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.

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.

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.

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.

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