Numerous experimental measurements in spatially complex systems have revealed anomalous diffusion in which The mean square displacement scales as a fractional order power law in time.
Over the past two decades a new mathematical description has been formulated, linked together by tools of fractional calculus: fractional constitutive laws.
probabilistic models based on continuous time random walks and generalized central limit theorems.( CLT is always very important ! )
Remark:
Over the past decade a new theoretical framework has been developed to model anomalous diffusion. The new framework is based around the physics of continuous time random walks and the mathematics of fractional calculus.
One can ask what would be a differential having as its exponent a fraction. Although this seems removed from Geometry . . . it appears that one day these paradoxes will yield useful consequences.
—Gottfried Leibniz (1695)
Fraction calculus can See my blog http://www.cnblogs.com/zhangwenbiao/
More: Oldham & Spanier (1974),Miller & Ross (1993), I.Podlubny (1999)
(Your must know Which papers ? Google scholar W.B. Zhang 2014-5-19)
Remark:
We must pose the examples of anomalous diffusion of papers that have print.
² Fractional kinetics motivated by biological systems
² Molecular crowding affects diffusion and binding of nuclear proteins in heterochromatin and reveals the fractal organization of chromatin
² Generalized Langevin Equation with Fractional Gaussian Noise:Subdiffusion within a Single Protein Molecule S. C. Kou and X. Sunney Xie(P.R. L)
l fractional Langevin equations, fractional Brownian motions
² Brownian motion of molecules: the classical theory
² Langevin-Vladimirsky approachto Brownian motion with memory
² Benoˆıt Mandelbrot and Fractional Brownian Motion
² CONTINUOUS TIME LONG MEMORY MODELS:A FRACTIONAL BROWNIAN MOTION AND FRACTIONAL GAUSSIAN NOISE
² Stochastic Analysis of the Fractional Brownian Motion
² Fractional Brownian motion: stochastic calculus and applications
² Brownian Motion: Langevin Equation
² ON THE THEORY OF THE BROKNIAN MOTION
² The origin of the Langevin equation and the calculation of the mean squared displacement: Let’s set the record straight
² The Fractional Langevin Equation: Brownian Motion Revisted
² BROWNIAN DYNAMICS SIMULATIONS OF POLYMERS AND SOFT MATTER
² .......
l fractional diffusion, fractional Fokker-Planck equations
² Non-Gaussian Statistics and Anomalous Diffusion in Porous Media
² An Introduction to Fractional Diffusion B.I. Henry, T.A.M. Langlands
² From Diffusion to Anomalous Diffusion: A Century After Einstein
² Master, Fokker-Planck and Langevin equations
² Remarks on Fractional Diffusion Equations Michael Taylor
² Diffusion on fractals and space fractional diffusion equations
² Mathematical analysis for fractional diffusion equations: forward problems and inverse problems
² THE RANDOM WALK:S GUIDE TO ANOMALOUS DIFFUSION: A FRACTIONAL DYNAMICS APPROACH ,Ralf.Metzler and Joseph.Klafter
² The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, Ralf Metzler and Joseph Klafter
² 基于连续时间无规行走模型研究反常扩散 林 方 包景东
² MOXE: A Model of Gas Exchange for Hyperpolarized Xe Magnetic Resonance of the Lung
² Anomalous Transport: Foundations and Applications Edited by Rainer Klages, Günter Radons, and Igor M. Sokolov
² Subordinated diffusion and continuous time random walk asymptotics Bartłomiej Dybiec and Ewa Gudowska-Nowak
² Directed transport driven by Lévy flights coexisting with subdiffusion
² ........
l Fractional reaction-diffusion
² Fractional reaction-diffusion equation for species growth and dispersal
NUMERICAL SOLUTIONS FOR FRACTIONAL REACTION-DIFFUSION EQUATIONS
Fractional Reaction-Diffusion Problems
Nonlinear analysis of a fractional reaction diffusion model for tumour invasion
Reaction-subdiffusion equations(A->B)
Anomalous subdiffusion with multispecies linear reaction dynamics
Reaction front in an A+B->C reaction-subdiffusion process
Some Properties of the A + B->C Reaction-Diffusion System with Initially Separated Components
Front propagation in A+B->2A reaction under subdiffusion
Reaction-subdiffusion model of morphogen gradient formation
Application of Fractional Calculus to Reaction-Subdiffusion Processes and Morphogen Gradient Formation
Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations
Fractional kinetics motivated by biological systems
Subdiffusive reaction-diffusion equations Robert Hahn
Mesoscopic description of reactions for anomalous diffusion: a case study
Stationary Fronts in anAB!0Reaction under Subdiffusion Daniela Froemberg and Igor M. Sokolov (P.R.L)
基于分数阶微积分的生化反应动力学 张文彪
同伦分析方法: 一种新的求解非线性问题的近似解析方法
Non-Markovian random walks and nonlinear reactions: Subdiffusion and propagating fronts Sergei Fedotov
NEW SOLUTION AND ANALYTICAL TECHNIQUES OF THE IMPLICIT NUMERICAL METHOD FOR THE ANOMALOUS SUB-DIFFUSION EQUATION Properties of the reaction front in a reaction-subdiffusion process ,Katja Lindenberg and S. B. Yuste
Kinetic equations for reaction-subdiffusion systems: Derivation and stability analysis, A. Yadav and Werner Horsthemke
Stochastic Modeling in Systems Biology,Jinzhi Lei
Analytical solutions, moments, and their asymptotic behaviors
for the time-space fractional cable equation