Physics Letters A 372 (2008) 5968–5972 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Fractional diffusion-wave problem in cylindrical coordinates Necati Özdemir ∗ , Derya Karadeniz Department of Mathematics, Faculty of Science and Arts, Balıkesir University, Cagis Campus, 10145 Balıkesir, Turkey article info abstract Article history: Received 12 May 2008 Received in revised form 22 July 2008 Accepted 23 July 2008 Available online 31 July 2008 Communicated by R. Wu Keywords: Fractional derivative Fractional diffusion-wave system Axial symmetry Grünwald–Letnikov approach Cylindrical coordinates In this Letter, we present analytical and numerical solutions for an axis-symmetric diffusion-wave equation. For problem formulation, the fractional time derivative is described in the sense of Riemann– Liouville. The analytical solution of the problem is determined by using the method of separation of variables. Eigenfunctions whose linear combination constitute the closed form of the solution are obtained. For numerical computation, the fractional derivative is approximated using the Grünwald– Letnikov scheme. Simulation results are given for different values of order of fractional derivative. We indicate the effectiveness of numerical scheme by comparing the numerical and the analytical results for α = 1 which represents the order of derivative.  2008 Published by Elsevier B.V. 1. Introduction The awareness of the Fractional Diffusion-Wave Equation (FDWE) has grown during the last decades. These equations pro- vide more accurate models of systems and processes under con- sideration. For this reason, there has been an increasing interest to investigate, in general, the response of the systems, and in partic- ular, the analytical and numerical solutions of FDWE. A FDWE is a linear partial integro-differential equation obtained from the classical diffusion or wave equation by replacing the first or second-order time derivative by a fractional derivative of order α > 0, see Mainardi [1]. Mainardi [2] presented the fundamental solutions of the ba- sic Cauchy and Signalling problems for the evolution of FDWE. The solutions of central-symmetric signalling, source and Cauchy problems for fractional diffusion equation in a spatially three- dimensional sphere were studied by Povstenko [3]. Wyss [4] de- rived the solution of the Cauchy and Signalling problems in terms of H -functions using the Mellin transform. Agrawal [5,6] obtained the fundamental solutions of a FDWE which contains a fourth order space derivative and a fractional or- der time derivative. The solution of a FDWE defined in a bounded space domain was also considered by Agrawal [7]. Mainardi [8] obtained fundamental solutions for a FDWE and the solutions for fractional relaxation oscillations by using the Laplace transform method. The Green’s function and propagator functions in multi- * Corresponding author. Tel.: +90 266 6121000, ext: 215; fax: +90 266 6121215. E-mail addresses: nozdemir@balikesir.edu.tr (N. Özdemir), fractional_life@hotmail.com (D. Karadeniz). dimensions which are obtained for the solution of a general initial value problem for the time-fractional diffusion-wave equation with source term and for the anisotropic space–time fractional diffusion equation were researched by Hanyga [9,10]. Mainardi, Luchko and Pagnini [11] dealt with the fundamental solution of the space–time fractional diffusion equation. Agrawal [12] presented stochastic analysis of FDWEs defined in one dimension whereas very little work has been done in the area of stochastic analysis of fractional order engineering systems. In this Letter, the analytical and numerical solutions of an axis-symmetric FDWE in cylindrical coordinates are studied. More recently, the solution of an axis-symmetric fractional diffusion- wave equation in polar coordinates has been presented in [13]. El-Shahed [14] considered the motion of an electrically conducting, incompressible and non-Newtonian fluid in the presence of a mag- netic field acting along the radius of a circular pipe. Furthermore, El-Shahed selected a cylindrical polar coordinate system with z- axis in the direction of motion and considered the flow as axially symmetric. Several axial-symmetric problems for a plane in cylin- drical coordinates and central-symmetric problems for an infinite space in spherical coordinates were presented in [15–17]. Radial diffusion in a cylinder of radius R was considered by Narahari Achar and Hanneken [18]. Povstenko [19] developed the results of Narahari Achar and Hanneken. The main problem considered in [19] is similar to our work. However, the formulation of problem here differs with [19] in some respects. Firstly, Povstenko [19] for- mulates the problem by using polar coordinates in terms of Caputo fractional derivative and finds only the closed form analytic so- lution, whereas this Letter considers the problem with cylindrical coordinates in Riemann–Liouville (RL) sense, and also presents nu- 0375-9601/$ – see front matter  2008 Published by Elsevier B.V. doi:10.1016/j.physleta.2008.07.054