Stochastic dynamics of nonlinear systems with a fractional power-law nonlinear term: The fractional calculus approach

2011 ◽  
Vol 26 (1) ◽  
pp. 101-108 ◽  
Author(s):  
Giulio Cottone ◽  
Mario Di Paola ◽  
Salvatore Butera
Keyword(s):  
Nonlinear Systems ◽  
Fractional Calculus ◽  
Power Law ◽  
Nonlinear Term ◽  
Stochastic Dynamics ◽  
Fractional Power

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.

Some misinterpretations and lack of understanding in differential operators with no singular kernels

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 594-612 ◽  
Author(s):  
Abdon Atangana ◽  
Emile Franc Doungmo Goufo
Keyword(s):  
Fractional Calculus ◽  
Power Law ◽  
Initial Value Problems ◽  
Differential Operators ◽  
Integral Operators ◽  
Initial Value ◽  
Complex Processes ◽  
Singular Kernels ◽  
Fractional Differential ◽  
Theoretical Results

AbstractHumans are part of nature, and as nature existed before mankind, mathematics was created by humans with the main aim to analyze, understand and predict behaviors observed in nature. However, besides this aspect, mathematicians have introduced some laws helping them to obtain some theoretical results that may not have physical meaning or even a representation in nature. This is also the case in the field of fractional calculus in which the main aim was to capture more complex processes observed in nature. Some laws were imposed and some operators were misused, such as, for example, the Riemann–Liouville and Caputo derivatives that are power-law-based derivatives and have been used to model problems with no power law process. To solve this problem, new differential operators depicting different processes were introduced. This article aims to clarify some misunderstandings about the use of fractional differential and integral operators with non-singular kernels. Additionally, we suggest some numerical discretizations for the new differential operators to be used when dealing with initial value problems. Applications of some nature processes are provided.

scholarly journals Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
J. F. Gómez Aguilar ◽  
T. Córdova-Fraga ◽  
J. Tórres-Jiménez ◽  
R. F. Escobar-Jiménez ◽  
V. H. Olivares-Peregrino ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Disordered Systems ◽  
Fourier Method ◽  
Mean Square Displacement ◽  
Fractional Power ◽  
Transport Processes ◽  
Diffusion Equations ◽  
Alternative Representation ◽  
Dimensional Parameters ◽  
Full Solution

The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus approach; the spatial-time derivatives of fractional order are approximated using the Caputo-type derivative in the range(0,2]. In this alternative representation we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space-time derivatives into the fractional Cattaneo-Vernotte equation. Finally, consider the Dirichlet conditions, the Fourier method was used to find the full solution of the fractional Cattaneo-Vernotte equation in analytic way, and Caputo and Riesz fractional derivatives are considered. The advantage of our representation appears according to the comparison between our model and models presented in the literature, which are not acceptable physically due to the dimensional incompatibility of the solutions. The classical cases are recovered when the fractional derivative exponents are equal to1.

AC Loss Modeling in Ba0.5Sr0.5TiO3 Using Dielectric Relaxation

2004 ◽  
Vol 833 ◽  
Author(s):  
Nadia K. Pervez ◽  
Jiwei Lu ◽  
Susanne Stemmer ◽  
Robert A. York
Keyword(s):  
Dielectric Loss ◽  
Dielectric Relaxation ◽  
Power Law ◽  
Fractional Power ◽  
Ac Loss ◽  
Dielectric Susceptibility ◽  
Network Analyzer ◽  
Q Factor ◽  
Relaxation Model ◽  
Q Factors

ABSTRACTIn universal relaxation, a material's complex dielectric susceptibility follows a fractional power law f1-n where 0 < n < 1 over multiple decades of frequency. In a variety of materials, including Ba0.5Sr0.5Ti03, dielectric relaxation has been observed to follow this universal relaxation model with values of n close to 1. In this work we have shown that the universal relaxation model can be used to calculate dielectric loss even when n is very close to 1. Our calculated Q-factors agree with measured values at 1 MHz; this agreement suggests that this technique may be used for higher frequencies where network analyzer measurements and electrode parasitics complicate Q-factor determination.

The fractional power law of wind wave growth in deep water for short fetch

2002 ◽  
Vol 1 (2) ◽  
pp. 105-108 ◽  
Author(s):  
Guan Changlong ◽  
Sun Qun ◽  
Philippe Fraunie
Keyword(s):  
Power Law ◽  
Deep Water ◽  
Wind Wave ◽  
Fractional Power ◽  
Wave Growth

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 Emergence of power laws in the pharmacokinetics of paclitaxel due to competing saturable processes.

2008 ◽  
Vol 11 (3) ◽  
pp. 77 ◽  
Author(s):  
Jack A Tuszynski ◽  
Rebeccah E. Marsh ◽  
Michael B. Sawyer ◽  
Kenneth J.E. Vos
Keyword(s):  
Power Law ◽  
Fractional Power ◽  
Area Under The Curve ◽  
Power Laws ◽  
Dose Dependence ◽  
Power Exponent ◽  
Least Squares Regression ◽  
Time Data ◽  
Concentration Time ◽  
Wide Range

Purpose: This study presents the results of power law analysis applied to the pharmacokinetics of paclitaxel. Emphasis is placed on the role that the power exponent can play in the investigation and quantification of nonlinear pharmacokinetics and the elucidation of the underlying physiological processes. Methods: Forty-one sets of concentration-time data were inferred from 20 published clinical trial studies, and 8 sets of area under the curve (AUC) and maximum concentration (Cmax) values as a function of dose were collected. Both types of data were tested for a power law relationship using least squares regression analysis. Results: Thirty-nine of the concentration-time curves were found to exhibit power law tails, and two dominant fractal exponents emerged. Short infusion times led to tails with a single power exponent of -1.57 ± 0.14, while long infusion times resulted in steeper tails characterized by roughly twice the exponent. The curves following intermediate infusion times were characterized by two consecutive power laws; an initial short slope with the larger alpha value was followed by a crossover to a long-time tail characterized by the smaller exponent. The AUC and Cmax parameters exhibited a power law dependence on the dose, with fractional power exponents that agreed with each other and with the exponent characterizing the shallow decline. Computer simulations revealed that a two- or three-compartment model with both saturable distribution and saturable elimination can produce the observed behaviour. Furthermore, there is preliminary evidence that the nonlinear dose-dependence is correlated with the power law tails. Conclusion: Assessment of data from published clinical trials suggests that power laws accurately describe the concentration-time curves and non-linear dose-dependence of paclitaxel, and the power exponents provide insight into the underlying drug mechanisms. The interplay between two saturable processes can produce a wide range of behaviour, including concentration-time curves with exponential, power law, and dual power law tails.

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