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.

scholarly journals Applications of Distributed-Order Fractional Operators: A Review

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 110
Author(s):  
Wei Ding ◽  
Sansit Patnaik ◽  
Sai Sidhardh ◽  
Fabio Semperlotti
Keyword(s):  
Fractional Calculus ◽  
Real World ◽  
Transport Processes ◽  
Model Systems ◽  
Variable Order ◽  
Fractional Operators ◽  
Research Activity ◽  
Road Map ◽  
Real World Applications ◽  
Distributed Order

Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

scholarly journals NUMERICAL SIMULATIONS FOR THE VARIABLE ORDER TWO-DIMENSIONAL REACTION SUB-DIFFUSION EQUATION: LINEAR AND NONLINEAR

Fractals ◽  
2021 ◽  
pp. 2240019
Author(s):  
MOHAMED ADEL
Keyword(s):  
Diffusion Equation ◽  
Numerical Technique ◽  
Weighted Average ◽  
Transport Processes ◽  
Diffusion Equations ◽  
Variable Order ◽  
High Concentration ◽  
Fractal Media ◽  
Linear And Nonlinear ◽  
The Stability

The applications and the fields that use the anomalous sub-diffusion equations cannot be easily listed due to their wide area. Sure, one of the main physical reasons for using and researching fractional order diffusion equations is to explain anomalous diffusion that occurs in transport processes through complex and/or disordered structures, such as fractal media. One of the important applications is their use in chemical reactions, where a single material continues to shift from a high concentration area to a low concentration area until the concentration across the space is equal. The mathematical model that describes these physical-chemical phenomena is called the reaction sub-diffusion equation. In our study, we try to solve the 2D variable order version of these equations (2DVORSE) (linear and nonlinear) by using an accurate numerical technique which is the variable weighted average finite difference method (WAFDM). We will analyze the stability of the resulting scheme by using a modified suitable version of the John Von Neumann procedure. Specific stability conditions that occur or some parameters in the resulting schemes are derived and checked. At the end of the study, numerical examples are simulated to check the stability and the accuracy of the proposed technique.

GROWTH KINETICS OF A NONEQUILIBRIUM ISING MODEL

2000 ◽  
Vol 14 (27n28) ◽  
pp. 983-993
Author(s):  
C. M. SOUKOULIS ◽  
M. C. TRINGIDES ◽  
M. J. VELGAKIS
Keyword(s):  
Time Dependence ◽  
Growth Kinetics ◽  
Disordered Systems ◽  
Spin Exchange ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Domain Boundaries ◽  
The Mean ◽  
Diffusive Processes ◽  
Kinetics Of

The growth kinetics of a system with spin-exchange dynamics has been investigated by means of computer simulations. These systems are used to simulate diffusive processes and kinetic phenomena far away from equilibrium. As a measure of growth we have studied the mean square displacement <R2> of tagged particles. It is found that <R2>t1-n follows a sublinear time dependence, which is explained in terms of the changes of the distribution of atoms between sites within the ordered region and sites of the domain boundaries. The time-dependence of the diffusion coefficient has been derived. The analogies to the relaxation in disordered systems, such as in a:Si–H and in O/W(110), is discussed.

scholarly journals Fractional thermal diffusion and the heat equation

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Francisco Gómez ◽  
Luis Morales ◽  
Mario González ◽  
Victor Alvarado ◽  
Guadalupe López
Keyword(s):  
Time Domain ◽  
Time Derivative ◽  
Space Time ◽  
Mathematical Representation ◽  
Alternative Representation ◽  
Complex Processes ◽  
Temporal Derivative ◽  
Dimensional Parameters ◽  
Non Local ◽  
Fractional Space

AbstractFractional calculus is the branch of mathematical analysis that deals with operators interpreted as derivatives and integrals of non-integer order. This mathematical representation is used in the description of non-local behaviors and anomalous complex processes. Fourier’s lawfor the conduction of heat exhibit anomalous behaviors when the order of the derivative is considered as 0 < β,ϒ ≤ 1 for the space-time domain respectively. In this paper we proposed an alternative representation of the fractional Fourier’s law equation, three cases are presented; with fractional spatial derivative, fractional temporal derivative and fractional space-time derivative (both derivatives in simultaneous form). In this analysis we introduce fractional dimensional parameters σ

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.

A Novel Finite Element Method for a Class of Time Fractional Diffusion Equations

2011 ◽  
Author(s):  
H. G. Sun ◽  
W. Chen ◽  
K. Y. Sze
Keyword(s):  
Fractional Diffusion ◽  
Transport Processes ◽  
Anomalous Transport ◽  
Diffusion Equations ◽  
Time Range ◽  
Main Task ◽  
Fractional Diffusion Equations ◽  
Long Time ◽  
Range Computation ◽  
Good Agreement

Anomalous transport of contaminants in groundwater or porous soil is a research focus in hydrology and soil science for decades. Because fractional diffusion equations can well characterize early breakthrough and heavy tail decay features of contaminant transport process, they have been considered as promising tools to simulate anomalous transport processes in complex media. However, the efficient and accurate computation of fractional diffusion equations is a main task in their applications. In this paper, we introduce a novel numerical method which captures the critical Mittag-Leffler decay feature of subdiffusion in time direction, to solve a class of time fractional diffusion equations. A key advantage of the new method is that it overcomes the critical problem in the application of time fractional differential equations: long-time range computation. To illustrate its efficiency and simplicity, three typical academic examples are presented. Numerical results show a good agreement with the exact solutions.

scholarly journals Classical Solutions to the Initial-Boundary Value Problems for Nonautonomous Fractional Diffusion Equations

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Jia Mu ◽  
Yang Liu ◽  
Huanhuan Zhang
Keyword(s):  
Fractional Power ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Classical Solutions ◽  
Operator Family ◽  
Initial Boundary Value Problems ◽  
Fractional Diffusion Equations ◽  
Theoretical Support ◽  
Initial Boundary Value

In this paper, we investigate a class of nonautonomous fractional diffusion equations (NFDEs). Firstly, under the condition of weighted Hölder continuity, the existence and two estimates of classical solutions are obtained by virtue of the properties of the probability density function and the evolution operator family. Secondly, it focuses on the continuity and an estimate of classical solutions in the sense of fractional power norm. The results generalize some existing results on classical solutions and provide theoretical support for the application of NFDE.

scholarly journals Analytical solution of the time fractional diffusion equation and fractional convection-diffusion equation

2018 ◽  
Vol 65 (1) ◽  
pp. 82 ◽  
Author(s):  
Francisco Gomez ◽  
Victor Morales ◽  
Marco Taneco
Keyword(s):  
Diffusion Equation ◽  
Analytical Solutions ◽  
Fourier Transforms ◽  
Time Derivative ◽  
Fractional Diffusion ◽  
Transport Processes ◽  
Diffusion Equations ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Diffusion And Convection

In this paper, we obtain analytical solutions for the time-fractional diffusion and time-fractional convection-diffusion equations. These equations are obtained from the standard equations by replacing the time derivative with a fractional derivative of order $\alpha$. Fractional operators of type Liouville-Caputo, Atangana-Baleanu-Caputo, fractional conformable derivative in Liouville-Caputo sense and Atangana-Koca-Caputo were used to model diffusion and convection-diffusion equation. The Laplace and Fourier transforms were applied to obtain the analytical solutions for the fractional order diffusion and convection-diffusion equations. The solutions obtained can be useful to understand the modeling of anomalous diffusive, subdiffusive systems and super-diffusive systems, transport processes, random walk and wave propagation phenomenon.

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