Abstract
In this work, we consider the (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity. Solitary wave solutions, soliton wave solutions, elliptic wave solutions, and periodic (hyperbolic) wave rational solutions are obtained by means of the unified method. The solutions showed that this method provides us with a powerful mathematical tool for solving nonlinear conformable fractional evolution equations in various fields of applied sciences.
1 Introduction
Nonlinear fractional partial differential equations (FPDEs) show a rich variety of nonlinear phenomena, arise in many physical and engineering applications like geophysical fluid mechanics, fluid mechanics, plasma physics, super-conductivity, and optics. Therefore, seeking exact solutions of nonlinear FPDEs specifically the nonlinear fractional evolution equations (NLFEEs) plays an important duty in the study of nonlinear physical phenomena [1, 2, 3, 4, 5, 6, 7, 8, 9]. It is significant to find new solutions, since either new exact solutions or numerical approximate solutions may provide more information for understanding the physical phenomena. For an overview and recent developments of the local approach to fractional calculus we refer the reader to [10, 11] and references therein.
Very recently, Khalil et al. [12] suggested a conformable fractional derivative. The new fractional derivative is very interesting and is getting an increasing of interest [13, 14, 15, 16, 17]. The conformable fractional derivatives didn’t have a physical meaning as the Caputo or Riemann-Liouville derivatives. This situation is a general open problem for fractional calculus. Despite this many physical applications of conformable fractional derivative appear in the literature. Dazhi Zhao and Maokang Luo generalized the conformable fractional derivative and give the physical interpretation of generalized conformable derivative [18]. In addition, with the help of this fractional derivative and some important formulas, one can convert conformable fractional partial differential equations into integer-order differential equations by traveling wave transformation [19]. Later on, many researchers established exact traveling wave solutions of various non-linear fractional evolution equations via this fractional derivative. For example, Eslami [20] solved nonlinear fractional coupled nonlinear Schrodinger equations by using the Kudryashov method. Kaplan [21] proposed the modified simple equation method and the exponential rational function method to solve the nonlinear conformable time-fractional Boussinesq equation. Korkmaz [22] applied modified Kudryashov method to obtain the exact solutions of the the (3+1) conformable time-fractional Jimbo-Miwa, Zakharov-Kuznetsov and Modified Zakharov-Kuznetsov equations. Aminikhah et al. [23] used the sub equation method to obtain the exact solutions of the fractional (1+1) and (2+1) regularized long-wave equations which arise in several physical applications, including ion sound waves in plasma. Rezazadeh et al. [24, 25] concerned about the same method for obtaining traveling wave solutions for the conformable fractional generalized Kuramoto-Sivashinsky equation and fractional Zakharov-Kuznetsov equation with dual-power law nonlinearity. Tariq et al. [26] investigated the new exact solutions of a nonlinear evolution equation that appear in mathematical physics, specifically Cahn-Allen equation by applying tanh method. Akbulut et al. [27] obtained exact solutions for (2+1)-dimensional time-fractional Zoomeron equation and the time-fractional third order modified KdV equation via the auxiliary equation method. Ekici et al. [28] proposed the first integral method to study the optical solitons with fractional temporal evolution in presence of Hamiltonian perturbation terms governed by three types of nonlinearity. Cenesiz et al. [29] obtained some exact solutions for time-fractional Burgers equation, modified Burgers equation and Burgers-Korteweg-de Vries equation via the same method. Kurt et al. [30] established some traveling wave solutions for fractional Nizhnik-Novikov-Veselov and fractional Klein-Gordon equations via the Exp-Function Method. Tasbozan et al., [31] solved nonlinear fractional Boussinesq and combined KdV-mKdV equations by using Jacobi elliptic function expansion method. Eslami et al. [32, 33] proposed the first integral method and functional variable method to solve the space-time fractional Schrfiodinger-Hirota equation and the space-time fractional modified KDV-Zakharov-Kuznetsov equation and fractional Bogoyavlenskii equations, respectively. Hosseini et al. [34] used the ansatz method to obtain the exact solutions of the fractional Klein-Gordon equations with quadratic and cubic nonlinearities. Eslami [35] applied G′/G−expansion method to obtain the exact solutions of the space-time fractional (2+1)- dimensional dispersive long wave equations. Cenesiz et al. [36] applied the functional variable method to obtain the exact solutions of fractional modified KdV-ZK equation and Maccari system. Kaplan et al. [37] solved (2+1)-dimensional conformable time-fractional Zoomeron equation and the conformable space-time fractional EW equation by using modified simple equation method.
The aim of this paper is to construct exact traveling wave solutions of the (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law non-linearity (for α = 1, see [38])
by using the unified method.
The unified method (UM) [39, 40, 41, 42, 43, 44, 45,] and its generalized form (GUM) [46, 47, 48, 49, 50, 51], the generalized unified method, that introduce a simple algorithm to find the exact solutions and multi-wave solutions respectively in nonlinear evolution equations and nonlinear conformable fractional evolution equations both with constant and variable coefficients.
This paper is organized as follows: In section 2, we recall some basic definitions of the conformable fractional derivative. In section 3, the key idea of our method is described. section 4 is devoted to the application of the UM for (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation. Conclusions are outlined in section 5.
2 Conformable fractional derivative
Here, we introduce some basic properties and definitions of the conformable fractional calculus theory which can be found in [12, 13].
Definition 2.1
Let f : (0,∞) ➝ ℝ, then, the conformable fractional derivative of f of order α is defined as [12]
for all t > 0, α ∈ (0, 1).
If f is α-differentiable in some (0, a), a > 0 and
The new definition satisfies the properties which given in the following theorem.
Theorem 1
Let α ∈ (0, 1], and f , g be α-differentiable at a point t, then [12]
(i) tDα(af + bg) = a tDα f + b tDαg, for all a, b ∈ R.
(ii) tDα(tμ) = μ tμ−α, for all μ ∈ R.
(iii) tDα fg = f tDαg + g tDα f.
(iv)
In addition, if f is differentiable, then
In [13], T. Abdeljawad established the chain rule for conformable fractional derivatives as in the following theorem.
Theorem 2
Let f : (0,∞) ➝ ℝ be a function such that f is differentiable and also α-differentiable. Let g be a function defined in the range of f and also differentiable; then, one has the following rule
3 The description of the UM
In this section we describe the UM for finding exact solutions of nonlinear conformable fractional evolution equations.
Consider the following nonlinear conformable fractional evolution equation in two variables and a dependent variable u as
where F is a polynomial in its arguments in which the highest order derivatives and nonlinear terms are involved.
To solve Eq. (4), we take the traveling wave transformation
This enables us to use the following changes
Substituting Eq. (5) into Eq. (4) yields a nonlinear ordinary differential equation as following
where
The
3.1 The polynomial function solution
To get the polynomial function solutions of Eq. (6), UM suggests that
where pi, ci, α0, and αs are arbitrary constants to be determined later. It is worth to be noticing that, n and k are determined from the balance equation by the criteria given in [39, 40, 41, 42, 43, 44, 45]. Also, a second condition (the consistency condition), which asserts that the arbitrary functions in Eq. (7) could be consistently determined, is used.
When p = 1, (7) solves to elementary solutions (explicit or implicit) while when p = 2, it solves to elliptic solutions.
3.2 The rational function solution
To get the rational function solutions of Eq. (6), UM suggests that
where pi , qi, ci, α0, and αs are arbitrary constants to be determined later. It is worth to be noticing that, n, r and k are determined from the balance equation by the criteria given in [39, 40, 41, 42, 43, 44, 45]. Also, a second condition (the consistency condition), which asserts that the arbitrary functions in Eq. (8) could be consistently determined, is used.
When p = 1, (8) solves to elementary solutions (explicit or implicit) while when p = 2, it solves to elliptic solutions.
3.3 Steps of computation
When substituting from Eq. (7) (or Eq. (8)) into Eq. (6), we get the principle equations and the following steps are done.
1-Solving the principle equations.
2-Solving the auxiliary equations.
3-Finding the exact solution.
4-We check that the obtained solution satisfies Eq. (4).
4 The (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity
In Eq.(1), a and b are real valued constants. The first term is the evolution term, while the coefficients of a and b respectively, are the nonlinearity and dispersion. Also the parameter n is the power law nonlinearity parameter. Solitons are the result of a delicate balance between dispersion and nonlinearity. Eq.(1) typically appears in the study of plasma physics. B. T. Matebese et al. [52] solved the (3+1) dimensional Zakharov-Kuzetsov equation by G0/G−expansion method, extended tanh-function method and ansatz metod. Furthermore, Aminikhah et al. [53] proposed the functional variable method to solve this equation when α = 1. The special case where n = 1 and α = 1 gives the (3+1) dimensional Zakharov-Kuzetsov equation [54]. Let
with puting the relation (9) and its derivatives in to the Eq.(1)
By using the transformation
4.1 The polynomial solutions
To find the polynomial solutions of the fractional (3+1)-Zakharov-Kuznetsov equation with power law nonlinearity, we assume that
where pi and bi are arbitrary constants. By considering the homogeneous balance between V Vξξ and V3 in Eq.(12), we get n = 2(k − 1), k = 2, 3, . . . .
Here, we confine ourselves to find these solutions when k = 2 and p = 1 or p = 2. So, we suppose that the polynomial solution of the ODE (12) has the form
4.1.1 The solitary wave solution
To obtain these solutions, we put p = 1 in the auxiliary equation given by (14). From Eq.(14) when p = 1, we have
When we use Eq.(15) into Eq.(12) and equating the coefficients of ϕ(ξ) to zero we get a system of algebraic equations. By solving this algebraic system of equations with the help of MATHEMATICA or MAPLE, it yields the following:
where
By solving the auxiliary equation ϕ′(ξ) = b0+b1ϕ(ξ)+ b2ϕ2(ξ) and substituting together with (16) into Eq.(12), we get the solution of Eq.(1) namely
where
Fig. 1 depicts the 3D and 2D charts of the solution given by u1(x, y, z, t) with the parameters a = 0.1, b = 0.2, n = 2, and R = 1.
(a)-(d) The wave solution u1(x, y, z, t) given by (17) in 3D- and 2D-plots when y = z = 0.
4.1.2 The soliton wave solution
Here, we put p = 2 in the auxiliary equation given by (14). From Eq.(14) when p = 2, we have
By substituting from (18) into Eq.(12) and by a similar way as we did in the last case, we get
By solving the auxiliary equation
where
Fig. 2 depicts 3D and 2D charts of the solution given by u2(x, y, z, t) with the parameters a = 0.1, b = −0.2, b1 = 1, and b2 = 1.
(a)-(d) The wave solution u2(x, y, z, t) given by (20) in 3D- and 2D-plots when y = z = 0.
4.1.3 The elliptic wave solution
In this section we find the complex elliptic wave solution. To this end, we put p = 2 in the auxiliary equation given by (14). From Eq.(14) when p = 2, we have
By substituting from (21) into Eq. (12) and by a similar way as we did in the last two cases, we get
where, b i , i = 0, 2, 4 are arbitrary constants. For particular values of bi we get different solutions in Jacobi elliptic functions. According to the classification in [55], namely
the auxiliary function takes the form ϕ(ξ) = k cn(ξ , k) + dn(ξ , k) and the solution of Eq. (1) will be in the form
where
Fig. 3 depicts the 3D and 2D charts of the solution given by u3(x, y, z, t) with the parameters a = 0.1, b = 0.2, and k = 0.5.
(a)-(d) The wave solution u3(x, y, z, t) given by (24) in 3D- and 2D-plots when y = z = 0.
4.2 The rational solutions
To find the rational solutions of the fractional (3+1)-Zakharov-Kuznetsov equation with power law nonlinearity, we assume that
where pi , qi and bi are arbitrary constants. By considering the homogeneous balance between V V″ and V′ in Eq. (12), we get n − r = 2(k − 1), k = 1, 2, 3, , . . . .
Here, we find these solutions when k = 1 (so n = r) and p = 2. So from (25), we have two cases as follow
4.2.1 Case 1: periodic type
In this case, we assume that
Similarly, when we use Eq. (26) in Eq. (12), we obtain a system of algebraic equations from the coefficients of polynomial of ϕ(ξ). By solving this algebraic system of equations, we get
By solving the auxiliary equation ϕ′(ξ) =
where
Fig. 4 depicts the 3D and 2D charts of the solution given by u4(x, y, z, t) with the parameters a = 0.1, b = −0.2, and b2 = 1.
(a)-(d) The wave solution u4(x, y, z, t) given by (28) in 3D- and 2D-plots when y = z = 0.
4.2.2 Case 2: soliton type
Here, we assume that
When we use Eq. (29) in Eq. (12), we obtain a system of algebraic equations from the coefficients of polynomial of ϕ(ξ). By solving this algebraic system of equations, we get
where
By solving the auxiliary equation ϕ′ (ξ) =
where
Fig. 5 depicts the 3D and 2D charts of the solution given by u5(x, y, z, t) with the parameters a = 0.1, b = −0.2, H = 1, and b2 = 1.
(a)-(d) The wave solution u5(x, y, z, t) given by (31) in 3D- and 2D-plots when y = z = 0.
5 Conclusion
In this work, we have constructed exact traveling wave solutions for nonlinear conformable fractional evolution equations by using unified method. This method allows us to solve more nonlinear conformable fractional evolution equations in mathematical physics via the (3+1) dimensional conformable fractional Zakharov-Kuzetsov equation with power law nonlinearity. As a result, many new types of exact traveling wave solutions are obtained. These solutions include the solitary wave solutions, soliton wave solutions, elliptic wave solutions, and periodic (hyperbolic) wave rational solutions.
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