Abstract
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.
Highlights
During recent years, fractional differential equations (FDEs) have attracted much attention due to their numerous applications in areas of physics, biology, and engineering [1,2,3]
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 [4,5,6,7]
The most important advantage of using FDEs is their nonlocal property, which means that the state of a system depends upon its current state and upon all of its historical states [8, 9]
Summary
Fractional differential equations (FDEs) have attracted much attention due to their numerous applications in areas of physics, biology, and engineering [1,2,3]. 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 [26,27,28], fractional generalized Burgers’ fluid [29], and fractional heat- and wave-like equations [30], and so forth. We introduce the fractional Riccati expansion method to construct many exact traveling wave solutions of nonlinear FDEs with the modified Riemann-Liouville derivative defined by Jumarie.
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