%0 Journal Article
%T Solution of Nonlinear Space-Time Fractional Differential Equations Using the Fractional Riccati Expansion Method
%A Emad A.-B. Abdel-Salam
%A Eltayeb A. Yousif
%J Mathematical Problems in Engineering
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/846283
%X The fractional Riccati expansion method is proposed to solve fractional differential equations. To illustrate the effectiveness 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. These 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 differential equations. Among these solutions, some are found for the first time. The periodic and kink solutions are founded as special case. 1. Introduction During recent years, fractional differential equations (FDEs) have attracted much attention due to their numerous applications in areas of physics, biology, and engineering [1¨C3]. Many important phenomena in non-Brownian motion, signal processing, systems identification, control problem, viscoelastic materials, polymers, and other areas of science are well described by fractional differential equation [4¨C7]. The 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 differential equations, fractional ordinary differential systems, and fractional partial differential equations have been discussed. Many powerful methods have been established and developed to obtain numerical and analytical solutions of FDEs, such as finite difference method [13], finite element method [14], Adomian decomposition method [15, 16], differential transform method [17], variational iteration method [18¨C20], 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. Thanks to the efforts of many researchers, several FDEs have been investigated and solved, such as the impulsive fractional differential equations [25], space- and time-fractional advection-dispersion equation [26¨C28], fractional generalized BurgersĄŻ fluid [29], and fractional heat- and wave-like equations [30], and so forth. The
%U http://www.hindawi.com/journals/mpe/2013/846283/