infinite dimensional models: infinite as distributed with a distribution in an infinite domain. This infinite domain or the infinitely large time constants of the impulse response reveal a property intrinsic to fractional models: their infinite memory. Solutions to generate fractional behaviors without infinite memory are finally proposed.

scholarly journals A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed A. Aba Oud ◽  
Aatif Ali ◽  
Hussam Alrabaiah ◽  
Saif Ullah ◽  
Muhammad Altaf Khan ◽  
...  
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Model Parameters ◽  
Disease Dynamics ◽  
Threshold Parameter ◽  
Fractional Model ◽  
Infected People ◽  
Fractional Models ◽  
Disease Epidemics ◽  
The Impact

AbstractCOVID-19 or coronavirus is a newly emerged infectious disease that started in Wuhan, China, in December 2019 and spread worldwide very quickly. Although the recovery rate is greater than the death rate, the COVID-19 infection is becoming very harmful for the human community and causing financial loses to their economy. No proper vaccine for this infection has been introduced in the market in order to treat the infected people. Various approaches have been implemented recently to study the dynamics of this novel infection. Mathematical models are one of the effective tools in this regard to understand the transmission patterns of COVID-19. In the present paper, we formulate a fractional epidemic model in the Caputo sense with the consideration of quarantine, isolation, and environmental impacts to examine the dynamics of the COVID-19 outbreak. The fractional models are quite useful for understanding better the disease epidemics as well as capture the memory and nonlocality effects. First, we construct the model in ordinary differential equations and further consider the Caputo operator to formulate its fractional derivative. We present some of the necessary mathematical analysis for the fractional model. Furthermore, the model is fitted to the reported cases in Pakistan, one of the epicenters of COVID-19 in Asia. The estimated value of the important threshold parameter of the model, known as the basic reproduction number, is evaluated theoretically and numerically. Based on the real fitted parameters, we obtained $\mathcal{R}_{0} \approx 1.50$ R 0 ≈ 1.50 . Finally, an efficient numerical scheme of Adams–Moulton type is used in order to simulate the fractional model. The impact of some of the key model parameters on the disease dynamics and its elimination are shown graphically for various values of noninteger order of the Caputo derivative. We conclude that the use of fractional epidemic model provides a better understanding and biologically more insights about the disease dynamics.

scholarly journals Memory effects on the proliferative function in the cycle-specific of chemotherapy

2021 ◽  
Author(s):  
Najma Ahmed ◽  
Dumitru Vieru ◽  
Fiazud Din Zaman
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Classical Model ◽  
Cell Mass ◽  
Time Derivative ◽  
Fractional Model ◽  
Fractional Differential ◽  
Mathcad Software

A generalized mathematical model of the breast and ovarian cancer is developed by considering the fractional differential equations with Caputo time-fractional derivatives. The use of the fractional model shows that the time-evolution of the proliferating cell mass, the quiescent cell mass, and the proliferative function are significantly influenced by their history. Even if the classical model, based on the derivative of integer order has been studied in many papers, its analytical solutions are presented in order to make the comparison between the classical model and the fractional model. Using the finite difference method, numerical schemes to the Caputo derivative operator and Riemann-Liouville fractional integral operator are obtained. Numerical solutions to the fractional differential equations of the generalized mathematical model are determined for the chemotherapy scheme based on the function of "on-off" type. Numerical results, obtained with the Mathcad software, are discussed and presented in graphical illustrations. The presence of the fractional order of the time-derivative as a parameter of solutions gives important information regarding the proliferative function, therefore, could give the possible rules for more efficient chemotherapy.

Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models

2020 ◽  
Vol 26 (17-18) ◽  
pp. 1445-1462 ◽  
Author(s):  
Ehsan Dadkhah Khiabani ◽  
Hosein Ghaffarzadeh ◽  
Babak Shiri ◽  
Javad Katebi
Keyword(s):  
Seismic Analysis ◽  
Spline Interpolation ◽  
Collocation Methods ◽  
Response Analysis ◽  
Spline Collocation ◽  
Elastic Materials ◽  
Recent Developments ◽  
Fractional Models ◽  
Structural Responses ◽  
Fractional Differential

The visco-elastic dampers can be economically designed for response reduction of dynamical systems. Recent developments in fractional calculus have affected the modeling of visco-elastic materials. The fractional models for visco-elastic dampers, which are parsimonious, require fewer parameters in comparison with other models. In this paper, we use the visco-elastic dampers for control of structural responses. The visco-elastic damper utilizes three important parameters including damping coefficient, stiffness, and fractional order. The dynamic model of the building with visco-elastic dampers can be acquired by a system of fractional differential equations, which includes both fractional and ordinary derivatives. We apply spline collocation methods for obtaining the numerical solution of this complex system. The collocation method is implemented in a graded mesh. The discrete data of earthquake are smoothed by spline interpolation of higher degree. The introduced method is used to simulate the response of a structure with active and passive controllers. A comparison of using different dampers is considered. Also, [Formula: see text] norm of the transform function is used for response analysis. The results of simulations for four-story and 10-story buildings show significant reductions in structural responses. Moreover, our analysis presents that the large variety of values can be used for parameters of the visco-elastic damper to provide flexibility in designing appropriate visco-elastic-dampers.

Advances in System Identification Using Fractional Models

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Rachid Malti ◽  
Stephane Victor ◽  
Alain Oustaloup
Keyword(s):  
System Identification ◽  
Least Squares ◽  
Prior Knowledge ◽  
Time Domain ◽  
Simultaneous Estimation ◽  
Thermal System ◽  
Fractional Model ◽  
Output Error ◽  
Equation Error ◽  
Fractional Models

This paper presents an up to date advances in time-domain system identification using fractional models. Both equation-error- and output-error-based models are detailed. In the former models, prior knowledge is generally used to fix differentiation orders; model coefficients are estimated using least squares. The latter models allow simultaneous estimation of model coefficients and differentiation orders using nonlinear programing. As an example, a thermal system is identified using a fractional model and is compared to a rational one.

Another Note on Stability of Sliding Mode Dynamics in Suppression of Fractional Chaotic Systems

2015 ◽  
Vol 743 ◽  
pp. 303-306
Author(s):  
J. Yuan ◽  
B. Shi ◽  
Yan Wang
Keyword(s):  
Stability Analysis ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Fractional Systems ◽  
Infinite Dimensional ◽  
Fractional Integrator ◽  
Fractional Differential ◽  
Continuous Frequency ◽  
The Stability ◽  
Sliding Mode Dynamics

This paper revisits the stability analysis of sliding mode dynamics in suppression of a classof fractional chaotic systems by a different approach. Firstly, we convert fractional differential equationsinto infinite dimensional ordinary differential equations based on the continuous frequency distributedmodel of the fractional integrator. Then we choose a Lyapunov function candidate to proposethe stability analysis. The result applies to both the commensurate fractional systems and the incommensurateones.

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