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Korkmaz Alper. Exact Solutions to (3+1) Conformable Time Fractional Jimbo–Miwa, Zakharov–Kuznetsov and Modified Zakharov–Kuznetsov Equations. Communications in Theoretical Physics, 2017, 67(5): 479  
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Exact Solutions to (3+1) Conformable Time Fractional Jimbo–Miwa, Zakharov–Kuznetsov and Modified Zakharov–Kuznetsov Equations
Korkmaz Alper *
Department of Mathematics, Çankırı Karatekin University, Çankırı, Turkey

 

† Corresponding author. E-mail: korkmazalper@yandex.com.tr

Abstract
Abstract

Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE. The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo–Miwa, the Zakharov–Kuznetsov, and the modified Zakharov–Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.

PACS: 02.30.Jr;02.70.Wz;47.35.Fg
Keyword:fractional (3+1)-dimensional Jimbo-Miwa equation;fractional modified Zakharov-Kuznetsov equation;modified Kudryashov method;fractional Zakharov-Kuznetsov equation;exact solutions
1 Introduction

The particular integer ordered form of the time fractional Jimbo–Miwa (JM) equation of the type

where is the α-th order fractional derivative operator with respect to τ is proposed in Ref. [1] as the second member of an integrable systems of the KP-hierarchy even though it is not successful to pass the integrability tests.[2–3] Some multi-soliton solutions are determined by bilinear form with an homoclinic test method.[3] These multi-soliton solutions are different from the multi-solitons described by using homogeneous balance and a Bäcklund transformation.[4] Some exact solutions in the forms of solitary waves, periodic waves and variable separation solution to the JM equation are derived by the aid of improved mapping approach.[5] The generalized method can be used to generate non traveling-solitonic and moving solitary wave type exact solutions.[6] Öziş and Aslan also determine some exact solutions to the JM equation by using the method of exp-function.[7] The exact solutions to space-time fractional version of the JM equation are proposed in Ref. [8] by using generalized Bernoulli equation method.

The time fractional Zakharov–Kuznetsov (KZ) equation is of the form

where p, q, r, and s are real parameters. The original form of the equation has been proposed to define three dimensional ion-sound solitons in a magnetized plasma with a low pressure.[9] The three conservation laws describing mass, momentum and center mass are also formulated in the same study. The ZK equation also admits the ellipsoidal and plane type solitons.[10] Some solutions of the cnoidal, periodic, singular periodic, solitary wave and non topological soliton forms are constructed by using the extended hyperbolic tangent, and ansatz methods.[11] The traveling wave solitons in various forms are derived in Ebadi et al.ʼs study by implementing the exp-function, modified F-expansion and methods.[12] Zhang and Zhou[13] obtains kink, antikink, solitary wave and periodic solutions to the ZK equation in general form by employing the bifurcation theory.

When the nonlinearity of the ZK equation is increased to three, the equation is named as the modified ZK (mZK) equation and the conformable time fractional form of the equation becomes

Liang[14] derived some exact solutions in forms of some trigonometric and hyperbolic functions to the mZK equation by using modified simple equation method. More than twenty exact solutions of the mZK equation are derived by using enhanced method in Ref. [15]. The fractional forms of both equations are solved exactly by using improved sub equation adapted for fractional cases.[16]

The related literature contains various methods from Lie symmetries to expansion, ansatz methods or first integral methods for the exact solutions of the nonlinear PDEs.[17–20] The PDE systems can also be solved exactly by using smart techniques like Riemann–Hilbert or Bäcklund transformation methods. These solutions can be in multi soliton solution form.[21–23] Ansatz methods are another technique to construct exact solutions to PDEs even fractional one.[24–26]

In the present study, finite series of a rational exponential function types solutions are derived for the three dimensional fractional PDEs in conformable sense listed above. All the solutions are expressed explicitly. Before explaining the used procedure, some significant descriptions and calculus properties of the conformable derivative are summarized below.

2 Preliminaries

Consider a function defined in the positive half space and α be number . Then, the conformable derivative of ζ for is defined as

for .[27] Even though this definition of the fractional derivative is pretty new, various important properties such as derivative of multiplication and division are defined clearly. The fundamental properties of the conformable derivative required to solve fractional PDEs are summarized below.

3 Modified Kudryashov Method

Let P be

where and be the fractional derivative order. The transformation

converts (6) to an ODE for new variable ξ

where the prime () indicates the derivative operator of η with respect to ξ.[31]

Assume that Eq. (8) has a solution of the form

for a finite n with all and . The procedure start by determining the degree of the polynomial type series n by balancing the non linear term and the highest order derivative term. The function H is required to satisfy the first-order ODE

Thus, is determined as , where d and A are non-zero constants with the conditions and .

Substituting the predicted solution (9) and its derivatives into Eq. (8) give a polynomial of . All the coefficients of the powers of and the constant term are equated to zero. The resultant algebraic equation system is solved for a0, a1, a2, …, an and the other constants used in the wave transformation (7). This method is explained in details in Ref. [32].

4 Solutions to (3+1)-Dimensional JM Equation

The JM equation given in Eq. (1) is reduced to

by using the compatible form of the transformation (7). Rearrangement of the last equation by integrating once gives

where K stands for the constant of integration. The balance between the non linear and the highest ordered terms gives . Accordingly, the predicted solution (9) should be

where a0 and a1 are the constants to be determined. Substituting this solution and its derivatives into Eq. (1) leads

It is clear that an should be nonzero. Hence, the coefficients of all powers of and K should be zero. Thus, the algebraic system of equations

Solving Eq. (15) for a0, a1, a, b, c, ν gives

for arbitrary choices of and . The formed solution

gives

for arbitrarily chosen a0, a, b, ν. It must be noted that the system (15) has three more solutions for but at least one of a, b, c, ν are zero in those solution sets. That is why these solutions are not reported here.

5 Solutions to (3+1)-Dimensional ZK Equation

The wave transformation (7) reduces the fractional ZK equation (2) to

where denotes . Integrating both sides of this equation converts it to

with integration constant K. The balance of η 2 and gives the compatible n as 2. Hence, the predicted solution must be in the form . Substituting this solution into Eq. (20) yields

The solution of this system for gives two different solutions as

for arbitrary constants a0, a, c, and K. Thus, the solution of Eq. (20) is determined as

arbitrary a0 and K. The solution of the conformable time fractional ZK equation (2) is expressed as

where a0, a, c arbitrary are as given in Eq. (22).

6 Solutions to (3+1)-Dimensional mZK Equation

The modified form of the ZK equation in the time fractional form (3) is reduced to

where denotes . Integrating this equation once converts it to

with the integration constant K. The balance between η 3 and gives . Thus, the solution is formed as

for a nonzero a1. Substituting this predicted solution and its derivative into Eq. (26), a polynomial equation of of the form

is obtained. This algebraic system has two different solutions satisfying the condition and can be written in the form

for arbitrarily chosen a, b, c, and . Thus, the solution to Eq. (26) is constructed as

with the condition , . Hence, the solutions of Eq. (3) are of the form

7 Conclusion

The method of Kudryashov in modified form is implemented to derive the exact solutions to (3+1)-dimensional conformable time fractional JM, ZK, and mZK equations. The valid and compatible traveling wave transformation reduces these equations to integer ordered ODEs. The predicted solution of the finite series form of a rational exponential function is substituted into the resultant ODEs. The algebraic operations are used to determine the relations between the coefficients originated from both the equations and the transformation. Once these relations are determined, the traveling wave type solutions in three dimensions are developed explicitly.

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