Research Article Exact Solutions of the Space Time Fractional Symmetric Regularized Long Wave Equation Using Different Methods Özkan Güner 1 and Dursun Eser 2 1 Department of Management Information Systems, School of Applied Sciences, Dumlupınar University, 43100 Kutahya, Turkey 2 Department of Mathematics-Computer, Art-Science Faculty, Eskisehir Osmangazi University, 26480 Eskisehir, Turkey Correspondence should be addressed to Dursun Eser; deser@ogu.edu.tr Received 4 April 2014; Accepted 22 June 2014; Published 22 July 2014 Academic Editor: Hossein Jafari Copyright © 2014 ¨ O. G¨ uner and D. Eser. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We apply the functional variable method, exp-function method, and (  /)-expansion method to establish the exact solutions of the nonlinear fractional partial diferential equation (NLFPDE) in the sense of the modifed Riemann-Liouville derivative. As a result, some new exact solutions for them are obtained. Te results show that these methods are very efective and powerful mathematical tools for solving nonlinear fractional equations arising in mathematical physics. As a result, these methods can also be applied to other nonlinear fractional diferential equations. 1. Introduction Fractional calculus is a feld of mathematics that grows out of the traditional defnitions of calculus. Fractional calculus has gained importance during the last decades mainly due to its applications in various areas of physics, biology, mathematics, and engineering. Some of the current application felds of fractional calculus include fuid fow, dynamical process in self-similar and porous structures, electrical networks, prob- ability and statistics, control theory of dynamical systems, systems identifcation, acoustics, viscoelasticity, control the- ory, electrochemistry of corrosion, chemical physics, fnance, optics, and signal processing [1–3]. Tere are several defnitions of the fractional derivative which are generally not equivalent to each other. Some of these defnitions are Sun and Chen’s fractal derivative [4, 5], Cresson’s derivative [6, 7], Gr¨ unwald-Letnikov’s frac- tional derivative [8], Riemann-Liouville’s derivative [8], and Caputo’s fractional derivative [9]. But the Riemann-Liouville derivative and the Caputo derivative are the most used ones. Lately, both mathematicians and physicists have devoted considerable efort to the study of explicit solutions to nonlinear fractional diferential equations. Many power- ful methods have been presented. Among them are the fractional (  /)-expansion method [10–13], the fractional exp-function method [14–16], the fractional frst integral method [17, 18], the fractional subequation method [19–22], the fractional functional variable method [23], the fractional modifed trial equation method [24, 25],andthe fractional simplest equation method [26]. Te paper suggests the functional variable method, the exp-function method, the (  /)-expansion method, and fractional complex transform to fnd the exact solutions of nonlinear fractional partial diferential equation with the modifed Riemann-Liouville derivative. Tis paper is organized as follows. In Section 2, basic defnitions of Jumarie’s Riemann-Liouville derivative are given; in Section 3, description of the methods for FDEs is given. Ten, in Section 4, these methods have been applied to establish exact solutions for the space-time fractional sym- metric regularized long wave (SRLW) equation. Conclusion is given in Section 5. 2. Jumarie’s Modified Riemann-Liouville Derivative Recently, a new modifed Riemann-Liouville derivative is proposed by Jumarie [27, 28]. Tis new defnition of frac- tional derivative has two main advantages: frstly, comparing Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2014, Article ID 456804, 8 pages http://dx.doi.org/10.1155/2014/456804