# New Classifications of All Single Traveling Wave Solutions of the Fractional Approximate Long Water Wave Equations by Using the Complete Discrimination System

Authors: Peng Li; Zhao Li
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

First, our work is to transform the space-time fractional approximate long water wave equations into nonlinear ordinary differential equations via the traveling wave transformation in the sense of conformable fractional derivative. Second, we simplify the nonlinear ordinary differential equations into an ordinary differential equation with only one variable by integration and some transformations. Finally, we can further get all single traveling wave solutions of the space-time fractional approximate long water wave equations by the complete discrimination system for the four-order polynomial method; these solutions include the hyperbolic function solutions, rational function solutions, and implicit solutions.


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## Body

## 1. Introduction

The exact traveling wave solutions of fractional partial differential equations have attracted much attention from mathematicians and engineering experts [1–10]. In recent years, some methods for constructing fractional partial differential equations have been proposed, such as the fractional double function method [11], improved fractional subequation method [12], G′/G1/G-expansion method [13], plane dynamic system analysis method [14], and complete discrimination method of polynomial [15]. The complete discrimination method of polynomials was first proposed by Liu [16]. Using this method, the classification of all single traveling wave solutions of fractional partial differential equations can be constructed.In this study, we consider the space-time fractional approximate long water wave equations [17]:(1)Dtαu−uDxβu−Dxβv+γDxβDxβu=0,Dtαv−Dxβuv−γDxβDxβv=0,t>0,0<α,β≤1,where u=ux,t, v=vx,t. Dtα and Dxβ represent the conformable fractional derivative. (1) is a very important shallow water waves model, which is usually used to describe the propagation of shallow water waves.The work of this study is as follows. In Section2, we give the construction steps of traveling wave solutions for fractional partial differential equations. In Section 3, we apply the method in Section 2 to the space-time fractional approximate long water wave equations. In Section 4, we give a conclusion.


## 2. Preliminaries

First, we consider the following integrated fractional partial differential equations:(2)Pu,ut,ux,Dtαu,Dxβu,Dxβuv,Dtt2αu,Dxx2βu,…=0,Qv,vt,vx,Dtαv,Dxβv,Dxβuv,Dtt2αv,Dxx2βv,…=0,where u=ut,x, v=vt,x, P and Q are the two polynomial in u, v, and its various fractional derivatives.Step 1.
Consider the traveling wave transformation as follows:(3)ut,x=Uξ,vt,x=Vξ,ξ=atαα+bxββ,where a and b are the constants. Using traveling wave transformation, (2) can be simplified to(4)PU,U′,U″,…=0,and satisfy the relation(5)QU,U′,U″,V,V′,V″,…,=0,where P is a polynomial of U and its derivatives. Q is a polynomial of U, the derivative of U and the derivative of V.Step 2.
The above nonlinear ordinary differential equation (4) can be reduced to the following ordinary differential form:(6)U′2=U4+pU2+qU+r.
Here, the integral form of (6) is(7)∫dUU4+pU2+qU+r=±ξ−ξ0,where ξ0 is an arbitrary constant. Here, we assume that the fourth-order polynomial is(8)FU=U4+pU2+qU+r,and its complete discrimination system is(9)D1=4,D2=−p,D3=−2p3+8pr−9q2,E2=9p2−32pr,D4=−p3q2+4p4r+36pq2r−32p2r2−274q4+64r3.
Next, we can obtain the classification of solutions of (6) according to polynomial complete discrimination system (9) ([18–23]).


## 3. Single Traveling Wave Solutions of System (1)

Consider the traveling wave transformation as follows:(10)ut,x=Uξ,vt,x=Vξ,ξ=atαα+bxββ,where a and b are the constants.Substituting (10) into (1), we obtain the ordinary differential equations as follows:(11)aU′−bUU′−bV′+γb2U″=0,aV′−bUV′+VU′−γb2V″=0.Integrating the second equation of (11) with respect to ξ, we have(12)V=abU−c1b−U22+γbU′,where c1 is the integration constant. Substituting (12) into the first equation of (11), we obtain(13)−γ2b3U″+b2U3−3a2U2+c1+a2bU−c2=0,where c2 is the integration constant.Multiplying both ends of (13) by U′ and integrating once, we obtain(14)U′2=14γ2b2U4−aγ2b3U3+c1b+a2γ2b4U2−2c2U+2c3,where c3 is the integration constant.In order to give the classification of traveling wave solutions of (1), we need to assume that the coefficient of U3 is zero. So, we make the following transformations:(15)ψ=14γ2b21/4U−ab,ξ1=14γ2b21/4ξ.Substituting (15) into (14), it becomes(16)ψξ12=ψ4+pψ2+qψ+r,where p=2|γb|c1b+a2/γ2b4−3/|γ‖b|3, q=4γ2b21/4−2a3/γ2b5+2aa2+bc1/γ2b5−2c2, r=−3a4/4γ2b6+a2a2+bc1/γ2b6−2ac2/b+2c3.The solution of (16) can be written in the following integral form:(17)∫dφψ4+pψ2+qψ+r=ξ1−ξ0,where ξ0 is an arbitrary constant.Case 1.
IfD2<0, D3=0, D4=0,(18)Fψ=ψ−l2+s22,where s>0.
Substituting (18) into (17) and combining (14), we can get the solution of (1) as(19)u1t,x=44γ2b2stans44γ2b2atαα+bxββ−sξ0+l44γ2b2+ab.
For example, whena=1,b=1,c1=−1, c3=−1/8, γ=1, and ξ0=0, we obtain u1t,x=1+tantα/α+xβ/β. Therefore, we draw the three-dimensional diagram of solution u1t,x through Maple software, as shown in Figure 1.Figure 1
The solutionu1t,x of equation (19) with a=1, b=1, c1=−1, c2=−1, c3=−1/8, γ=1, and ξ0=0.Case 2.
IfD2=0, D3=0, D4=0,(20)Fψ=ψ4.
Substituting (20) into (17) and combining (14), we can get the solutions of (1) as(21)u2t,x=−44γ2b244γ2b2atα/α+bxβ/β−ξ0+ab.Case 3.
IfD2>0, D3=0, D4=0, E2>0,(22)Fψ=ψ−θ2ψ−ϑ2,where θ>ϑ.
Next, substituting (22) into (17) and combining (14), when ψ>θ or ψ<ϑ, we can get the solution of equation (10) as(23)u3t,x=θ+ϑ44γ2b22+44γ2b2ϑ−θ2cothθ−ϑ44γ2b2atα/α+bxβ/β−ξ02.
Whenϑ<ψ<θ, we can attain(24)u4t,x=θ+ϑ44γ2b22+44γ2b2ϑ−θ2tanhθ−ϑ44γ2b2atα/α+bxβ/β−ξ02.
For example, whena=1,b=1, c1=−1/2, c2=−1/2, c3=1/8, γ=1, and ξ0=0, we obtain u4t,x=−tanhtα/α+xβ/β. Therefore, we draw the kink solitary wave solution u4t,x through Maple software, as shown in Figure 2.Figure 2
The solutionu4t,x of equation (24) with a=1,b=1, c1=−1/2, c2=−1/2, c3=1/8, γ=1, and ξ0=0.Case 4.
IfD2>0, D3>0, D4=0,(25)Fψ=ψ−θ12ψ−θ2ψ−θ3,where θ1, θ2, and θ3 are the real numbers, and θ2>θ3.
Ifθ1>θ2 and ψ<θ2 or θ1<θ3 and ψ<θ3, the implicit analytical solution of (13) is obtained:(26)±ξ1−ξ0,=∫dψψ−θ12ψ−θ2ψ−θ3,=1θ1−θ2ρ1−θ3lnψ−θ2θ1−θ3−θ1−θ2ψ−θ32|ψ−θ1|.
Ifθ1>θ2 and ψ<θ3 or when θ1<θ3 and ψ<θ2, the implicit analytical solution of (13) is obtained:(27)±ξ1−ξ0,=∫dψψ−θ12ψ−θ2ψ−θ3,=1θ1−θ2θ1−θ3lnψ−θ2θ3−θ1−θ2−θ1ψ−θ32ψ−θ1.
Ifθ2>θ1>θ3, the implicit analytical solution of (13) is obtained:(28)±ξ1−ξ0,=∫dψψ−θ12ψ−θ2ψ−θ3,=1θ2−θ1θ1−θ3arcsinψ−θ2θ1−θ3+θ1−v2ψ−θ3ψ−θ1θ2−θ3.Case 5.
IfD2>0, D3=0, D4=0, E2=0,(29)Fψ=ψ−ν3ψ−ω,where ν and ω are the real numbers.
Substituting (29) into (17) and combining (14), when ψ>ν and ψ>ω or when ψ<ν and ψ<ω, we can yield(30)u5t,x=4ν−ω44γ2b2ω−ν244γ2b2atα/α+bxβ/β−ξ02−4+ν44γ2b2+ab.Case 6.
IfD2D3<0, D4=0,(31)Fψ=ψ−ϑ12ψ−ϑ22+ϑ32,where ϑ1, ϑ2, and ϑ3 are the real numbers.
Substituting (31) into (14), we obtain the solution of equation (1):(32)u6t,x=4γ2b24e±ϑ1−ϑ22+ϑ324γ2b24atα/α+b+xβ/β−ξ0−ε+ϑ1−ϑ22+ϑ322−εe±ϑ1−ϑ22+ϑ324γ2b24atα/α+bxβ/β−ξ0−ε2−1+ab,where ε=ϑ1−2ϑ2/ϑ1−ϑ22+ϑ32.Case 7.
IfD2>0, D3>0, D4>0,(33)Fψ=ψ−ρ1ψ−ρ2ψ−ρ3ψ−ρ4,where ρ1, ρ2, ρ3, and ρ4 are the real numbers and ρ1>ρ2>ρ3>ρ4.
Then, we can obtain the solution of (1):(34)u7t,x=44γ2b2ρ2ρ1−ρ4sn2ρ1−ρ3ρ2−ρ4/244γ2b2atα/α+bxβ/β−ξ0,m−ρ1ρ2−ρ4ρ1−ρ4sn2ρ1−ρ3ρ2−ρ4/244γ2b2atα/α+bxβ/β−ξ0,m−ρ2−ρ4+ab,(35)u8t,x=44γ2b2ρ4ρ2−ρ34sn2ρ1−ρ3ρ2−ρ4/244γ2b2atα/α+bxβ/β−ξ0,m−ρ2ρ2−ρ4ρ2−ρ3sn2ρ1−ρ3ρ2−ρ4/244γ2b2atα/α+bxβ/β−ξ0,m−ρ2−ρ4+ab,where m2=ρ1−ρ4ρ2−ρ3/ρ1−ρ3ρ2−ρ4.Case 8.
IfD2D3≥0, D4<0,(36)Fψ=ψ−s1ψ−s2ψ−s32+s42,where s1, s2, s3, and s4 are the real constants, s1>s2 and s4>0.
Then, we can give the solution of (1):(37)u9t,x=44γ2b2ε1cn2∓2s4m1s1−s2/2mm144γ2b2atα/α+bxβ/β−ξ0,m+ε2ε3cn2∓2s4m1s1−s2/2mm144γ2b2atα/α+bxβ/β−ξ0,m+ε4+ab,where ε1=1/2s1+s2ε3−1/2s1+s2ε4, ε2=1/2s1+s2c4−1/2s1+s2ε3, ε3=s1−s3−s4/m1, ε4=s1−s3−s4m1, E=s42+s1−s3s2−s3/s4s1−s2, m1=E±E2+1.Case 9.
IfD2D3≤0, D4>0,(38)Fψ=ψ−χ12+l22ψ−χ22+l22,where χ1, χ2, l1, and l2 are the real constants, and l1≥l2>0.
So, we can give the solution of (1):(39)u10t,x=44γ2b2ε1snμ44γ2b2atα/α+bxβ/β−ξ0,m2+ε2cnμ44γ2b2atα/α+bxβ/β−ξ0,m2ε3snμ44γ2b2atα/αtα/α+bxβ/β−ξ0,m2+ε4cnμ44γ2b2atα/α+bxβ/β−ξ0,m2+ab,where ε1=χ1ε3+l1ε4, ε2=χ1ε4−l1ε3, ε3=−l1−l2/m1, ε4=χ1−χ2, m2=m12−1/m12, E=χ1−χ22+l12+l22/2l1l2, m1=E+E2−1, and μ=l2ε32+ε42m12ε32+ε42/ε32+ε42.Remark 1.
In this section, we obtain all single traveling wave solutionsut,x of (1) by using the complete discriminant of fourth-order polynomials. Accordingly, we can obtain the solutions vt,x of equation (1) through the relationship (12) and the solutions u1t,x∼u10t,x.


## 4. Conclusion

The construction of exact solutions of fractional partial differential equations has always been a very important research field. In this study, the fractional partial differential equations are simplified into an ordinary differential equation with only one variable by integration and some transformations. In this way, through the complete discriminant system of polynomials, we can easily obtain the classification of the exact solutions of the ordinary differential equation. We apply the method to the classification of the single traveling wave solutions of the fractional approximate long water wave equations.



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