Nonlinear Analysis: Real World Applications ( ) – Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa On electromagnetic wave propagation in fractional space M. Zubair a,∗ , M.J. Mughal a , Q.A. Naqvi b a Faculty of Electronic Engineering, GIK Institute of Engineering Sciences and Technology, Topi, Swabi, 23640, Khyber Pakhtunkhwa, Pakistan b Electronics Department, Quaid-i-Azam University, Islamabad 45320, Pakistan article info Article history: Received 20 January 2011 Accepted 17 April 2011 Keywords: Fractional space Helmholtz equation Wave propagation Fractal media abstract The vector Helmholtz equation in fractional space can describe the complex phenomenon of wave propagation in fractal media. With this view, a generalized Helmholtz equation for wave propagation in fractional space is established and its analytical solution is obtained. The special case for integer space is recovered and the results are in exact agreement with those obtained in the literature. © 2011 Elsevier Ltd. All rights reserved. 1. Introduction The concept of fractional dimensional space is effectively used in many areas of physics to describe the effective parameters of physical systems [1–12]. This approach can be used to replace the real confining structure with an effective space, where the measurement of its confinement is given by the fractional dimension [6,7]. Such confinement can be described in low dimensional systems which can have different degrees of confinement in different orthogonal directions [8]. Fractional calculus is used by different authors to describe fractional solutions to many electromagnetic problems as well as fractional dimensional space [13–26]. Stillinger [8] has developed a formalism for constructing a generalization of an integer dimensional Laplacian operator into a non-integer dimensional space. Palmer and Stavrinou [12] generalized the results of Stillinger to n orthogonal coordinates. Also it is known that the experimental measurement of the dimension D of our real world is given by D = 3 ± 10 −6 , not exactly 3 [8,27]. The formulation of Schrödinger wave mechanics in D-dimensional space is also provided in [8,12]. Some applications of the concept of fractional space in electromagnetic research include the description of fractional multipoles in fractional space [7] and the study of electromagnetic fields in fractional space by solving Poisson’s equation in D-dimensional space with 2 < D ≤ 3[9]. Also the scattering phenomenon in fractal media is discussed in [28]. The Helmholtz or wave equation has a very important role in many areas of physics. It has a fundamental meaning in classical as well as quantum field theory. With this view, one is strongly motivated to discuss solutions of the wave equation in all possible situations. The vector Helmholtz equation in fractional space can describe complex phenomena of wave propagation in fractal media. In the fractal media, some of the regions or domains are not filled with medium particles and these unfilled domains are called porous domains. Fractal models of media are becoming popular due to the fact that we require a small number of parameters that define a medium of great complexity and rich structure [11]. The fractal media in a Euclidean space can be considered as continuous media in a fractional space, so the phenomenon of wave propagation in a fractal medium can be studied by considering the fractal medium as an ordinary continuous medium in a D-dimensional fractional space, where D is the fractional dimension of that fractal medium, best calculated by the box counting method [13]. ∗ Corresponding author. Tel.: +92 300 8503480. E-mail address: zubair_wah@hotmail.com (M. Zubair). 1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2011.04.010