Physica A 390 (2011) 3397–3403 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Radial anomalous diffusion in an annulus Shaowei Wang a,∗ , Moli Zhao b , Xicheng Li c a Department of Engineering Mechanics, School of Civil Engineering, Shandong University, Jinan 250061, PR China b Geotechnical and Structural Engineering Research Center, Shandong University, Jinan, 250061, PR China c Department of Engineering Mechanics, Hohai University, Nanjing 210098, PR China article info Article history: Received 11 February 2011 Received in revised form 17 May 2011 Available online 26 May 2011 Keywords: Radial diffusion Fractional derivative Hankel transform Laplace transform Moment abstract In this study, time fractional radial diffusion has been modeled in cylindrical coordinates in order to analyze the anomalous diffusion in an annulus. By using an integral transform technique, the analytical solution of the concentration distribution formula is obtained. The establishing of the concentration distribution is found to be controlled by the fractional derivative α, and the influences of α on the concentration field, the total amount diffused and the quantity of mass passing through the inner wall are presented graphically and studied in detail. Asymptotic expressions for the exact solutions are developed in order to explain the numerical results at small and large time, respectively, and the physical mechanism explanation for the paradoxical behavior shown in the numerical results is given. © 2011 Elsevier B.V. All rights reserved. 1. Introduction The diffusion equations of fractional order are used to describe phenomena of anomalous diffusion usually met in transport processes through complex and/or disorder systems including fractal media [1]. An anomalous subdiffusion equation, in which the first-order time derivative has been replaced by a generalized derivative of order 0 <α< 1, is a dispersion of particles whose mean square displacement is given by a power law ⟨r 2 (t )⟩∼ t α , and a number of authors have extensively studied the anomalous diffusion in many branches of physics [2–9]. The applications of this theory have grown rapidly in the past few decade, and have attracted a fairly broad research activity. A topic review given by Metzler and Klafter summarized the development of anomalous diffusion in various fields [10]. Tan et al. developed an anomalous subdiffusion model in order to explore Ca 2+ spark formation in cardiac myocytes [11]. Recently, many researchers have studied the time fractional radial diffusion in circular cylindrical coordinates [12–15]; the exact solutions have been obtained with the help of the Laplace transform. However, they did not explain why the effects of the fractional derivative parameter are different for the solutions at small and large time. In this paper, we consider the radial diffusion in an annulus. The purpose of this paper is threefold: (i) we consider the first kind of boundary condition and generalize the result to the model; (ii) we carry out numerical analysis of the results obtained and discuss how the fractional derivative affects the establishing of the concentration field; (iii) we give the physical explanation for the behavior shown in the numerical results. ∗ Corresponding author. E-mail address: shaoweiwang@126.com (S. Wang). 0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.05.022