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 Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1866
Author(s):  
Mohamed Jleli ◽  
Bessem Samet ◽  
Calogero Vetro
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
A Priori Estimates ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Higher Order ◽  
Differential Inequalities ◽  
Structure Of Solutions ◽  
Nonlinear Capacity ◽  
Fractional Differential

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

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 The Numerical Solutions of a Two-Dimensional Space-Time Riesz-Caputo Fractional Diffusion Equation

2011 ◽  
Vol 1 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Necati OZDEMIR ◽  
Derya AVCI ◽  
Beyza Billur ISKENDER
Keyword(s):  
Diffusion Equation ◽  
Anomalous Diffusion ◽  
Numerical Solutions ◽  
Fourier Transforms ◽  
Dynamic Properties ◽  
Dimensional Space ◽  
Space Time ◽  
Two Dimensional ◽  
Dimensional Space Time ◽  
Fractional Differential

This paper is concerned with the numerical solutions of a two dimensional space-time fractional differential equation used to model the dynamic properties of complex systems governed by anomalous diffusion. The space-time fractional anomalous diffusion equation is defined by replacing second order space and first order time derivatives with Riesz and Caputo operators, respectively. By using Laplace and Fourier transforms, a general representation of analytical solution is obtained as Mittag-Leffler function. Gr\"{u}nwald-Letnikov (GL) approximation is also used to find numerical solution of the problem. Finally, simulation results for two examples illustrate the comparison of the analytical and numerical solutions and also validity of the GL approach to this problem.

Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas

2018 ◽  
Vol 21 (6) ◽  
pp. 1493-1505
Author(s):  
Takahiro Yajima ◽  
Shunya Oiwa ◽  
Kazuhito Yamasaki
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Tangent Vector ◽  
Plane Curve ◽  
Initial Time ◽  
Composite Function ◽  
Long Period ◽  
Fractional Differential ◽  
Order Tangent ◽  
Over Time

Abstract This paper discusses a construction of fractional differential geometry of curves (curvature of curve and Frenet-Serret formulas). A tangent vector of plane curve is defined by the Caputo fractional derivative. Under a simplification for the fractional derivative of composite function, a fractional expression of Frenet frame of curve is obtained. Then, the Frenet-Serret formulas and the curvature are derived for the fractional ordered frame. The different property from the ordinary theory of curve is given by the explicit expression of arclength in the fractional-order curvature. The arclength part of the curvature takes a large value around an initial time and converges to zero for a long period of time. This trend of curvature may reflect the memory effect of fractional derivative which is progressively weaken for a long period of time. Indeed, for a circle and a parabola, the curvature decreases over time. These results suggest that the basic property of fractional derivative is included in the fractional-order curvature appropriately.

scholarly journals Modeling the Dependence of Barometric Pressure with Altitude Using Caputo and Caputo–Fabrizio Fractional Derivatives

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Muath Awadalla ◽  
Yves Yameni Noupoue Yannick ◽  
Kinda Abu Asbeh
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Error Rate ◽  
Fractional Derivatives ◽  
Classical Method ◽  
Main Tool ◽  
Barometric Pressure ◽  
Proposed Model ◽  
Fractional Differential ◽  
The Relationship

This work is dedicated to the study of the relationship between altitude and barometric atmospheric pressure. There is a consistent literature on this relationship, out of which an ordinary differential equation with initial value problems is often used for modeling. Here, we proposed a new modeling technique of the relationship using Caputo and Caputo–Fabrizio fractional differential equations. First, the proposed model is proven well-defined through existence and uniqueness of its solution. Caputo–Fabrizio fractional derivative is the main tool used throughout the proof. Then, follow experimental study using real world dataset. The experiment has revealed that the Caputo fractional derivative is the most appropriate tool for fitting the model, since it has produced the smallest error rate of 1.74% corresponding to the fractional order of derivative α  = 1.005. Caputo–Fabrizio was the second best since it yielded an error rate value of 1.97% for a fractional order of derivative α  = 1.042, and finally the classical method produced an error rate of 4.36%.

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