Physica A 388 (2009) 806–810 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Some results for a fractional diffusion equation with radial symmetry in a confined region E.K. Lenzi a,∗ , L.R. da Silva b , A.T. Silva a , L.R. Evangelista a , M.K. Lenzi c a Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá, Avenida Colombo, 5790 - 87020-900 Maringá, Paraná, Brazil b Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN, Brazil c Departamento de Engenharia Química, Universidade Federal do Paraná, Setor de Tecnologia - Jardim das Américas, Caixa Postal 19011, 81531-990, Curitiba - Paraná, Brazil article info Article history: Received 7 August 2008 Available online 30 November 2008 PACS: 02.50.-r 05.40.-a 05.90.+m Keywords: Fractional diffusion equation Anomalous diffusion Green function abstract We investigate an N -dimensional fractional diffusion equation with radial symmetry by taking a spatial and time dependent diffusion coefficient into account, i.e.,  D(r , t ) = D(t )r −η with D(t ) = Dδ(t ) + D(t ). The equation is considered in a confined region and subjected to time dependent boundary conditions which may be related to inhomogeneous characteristics of the surfaces confining the system. The results show an anomalous spreading of the solutions and an unusual behavior of the survival probability. © 2008 Elsevier B.V. All rights reserved. 1. Introduction The applications of the fractional diffusion equations [1–6] to anomalous diffusion and the connection to the other formalisms such as continuous random walk formalism [7], Langevin approach [8] and master equation [9] have motivated the study of these equations in order to comprehend their applications to physical contexts where non-conventional dynamical behavior can be found. In this direction, we consider the following fractional diffusion equation ∂ γ ∂ t γ ρ( r , t ) =  t 0 d t ∇· (  D(r , t − t )∇ρ(r , t ) ) (1) by taking the N -dimensional case with radial symmetry into account (∇· (  D(r , t )∇···) ≡ r 1−N ∂ r (r N −1  D(r , t )∂ r ···)), 0 <γ ≤ 1 (subdiffusive case), the diffusion coefficient given by  D( r , t ) = D(t )r −η where D(t ) is an arbitrary time dependent function, and the fractional time derivative considered here is the Caputo derivative [10]. Eq. (1) has as particular cases several situations such as the ones worked out in [11,12] for the two- and three-dimensional cases in a confined region, and the results presented in [13] for cylindrical symmetry. In this manner, many cases presented in the literature are extended to a broad context which may present, for example, different regimes of spreading of the solution or connection to the fractional equation of distributed order for a suitable choice of D(t ). ∗ Corresponding author. Tel.: +55 04432614330; fax: +55 04432634623. E-mail address: eklenzi@dfi.uem.br (E.K. Lenzi). 0378-4371/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2008.11.030