The capture of particles on absorbing traps in the medium with anomalous diffusion: The effective fractional order diffusion equation and the slow temporal asymptotic of survival probability

2020 ◽  
Vol 550 ◽  
pp. 124487
Author(s):  
V.E. Arkhincheev
Keyword(s):  
Diffusion Equation ◽  
Fractional Order ◽  
Anomalous Diffusion ◽  
Survival Probability

scholarly journals Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 216
Author(s):  
Rafail K. Gazizov ◽  
Stanislav Yu. Lukashchuk
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Anomalous Diffusion ◽  
Higher Order ◽  
Recursion Operators ◽  
Fractional Differential ◽  
Time Fractional Derivative ◽  
Linear Fractional Differential Equations ◽  
Infinite Sequences

Higher-order symmetries are constructed for a linear anomalous diffusion equation with the Riemann–Liouville time-fractional derivative of order α∈(0,1)∪(1,2). It is proved that the equation in question has infinite sequences of nontrivial higher-order symmetries that are generated by two local recursion operators. It is also shown that some of the obtained higher-order symmetries can be rewritten as fractional-order symmetries, and corresponding fractional-order recursion operators are presented. The proposed approach for finding higher-order symmetries is applicable for a wide class of linear fractional differential equations.

scholarly journals Caputo Fractional Derivative and Quantum-Like Coherence

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 211
Author(s):  
Garland Culbreth ◽  
Mauro Bologna ◽  
Bruce J. West ◽  
Paolo Grigolini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Caputo Fractional Derivative ◽  
Quantum Coherence ◽  
Time Derivative ◽  
The Other ◽  
Diffusion Equations ◽  
Self Organization ◽  
Ordinary Time

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

scholarly journals Studies of anomalous diffusion in the human brain using fractional order calculus

2010 ◽  
Vol 63 (3) ◽  
pp. 562-569 ◽  
Author(s):  
Xiaohong Joe Zhou ◽  
Qing Gao ◽  
Osama Abdullah ◽  
Richard L. Magin
Keyword(s):  
Human Brain ◽  
Fractional Order ◽  
Anomalous Diffusion ◽  
Fractional Order Calculus

scholarly journals Supplement of differential equations of fraction order for forecasting of financial markets

2018 ◽  
Vol 170 ◽  
pp. 01075
Author(s):  
Sergey Erokhin ◽  
Olga Roshka
Keyword(s):  
Differential Equation ◽  
Diffusion Equation ◽  
Fractional Order ◽  
Capital Markets ◽  
Equity Markets ◽  
Planck Equation ◽  
Theoretical Physics ◽  
Advection Diffusion Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck

In this paper, the analysis of capital markets takes place using the advection-diffusion equation. It should be noted that the methods used in modern theoretical physics have long been used in the analysis of capital markets. In particular, the Fokker-Planck equation has long been used in finding the probability density function of the return on equity. Throughout the study, a number of authors have considered the supplement of the Fokker-Planck equation in the forecasting of equity markets, as a differential equation of second order. In this paper, the first time capital markets analysis is performed using the fractional diffusion equation. The rationale is determined solely by the application nature, which consists in generation of trading strategy in equity markets with the supplement of differential equation of fractional order. As the subject for studies, the differential operator of fractional order in partial derivatives was chosen – the Fokker-Planck equation. The general solutions of equation are the basis for the forecast on the exchange rate of equities included in the Dow Jones Index Average (DJIA).

Anomalous diffusion equation using a new general fractional derivative within the Miller–Ross kernel

2020 ◽  
Vol 34 (27) ◽  
pp. 2050289
Author(s):  
Yiying Feng ◽  
Jiangen Liu
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Fractional Integral ◽  
Liouville Type

In view of the generalization of Miller–Ross kernel in the sense of Riemann–Liouville type, we propose the new definitions of the general fractional integral (GFI) and general fractional derivative (GFD) to discuss the anomalous diffusion equation, which is distinct from those classic calculus operators. The obtained analytical solution of the application described in the graph is effective and accurate making the use of Laplace transform.

scholarly journals On a Fractional Master Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Anitha Thomas
Keyword(s):  
Wave Equation ◽  
Diffusion Equation ◽  
Master Equation ◽  
Fractional Order ◽  
Caputo Derivative ◽  
Analytic Solution ◽  
Two Dimensions ◽  
Integer Order ◽  
Laplace Equations ◽  
Independent Form

A fractional order time-independent form of the wave equation or diffusion equation in two dimensions is obtained from the standard time-independent form of the wave equation or diffusion equation in two-dimensions by replacing the integer order partial derivatives by fractional Riesz-Feller derivative and Caputo derivative of order and respectively. In this paper, we derive an analytic solution for the fractional time-independent form of the wave equation or diffusion equation in two dimensions in terms of the Mittag-Leffler function. The solutions to the fractional Poisson and the Laplace equations of the same kind are obtained, again represented by means of the Mittag-Leffler function. In all three cases, the solutions are represented also in terms of Fox's -function.

scholarly journals Unique continuation principle for the one-dimensional time-fractional diffusion equation

2019 ◽  
Vol 22 (3) ◽  
pp. 644-657 ◽  
Author(s):  
Zhiyuan Li ◽  
Masahiro Yamamoto
Keyword(s):  
Diffusion Equation ◽  
Anomalous Diffusion ◽  
Function Method ◽  
Unique Continuation ◽  
Fractional Diffusion ◽  
Uniqueness Of Solutions ◽  
One Dimensional ◽  
Continuation Principle ◽  
Time Fractional Diffusion Equation ◽  
The One

Abstract This paper deals with the unique continuation of solutions for a one-dimensional anomalous diffusion equation with Caputo derivative of order α ∈ (0, 1). Firstly, the uniqueness of solutions to a lateral Cauchy problem for the anomalous diffusion equation is given via the Theta function method, from which we further verify the unique continuation principle.

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