scholarly journals REVIEW OF SOME PROMISING FRACTIONAL PHYSICAL MODELS

2013 ◽  
Vol 27 (09) ◽  
pp. 1330005 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Fractional Calculus ◽  
Power Law ◽  
Particle Interaction ◽  
Discrete Systems ◽  
Physical Models ◽  
Long Term Memory ◽  
Fractional Dynamics ◽  
Fractal Media ◽  
Systems With Memory ◽  
With Memory

Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law nonlocality, power-law long-term memory or fractal properties by using integrations and differentiation of non-integer orders, i.e., by methods in the fractional calculus. This paper is a review of physical models that look very promising for future development of fractional dynamics. We suggest a short introduction to fractional calculus as a theory of integration and differentiation of noninteger order. Some applications of integro-differentiations of fractional orders in physics are discussed. Models of discrete systems with memory, lattice with long-range inter-particle interaction, dynamics of fractal media are presented. Quantum analogs of fractional derivatives and model of open nano-system systems with memory are also discussed.

scholarly journals Generalized Memory: Fractional Calculus Approach

2018 ◽  
Vol 2 (4) ◽  
pp. 23 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Power Law ◽  
Memory Function ◽  
Time Interval ◽  
Infinite Time ◽  
Suggested Approach ◽  
Infinite Time Interval ◽  
History Of ◽  
With Memory

The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time.

scholarly journals Quantum Maps with Memory from Generalized Lindblad Equation

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 544
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Discrete Time ◽  
Power Law ◽  
Quantum Dynamics ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Periodic Sequence ◽  
Lindblad Equation ◽  
Open Quantum ◽  
Systems With Memory ◽  
With Memory

In this paper, we proposed the exactly solvable model of non-Markovian dynamics of open quantum systems. This model describes open quantum systems with memory and periodic sequence of kicks by environment. To describe these systems, the Lindblad equation for quantum observable is generalized by taking into account power-law fading memory. Dynamics of open quantum systems with power-law memory are considered. The proposed generalized Lindblad equations describe non-Markovian quantum dynamics. The quantum dynamics with power-law memory are described by using integrations and differentiation of non-integer orders, as well as fractional calculus. An example of a quantum oscillator with linear friction and power-law memory is considered. In this paper, discrete-time quantum maps with memory, which are derived from generalized Lindblad equations without any approximations, are suggested. These maps exactly correspond to the generalized Lindblad equations, which are fractional differential equations with the Caputo derivatives of non-integer orders and periodic sequence of kicks that are represented by the Dirac delta-functions. The solution of these equations for coordinates and momenta are derived. The solutions of the generalized Lindblad equations for coordinate and momentum operators are obtained for open quantum systems with memory and kicks. Using these solutions, linear and nonlinear quantum discrete-time maps are derived.

scholarly journals The Memory Effect on Fractional Calculus: An Application in the Spread of Covid-19

2020 ◽  
Author(s):  
Laécio Carvalho de Barros ◽  
Michele Martins Lopes ◽  
Francielle Santo Pedro Simões ◽  
Estevão Esmi ◽  
José Paulo Carvalho dos Santos ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Fractional Calculus ◽  
Memory Effect ◽  
Compartmental Model ◽  
Statistical Approach ◽  
Hysteresis Phenomenon ◽  
Fractional Operators ◽  
Main Interest ◽  
Systems With Memory ◽  
With Memory

Abstract Fractional calculus has been widely used in mathematical modeling of evolutionary systems with memory effect on dynamics. The main interest of this work is, through a statistical approach, to attest how the hysteresis phenomenon, which describes the memory effect present in biological systems, can be treated by fractional calculus, and to analyze the contribution of the historical values of a function in the evaluation of fractional operators according their order. In order to illustrate the efficiency of this non-integer order calculus, we consider the SIR (Susceptible-Infected-Recovered) compartmental model which is widely used in epidemiology. We employ SIR models to model the dynamics, with and without memory, of the spread of Covid-19 in some countries.

scholarly journals The Memory Effect on Fractional Calculus: An Application in the Spread of Covid-19

2021 ◽  
Author(s):  
Laécio Carvalho de Barros ◽  
Michele Martins Lopes ◽  
Francielle Santo Pedro Simões ◽  
Estevão Esmi ◽  
José Paulo Carvalho dos Santos ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Fractional Calculus ◽  
Memory Effect ◽  
Compartmental Model ◽  
Statistical Approach ◽  
Hysteresis Phenomenon ◽  
Fractional Operators ◽  
Main Interest ◽  
Systems With Memory ◽  
With Memory

Abstract Fractional calculus has been widely used in mathematical modeling of evolutionary systems with memory effect on dynamics. The main interest of this work is to attest, through a statistical approach, how the hysteresis phenomenon, which describes a type of memory effect present in biological systems, can be treated by fractional calculus. We also analyse the contribution of the historical values of a function in the evaluation of fractional operators according to their order. In order to illustrate the efficiency of this non-integer order calculus, we consider the SIR (Susceptible-Infected-Recovered) compartmental model which is widely used in epidemiology. We employ this compartmental model to study the dynamics of the spread of Covid-19 in some countries, one version with memory and one without memory.

scholarly journals Quantum Szilard engine for the fractional power-law potentials

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Ekrem Aydiner
Keyword(s):  
Quantum Information ◽  
Fractional Calculus ◽  
Power Law ◽  
Fractional Power ◽  
Thermodynamic Cycle ◽  
Partition Functions ◽  
Fractional Dynamics ◽  
Energy Eigenvalues ◽  
Positive Work ◽  
Degenerate Cases

AbstractIn this study, we consider the quantum Szilárd engine with a single particle under the fractional power-law potential. We suggest that such kind of the Szilárd engine works a Stirling-like cycle. We obtain energy eigenvalues and canonical partition functions for the degenerate and non-degenerate cases in this cycle process. By using these quantities we numerically compute work and efficiency for this thermodynamic cycle for various power-law potentials with integer and non-integer exponents. We show that the presented simple engine also yields positive work and efficiency. We discuss the importance of fractional dynamics in physics and finally, we conclude that fractional calculus should be included in the fields of quantum information and thermodynamics.

scholarly journals And I say to myself: “What a fractional world!”

2011 ◽  
Vol 14 (4) ◽  
Author(s):  
J. Tenreiro Machado
Keyword(s):  
Global Warming ◽  
Fourier Transform ◽  
Fractional Calculus ◽  
Complex Systems ◽  
Power Law ◽  
Deoxyribonucleic Acid ◽  
Warming Trend ◽  
Fractional Dynamics ◽  
The Fourier Transform ◽  
Musical Rhythms

AbstractThis paper discusses several complex systems in the perspective of fractional dynamics. For prototype systems are considered the cases of deoxyribonucleic acid decoding, financial evolution, earthquakes events, global warming trend, and musical rhythms. The application of the Fourier transform and of the power law trendlines leads to an assertive representation of the dynamics and to a simple comparison of their characteristics. Moreover, the gallery of different systems, both natural and man made, demonstrates the richness of phenomena that can be described and studied with the tools of fractional calculus.

scholarly journals Analysis of the spread of COVID-19 in Morocco based on a SEIR epidemic model

2021 ◽  
Vol 10 (2) ◽  
pp. 72-78
Author(s):  
Moulay Rchid Sidi Ammi ◽  
Mostafa Tahiri
Keyword(s):  
Mathematical Modeling ◽  
Fractional Calculus ◽  
Memory Effect ◽  
Epidemic Model ◽  
Evolutionary Systems ◽  
Integer Order ◽  
Systems With Memory ◽  
With Memory

Fractional calculus has been widely used in mathematical modeling of evolutionary systems with memory effect on dynamics. In order to illustrate the efficiency of this non-integer order calculus, we employ SEIR models to model the dynamics, with and without memory, of the spread of Covid-19 in Morocco country.

Linear discrete systems with memory: a generalization of the Langmuir model

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Dumitru Băleanu ◽  
Raoul Nigmatullin
Keyword(s):  
General Solution ◽  
Physical Meaning ◽  
Time Scale ◽  
Discrete Systems ◽  
Langmuir Model ◽  
Time Scale Calculus ◽  
Systems With Memory ◽  
With Memory ◽  
Linear Discrete Systems

AbstractIn this manuscript we analyzed a general solution of the linear nonlocal Langmuir model within time scale calculus. Several generalizations of the Langmuir model are presented together with their exact corresponding solutions.The physical meaning of the proposed models are investigated and their corresponding geometries are reported.

scholarly journals The Memory Effect on Fractional Calculus: An Application in the Spread of Covid-19

2021 ◽  
Author(s):  
Laécio Carvalho de Barros ◽  
Michele Martins Lopes ◽  
Francielle Santo Pedro Simões ◽  
Estevão Esmi ◽  
José Paulo Carvalho dos Santos ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Fractional Calculus ◽  
Memory Effect ◽  
Compartmental Model ◽  
Statistical Approach ◽  
Hysteresis Phenomenon ◽  
Fractional Operators ◽  
Main Interest ◽  
Systems With Memory ◽  
With Memory

Abstract Fractional calculus has been widely used in mathematical modeling of evolutionary systems with memory effect on dynamics. The main interest of this work is to attest, through a statistical approach, how the hysteresis phenomenon, which describes a type of memory effect present in biological systems, can be treated by fractional calculus. We also analyse the contribution of the historical values of a function in the evaluation of fractional operators according to their order. In order to illustrate the efficiency of this non-integer order calculus, we consider the SIR (Susceptible-Infected-Recovered) compartmental model which is widely used in epidemiology. We employ this compartmental model to study the dynamics of the spread of Covid-19 in some countries, one version with memory and one without memory.

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