Solitary and periodic wave solutions of higher-dimensional conformable time-fractional differential equations using the (G′G,1G)$( \frac{G'}{G},\frac{1}{G} ) $-expansion method

*Correspondence: farahaini@usm.my 1School of Mathematical Sciences, Universiti Sains Malaysia, USM Penang, Malaysia Full list of author information is available at the end of the article Abstract In this paper, the two variables ( ′ G , 1 G )-expansion method is applied to obtain new exact solutions with parameters of higher-dimensional nonlinear time-fractional differential equations (NTFDEs) in the sense of the conformable fractional derivative. To clarify the veracity of this method, it is implemented in nonlinear (2 + 1)-dimensional time-fractional biological population (BP) model and nonlinear (3 + 1)-dimensional KdV–Zakharov–Kuznetsov (KdV–ZK) equation with time-fractional derivative. When the parameters take some special values, the solitary and periodic solutions are obtained from the hyperbolic and trigonometric function solutions.


Introduction
Fractional differential equations (FDEs) can be viewed as the generalized type of the ordinary differential equations (ODEs). The FDEs have attracted the researchers' attention over the past two decades because the effects in ODEs are neglected. Oldham and Spanier [1] are the first researchers who have taken the FDEs into consideration. The search for the exact solutions of FDEs plays an important role in understanding the qualitative and quantitative features of many physical phenomena, which are described by these equations. For instance, the nonlinear oscillation of an earthquake can be modeled by derivatives of fractional order. Actually, the physical phenomena may not depend only on the time moment but also on the former time history, which can be successfully modeled utilizing the theory of fractional integrals and derivatives [2][3][4]. Fractional evolution equations play a significant role in various fields like engineering, biology, physics, signal processing, rheology, fluid flow, finance, electrochemistry, and so on [5][6][7][8][9]. Several efficient methods have recently been developed to get analytical solutions for FDEs. For example, the generalized tanh-coth method [10], the auxiliary equation method [11], the ( G G )-expansion method [12][13][14][15][16][17][18][19][20][21][22][23], the improved F-expansion method [24], the exponential rational function method [25][26][27], the simplest equation method [28], the modified simple equation method [29][30][31][32], the first integral method [33][34][35][36][37], the Kudryashov method [38][39][40][41][42][43], the D β t u = u 2 xx + u 2 yy + h u 2r , t > 0, 0 < β < 1, x, y ∈ R, (1.1) in which u denotes the density of population, h(u 2r) shows the population supply because of deaths and births and h, r are constants. When β → 1, the BP model assists us to understand the dynamical proceeding of population changes and provides valuable predictions. Recently, Zhang and Zhang [57], Lu [58], Bekir et al. [59], Bekir and Güner [13] and Manafian and Lakestani [10] have found the exact solutions of Eq. (1.1) using the fractional sub-equation method, the Bäcklund transformation of fractional Riccati equation, the exp-function method, the ( G G )-expansion method, and the generalized tanh-coth method, respectively. The comparison of the obtained results with the results obtained in [13,[57][58][59] will be discussed in the following sections of the paper. The second studied model is nonlinear (3 + 1)-dimensional KdV-ZK equation with time-fractional derivative [60]: When β → 1, the KdV-ZK equation is derived for plasma comprised of hot and cool electrons and fluid ions species. Recently, Sahoo et al. [60] have found the exact solutions of Eq. (1.2) utilizing the improved fractional sub-equation method, whereas Kaplan et al. [61] have obtained the exact solutions of Eq. (1.2) using the exp(-φ(ξ )) method. The study is organized as follows: In Sect. 2, the description of the conformable fractional derivative and its important properties are presented. In Sect. 3, the main ideas of the ( G G , 1 G )-expansion method are discussed. In Sect. 4, the new exact solutions for the conformable BP model and KdV-ZK equation with time-fractional derivative by the ( G G , 1 G )-expansion method are constructed. Finally, conclusions are presented in Sect. 5 of this paper.

Conformable fractional derivative and its important properties
The conformable fractional derivative of g of order β is defined as follows [62][63][64]: x , which g : [0, ∞) → R, t > 0 and β ∈ (0, 1). Some important properties of the above definition are given by 3 Key ideas of the ( G G , 1 G )-expansion method to the NTFDEs Li et al. [45] suggested the ( G G , 1 G )-expansion method as follows: For the auxiliary equation From Eqs. (3.1) and (3.2), we get The general solution of the ODE (3.1), in the following three distinct subcases: Case 1 If λ < 0, the general solution of the ODE (3.1) is and thus where A 1 , A 2 are two arbitrary constants and σ = A 2 1 -A 2 2 . Case 2 If λ > 0, the general solution of the ODE (3.1) is given by therefore, we have where σ = A 2 1 + A 2 2 . Case 3 If λ = 0, the general solution of the ODE (3.1) is and hence The main steps of the two variables ( G G , 1 G )-expansion method are described in the following steps.
Step 1 Assume that we have the following general NFDE By introducing the transformation where l, a, b and c are nonzero arbitrary constants. Equation (3.10) can be reduced into ODE in the following form: Step 2 Assume that the solution of ODE (3.12) can be expressed by a polynomial in φ, ψ as follows: (3.13) where α i (i = 0, . . . , m) and β i (i = 1, . . . , m) are constants, and m in (3.13) can be determined by utilizing the homogeneous balance between the nonlinear terms and the highest-order derivative in (3.12).
Step 3 Substituting (3.13) into Eq. (3.12) utilizing (3.3) and (3.5), we obtain a polynomial in ψ and φ, where the degree of ψ is not bigger than one. Setting the coefficients of φ i (i = 0, 1, . . .) and ψ j (j = 0, 1) to be zero, yields a set of algebraic equations, which can be solved with the help of Mathematica or Maple software package to obtain the values of α i , β i , l, a, b and c in which (λ < 0).

Applications of ( G G , 1 G )-expansion method to NTFDEs in mathematical physics
In this part, new exact solutions of the (2 + 1)-dimensional BP model and (3 + 1)dimensional KdV-ZK equation with time-fractional derivative are extricated by utilizing the ( G G , 1 G )-expansion method.

(2 + 1)-dimensional BP model with time-fractional derivative
Using the fractional traveling wave variable, To get the exact solution, we utilize the transformation in Eq. (4.2) to find a new equation, By utilizing the homogeneous balance principle, we get m = 1. Therefore, Eq. (1.1) has the formal solution where α 0 , α 1 and β 1 are constants. Case 1 When λ < 0 (hyperbolic function solutions) By substituting (4.5) into (4.4) and utilizing (3.3) and (3.5), we obtain a polynomial in ψ and φ. Equating the coefficients of the equation to zero, we get a system of algebraic equations for α 0 , α 1 , β 1 and l as follows: By solving the algebraic equations mentioned above utilizing the Maple software package, the following results are obtained.

Result 1
In particular, if we put A 1 = 0 and A 2 > 0 in Eq. (4.7), we get the solitary solution while if we set A 2 = 0 and A 1 > 0, then we get the solitary solution where ξ = x + y ± 24 √ -λr t β β .

Result 2
(4.10) Based on Result 2, an exact solution of Eq. (1.1) is obtained. We have In particular, if we put μ = 0, A 1 = 0 and A 2 > 0 in Eq. (4.11), we get the solitary solution but if we set μ = 0, A 2 = 0 and A 1 > 0, then we get the solitary solution where ξ = x + y ± 12 √ -λr t β β . Case 2 When λ > 0 (trigonometric function solutions) By substituting (4.5) into (4.4) and utilizing (3.3) and (3.7), we obtain a polynomial in ψand φ. Equating the coefficients of the equation to zero to obtain a set of algebraic equations for α 0 , α 1 , β 1 and l as follows: By solving the algebraic equations mentioned above utilizing the Maple software package, the following results are obtained.
Remark 1 By comparing our results with the results obtained by Lu [58], Zhang and Zhang [57], Bekir et al. [59], Bekir and Güner [13] and Manafian and Lakestani [10], we conclude that all our solutions of Eq. (1.1) are new and satisfy the equation.

Graphical presentation of some exact solutions
We presented some graphs to illustrate the behavior of exact solutions of Eqs. (1.1) and (1.2). Figures 1-5 show the solitary and periodic wave forms.

Conclusion
The ( G G , 1 G )-expansion method is used to discuss the exact solutions to NFDEs. The ( G G , 1 G ) is successfully implemented to solve two NTFDEs. As applications, new exact solutions  13), respectively, the ( G G , 1 G )-expansion method is reduced to the ( G G )-expansion method. Therefore, it can be concluded that the ( G G , 1 G )-expansion method is more general and efficient than the ( G G )-expansion method. In comparison with other methods, the key feature of this method is that it possesses all three types of solu- tions. Some diagrams have been given in three dimensions for fractional order to illustrate the behavior of the solutions when the parameters take some special values.