Physica A 388 (2009) 4586–4592 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Variable-order fractional differential operators in anomalous diffusion modeling HongGuang Sun a,b,∗ , Wen Chen a , YangQuan Chen b a Institute of Soft Matter Mechanics, Department of Engineering Mechanics, Hohai University, No.1 XiKang Road, Nanjing, Jiangsu 210098, China b Center for Self-Organizing and Intelligent Systems (CSOIS), Department of Electrical and Computer Engineering, Utah State University, 4160 Old Main Hill, Logan, UT 84322-4160, USA article info Article history: Received 14 May 2009 Received in revised form 15 June 2009 Available online 21 July 2009 PACS: 66.10.C- 05.10.Gg 02.60.Cb Keywords: Anomalous diffusion Variable-order fractional differential operator Probability density function Complex medium abstract The purpose of this paper is to offer a unified discussion of variable-order differential oper- ators in anomalous diffusion modeling. The characteristics of the new models, in contrast to constant-order fractional diffusion models, change with time, space, concentration or other independent quantities. We introduced a classification of variable-order fractional diffu- sion models based on the possible physical origins which prompt the variable-order. Some potential applications of the variable-order fractional diffusion models are also discussed. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Anomalous diffusion phenomena are observed in various physical, chemical and biological situations, which have motivated the development of new mathematical and physical models [1]. It has been widely studied both by statistical methods and differential equation models [2,3]. Fractional diffusion equations account for typical ‘‘anomalous’’ features which are observed in many systems, e.g. in the cases of dispersive transports in amorphous semiconductors, porous medium, colloid, proteins, biosystems or even in ecosystems [4,5]. In this study, we focus on the differential equation models of anomalous diffusion which govern many physical phenomena such as heat, mass or electron transfer; pollutants or liquid transport through porous media. Up until now, the constant-order (CO) fractional kinetic equations have been considered in most cases and received tremendous success in anomalous diffusion modeling and other fields [1,6–8]. However, it has become clear that further theoretical and numerical investigations are required in order to incorporate adequate tools for the description of more complicated (or more realistic) stochastic diffusion processes [9]. There are a large class of physical, biological and physiological diffusion phenomena that the CO fractional diffusion equation with constant coefficients is not equipped to characterize. The typical features of these phenomena are that they are complex to analysis and the diffusion behaviors ∗ Corresponding author at: Institute of Soft Matter Mechanics, Department of Engineering Mechanics, Hohai University, No.1 XiKang Road, Nanjing, Jiangsu 210098, China. Tel.: +86 13621586259. E-mail address: sunhongguang08@gmail.com (H. Sun). 0378-4371/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.07.024