FDA Express

In these lecture notes, we will analyze the behavior of random walk on disordered media by means of both probabilistic and analytic methods, and will study the scaling limits. We will focus on the discrete potential theory and how the theory is effectively used in the analysis of disordered media. The first few chapters of the notes can be used as an introduction to discrete potential theory.

Recently, there has been significant progress on the theory of random walk on disordered media such as fractals and random media. Random walk on a percolation cluster (‘the ant in the labyrinth’) is one of the typical examples. In 1986, H. Kesten showed the anomalous behavior of a random walk on a percolation cluster at critical probability. Partly motivated by this work, analysis and diffusion processes on fractals have been developed since the late eighties. As a result, various new methods have been produced to estimate heat kernels on disordered media. These developments are summarized in the notes.

More information on this book can be found by the following link:

http://www.springer.com/mathematics/probability/book/978-3-319-03151-4

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Topics in Fractional Differential Equations

Abbas, Said, Benchohra, Mouffak, N'Guérékata, Gaston M.

Book Description

During the last decade, there has been an explosion of interest in fractional dynamics as it was found to play a fundamental role in the modeling of a considerable number of phenomena; in particular the modeling of memory-dependent and complex media. Fractional calculus generalizes integrals and derivatives to non-integer orders and has emerged as an important tool for the study of dynamical systems where classical methods reveal strong limitations. This book is addressed to a wide audience of researchers working with fractional dynamics, including mathematicians, engineers, biologists, and physicists. This timely publication may also be suitable for a graduate level seminar for students studying differential equations.

Topics in Fractional Differential Equations is devoted to the existence and uniqueness of solutions for various classes of Darboux problems for hyperbolic differential equations or inclusions involving the Caputo fractional derivative. In this book, problems are studied using the fixed point approach, the method of upper and lower solution, and the Kuratowski measure of noncompactness. An historical introduction to fractional calculus will be of general interest to a wide range of researchers. Chapter one contains some preliminary background results. The second Chapter is devoted to fractional order partial functional differential equations. Chapter three is concerned with functional partial differential inclusions, while in the fourth chapter, we consider functional impulsive partial hyperbolic differential equations. Chapter five is concerned with impulsive partial hyperbolic functional differential inclusions. Implicit partial hyperbolic differential equations are considered in Chapter six, and finally in Chapter seven, Riemann-Liouville fractional order integral equations are considered. Each chapter concludes with a section devoted to notes and bibliographical remarks. The work is self-contained but also contains questions and directions for further research.

More information on this book can be found by the following link:

http://www.springer.com/mathematics/dynamical+systems/book/978-1-4614-4035-2

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Journals

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Fractional Calculus & Applied Analysis

Volume 17, Issue 3

Waveform relaxation method for fractional differential-algebraic equations

Xiao-Li Ding, Yao-Lin Jiang

Multiple positive solutions of nonlinear fractional differential equations with integral boundary value conditions

Wei Sun, Youyu Wang

Performance modeling of serial and parallel implementations of the fractional Adams-Bashforth-Moulton method

Wei Zhang, Wenjie Wei, Xing Cai

Eigenvalues for iterative systems of Sturm-Liouville fractional order two-point boundary value problems

Kapula Rajendra Prasad…

Solutions to fractional functional differential equations with nonlocal conditions

Shruti Dubey, Madhukant Sharma

Centennial jubilee of Academician Rabotnov and contemporary handling of his fractional operator

Yury A. Rossikhin, Marina V. Shitikova

Equilibrium of an elastic medium with after-effect

Yury Nikolaevich Rabotnov

Conceptual design of a selectable fractional-order differentiator for industrial applications

Emmanuel A. Gonzalez, Ľubomír Dorčák

Existence of solutions to boundary value problem for impulsive fractional differential equations

Gabriele Bonanno, Rosana Rodríguez-López…

Fractional generalizations of filtering problems and their associated fractional Zakai equations

Sabir Umarov, Frederick Daum, Kenric Nelson

The λ-cosine transforms with odd kernel and the hemispherical transform

Boris Rubin

Covariance measure and stochastic heat equation with fractional noise

Ciprian Tudor, Mounir Zili

Representation of complex powers of C-sectorial operators

Chuang Chen, Marko Kostić, Miao Li

Conjugate points for fractional differential equations

Paul Eloe, Jeffrey T. Neugebauer

Eigenvalue comparison for fractional boundary value problems with the Caputo derivative

Johnny Henderson, Nickolai Kosmatov

Asymptotic properties of solutions of the fractional diffusion-wave equation

Anatoly N. Kochubei

Characterization of weighted analytic Besov spaces in terms of operators of fractional differentiation

Alexey Karapetyants, Ferdos Kodzoeva

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Nonlinear Dynamics

Volume 77, Issue 1-2 (selected)

A piecewise beam element based on absolute nodal coordinate formulation

Zuqing Yu, Peng Lan, Nianli Lu

Simultaneous Impact of a Two-Link Chain

Dan B. Marghitu, Dorian Cojocaru

Pattern dynamics of an epidemic model with nonlinear incidence rate

Tao Wang

Decentralized sliding mode control of fractional-order large-scale nonlinear systems

Sajjad Shoja Majidabad, Heydar Toosian Shandiz, Amin Hajizadeh

Instability conditions for a class of switched linear systems with switching delays based on sampled-data analysis: applications to DC–DC converters

Chung-Chieh Fang

Global exponential stabilization for chaotic brushless DC motors with a single input

Du Qu Wei, Li Wan, Xiao Shu Luo, Shang You Zeng, Bo Zhang

Bifurcation analysis of a piecewise-linear impact oscillator with drift

Joseph Páez Chávez, Ekaterina Pavlovskaia, Marian Wiercigroch

Erratum to: Bifurcation analysis of a piecewise-linear impact oscillator with drift

Joseph Páez Chávez, Ekaterina Pavlovskaia, Marian Wiercigroch

Chaotic behavior in fractional-order memristor-based simplest chaotic circuit using fourth degree polynomial

Lin Teng, Herbert H. C. Iu, Xingyuan Wang, Xiukun Wang

Security evaluation of bilateral-diffusion based image encryption algorithm

Moting Su, Wenying Wen, Yushu Zhang

Generalized macroscopic traffic model with time delay

D. Ngoduy

Comments on “Particle swarm optimization with fractional-order velocity”

Xiangyin Zhang, Haibin Duan

Comment on “Particle swarm optimization with fractional-order velocity”

Ling-Yun Zhou, Shang-Bo Zhou, Muhammad Abubakar Siddiqu

Reply to: Comments on “Particle Swarm Optimization with Fractional-Order Velocity”

J. A. Tenreiro Machado, E. J. Solteiro Pires, Micael S. Couceiro

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Paper Highlight
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Discrete random walk models for space-time fractional diffusion

Rudolf Gorenflo, Francesco Mainardi, Daniele Moretti, Gianni Pagnini, Paolo Paradisi

Publication information: Rudolf Gorenflo, Francesco Mainardi, Daniele Moretti, Gianni Pagnini, Paolo Paradisi. Discrete random walk models for space-time fractional diffusion. Chemical Physics 284 (2002) 521-541.

http://www.sciencedirect.com/science/article/pii/S0301010402007140

Abstract

A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By space-time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order a∈(0, 2] and skewness q, and the first-order time derivative with a Caputo derivative of order b∈(0, 1]. Such evolution equation implies for the flux a fractional Fick’s law which accounts for spatial and temporal non-locality. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.

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Fractional diffusion and reflective boundary condition

Natalia Krepysheva, Liliana Di Pietro, Marie-Christine Néel

Publication information: Natalia Krepysheva, Liliana Di Pietro, Marie-Christine Néel. Fractional diffusion and reflective boundary condition. Physica A 368 (2006) 355-361.

http://www.sciencedirect.com/science/article/pii/S0378437106000380

Abstract

Anomalous diffusive transport arises in a large diversity of disordered media. Stochastic formulations in terms of continuous time random walks (CTRW) with transition probability densities presenting spatial and/or time diverging moments were developed to account for anomalous behaviours. Many CTRWs in infinite media were shown to correspond, on the macroscopic scale, to diffusion equations sometimes involving derivatives of non-integer order. A wide class of CTRWs with symmetric Levy distribution of jumps and finite mean waiting time leads, in the macroscopic limit, to space-time fractional equations that account for super diffusion and involve an operator, which is non-local in space. Due to non-locality, the boundary condition results in modifying the large-scale model. We are studying here the diffusive limit of CTRWs, generalizing Levy flights in a semi-infinite medium, limited by a reflective barrier. We obtain space-time fractional diffusion equations that differ from the infinite medium in the kernel of the fractional derivative w.r.t. space.

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