scholarly journals Anomalous Diffusion with an Irreversible Linear Reaction and Sorption-Desorption Process

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Maike A. F. dos Santos ◽  
Marcelo K. Lenzi ◽  
Ervin K. Lenzi
Keyword(s):  
Anomalous Diffusion ◽  
Diffusion Processes ◽  
Mean Square Displacement ◽  
Fractional Diffusion ◽  
Infinite Medium ◽  
Desorption Process ◽  
Fractional Diffusion Equations ◽  
Linear Reaction ◽  
The Mean ◽  
Adsorption Desorption

We investigate the diffusion of two different species in a semi-infinite medium considering the presence of linear reaction terms. The dynamics for these species is governed by fractional diffusion equations. We also consider the presence of an adsorption-desorption boundary condition. The solutions for this system are found in terms of the H function of Fox and by analyzing the behavior of the mean square displacement a rich class of diffusion processes is verified. In this sense, we show how the surface effects modify the bulk dynamics and promote an anomalous diffusion of system.

ANOMALOUS DIFFUSION AND GENERALIZED DIFFUSION EQUATIONS

2005 ◽  
Vol 05 (02) ◽  
pp. L275-L282 ◽  
Author(s):  
I. M. SOKOLOV ◽  
A. V. CHECHKIN
Keyword(s):  
Anomalous Diffusion ◽  
Diffusion Processes ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Operator ◽  
Fractional Diffusion Equations ◽  
Whole Number ◽  
Simple Scaling ◽  
Special Case ◽  
Distributed Order

Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. The forms of such equations might differ with respect to the position of the corresponding fractional operator in addition to or instead of the whole-number derivative in the Fick's equation. For processes lacking simple scaling the corresponding description may be given by distributed-order equations. In the present paper different forms of distributed-order diffusion equations are considered. The properties of their solutions are discussed for a simple special case.

Anomalous diffusion and the adsorption-desorption process in anisotropic media

2009 ◽  
Vol 85 (2) ◽  
pp. 28004 ◽  
Author(s):  
E. K. Lenzi ◽  
L. R. Evangelista ◽  
G. Barbero ◽  
F. Mantegazza
Keyword(s):  
Anomalous Diffusion ◽  
Anisotropic Media ◽  
Desorption Process ◽  
Adsorption Desorption

scholarly journals Anomalous diffusion and transport in heterogeneous systems separated by a membrane

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160502 ◽  
Author(s):  
E. K. Lenzi ◽  
H. V. Ribeiro ◽  
A. A. Tateishi ◽  
R. S. Zola ◽  
L. R. Evangelista
Keyword(s):  
Anomalous Diffusion ◽  
Heterogeneous System ◽  
Particle Distribution ◽  
Kinetic Equations ◽  
Heterogeneous Systems ◽  
Particle Dynamics ◽  
Fractional Diffusion ◽  
Semipermeable Membrane ◽  
Fractional Diffusion Equations ◽  
Rich Variety

Diffusion of particles in a heterogeneous system separated by a semipermeable membrane is investigated. The particle dynamics is governed by fractional diffusion equations in the bulk and by kinetic equations on the membrane, which characterizes an interface between two different media. The kinetic equations are solved by incorporating memory effects to account for anomalous diffusion and, consequently, non-Debye relaxations. A rich variety of behaviours for the particle distribution at the interface and in the bulk may be found, depending on the choice of characteristic times in the boundary conditions and on the fractional index of the modelling equations.

scholarly journals Lecture on the anomalous diffusion in Condensed Matter Physics

2018 ◽  
Vol 3 (2) ◽  
Author(s):  
M. Benhamou ◽  
Keyword(s):  
Anomalous Diffusion ◽  
Condensed Matter ◽  
Mean Square Displacement ◽  
Continuous Phase ◽  
Generalized Langevin Equation ◽  
Velocity Autocorrelation Function ◽  
Condensed Matter Physics ◽  
Mean Square ◽  
Pickering Emulsions ◽  
The Mean

Diffusion is a natural or artificial process that governs many phenomena in nature. The most known diffusion is the Brownian or normal motion, where the mean-square-displacement of the tracer (diffusive particle among others) increases as the square-root of time. It is not the case, however, for complex systems, where the diffusion is rather slow, because at small-scales, these media present an heterogenous structure. This kind of slow motion is called subdiffusion, where the associated mean-square-displacement increases in time, with a non trivial exponent, alpha, whose value is between 0 and 1. In this review paper, we report on new trends dealing with the study of the anomalous diffusion in Condensed Matter Physics. The study is achieved using a theoretical approach that is based on a Generalized Langevin Equation. As particular crowded systems, we choose the so-called Pickering emulsions (oil-in-water), and we are interested in how the dispersed droplets (protected by small solid charged nanoparticles) can diffuse in the continuous phase (water). Dynamic study is accomplished through the mean-square-displacement and the velocity-autocorrelation-function. Finally, a comparison with Molecular Dynamics data is made.

An efficient localized collocation solver for anomalous diffusion on surfaces

2021 ◽  
Vol 24 (3) ◽  
pp. 865-894 ◽  
Author(s):  
Zhuochao Tang ◽  
Zhuojia Fu ◽  
HongGuang Sun ◽  
Xiaoting Liu
Keyword(s):  
Anomalous Diffusion ◽  
Early Period ◽  
Fractional Diffusion ◽  
Least Square ◽  
Diffusion Equations ◽  
Moving Least Square ◽  
Fractional Diffusion Equations ◽  
Time Stepping ◽  
Temporal Discretization ◽  
Surface Time

Abstract This paper introduces an efficient collocation solver, the generalized finite difference method (GFDM) combined with the recent-developed scale-dependent time stepping method (SD-TSM), to predict the anomalous diffusion behavior on surfaces governed by surface time-fractional diffusion equations. In the proposed solver, the GFDM is used in spatial discretization and SD-TSM is used in temporal discretization. Based on the moving least square theorem and Taylor series, the GFDM introduces the stencil selection algorithms to choose the stencil support of a certain node from the whole discretization nodes on the surface. It inherits the similar properties from the standard FDM and avoids the mesh generation, which is available particularly for high-dimensional irregular discretization nodes. The SD-TSM is a non-uniform temporal discretization method involving the idea of metric, which links the fractional derivative order with the non-uniform discretization strategy. Compared with the traditional time stepping methods, GFDM combined with SD-TSM deals well with the low accuracy in the early period. Numerical investigations are presented to demonstrate the efficiency and accuracy of the proposed GFDM in conjunction with SD-TSM for solving either single or coupled fractional diffusion equations on surfaces.

scholarly journals Regional Controllability of Anomalous Diffusion Generated by the Time Fractional Diffusion Equations

2015 ◽  
Author(s):  
Fudong Ge ◽  
YangQuan Chen ◽  
Chunhai Kou
Keyword(s):  
Anomalous Diffusion ◽  
Minimum Energy ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Fractional Diffusion Equations ◽  
Regional Controllability

This paper is concerned with the investigation of the regional controllability of the time fractional diffusion equations. First, some preliminaries and definitions of regional controllability of the system under consideration are introduced, which promote the existence contributions on controllability analysis. Then we analyze the regional controllability with minimum energy of the time fractional diffusion equations on two cases: B ∈ L (Rm, L2 (Ω)) and B ∉ L (Lm, L2 (Ω)). In the end, two applications are given to illustrate our obtained results.

Solution of Fourth Order Diffusion Equations and Analysis Using the Second Moment

2020 ◽  
Vol 399 ◽  
pp. 10-20
Author(s):  
Jader Lugon Junior ◽  
João Flávio Vieira Vasconcellos ◽  
Diego Campos Knupp ◽  
Gisele Moraes Marinho ◽  
Luiz Bevilacqua ◽  
...  
Keyword(s):  
Fourth Order ◽  
Anomalous Diffusion ◽  
Mean Square Displacement ◽  
Practical Interest ◽  
Order Partial Differential Equation ◽  
New Approach ◽  
Fractional Equations ◽  
System Of Particles ◽  
Cancer Tumor ◽  
The Mean

The classical concept of diffusion characterized by Fick’s law is well suited for describing a wide class of practical problems of interest. Nevertheless, it has been observed that it is not enough to properly represent other relevant applications of practical interest. When in a system of particles their spreading is slower or faster than predicted by the classical diffusion model, such a phenomenon is referred to as anomalous diffusion. Time fractional, space fractional and even space-time fractional equations are widely used to model phenomena such as solute transport in porous media, financial modelling and cancer tumor behavior. Considering the effects of partial and temporary retention in dispersion processes a new analytical formulation was derived to simulate anomalous diffusion. The new approach leads to a fourth-order partial differential equation (PDE) and assumes the existence of two concomitant fluxes. This work investigates the behavior of the bi-flux approach in one dimensional (1D) medium evaluating the mean square displacement for different cases in order to classify the diffusion process in normal, sub-diffusive or super-diffusive.

Continuous time random walk models associated with distributed order diffusion equations

2015 ◽  
Vol 18 (3) ◽  
Author(s):  
Sabir Umarov
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Diffusion Processes ◽  
Stability Index ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Analytic Method ◽  
Fractional Diffusion Equations ◽  
The Stability ◽  
Distributed Order

AbstractIn this paper continuous time and discrete random walk models approximating diffusion processes associated with time-fractional and spacedistributed order differential equations are studied. Stochastic processes associated with the considered equations represent time-changed processes, where the time-change process is the inverse to a Levy’s stable subordinator with the stability index β ∈ (0, 1). In the paper the convergence of modeled continuous time and discrete random walks to time-changed processes associated with distributed order fractional diffusion equations are proved using an analytic method.

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