Traveling wave solutions of conformable time-fractional Zakharov–Kuznetsov and Zoomeron equations

doi:10.1016/j.cjph.2019.11.003

Chinese Journal of Physics

Volume 64, April 2020, Pages 194-202

https://doi.org/10.1016/j.cjph.2019.11.003 Get rights and content

Highlights

•
Conformable time-fractional Zakharov–Kuznetsov and Zoomeron equations were studied.
•
Modified trial equation method was applied to obtain traveling wave solutions.
•
Solitary wave, hyperbolic, and periodic solutions have been derived.
•
All solutions have been checked to verify the validity.
•
Graphs of solutions have been given to illustrate the behavior of the waves.

Abstract

To model physical phenomena more accurately, fractional order differential equations have been widely used. Investigating exact solutions of the fractional differential equations have become more important because of the applications in applied mathematics, mathematical physics, and other areas. In this work, by means of the trial solution method and complete discrimination system, exact traveling wave solutions of the conformable time-fractional Zakharov–Kuznetsov equation and conformable time-fractional Zoomeron equation have been obtained and also solutions have been illustrated. Finding exact solutions of these equations that are encountered in plasma physics, nonlinear optics, fluid mechanics, and laser physics can help to understand nature of the complex phenomena.

Introduction

Any phenomenon that changes with respect to time can be modeled by partial differential equations. In real life, we see that the relation between variables are not always linear, so nonlinear models are used to describe the phenomena. Recently, to construct more accurate models, fractional order differential equations have become to use more often. In fact, fractional derivatives have appeared in the late 17th century [1]. Due to the applications in mathematical physics, chemistry, bioengineering, nonlinear optics, optical fibers, control theory, finance, solid mechanics, signal processing and so on, fractional differential equations have take more and more attention. Various derivative definitions from different point of view such as Caputo, Grunwald–Letnikov, Riemann–Liouville and Riesz fractional derivative have been introduced so far [2], [3], [4], [5]. It is known that there are some setbacks of one definition to another and all definitions of fractional derivatives do not satisfy the product, quotient or chain rules. Recently, Khalil et al. [6] introduced a novel definition for fractional derivative to overcome some deficiencies of existing definitions. The proposed definition is called conformable fractional derivative and satisfies product and quotient rules. Also, Abdeljawad [7] contributed this derivative by presenting left and right conformable fractional derivative, fractional integration by parts formulas and chain rule. Atangana et al. [8] gave some new properties of the conformable derivative.

To find analytical solutions of partial and fractional order differential equations numerous methods have been proposed and applied so far [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. Among them, the trial equation method has been proposed by Liu [21], [22], [23] to investigate the exact solutions of differential equations by reducing the considered equation to solvable differential equation. Some authors [24], [25], [26], [27] applied the trial equation and the modified trial equation methods to some partial differential and fractional differential equations.

In this work, traveling wave solutions of conformable time-fractional differential equations have been investigated. To this aim, we first consider the conformable time-fractional Zakharov–Kuznetsov equation which illustrates the ion-acoustic waves of the weakly nonlinear type in strongly magnetized lossless plasma [28], [29]. The equation was first introduced by Zakharov and Kuznetsov [28]. Till now, some authors [30], [31], [32] applied exp $(- ϕ (ε)$ -expansion method, the modified simple equation method, the first integral method, and the modified Kudryashov method to investigate exact solutions of the equation.

We then consider the conformable time-fractional Zoomeron equation which was introduced by Calogero and Degasperis [33]. In the literature, (G’/G)-expansion method, the extended tanh method, the exp-function method, the sub-equation method, the generalized Kudryashov method and exponential rational function technique have been applied to this equation by some authors [34], [35], [36], [37], [38], [39].

This paper implemented the modified trial equation method to the conformable time-fractional Zakharov–Kuznetsov equation and the conformable time-fractional Zoomeron equation for the first time. By means of the method and the complete discrimination system, exact traveling wave solutions including solitary wave solutions, hyperbolic function solutions, and periodic solutions of these equations were successfully obtained. These equations play an important role in mathematical physics, so finding comprehensive analytical methods to solve them will be useful to understand the physical behavior.

Section snippets

Conformable fractional derivative (CFD)

The CFD was defined by Khalil et al. [6] as follows:

Definiton: Suppose that f: [0, ∞) →  $R$ is a function. Then, the conformable time-fractional derivative of f of order α is

$T_{α} (f) (t) = \lim_{τ \to 0} \frac{f (t + τ t^{1 - α}) - f (t)}{τ},$ for all t > 0 and α  ∈  (0,1).

Some useful properties of the conformable fractional derivative were given with theorems [6], [7], [8].

The modified trial equation method

The method can be described as follows [23]:

Step1. Consider time fractional partial differential equation $P (u, D_{t}^{α} u, u_{x}, u_{x x}, u_{x x x}, \dots) = 0 .$ Introducing the wave transformation $u (x, t) = U (ξ), ξ = x - λ (t^{α} / α),$ reduces Eq. (1) to a nonlinear ordinary differential equation of integer order as follows: $N (U, U^{'}, U^{″}, \dots) = 0 .$

Step 2. The trial equation can be taken as $U^{'} = \frac{F (U)}{G (U)} = \frac{\sum_{i = 0}^{n} a_{i} U^{i}}{\sum_{j = 0}^{l} b_{j} U^{j}},$ $U^{″} = \frac{F (U) [F^{'} (U) G (U) - F (U) G^{'} (U)]}{G^{3} (U)},$ where F(U) and G(U) are polynomials. Putting the Eqs. (4) and (5) into Eq.(3) gives an equation

(3+1)-dimensional conformable time-fractional Zakharov–Kuznetsov (FZK) equation

Consider the FZK equation in the form [28], [29], [30], [31]: $\frac{\partial^{α} u}{\partial t^{α}} + β u \frac{\partial u}{\partial x} + \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} u}{\partial z^{2}} = 0 .$ The (3+1)-dimensional FZK equation plays a significant role in mathematical physics. The equation describes weakly nonlinear wave process in dispersive and isotropic media such as water waves in shear flows or waves in magnetized plasma [32].

Using the transformation $u (x, y, z, t) = U (ξ), ξ = x + y + z - \frac{υ}{α} t^{α},$ Eq. (9) change into the form of an ordinary differential equation (ODE): $U^{″} (ξ) - \frac{υ}{3} U^{'} (ξ) + \frac{β}{3} U (ξ) U^{'} (ξ) = 0 .$

Conclusion

The modified trial solution method is implemented for conformable time-fractional differential equations. By using the conformable fractional derivative and the wave transformation, conformable time-fractional partial differential equations have been reduced to the ordinary differential equations. By applying the method and the complete discrimination system, exact traveling wave solutions including periodic solutions, solitary wave solutions, and hyperbolic function solutions of the

ORCID

Meryem Odabasi https://orcid.org/0000-0002-3025-3063

Declaration of Competing Interest

None.

Acknowledgment

This research is supported by Ege University, Scientific Research Project(BAP), Project Number: 17-TKMYO-002.

The authors would like to thank to the editor and the referees for their valuable contributions and comments to improve the paper.

References (39)

R. Khalil et al.
A new definition of fractional derivative
J. Comput. Appl. Math.
(2014)
T. Abdeljawad
On conformable fractional calculus
J. Comput. Appl. Math.
(2015)
K. Hosseini et al.
New exact traveling wave solutions of the Tzitzéica type equations using a novel exponential rational function method
Optik
(2017)
K. Hosseini et al.
Resonant optical solitons with perturbation terms and fractional temporal evolution using improved tan (ϕ(η)/2)-expansion method and exp function approach
Optik
(2018)
W.X. Ma et al.
Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation
Int. J. Nonlinear Mech.
(1996)
M. Odabasi et al.
A note on the traveling wave solutions of some nonlinear evolution equations
Optik
(2017)
P. Veeresha et al.
Solution for fractional Zakharov Kuznetsov equations by using two reliable techniques
Chin. J. Phys.
(2019)
E. Aksoy et al.
Soliton solutions of (2+1)-dimensional time-fractional zoomeron equation
Optik (Stuttg)
(2016)
A. Akbulut et al.
Auxiliary equation method for time-fractional differential equations with conformable derivative
Comput Math Appl.
(2018)
D. Kumar et al.
New analytical solutions of (2 + 1)-dimensional conformable timefractional zoomeron equation via two distinct techniques
Chin. J. Phys.
(2018)

R. Bertram

The development of fractional calculus 1695–1900

Hist. Math.

(1977)

I. Podlubny

Fractional Differential Equations

(1999)

A.A. Kilbas et al.

Theory and Applications of Fractional Differential Equations, North-HollandMathematical Studies

(2006)

K. Mille et al.

An Introduction to the Fractional Calculus and Fractional Differential Equations

(1993)

R.L. Magin

Fractional Calculus in Bioengineering

(2006)

A. Atangana et al.

New properties of conformable derivative

Open Math.

(2015)

M.J. Ablowitz et al.

Solitons, Non-linear Evolution Equations and Inverse Scattering Transform

(1991)

K.U. Tariq et al.

Explicit, periodic and optical solutions to the space-time fractional nonlinear schrodinger equation with quadratic-cubic nonlinearity

Mod. Phys. Lett. B

(2018)

U. Ali et al.

Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation

Interpolat. Approx. Sci. Comput.

(2017)

Cited by (0)

View full text

Traveling wave solutions of conformable time-fractional Zakharov–Kuznetsov and Zoomeron equations

Highlights

Abstract

Introduction

Section snippets

Conformable fractional derivative (CFD)

The modified trial equation method

(3+1)-dimensional conformable time-fractional Zakharov–Kuznetsov (FZK) equation

Conclusion

ORCID

Declaration of Competing Interest

Acknowledgment

J. Comput. Appl. Math.

J. Comput. Appl. Math.

Optik

Optik

Int. J. Nonlinear Mech.

Optik

Chin. J. Phys.

Optik (Stuttg)

Comput Math Appl.

Chin. J. Phys.

The development of fractional calculus 1695–1900

Hist. Math.

Fractional Differential Equations

Theory and Applications of Fractional Differential Equations, North-HollandMathematical Studies

An Introduction to the Fractional Calculus and Fractional Differential Equations

Fractional Calculus in Bioengineering

New properties of conformable derivative

Open Math.

Solitons, Non-linear Evolution Equations and Inverse Scattering Transform

Explicit, periodic and optical solutions to the space-time fractional nonlinear schrodinger equation with quadratic-cubic nonlinearity

Mod. Phys. Lett. B

Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation

Interpolat. Approx. Sci. Comput.