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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 [], fractional generalized Burgers' fluid [29], and fractional heat- and wave-like equations [30], and so forth.

The finding of a new mathematical algorithm to construct exact solutions of nonlinear FDEs is important and might have significant impact on future research. In this research paper, 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. 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.

The structure of this paper is as follows: in Section 2, we introduce some basic definitions and mathematical preliminaries of the fractional calculus theory. Section 3 describes the fractional Riccati expansion method for solving nonlinear FDEs. Finally, we give some conclusions and discussions. Fractional calculus is one of the generalizations of ordinary calculus. Generally speaking, there are two kinds of fractional derivatives. One of them is nonlocal fractional derivative, that is, Caputo derivative and Riemann-Liouville derivative which have been used successfully in various fields of science and engineering.

However, the Caputo derivative requires the function to be smooth and differentiable. Obviously, the nonlocal derivatives are not suitable for the investigation of the local behavior of fractional differentiable equations. The other one is the local fractional derivative, that is, Kolwankar-Gangal K-G derivative [31], Chen's fractal derivative [32] and Cresson's derivative [33].

One of the famous examples is the devi-stair curve, which can be described by a continuous but nowhere differentiable function. Recently, there is new development of continuous but nowhere differentiable functions [34]. This study is motivated by the need to propose a fractional Riccati expansion method to construct exact analytical solutions of nonlinear FDEs with the following modified Riemann-Liouville derivative defined by Jumarie [35]:. Owing to these merits, Jumarie's modified Riemann-Liouville derivative was successfully applied to the probability calculus [36], fractional Laplace problems [37], and fractional variational calculus [38].

Some useful formulas and results of Jumarie's modified Riemann-Liouville derivative were summarized in [35], three of them are. In this section, we outline the main steps of the fractional Riccati expansion method for solving nonlinear FDEs. For a given nonlinear FDE, say, in two variables x and t. Case 9. Step 3. Setting each coefficient of the polynomial to zero yields system of algebraic equations for fl0, flj, Step 4. Remark 1.

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Zhang and H. Zhang [23]. Remark 2. In [39], Professor He introduced the simple fractional complex transform to convert nonlinear FDEs into its nonlinear differential equations. The KdV equation is the earliest soliton equation that was firstly derived by Kor-teweg and de Vries to model the evolution of shallow water wave in The KdV-type equations have applications in shallow-water waves [40], optical solitons in the two cycle regime [41], density waves in traffic flow of two kinds of vehicles [42], short waves in nonlinear dispersive models [43], surface acoustic soliton in a system supporting long waves [44], quantum field theory, plasma physics, and solid-state physics.

The space-time fractional KdV equation is. Then, 13 is reduced to the following nonlinear FODE:. The general formula for the traveling wave solution of the space-time fractional KdV equation The remaining solutions can be obtained in a similar manner. Solutions given in 18 reduced to the well-known periodic and kink solutions of the KdV equation.

The RLW equation that describes approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems has been proposed by Benjamin et al. The RLW equation is considered as an alternative to the KdV equation, which is modeled to govern a large number of physical phenomena such as shallow waters and plasma waves [45]. The space-time fractional RLW equation has the form.

Space-Time Fractional Boussinesq Equation. The Boussinesq equation was first derived to describe the propagation of long waves in shallow water [46]. It also arises in many other applications of physical interest including one-dimensional nonlinear lattice waves, vibrations in a nonlinear string, and ion sound waves in a plasma [47, 48]. Space-time fractional Boussinesq equation has the form. The general formula of the travelling wave solution of the space-time fractional RLW equation 21 is. The general formula of the travelling wave solution of the fractional space-time Boussinesq equation 29 is.

The solutions 26 take the form of the well-known periodic and kink solutions of the RLW equation. The nonlinear Klein-Gordon equation is a good physical equation in the sense that it appears in many fields of applications [49]. For example, in relativistic quantum mechanics, it describes the processes involving particles of spin zero.

The nonlinear space-time fractional Klein-Gordon equation is. The general formula of the travelling wave solution of the fractional space-time Klein-Gordon equation 37 is. As a result, some exact analytical solutions are obtained including the generalized hyperbolic function and generalized trigonometric function solutions. To the best of our knowledge, some of the solutions obtained in this research paper have not been reported in the literature.

The fractional Riccati expansion method is more effective and simpler than other methods, and a number of solutions can be obtained simultaneously. Mathematical packages can be used to perform more complicated and tedious algebraic calculations. The fractional Riccati expansion method can be applied to other nonlinear FDEs. How to extend other methods used for solving differential equations, such as F-expansion method, Fan sub-equation method, auxiliary sub-equation method, and the projective Riccati equation method, to handle FDEs is worthy of study.

This is our task in the future. Binkowski, L Caron. Bricout, S. Caron, H. Derrien, M. Batella, A. Debailleul, F. Abbate, M. Caron, D. Da Costa, E. Boyaval, C. Li, F. Hapiot, M. Warenghem, N. Isaert, Y. Guyot, G. Boulon, P. Jalowiecki-Duhamel, A. Ponchel, C. Lamonier, A. Hamza Reguig, A. Pages, M. Ajjoun, J. Tournier, J. Desfeux, S. Bailleul, A. Prellier, A.

Surpateanou, Surface Science , , Caron, E. Monflier, Catalysis Today , 66, Mathivet, C. Meliet, Y. Mortreux, L. Monflier, Tetrahedron Lett. Boyaval, Magn. Boulon and P. Carette, Molecular Crystals and Liquids Crystals, , , Tilloy, J. Cabou, H. Gratchev, E.

Monflier, E. Nifantiev, XVth Int. Blach, S.

Saitzek, R. Da Costa, M. Warenghem, W. Prellier, B. J-F Blach, D. Bormann, R. Desfeux, D. Tondelier, B. Abramson, S. Tilloy, Y. Cabou, L. Tokarski, E. Mouton, D. Ruffin, B. Martel, E. Monflier, G. Ricart, C. Binkowski, H. J-F Henninot, M. Derrien, J. Warenghem and G. Abbate J.

Chemical Physics Letters

A : Pure Appl. Physical characterizations and determination of the chronium active species " P. Moriceau, B. Grzybowska, L. Barbaux Applied Catalysis A : General , Andrienko, F. Barbet, D. Kurioz, S. Kwon, Y. Reznikov and M. Cholesteryl Tetradecanoate or Nonanoate complexes ".

Old publications and proceedings

Cryst Li, J. Boyaval, M. Warenghem and P. Carette European Physical Journal D. Duhem, J. Douay Optics Comm.


Pages, A. Lazreg, A. Zaoui, M. Certier, C. Thiandoume, A. Bouanami, L. Svob, O. Gorochov et D. Bormann Thin solid films, , , Tournier et J. Faurie Appl. Razafimahatratra, M. Benatsou, M. Bouazaoui, W. Xie, A. Da Costa and M. Haghiri-Gosnet, J. Wolfman, B. Mercey, Ch. Simon, P.

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