Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 846283, 6 pages http://dx.doi.org/10.1155/2013/846283 Research Article Solution of Nonlinear Space-Time Fractional Differential Equations Using the Fractional Riccati Expansion Method Emad A.-B. Abdel-Salam 1,2 and Eltayeb A. Yousif 2,3 1 Department of Mathematics, Faculty of Science, Assiut University, New Valley Branch, El-Kharja 72511, Egypt 2 Department of Mathematics, Faculty of Science, Northern Border University, Arar 91431, Saudi Arabia 3 Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Khartoum, 11111 Khartoum, Sudan Correspondence should be addressed to Emad A.-B. Abdel-Salam; emad abdelsalam@yahoo.com Received 19 June 2013; Accepted 2 December 2013 Academic Editor: Andrzej Swierniak Copyright © 2013 E.-B. Abdel-Salam and E. A. Yousif. 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. Te fractional Riccati expansion method is proposed to solve fractional diferential equations. To illustrate the efectiveness of the method, space-time fractional Korteweg-de Vries equation, regularized long-wave equation, Boussinesq equation, and Klein- Gordon equation are considered. As a result, abundant types of exact analytical solutions are obtained. Tese solutions include generalized trigonometric and hyperbolic functions solutions which may be useful for further understanding of the mechanisms of the complicated nonlinear physical phenomena and fractional diferential equations. Among these solutions, some are found for the frst time. Te periodic and kink solutions are founded as special case. 1. Introduction During recent years, fractional diferential equations (FDEs) have attracted much attention due to their numerous applica- tions in areas of physics, biology, and engineering [1–3]. Many important phenomena in non-Brownian motion, signal pro- cessing, systems identifcation, control problem, viscoelastic materials, polymers, and other areas of science are well described by fractional diferential equation [4–7]. Te most important advantage of using FDEs is their nonlocal property, which means that the next state of a system depends not only upon its current state but also upon all of its historical states [8, 9]. Recently, the fractional functional analysis has been investigated by many researchers [10, 11]. For example, the properties and theorems of Yang-Laplace transforms and Yang-Fourier transforms [12] and their applications to the fractional ordinary diferential equations, fractional ordinary diferential systems, and fractional partial diferential equa- tions have been discussed. Many powerful methods have been established and developed to obtain numerical and ana- lytical solutions of FDEs, such as fnite diference method [13], fnite element method [14], Adomian decomposition method [15, 16], diferential transform method [17], variational iter- ation method [18–20], homotopy perturbation method [21, 22], the fractional sub-equation method [23], and generalized fractional subequation method [24]. How to extend the existing methods to solve other FDEs is still an interesting and important research problem. Tanks to the eforts of many researchers, several FDEs have been investigated and solved, such as the impulsive fractional diferential equations [25], space- and time-fractional advection-dispersion equation [26–28], fractional generalized Burgers’ fuid [29], and frac- tional heat- and wave-like equations [30], and so forth. Te fnding of a new mathematical algorithm to construct exact solutions of nonlinear FDEs is important and might have signifcant impact on future research. In this research paper, we introduce the fractional Riccati expansion method to con- struct many exact traveling wave solutions of nonlinear FDEs with the modifed Riemann-Liouville derivative defned by Jumarie. We use the fractional Riccati expansion method for solving the space-time fractional Korteweg-de Vries (KdV) equation, space-time fractional regularized long-wave (RLW) equation, space-time fractional Boussinesq equation, and space-time fractional Klein-Gordon equation.