# Exact Single Traveling Wave Solutions for Generalized Fractional Gardner Equations

Authors: Zhao Li; Tianyong Han; Chun Huang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, the classification of all single traveling wave solutions to generalized fractional Gardner equations is presented by utilizing the complete discrimination system method. Under the fractional traveling wave transformation, generalized fractional Gardner equations can be reduced to an ordinary differential equations. All possible exact traveling wave solutions are given through the complete discrimination system of the fourth-order polynomial. Moreover, graphical representations of different kinds of the exact solutions reveal that the method is of significance for searching the exact solutions to generalized fractional Gardner equations.


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## Body

## 1. Introduction

It is common knowledge that fractional partial differential equations (FPDEs) [1–9] have gained great attention because they have been widely used to model various complex physical phenomena in the domain of science and engineering. Therefore, it is of great significance to search the exact traveling wave solutions [10–24] of FPDEs in the research of nonlinear science, which can accurately reflect the propagation of nonlinear waves and better understand nonlinear physical phenomena. So far, many powerful methods have been established and developed to analyze the exact solutions to the FPDE, which include the G′/G-expansion method [25, 26], the integral bifurcations [27, 28], the Lie symmetry analysis method [29, 30], first integral method [31], modified trial equation method [32], the exp-function method [33], F-expansion method [34], and the Kudryashov method [35].In this paper, we shall consider the following generalized fractional Gardner equation [36]:(1)Dtαu+pux+qun+ru2nux+uxxx=0,n≥0,r<0,0<α≤1,where Dtαu is the conformable derivative of u depending on the variable t. ut,x represents the amplitude of the wave mode, and variables t and x represent the time and spatial variable, respectively. The coefficients p, q, and r are constants. Equation (1) can be usually used to describe the nonlinear propagation of ion-acoustic waves at an unmagnetized plasma. As we all know, equation (1) is a kind of very important FPDE. When the parameters of equation (1) are changed, it can be simplified to the following famous nonlinear FPDE. For example, when n=1, q≠0, and r=0, equation (1) becomes the fractional KdV equation; when n=1, q=0, and r≠0, equation (1) becomes the fractional mKdV equation; and when n=1, q≠0, and r≠0, equation (1) becomes a fractional KdV-mKdV equation. In [36], Reazadeh et al. obtained the hyperbolic and trigonometric function solutions to the generalized fractional Gardner equation by using modified Kudryashov method and hyperbolic function method, respectively. But, their research only focused on acquiring the hyperbolic and trigonometric function solutions. Motivated by the aforementioned discussion, in the paper, we will construct new exact traveling wave solution to the generalized fractional Gardner equations via the polynomial method.The complete discrimination system for the polynomial method was first proposed by Liu [37]. It is one of the most powerful methods to find the single traveling wave solutions to partial differential equations (PDEs). With the development of fractional calculus, the study of exact solution to FPDEs has been gaining more and more attention by many experts and scholars. Because of the complexity of fractional derivative, the exact solution of FPDE develops very slowly than PDE of integers. Many scholars [38, 39] have been trying to find new methods to construct the exact solutions of FPDEs. Recently, Khalil et al. [40] introduced the conformable fractional derivative. FPDEs can be reduced into nonlinear ordinary differential equations by the fractional traveling wave transformation. In the paper, we will find the exact solutions to the generalized fractional Gardner equation by the complete discrimination system of the polynomial method.The main objective of the paper is to draw support from the complete discrimination system to construct exact traveling wave solutions to the generalized fractional Gardner equation. In Section2, we review the definition of conformable derivative and introduce the complete discrimination system for constructing the exact traveling wave solutions of FPDE. Then, in Section 3, we discuss the exact solutions to the generalized fractional Gardner equation by using the complete discrimination system. Finally, we give a brief conclusion in Section 4.


## 2. Mathematical Preliminaries

### 2.1. The Conformable Derivative

The definition and properties of the conformable derivative are defined asDefinition 1.
Letf:0,∞⟶R. Then, the conformable derivative of f of order α is defined as(2)Dtαft=limε⟶0ft+εt1−α−ftε,∀t∈0,+∞,α∈0,1,and the function f is α-conformable differentiable at a point t if the limit in equation (2) exists.Remark 1.
The conformable derivative possesses many important properties (See [41, 42] and references therein). In recent years, several scholars [43–47] have developed and constructed exact solutions of FPDE in the sense of conformable derivative.


### 2.2. Description of the Method

Consider the following conformable FPDE:(3)Pu,Dtαu,ux,Dt2αu,uxx,uxxx,…=0,0<α≤1,where u=ut,x is an unknown function.Introducing the following fractional traveling wave transformation,(4)ut,x=uξ,ξ=kx−ωtαα,where k and ω are nonzero constants.Substituting equation (4) into equation (3), then equation (3) can be converted into the following integer-order ordinary differential equation:(5)Qu,u′,u″,…=0,where Q is a polynomial in u and its derivatives and notation ′ is the derivative with respect to ξ.Equation (5) is usually reduced to(6)u′2=Fu,where Fu is a polynomial.Then, integrating equation (6) once, we can obtain(7)±ξ−ξ0=∫duFu,where ξ0 is an integral constant.According to the above procedures, recent results have been reported via the complete discrimination system [48–50].


## 2.1. The Conformable Derivative

The definition and properties of the conformable derivative are defined asDefinition 1.
Letf:0,∞⟶R. Then, the conformable derivative of f of order α is defined as(2)Dtαft=limε⟶0ft+εt1−α−ftε,∀t∈0,+∞,α∈0,1,and the function f is α-conformable differentiable at a point t if the limit in equation (2) exists.Remark 1.
The conformable derivative possesses many important properties (See [41, 42] and references therein). In recent years, several scholars [43–47] have developed and constructed exact solutions of FPDE in the sense of conformable derivative.


## 2.2. Description of the Method

Consider the following conformable FPDE:(3)Pu,Dtαu,ux,Dt2αu,uxx,uxxx,…=0,0<α≤1,where u=ut,x is an unknown function.Introducing the following fractional traveling wave transformation,(4)ut,x=uξ,ξ=kx−ωtαα,where k and ω are nonzero constants.Substituting equation (4) into equation (3), then equation (3) can be converted into the following integer-order ordinary differential equation:(5)Qu,u′,u″,…=0,where Q is a polynomial in u and its derivatives and notation ′ is the derivative with respect to ξ.Equation (5) is usually reduced to(6)u′2=Fu,where Fu is a polynomial.Then, integrating equation (6) once, we can obtain(7)±ξ−ξ0=∫duFu,where ξ0 is an integral constant.According to the above procedures, recent results have been reported via the complete discrimination system [48–50].


## 3. Applications

Whenn=1, q≠0, and r≠0, equation (1) becomes the following fractional KdV-mKdv equation:(8)Dtαu+pux+qu+ru2ux+uxxx=0.Substituting equation (4) into equation (8), equation (8) can be converted to an ordinary differential equation as follows:(9)−kωu′+pku′+qu+ru2ku′+k3u″′=0.Integrating the above once with respect toξ, we obtain(10)k3u″+p−ωku+kq2u2+kr3u3=c0.Multiplying equation (9) on both sides by u′ and then integrating it with respect to ξ again,(11)k3u′2+p−ωk2u2+kq6u3+kr12u4=c0u+c1,where c0 and c1 are integral constants. Then, we obtain(12)u′2=−r12k2u4−q6k2u3−p−ω2k2u2+c0k3u+c1k3.Suppose thata4=−r/12k2, a3=−q/6k2, a2=−p−ω/2k2, a1=c0/k3, and a0=c1/k3; then,(13)u′2=a4u4+a3u3+a2u2+a1u+a0.Making the transformation, we obtain(14)Φ=a41/4u+a34a4,ξ1=a41/4ξ.Substituting transformation (14) into equation (13), it will be changed into(15)Φξ12=Φ4+b1Φ2+b2Φ+b3,where b1=a2/a4, b2=a33/8a42−a2a3/2a4+a1a4−1/4, and b3=−3a34/256a43+a2a32/16a42−a1a3/4a4+a0.Integrating equation (15) once, we obtain(16)±ξ1−ξ0=∫dΦΦ4+b1Φ2+b2Φ+b3,where ξ0 is the integration constant.LetGΦ=Φ4+b1Φ2+b2Φ+b3; then, we can obtain its complete discrimination system:(17)D1=4,D2=−b1,D3=−2b13+8b1b3−9b22,E2=9b12−32b1b3,D4=−b13b22+4b14b3+36b1b22b3−32b12b32−274b24+64b33.Integrating formula (16), we will obtain the exact traveling wave solutions of equation (8) under nine cases.Case 1.
D2<0, D3=0, and D4=0. GΦ has a pair of conjugate complex roots of multiplicities two, i.e.,(18)GΦ=Φ−β2+γ22,where γ>0. By using equation (16), we attain(19)ξ1−ξ0=∫dΦΦ−β2+γ2=1γarctanΦ−βγ.
Then, equation (19) is simplified to(20)Φξ1=γtanγξ1−ξ0+β,which is a trigonometric function solution. Namely, when b1>0, b2=0, and b3=b12/4; therefore, γ=b1/2 and then the solution of equation (15) is expressed as follows:(21)u1ξ=±a4−1/4γtanγa41/4ξ−ξ0−a34a4.
For instance, whenk=1, p=2, ω=4, q=−12, r=−12, c0=0, c1=11/16, and ξ0=0, we can obtain a trigonometric solution of equation (8) as follows. Under the given parameters, we draw the traveling wave 3D solution surfaces and corresponding 2D solution graphs for the obtained solution u1t,x in Figures 1 and 2:(22)u1t,x=12tan12x−2tαα−12.Figure 1
The graphics of solutions ofu1t,x for differential values of fractional parameter α.
(a)(b)(c)Figure 2
Two-dimensional graphic of solution ofu1t,x for differential values of fractional parameter α.Case 2.
D2=0, D3=0, and D4=0. GΦ has real roots of multiplicities four, namely,(23)GΦ=Φ4.
By using equation (16), we can obtain(24)ξ1−ξ0=∫dΦΦ2=−1Φ.
Then, we can obtain a rational function solution:(25)Φξ1=−1ξ1−ξ0.
Therefore, the solutions of equation (16) can be shown as(26)u2ξ=∓a−1/41a−1/4ξ−ξ0−a34a4.
For example, whenk=1, p=2, ω=2, q=0, r=−12, c0=0, c1=0, and ξ0=0, we can obtain a rational function solution of equation (8) as(27)u2t,x=1x−2tα/α.Case 3.
D2>0, D3=0, D4=0, and E2>0. GΦ has two real roots of multiplicities two, namely,(28)GΦ=Φ−β2Φ−γ2,where β>γ. By using equation (16), we can obtain(29)±ξ1−ξ0=∫dΦΦ−βΦ−γ=1β−γlnΦ−βΦ−γ.
WhenΦ>β or Φ<γ, we can obtain the solution of equation (16) as follows:(30)Φξ1=γ−βeβ−γξ1−ξ0−1+γ=γ−β2cothβ−γξ1−ξ02−1+γ.
Then, we have(31)u3ξ=γ−βa4−1/42cothβ−γa41/4ξ−ξ02−1+γ−a34a4.
Whenγ<Φ<β, we can gain the solution of equation (16) as follows:(32)Φξ1=γ−β−eβ−γξ1−ξ0−1+γ=γ−β2tanhβ−γξ1−ξ02−1+γ,
Similarly, we can have(33)u4ξ=γ−βa4−1/42tanhβ−γa41/4ξ−ξ02−1+γ−a34a4.
We can see that equations (31) and (33) are two solitary wave solutions. Especially, when k=1, p=4, ω=2, q=−12, r=−12, c0=−2, c1=11/16, and ξ0=0, we have b1=−1, b2=0, b3=1/4, and −1<Φ<1, and then we can obtain a hyperbolic solution of equation (8) as follows. Under the given parameters, we draw the traveling wave 3D solution surface and corresponding 2D solution graphs for the obtained solution u4t,x in Figure 3:(34)u4t,x=tanhx−2tαα−12.Figure 3
The graphics of solutions ofu4t,x.
(a)(b)Case 4.
D2>0, D3>0, and D4=0. GΦ has two real roots and real roots with multiplicities two, namely,(35)GΦ=Φ−β12Φ−β2Φ−β3,where βi(i=1,2,3) are real numbers and β2>β3.
Whenβ1>β2 and Φ<β2 or when β1<β3 and Φ<β3, the following formula can be obtained:(36)±ξ1−ξ0=∫dΦΦ−β12Φ−β2Φ−β3=1β1−β2β1−β3lnΦ−β2β1−β3−β1−β2Φ−β32Φ−β1.
Whenβ1>β2 and Φ<β3 or when β1<β3 and Φ<β2, we can obtain the solution of equation (28):(37)±ξ1−ξ0=∫dΦΦ−β12Φ−β2Φ−β3=1β1−β2β1−β3lnΦ−β2β3−β1−β2−β1Φ−β32Φ−β1.
Whenβ2>β1>β3, we can have the solution of equation (16):(38)±ξ1−ξ0=∫dΦΦ−β12Φ−β2Φ−β3=1β2−β1β1−β3arcsinΦ−β2β1−β3+β1−β2Φ−β3Φ−β1β2−β3.
Especially, whenk=1, p=4, ω=2, q=−12, r=−12, c0=−2, c1=−9/16, and ξ0=0, we have b1=−1, b2=0, and b3=0. By taking them into (38), then we can obtain(39)u5t,x=sinx−2tαα−12.Case 5.
D2>0, D3=0, D4=0, and E2=0. GΦ has real roots of multiplicities three and real roots with multiplicities one, namely,(40)GΦ=Φ−β3Φ−γ,where β and γ are real numbers. Using formula (16) can yield(41)±ξ1−ξ0=∫dΦΦ−β3Φ−γ=2γ−βΦ−γΦ−β.
WhenΦ>β and Φ>γ or when Φ<β and Φ<γ, the solution of equation (16) is given by(42)Φξ1=4β−γγ−β2ξ1−ξ02−4+β.
Then, we can obtain a rational solution of equation (8) as(43)u6ξ=4a−1/4β−γa1/4ξ−ξ02γ−β2−4+β−a34a4.Case 6.
D2D3<0 and D4=0. GΦ has real roots of multiplicities two and a pair of conjugate complex roots, namely,(44)GΦ=Φ−β12Φ−β22+β32,where βi(i=1,2,3) are real numbers. By using equation (16), we can obtain(45)±ξ1−ξ0=∫dΦΦ−β1Φ−β22+β32=1β1−β22+β32lnc1Φ+c2−Φ−β22+β32Φ−β1,where c1=β1−2β2/β1−β22+β32 and c2=β1−β22+β32−β1β1−2β2/β1−β22+β32. Thus, we obtain the solution of equation (16):(46)Φξ1=e±β1−β22+β32ξ1−ξ0−c1+β1−β22+β322−c1e±β1−β22+β32ξ1−ξ0−c12−1.
Hence,(47)u7ξ=e±β1−β22+β32a41/4ξ−ξ0−c1+β1−β22+β322−c1a41/4e±β1−β22+β32a41/4ξ−ξ0−c12−1−a34a4,which is a solitary wave solution.Case 7.
D2>0, D3>0, and D4>0. GΦ has four distinct real roots, namely,(48)GΦ=Φ−β1Φ−β2Φ−β3Φ−β4,where β1, β2, β3, and β4 are real numbers and β1>β2>β3>β4.
WhenΦ>β1 or Φ<β4, we make the following transformation:(49)Φ=β2β1−β4sin2θ−β1β2−β4β1−β4sin2θ−β2−β4.
Whenβ3<Φ<β2, similarly(50)Φ=β4β2−β3sin2θ−β3β2−β4β2−β3sin2θ−β2−β4.
By using equation (16), we obtain(51)±ξ1−ξ0=∫dΦΦ−β1Φ−β2Φ−β3Φ−β4=2β1−β3β2−β4∫dθ1−m2sin2θ,where m2=β1−β4β2−β3/β1−β3β2−β4.
From equation (50) and the definition of Jacobian elliptic sine function, we obtain(52)sinθ=snβ1−β3β2−β42ξ1−ξ0,m.
Combining equation (51) with expression (48), we can gain the solutions of equation (16):(53)Φξ1=β2β1−β4sn2β1−β3β2−β4/2ξ1−ξ0,m−β1β2−β4β1−β4sn2β1−β3β2−β4/2ξ1−ξ0,m−β2−β4,and then we can give the solution of equation (8):(54)u8ξ=β2β1−β4a4−1/4sn2β1−β3β2−β4/2a41/4ξ−ξ0,m−β1β2−β4β1−β4sn2β1−β3β2−β4/2a41/4ξ−ξ0,m−β2−β4−a34a4.
Similarly, combining equation (51) with expression (49), we can obtain the solution of equation (8):(55)u9ξ=β4β2−β3a4−1/4sn2β1−β3β2−β4/2a41/4ξ−ξ0,m−β3β2−β4β2−β3sn2β1−β3β2−β4/2a41/4ξ−ξ0,m−β2−β4−a34a4.Case 8.
D2D3≥0 and D4<0. GΦ has two different real roots and a pair of conjugate complex roots, namely,(56)GΦ=Φ−β1Φ−β2Φ−β32+β42,where β1, β2, β3, and β4 are real constants, β1>β2, and β4>0.
Making the following transformation, we obtain(57)Φ=c1cosθ+c2c2cosθ+c4,where c1=1/2β1+β2c3−1/2β1−β2c4, c2=1/2β1+β2c4−1/2β1−β2c3, c3=β1−β3−β4/m1, c4=β1−β3−β4m1, E=β42+β1−β3β2−β3/β4β1−β2, and m1=E±E2+1.
By using equation (16), we obtain:(58)ξ1−ξ0=∫dΦ±Φ−β1Φ−β2Φ−β32+β42=2m1m2∓2β4m1β1−β2∫dθ1−m22sin2θ,where m22=2/1+m12.
From equation (57) and the definition of Jacobian elliptic function, we obtain(59)cosθ=cn∓2β4m1β1−β22m1m2ξ1−ξ0,m2.
Combining equation (58) with expression (56), we can gain the solutions of equation (16):(60)Φξ1=c1cn∓2β4m1β1−β2/2m1m2ξ1−ξ0,m2+c2c3cn∓2β4m1β1−β2/2m1m2ξ1−ξ0,m2+c3,and then we can give the solution of equation (8):(61)u10ξ=a4−1/4c1cn∓2β4m1β1−β2/2m1m2a41/4ξ−ξ0,m2+c2c3cn∓2β4m1β1−β2/2m1m2a41/4ξ−ξ0,m2+c3,which is an elliptic double periodic function solution.Case 9.
D2D3≤0 and D4>0. GΦ has two pairs of conjugate complex roots, namely,(62)GΦ=Φ−β12+l22Φ−β22+l22,where β1, β2, l1, and l2 are real constants and l1≥l2>0.
Making the following transformation, we obtain(63)Φ=c1tanθ+c2c2tanθ+c4,where c1=β1c3+l1c4, c2=β1c4−l1c3, c3=−l1−l2/m1, c4=β1−β2, E=β1−β22+l12+l22/2l1l2, and m1=E+E2−1.
By using equation (16), we obtain(64)ξ1−ξ0=∫dΦΦ−β12+l22Φ−β22+l22=c32+c42l2c32+c42m12c32+c42∫dθ1−m22sin2θ,where m22=m12−1/m12.
From equation (62) and the definition of Jacobian elliptic function, we obtain(65)sinθ=snl2c32+c42m12c32+c42c32+c42ξ1−ξ0,m2,(66)cosθ=cnl2c32+c42m12c32+c42c32+c42ξ1−ξ0,m2.
Combining equations (64) and (65) with expression (62), we can gain the solutions of equation (16):(67)Φξ1=c1snμξ1−ξ0,m2+c2cnμξ1−ξ0,m2c3snμξ1−ξ0,m2+c4cnμξ1−ξ0,m2,where μ=l2c32+c42m12c32+c42/c32+c42. Then, we can give the solution of equation (8):(68)u11ξ=a4−1/4c1snμa41/4ξ−ξ0,m2+a4−1/4c2cnμa41/4ξ−ξ0,m2c3snμa41/4ξ−ξ0,m2+c4cnμa41/4ξ−ξ0,m2−a34a4,which is an elliptic double periodic function solution.


## 4. Conclusion

By using the complete discrimination system method, we obtain exact traveling wave solutions to generalized fractional Gardner equations under the given parameter conditions. Many exact solutions have been obtained, which include hyperbolic function solutions, Jacobi elliptic function solutions, trigonometric function solutions, and rational function solutions. Compared with the previous work, the solution obtained in the paper has not been reported. Furthermore, the method we employ here can be used to analyze the exact solutions to other FPDEs.



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