Exp-function method for the conformable space-time fractional STO, ZKBBM and coupled Boussinesq equations

Abstract In the present paper, new analytical solutions for the conformable space-time fractional Sharma-Tasso-Olever (STO), Zakharov Kuznetsov Benjamin Bona Mahony (ZKBBM) and coupled Boussinesq equations are obtained by using the Exp-function method. The obtained traveling wave solutions are presented by exponential functions. Simulations of the obtained solutions are given at the end of the paper.

In the solitary wave theory, traveling waves are particularly interesting. They appear in many areas such as elastic media, plasmas, solid state physics, condensed matter physics, electrical circuits, optical fibers, chemical kinematics, fluids, bio-genetics, etc. Three types of traveling waves are given as (Kichenassamy & Olver, 1992;Whitham, 1999): the solitary waves, which are localized traveling waves, asymptotically zero at large distances, the periodic solutions, the kink waves which rise or descend from one asymptotic state to another. Solitonic solutions of nonlinear partial differential equations have been investigated in (Dai & Wang, 2008;Dai & Xu, 2015;Ding et al., 2017;Wang, Zhang, & Dai, 2016).
In general, STO, ZKBBM and coupled Boussinesq equations have been studied for the case of time fractional in the literature. For the space-time fractional STO, ZKBBM and coupled Boussinesq equations, Jumarie's modified Riemann-Liouville derivatives have been used. In this paper, we consider space-time fractional STO, ZKBBM and coupled Boussinesq equations. Here, fractional derivatives are defined in conformable sense. Applying Exp-function method we have obtain analytic solutions including exponential functions for conformable space-time fractional STO, ZKBBM and coupled Boussinesq equations.

Description of conformable fractional derivative and its properties
For a function f : ð0; 1Þ ! R; the conformable fractional derivative of f of order 0 < a < 1 is defined as (see, for example (Khalil et al., 2014)) Some important properties of the the conformal fractional derivative are as follows: 3. Analytic solutions to the conformable space-time fractional STO equation Conformable space-time fractional STO equation is denoted by (Sontakke & Shaikh, 2016;Taghizadeh et al., 2013) x is the spatial coordinate in the propagation direction and t is the temporal coordinates, which occur in different contexts in mathematical physics. The dissipative u xxx term provides damping at small scales, and the nonlinear term u 2 u x stabilizes by transferring energy between large and small scales.
Using the following transformation for Eq. (3) where k and m are non zero arbitrary constants, and integrating resulting equation with zero constant we have kU þ 3cm 2 UU 0 þ cmU 3 þ cm 3 U 00 ¼ 0: According to Exp-function method, the solution of Eq. (5) can be expressed in the following form where t, s, h and l are positive integers which are known to be further determined, a i and b j are unknown constants. Substituting Eq. (6) into Eq. (5) and balancing in the obtained equation, we get r ¼ s ¼ By substituting Eq. (7) into Eq. (5), and collecting all the terms with the same power of e s ðs ¼ 3; 2; 1; 0; À1; À2; À3Þ; we can obtain a set of algebraic equations for the unknowns a 0 , a 1 , a À1 ; b 0 , b 1 , b À1 ; k, m: Solving the algebraic equations in the Mathematica, we obtain the following set of solutions: Case 1: where a 1 and a 0 are free parameters.
Case 2: where a 1 and a 0 is free parameter.
where a, b are real-valued constants. It is well known that ZK (Zakharov Kuznetsov) equation models are weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasmas. ZK-BBM equation is the conjunction of ZK equation and BBM (Benjamin-Bona-Mahony) equation that models shallow water waves. Substituting Eq. (4) into Eq. (10) and integrating resulting equation with zero constant we have k þ m ð ÞUÀamU 2 Àbm 2 kU 00 ¼ 0: Substituting Eq. (6) into Eq. (11) and balancing in the obtained equation, we get r ¼ s ¼ h ¼ l ¼ 1; so Eq. (6) reduces to Eq. (7). By substituting Eq. (7) into Eq. (11), and collecting all the terms with the same power of e s ðs ¼ 3; 2; 1; 0; À1; À2; À3Þ; we can obtain a set of algebraic equations for the unknowns a 0 , a 1 , a À1 ; b 0 , b 1 , b À1 ; k, m: Solving the algebraic equations in the Mathematica, we obtain the following set of solution: where b 1 and b 0 are free parameters.

Analytic solutions to the conformable space-time fractional coupled Boussinesq equations
Finally, we consider the conformable space-time fractional coupled Boussinesq equations ) Boussinesq type equations can be considered as the first model for nonlinear, dispersive wave propagation and describe the surface water waves whose horizontal scale is much larger than the depth of the water (Madsen, Murray, & Sorensen, 1991). Substituting Eq. (4) into Eqs. (13)-(14) we obtain the following differential equations  kV 0 þ km U 2 ð Þ 0 Àlm 3 U 000 ¼ 0: Integrating Eqs. (15)-(16) and using V ¼ À k m U we have À k 2 m U þ kmU 2 Àlm 3 U 00 ¼ 0: By balancing in Eq. (17), we set r ¼ s ¼ h ¼ l ¼ 1; so Eq. (6) reduces to form of the Eq. (7). By substituting Eq. (7) into Eq. (17) and collecting all the terms with the same power of e s ðs ¼ 3; 2; 1; 0; À1; À2; À3Þ; we can obtain a set of algebraic equations for the unknowns a 0 , a 1 , a À1 ; b 0 , b 1 , b À1 ; k, m: Solving the algebraic equations in the Mathematica, we obtain the following set of solution: where b 1 and b 0 are free parameters.
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Conclusion
In this paper, Exp-function method has been applied to the conformable space-time fractional STO, ZKBBM and coupled Boussinesq equations. The method can be used directly without requiring linearization, discretization or perturbation. New solitary wave solutions for conformable space-time fractional STO, ZKBBM and coupled Boussinesq equations have been obtained. It has been checked that all of the obtained solutions satisfy the corresponding equations.