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.

SOME NEW PROPERTIES CONCERNING THE q-DEFORMED CALCULUS AND THE q-DEFORMATION OF THE DIFFUSION EQUATION

2013 ◽  
Vol 28 (22) ◽  
pp. 1350100 ◽  
Author(s):  
WON SANG CHUNG ◽  
MIN JUNG
Keyword(s):  
Diffusion Equation ◽  
Time Derivative ◽  
Diffusion Equations ◽  
Ordinary Time

In this paper, we introduce some new properties concerning the q-derivative and q-integral. Using these properties, we construct the q-deformation of the diffusion equation. Two types of the q-deformed diffusion equations are possible. One is expressed in terms of the ordinary time derivative and another is expressed in terms of the q-derivative.

NUMERICAL SIMULATIONS FOR SPACE–TIME FRACTIONAL DIFFUSION EQUATIONS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341001 ◽  
Author(s):  
LEEVAN LING ◽  
MASAHIRO YAMAMOTO
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Long Time ◽  
Liouville Fractional Derivative ◽  
Time Fractional Diffusion Equation

We consider the solutions of a space–time fractional diffusion equation on the interval [-1, 1]. The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann–Liouville fractional derivative of order between one and two, and the first-order time derivative by a Caputo fractional derivative of order between zero and one. As the fundamental solution of this fractional equation is unknown (if exists), an eigenfunction approach is applied to obtain approximate fundamental solutions which are then used to solve the space–time fractional diffusion equation with initial and boundary values. Numerical results are presented to demonstrate the effectiveness of the proposed method in long time simulations.

scholarly journals A novel series method for fractional diffusion equation within Caputo fractional derivative

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 695-699 ◽  
Author(s):  
Sheng-Ping Yan ◽  
Wei-Ping Zhong ◽  
Xiao-Jun Yang
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Series Solution ◽  
Expansion Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Series Method ◽  
Time Fractional Diffusion Equation ◽  
Series Expansion Method

In this paper, we suggest the series expansion method for finding the series solution for the time-fractional diffusion equation involving Caputo fractional derivative.

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 Analytical Solutions of a Class of Fluids Models with the Caputo Fractional Derivative

2022 ◽  
Vol 6 (1) ◽  
pp. 35
Author(s):  
Ndolane Sene
Keyword(s):  
Diffusion Equation ◽  
Prandtl Number ◽  
Fractional Derivative ◽  
Analytical Solutions ◽  
Caputo Fractional Derivative ◽  
Caputo Derivative ◽  
Fractional Diffusion ◽  
Fluid Models ◽  
Fractional Operators ◽  
Fractional Heat Equation

This paper studies the analytical solutions of the fractional fluid models described by the Caputo derivative. We combine the Fourier sine and the Laplace transforms. We analyze the influence of the order of the Caputo derivative the Prandtl number, the Grashof numbers, and the Casson parameter on the dynamics of the fractional diffusion equation with reaction term and the fractional heat equation. In this paper, we notice that the order of the Caputo fractional derivative plays the retardation effect or the acceleration. The physical interpretations of the influence of the parameters of the model have been proposed. The graphical representations illustrate the main findings of the present paper. This paper contributes to answering the open problem of finding analytical solutions to the fluid models described by the fractional operators.

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations: A Second-Order Scheme

2017 ◽  
Vol 22 (4) ◽  
pp. 1028-1048 ◽  
Author(s):  
Yonggui Yan ◽  
Zhi-Zhong Sun ◽  
Jiwei Zhang
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Computational Cost ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fast Evaluation ◽  
Fractional Diffusion Equations ◽  
Long Time ◽  
Order Scheme ◽  
Speed Up

AbstractThe fractional derivatives include nonlocal information and thus their calculation requires huge storage and computational cost for long time simulations. We present an efficient and high-order accurate numerical formula to speed up the evaluation of the Caputo fractional derivative based on theL2-1σformula proposed in [A. Alikhanov,J. Comput. Phys., 280 (2015), pp. 424-438], and employing the sum-of-exponentials approximation to the kernel function appeared in the Caputo fractional derivative. Both theoretically and numerically, we prove that while applied to solving time fractional diffusion equations, our scheme not only has unconditional stability and high accuracy but also reduces the storage and computational cost.

scholarly journals Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Karel Van Bockstal
Keyword(s):  
Weak Solution ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Differential Operators ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Variable Order ◽  
Unique Weak Solution ◽  
Bounded Lipschitz Domain

AbstractIn this contribution, we investigate an initial-boundary value problem for a fractional diffusion equation with Caputo fractional derivative of space-dependent variable order where the coefficients are dependent on spatial and time variables. We consider a bounded Lipschitz domain and a homogeneous Dirichlet boundary condition. Variable-order fractional differential operators originate in anomalous diffusion modelling. Using the strongly positive definiteness of the governing kernel, we establish the existence of a unique weak solution in $u\in \operatorname{L}^{\infty } ((0,T),\operatorname{H}^{1}_{0}( \Omega ) )$ u ∈ L ∞ ( ( 0 , T ) , H 0 1 ( Ω ) ) to the problem if the initial data belongs to $\operatorname{H}^{1}_{0}(\Omega )$ H 0 1 ( Ω ) . We show that the solution belongs to $\operatorname{C} ([0,T],{\operatorname{H}^{1}_{0}(\Omega )}^{*} )$ C ( [ 0 , T ] , H 0 1 ( Ω ) ∗ ) in the case of a Caputo fractional derivative of constant order. We generalise a fundamental identity for integro-differential operators of the form $\frac{\mathrm{d}}{\mathrm{d}t} (k\ast v)(t)$ d d t ( k ∗ v ) ( t ) to a convolution kernel that is also space-dependent and employ this result when searching for more regular solutions. We also discuss the situation that the domain consists of separated subdomains.

On the Approximate Solution of Caputo-Riesz-Feller Fractional Diffusion Equation

2020 ◽  
pp. 224-244
Author(s):  
Samir Shamseldeen ◽  
Ahmed Elsaid ◽  
Seham Madkour
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Optimal Homotopy Analysis Method ◽  
Homotopy Analysis ◽  
Fractional Time Derivative ◽  
Complex Exponential ◽  
Fractional Equation

In this work, a space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative is introduced. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. Also, a very useful Riesz-Feller fractional derivative is proved; the property is essential in applying iterative methods specially for complex exponential and/or real trigonometric functions. The analytic series solution of the problem is obtained via the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to highlight the effect of changing the fractional derivative parameters on the behavior of the obtained solutions. The results in this work are originally extracted from the author's work.

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