Review Article Volume 2 Issue 5
Mathematics Department, Zagazig University, Egypt
Correspondence: Elsayed ME Zayed, Department of Mathematics, Faculty of Sciences, Zagazig University, Zagazig, Egypt
Received: January 24, 2018 | Published: October 9, 2018
Citation: Zayed EME, Shohib RMA. The unified sub-equation method and its applications to conformable space-time fractional fourth-order pochhammer-chree equation. Phys Astron Int J. 2018;2(5):451-464. DOI: 10.15406/paij.2018.02.00124
In this article, we apply the unified sub-equation method proposed by Lu Bin and Zhang Hong Qing to construct many new Jacobi elliptic function solutions, solitons and other solutions for the conformable space-time fractional fourth-order Pochhammer-Chree equation. This method is direct and more powerful than the projective Riccati equation method. The solitons and other solutions of this equation can be found from the Jacobi elliptic solutions when its modulasm→1 or m→0 respectively. Comparing our new results with the well-known results is given.
Keywords: unified sub-equation method, jacobi elliptic function solutions, dark, singular and bright solitons, periodic solutions, the conformable space-time fractional fourth-order pochhammer-chree equation
02.03.Jr; 04.20.Jb; 05.45.Yv.
When the nonlinear partial differential equations (PDEs) are analyzed, one of the most important equation is the construction of the exact solutions of those equations. Searching for the exact solutions of those equations plays an important role in the study of nonlinear physical phenomena. Nonlinear wave phenomena appear in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics, geochemistry, thermodynamics, soli mechanics, civil engineering, and non-Newtonian fluids to the natural sciences including population acology, inflectious disease epidemiology, natural networks and so on. Throughout the past few decades, a particular attention has been given to the problem of finding the exact solutions of these nonlinear PDEs. By virtue of these solutions, one may give better insight into the physical aspects of the nonlinear models studies. In recent years, quite a few methods for constructing explicit and solitary wave solutions of the nonlinear PDEs have been presented. A variety of powerful methods, such as the inverse scattering method,1 the Hirota method,2 the Bäcklund transform method,3,4 the Painlevé expansion method,5 the exp-function method,6,7 the sub-ODE method,8-10 the Jacobi elliptic function method,11,12 the sine-cosine function method,13,14 the (G'/G) -expansion method,15-17 the modified simple equation method,18,19 the Kudryashov method,20,21 the multiple exp–function method,22,23 the homogeneous balance method,24 the auxiliary equation method,25,26 the extended auxiliary equation method,27-29 the soliton ansatz method,30-33 the new mapping method,34,35 the first integral method,36,37 the (G'/G,1/G) -expansion method38,39 and an unified sub-equation method,40 the projective Riccati equation method41,42 and so on. Motivated by the projective Riccati equation method proposed by Conte & Musette43 and developed by Yan,44 we present a new direct algebraic method proposed by Bin L & Hong-Qing Z40 to obtain many new double periodic wave solutions of nonlinear PDEs which cannot be acquired by using the projective Riccati equation method.
The objective of this article is to apply a unified sub-equation method combined with the conformable space-time fractional derivatives40 for finding many new Jacobi elliptic function solutions, solitons and other solutions of the following nonlinear conformable space-time fractional fourth-order Pochhammer-Chree equation:
∂2αu∂t2α−∂2α∂t2α(∂2βu∂x2β)−∂2β∂x2β(α1u+β1un+1+γ1u2n+1)=0, n≥1, (1.1)
Where 0<α,β≤1 and u(x,t) is a real function, while α1 , β1 and γ1 are arbitrary constants. Equation (1.1) represents a nonlinear model of longitudinal wave propagation of elastic rods.45,46 Here the exponent n≥1 is the power law nonlinearity parameter. When α=β=1, Equation (1.1) has been discussed in42 using the generalized projective Riccati equation method, in47 using the extended (G'/G) -expansion method, in48 using the (G'/G) -expansion method, in49 using the tanh-coth and the sine-cosine methods, and in50 using the exp-function method.
This article is organized as follows: In Section 2, the description of conformable fractional derivative is given. In Section 3, the description of the unified sub-equation method combined with the conformable space-time fractional derivatives is obtained. In Section 4, we apply this method to the conformable space-time fractional fourth-order Pochhammer-Chree equation (1.1). In Section 5, we present the graphical representations for some solutions of Equation (1.1). In Section 6, conclusions are obtained. To the best of our knowledge. Equation (1.1) has not been previously considered in literature using the method of Section 3.
Khalil et al.51 introduced a novel definition of fractional derivative named the conformable fractional derivative, which can rectify the deficiencies of the other definitions.
Definition1: Suppose f:[0,∞)→R is a function. Then, the conformable fractional derivative of f of order α is defined as
Tα(f)(t)=limτ→0f(t+τt1−α)−f(t)τ, (2.1)
For all t>0 and α∈(0,1]. Several properties of the conformable fractional derivative are given below as in51−53
Thereom 1: Suppose α∈(0,1], and f and g are α− differentiable at t>0. Then
Tα(af+bg)=aTα(f)+bTα(g), ∀a,b∈R. (2.2)
Tα(tμ)=μtμ−α, ∀μ∈R. (2.3)
Tα(fg)=fTα(g)+gTα(f), (2.4)
Tα(fg)=(gTα(f)−fTα(g)g2), (2.5)
Furthermore, if f is differentiable; then
Tα(f)(t)=t1−αdfdt(t). (2.6)
Thereom 2: Suppose f:[0,∞)→R is a differentiable function and also α− differentiable. Let g be a function defined in the range of f and also differentiable. Then
Tα(f0g)(t)=t1−αg'(t)f'(g(t)). (2.7)
Consider the following nonlinear PDE:
F(u,∂αu∂tα,∂βu∂xβ,∂2αu∂t2α,∂2βu∂x2β,.....)=0, 0<α,β≤1 (3.1)
Where F is a polynomial in u(x,t) and its partial derivatives, in which the highest order derivatives and the nonlinear terms are involved. In the following, we give the main steps of this method:
Step 1: We use the conformable space-time wave transformation:
u(x,t)=u(ξ), ξ=xββ−c1tαα, (3.2)
Where c1 is a constant, to reduce Equation (3.1) to the following ODE:
P(u,u',u'',.....)=0, (3.3)
Where P is a polynomial in u(ξ) and its total derivatives, such that '=ddξ .
Step 2: We assume that Equation (3.3) has the formal solution:
u(ξ)=a0+N∑i=1fi−1(ξ)[aif(ξ)+big(ξ)], (3.4)
Where a0 , ai , bi (i=1,.....,N) are constants to be determined later, such that aN≠0 or bN≠0, while f(ξ) and g(ξ) satisfy the auxiliary ODEs:
f'(ξ)=f(ξ)g(ξ), (3.5)
g'(ξ)=q+g2(ξ)+rf−2(ξ), (3.6)
g2(ξ)=−[q+r2f−2(ξ)+cf2(ξ)] (3.7)
Where q , r and c are constants.
Step 3: We determine the positive integer N in (3.4) by using the homogeneous balance between the highest order derivatives and the nonlinear terms in Equation (3.3). More precisely, we define the degree of u(ξ) as D[u(ξ)]=N , which gives rise to the degree of other expressions as follows:
D[up1(ξ)(dq1u(ξ)dξq1)s1]=NP1+s1(q1+N). (3.8)
From (3.8) we can get the value of N in (3.4). In some nonlinear equations, the balance number N is not a positive integer. In this case, we make the following transformations:
(a) When N=q1p1, where q1p1 is a fraction in the lowest terms, we let
u(ξ)=[v(ξ)]q1p1, (3.9)
When N is a negative number, we let
u(ξ)=[v(ξ)]N, (3.10)
And substitute (3.9) or (3.10) into Equation (3.3) to get a new equation in terms of the function v(ξ) with a positive integer balance number.
Step 4: We substitute (3.4) along with (3.5)-(3.7) into Equation (3.3) and collect all terms of the same order of fi(ξ) gj(ξ) (i,j=0,1,.....) and set them to zero, yield a set of algebraic equations which can be solved by using the Maple or Mathematical to find a0 , ai , bi , c1 , q , r , c .
Step 5: It is well-known40 that (3.5), (3.6) have the following Jacobi elliptic function solutions:
f1(ξ)=1sn(ξ,m) , g1(ξ)=−cn(ξ,m)dn(ξ,m)sn(ξ,m).
f2(ξ)=1cn(ξ,m) , g2(ξ)=sn(ξ,m)dn(ξ,m)cn(ξ,m).
f3(ξ)=1dn(ξ,m), g3(ξ)=m2sn(ξ,m)cn(ξ,m)dn(ξ,m).
f4(ξ)=sn(ξ,m) , g4(ξ)=cn(ξ,m)dn(ξ,m)sn(ξ,m) .
f5(ξ)=cn(ξ,m), g5(ξ)=−sn(ξ,m)dn(ξ,m)cn(ξ,m).
f6(ξ)=dn(ξ,m), g6(ξ)=−m2sn(ξ,m)cn(ξ,m)dn(ξ,m).
f7(ξ)=cn(ξ,m)sn(ξ,m), g7(ξ)=−dn(ξ,m)sn(ξ,m)cn(ξ,m).
f8(ξ)=dn(ξ,m)sn(ξ,m), g8(ξ)=−cn(ξ,m)sn(ξ,m)dn(ξ,m).
f9(ξ)=sn(ξ,m)cn(ξ,m), g9(ξ)=dn(ξ,m)sn(ξ,m)cn(ξ,m).
f10(ξ)=dn(ξ,m)cn(ξ,m), g10(ξ)=(1−m2)sn(ξ,m)cn(ξ,m)dn(ξ,m).
f11(ξ)=sn(ξ,m)dn(ξ,m), g11(ξ)=cn(ξ,m)sn(ξ,m)dn(ξ,m).
f12(ξ)=cn(ξ,m)±1sn(ξ,m) , g12(ξ)=∓ds(ξ,m).
f13(ξ)=dn(ξ,m)msn(ξ,m)±1, g13(ξ)=∓mcd(ξ,m).
f14(ξ)=mcn(ξ,m)+dn(ξ,m) , g14(ξ)=−msn(ξ,m).
f15(ξ)=cn(ξ,m)±dn(ξ,m)sn(ξ,m) , g15(ξ)=∓ns(ξ,m).
f16(ξ)=sn(ξ,m)msn2(ξ,m)−1 , g16(ξ)=−cs(ξ,m)dn(ξ,m)[msn2(ξ,m)+1msn2(ξ,m)−1].
f17(ξ)=sn(ξ,m)msn2(ξ,m)+1 , g17(ξ)=−cs(ξ,m)dn(ξ,m)[msn2(ξ,m)−1msn2(ξ,m)+1].
f18(ξ)=cn(ξ,m)sn(ξ,m)±1 , g18(ξ)=∓dc(ξ,m).
f19(ξ)=dn(ξ,m)±1sn(ξ,m) , g19(ξ)=∓cs(ξ,m).
f20(ξ)=sn(ξ,m)1±cn(ξ,m) , g20(ξ)=±ds(ξ,m).
f21(ξ)=sn(ξ,m)cn(ξ,m)±dn(ξ,m) , g21(ξ)=±ns(ξ,m).
Where sn(ξ,m), cn(ξ,m) and dn(ξ,m) are Jacobi elliptic sine function, Jacobi elliptic cosine function, Jacobi elliptic function of the third kind respectively, and m denotes the modulus of Jacobi elliptic functions, where 0≤m≤1 . It is well-known54−56 that the Jacobi-elliptic functions satisfy the following relations:
cn2(ξ,m)=1−sn2(ξ,m),dn2(ξ,m)=1−m2sn2(ξ,m),
sn'(ξ,m)=cn(ξ,m)dn(ξ,m),cn'(ξ,m)=−sn(ξ,m)dn(ξ,m),
dn'(ξ,m)=−m2sn(ξ,m)cn(ξ,m).
ns(ξ,m)=1sn(ξ,m),nc(ξ,m)=1cn(ξ,m),nd(ξ,m)=1dn(ξ,m),
sc(ξ,m)=sn(ξ,m)cn(ξ,m),sd(ξ,m)=sn(ξ,m)dn(ξ,m),cs(ξ,m)=cn(ξ,m)sn(ξ,m),
cd(ξ,m)=cn(ξ,m)dn(ξ,m),ds(ξ,m)=dn(ξ,m)sn(ξ,m),dc(ξ,m)=dn(ξ,m)cn(ξ,m),
The Jacobi elliptic functions degenerate into hyperbolic functions when m→1 as follows:
sn(ξ,1)→tanh(ξ),cn(ξ,1)→sech(ξ),dn(ξ,1)→sech(ξ),ns(ξ,1)→coth(ξ),dc(ξ,1)→1,
ds(ξ,1)→cosech(ξ),sc(ξ,1)→sinh(ξ),sd(ξ,1)→sinh(ξ),cs(ξ,1)→cosech(ξ),
And into trigonometric functions when m→0 as follows:
sn(ξ,0)→sin(ξ),cn(ξ,0)→cos(ξ),dn(ξ,0)→1,ns(ξ,0)→cosec(ξ),cs(ξ,0)→cot(ξ),
ds(ξ,0)→cosec(ξ),sc(ξ,0)→tan(ξ),sd(ξ,0)→sin(ξ),dc(ξ,0)→sec(ξ)
Step 6: We substitute the values a0 , ai bi and the solutions (1)−(21) given in step 5 into (3.4), to get the exact solutions of Equation (3.1).
In this section, we apply the method of section 3, to solve Equation (1.1).To this aim, we first use the conformable space-time wave transformation:
u(x,t)=u(ξ), ξ=xββ−c1tαα, (4.1)
Where c1 is a non zero constant and 0<α,β≤1 , to reduce Equation (1.1) into the following nonlinear ordinary differential equation (ODE):
c21u''−c21u''''−(α1u+β1un+1+γ1u2n+1)''=0, (4.2)
Integrating Equation (4.2) twice with respect to ξ and vanshing the constants of integration, we get
(c21−α1)u−c21u''−β1un+1−γ1u2n+1=0, (4.3)
Balancing u'' with u2n+1 , we get N=1n . According to (3.9) we use the transformation:
u(ξ)=v1n(ξ), (4.4)
Where v(ξ) is a new function of ξ , to reduce Equation (4.3) into the new ODE:
(c21−α1)n2v2−c21nvv''−c21(1−n)v'2−β1n2v3−γ1n2v4=0, (4.5)
Balancing vv'' with v4 in Equation (4.5), we get N=1 . According to the form (3.4), Equation (4.5) has the formal solution:
v(ξ)=a0+a1f(ξ)+b1g(ξ), (4.6)
Where a0 , a1 , b1 are constants to be determined, such that a1≠0 or b1≠0 . Substituting (4.6) along with (3.5)−(3.7) into Equation (4.5) and collecting all terms of the same order of fi(ξ) , gj(ξ),(i,j=0,1,2,...) and setting them to zero, we have the following algebraic equations:
{f4(ξ):6n2γ1a21b21c+c21a21c−c21b21c2+nc21a21c−n2γ1a41−n2γ1b41c2−c21nb21c2=0, f−4(ξ):c21r2b21+c21nr2b21+n2γ1r2b41=0, f3(ξ):−n2β1a31+3n2β1a1b21c+2nc21a0a1c+12n2γ1a0a1b21c−4n2γ1a0a31=0, f2(ξ):c21a21q−n2c21b21c+6n2γ1a20b21c−6n2γ1a20a21+n2α1b21c−3n2β1a0a21+n2 c21a21 −n2 α1a21 :−2n2γ1b41qc+6n2γ1a21b21q−2nc21b21qc+3n2β1a0b21c=0, f(ξ): 3n2β1a1qb21−4n2γ1a30a1+ nc21a0qa1+2n2c21a1a0+12n2γ1a0a1qb21 :−3n2β1a20a1−2n2α1a0a1=0,f−1(ξ): 3n2β1a1rb21+12n2γ1a0a1rb21=0, (4.7) f0(ξ): 2c21rb21c−2n2c21b21q−nc21a21r+2n2α1b21q−2n2γ1b41q2−6nc21rb21c+12n2γ1a20b21q :6n2β1a0b21q+6n2γ1a21b21r−2n2γ1b41rc+2n2c21a20−2n2α1a20−2n2β1a30−2n2γ1a40+c21a21r=0,f2(ξ)g(ξ):2nc21a0b1c−3n2β1a21b1+n2β1b31c+4n2γ1a0b31c−12n2γ1a0a21b1=0, f−2(ξ)g(ξ): 2nc21a0rb1+n2β1rb31+4n2γ1a0rb31=0, f3(ξ)g(ξ): 2nc21a1b1c+2c21a1b1c−4n2γ1a31b1+4n2γ1a1b31c=0, g(ξ): 2n2c21a0b1+n2β1b31q−2n2α1a0b1−4n2γ1a30b1+4n2γ1a0b31q−3n2β1a20b1=0, f(ξ)g(ξ):−2n2α1a1b1+2n2c21a1b1+4n2γ1a1b31q−6n2β1a0a1b1−12n2γ1a20a1b1+nc21b1a1q=0, f−1(ξ)g(ξ):2nc21a1rb1+2n2γ1a1rb31−c21a1rb1=0, }
According to Step5 of Section 3, we have the following results:
Result 1: If we substitute q=(1+m2),r=−2m2,c=−1 into the algebraic equations (4.7) and use the Maple, we have
{ m=m,n=1,β1=0,a0=0,a1=0,b1=√2α1γ1(1+2m2),c1=√−α1(1+2m2),α1<0,γ1<0 } (4.8)
or{m=m,n=2,α1=1564β21(m2−1)γ1m2,a0=−3β18γ1,a1=3β18mγ1,b1=0,c1=β14m√−3γ1,γ1<0} (4.9)
Substituting (4.8) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)=−√2α1γ1(1+2m2)(cn(ξ,m)dn(ξ,m)sn(ξ,m)), α1<0,γ1<0 (4.10)
Where ξ=xββ−(√−α1(1+2m2))tαα . If m→0 , then we have the periodic solution:
u(x,t)=−√2α1γ1[cot(xββ−√−α1tαα)], (4.11)
while if m→1 , then we have the solitary wave solution:
u(x,t)=−√2α13γ1[coth(xββ−√−α13tαα)−tanh(xββ−√−α13tαα)]. (4.12)
Substituting (4.9) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)={−3β18γ1(1−1msn(ξ,m))}12, γ1<0,β1>0 (4.13)
where ξ=xββ−(β14m√−3γ1)tαα . If m→1 , then we have the singular soliton solution:
u(x,t)={−3β18γ1[1−coth(xββ−β14√−3γ1tαα)]}12. (4.14)
Result 2: If we substitute q=(1−2m2), r=2m2, c=(m2−1) into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=2,a0=−3β18γ1,a1=3β18γ1,c1=β14√3γ1(m2−1),α1=3β21(4m2−1)64γ1(m2−1),b1=0,γ1(m2−1)>0,} (4.15)
Substituting (4.15) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)={−3β18γ1(1−1cn(ξ,m))}12, γ1β1<0, (4.16)
where ξ=xββ−(β14√3γ1(m2−1))tαα. If m→0 , then we have the periodic solution:
u(x,t)={−3β18γ1[1−sec(xββ−β14√−3γ1tαα)]}12, β1γ1<0. (4.17)
Result 3: If we substitute q=(−2+m2), r=2, c=(1−m2) into the algebraic equations (4.7) and use the Maple, we have
{ m=m,n=1,β1=0,a0=0,a1=0,b1=√2α1γ1(2m2−5),c1=√−α1(2m2−5),α1>0,γ1<0,} (4.18)
Substituting (4.18) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)=m2√2α1γ1(2m2−5)(sn(ξ,m)cn(ξ,m)dn(ξ,m)), α1>0,γ1<0 (4.19)
where ξ=xββ−(√−α1(2m2−5))tαα. If m→1 , then we have the dark soliton solution:
u(x,t)=√−2α13γ1[tanh(xββ−√α13tαα)]. (4.20)
Result 4: If we substitute q=(1+m2), r=−2, c=−m2 into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=2,a0=−3β18γ1,b1=0,a1=−3β18γ1,α=1564β21(m2−1)γ1m2,c1=β14m√−3γ1,γ1<0 } (4.21)
Substituting (4.21) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)={−3β18γ1(1+sn(ξ,m))}12, γ1<0,β1>0 (4.22)
where ξ=xββ−(β14m√−3γ1)tαα. If m→1 , then we have the dark soliton solution:
u(x,t)={−3β18γ1[1+tanh(xββ−β14√−3γ1tαα)]}12. (4.23)
Result 5: If we substitute q=(1−2m2), r=(−2+2m2),c=m2 into the algebraic equations (4.7) and use the Maple, we have
{n=1,a0=−β13γ1,a1=0,b1=−β13γ1√2m2−1,c1=β13√−12γ1(2m2−1),α1=β21(8m2−5)18γ1(2m2−1),γ1(2m2−1)<0.} (4.24)
Substituting (4.24) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)=−β13γ1[1−1√2m2−1(sn(ξ,m)dn(ξ,m)cn(ξ,m))] (4.25)
where ξ=xββ−(β13√−12γ1(2m2−1))tαα, γ1>0 or γ1<0. If m→1 , then we have the dark soliton solution:
u(x,t)=−β13γ1[1−tanh(xββ−β13√−12γ1tαα)]. γ1<0. (4.26)
Result 6: If we substitute q=(−2+m2), r=(2−2m2), c=1 into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=2,α1=364β21(m2+8)γ1,a0=−3β18γ1,a1=−3β18γ1,b1=0,c1=β14√3γ1,γ1>0 } (4.27)
Substituting (4.27) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)={−3β18γ1(1+dn(ξ,m))}12, γ1>0,β1<0. (4.28)
where ξ=xββ−(β14√3γ1)tαα. If m→1 , then we have the bright soliton solution:
u(x,t)={−3β18γ1[1+sech(xββ−β14√3γ1tαα)]}12. (4.29)
Result 7: If we substitute q=(−2+m2), r=(−2+2m2), c=−1 into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=2,b1=3β18m2γ1(−1+√1−m2),a0=−3β18γ1,c1=β14m2√3γ1(m2−2+2√1−m2) ,α1=3β2116m4γ1((m2−2+2√1−m2)(−m4−3m2+4+(m2−4)√1−m2)(−1+√1−m2)2), a1=0,(m2−2+2√1−m2)γ1>0,} (4.30)
Substituting (4.30) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)={−3β18γ1[1+(−1+√1−m2m2)(dn(ξ,m)sn(ξ,m)cn(ξ,m))]}12, γ1β1<0. (4.31)
where ξ=xββ−(β14m2√3γ1(m2−2+2√1−m2))tαα. If m→1 , then we have the same solitary wave solution (4.14).
Result 8. If we substitute q=(1−2m2), r=(2m2−2m4), c=−1 into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=1,β1=0,a0=0,b1=0,a1=√α1γ1(m2−1),c1=√−α12(m2−1),α1(m2−1)<0,γ1<0} (4.32)
Substituting (4.32) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)=√α1γ1(m2−1)(dn(ξ,m)sn(ξ,m)), (4.33)
where ξ=xββ−(√−α12(m2−1))tαα. If m→0 , then we have the periodic solution:
u(x,t)=√−α1γ1[cosec(xββ−√α12tαα)]. (4.34)
Result 9: If we substitute q=(−2+m2), r=−2, c=(−1+m2) into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=1,β1=0,a0=0,a1=0,b1=√2α1γ1(2m2−5),c1=√−α1(2m2−5),α1>0,γ1<0} (4.35)
Substituting (4.35) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)=√2α1γ1(2m2−5)(dn(ξ,m)sn(ξ,m)cn(ξ,m)), α1>0,γ1<0 (4.36)
where ξ=xββ−(√−α1(2m2−5))tαα. If m→1 , then we have the singular soliton solution:
u(x,t)=√−2α13γ1[coth(xββ−√α13tαα)], (4.37)
while if m→0 , then we have the periodic solution:
u(x,t)=√−2α15γ1[cosec(xββ−√α15tαα)sec(xββ−√α15tαα)]. (4.38)
Result 10: If we substitute q=(1+m2), r=−2m2, c=−1 into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=2,α1=−3β2164γ1(m2−1),a0=−3β18γ1,a1=−3β18γ1,b1=0,c1=β14√−3γ1,γ1<0} (4.39)
Substituting (4.39) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)={−3β18γ1[1+dn(ξ,m)cn(ξ,m)]}12, β1>0,γ1<0 (4.40)
where ξ=xββ−(β14√−3γ1)tαα. If m→0 , then we have the periodic solution:
u(x,t)={−3β18γ1[1+sec(xββ−β14√−3γ1tαα)]}12. (4.41)
Result 11: If we substitute q=(1−2m2), r=−2, c=(m2−m4) into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=1,β1=0,a0=0,a1=0,b1=√−2α1γ1(4m2−1),c1=√α1(4m2−1),(4m2−1)α1>0,γ1<0,} (4.42)
Substituting (4.42) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)=√−2α1γ1(4m2−1)[cn(ξ,m)sn(ξ,m)dn(ξ,m)], ξ=xββ−(√α1(4m2−1))tαα, (4.43)
If m→0 , then we have the same periodic solution (4.11), while if m→1 , then we have the same singular soliton solution (4.37).
Result 12: If we substitute q=12(−1+2m2),r=−12,c=−14 into the algebraic equations (4.7) and use the Maple, we have:
{m=m,n=1,a0=0,a1=0,β1=0,b1=√α1γ1(m2−1),c1=√−α12(m2−1),α1(m2−1)<0,γ1<0} (4.44)
Substituting (4.44) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)=∓[√α1γ1(m2−1)ds(ξ,m)], (3.45)
where ξ=xββ−(√−α12(m2−1))tαα. If m→0 , then we have the same periodic solution (4.34).
Result 13: If we substitute q=−12(m2+1),r=12(1−m2),c=14(1−m2) into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=1,a0=0,b1=0,β1=0,a1=√α1γ1,c1=√−2α1(m2−1),α1(m2−1)<0,γ1α1>0} (4.46)
Substituting (4.46) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)=√α1γ1[dn(ξ,m)msn(ξ,m)±1], (4.47)
where ξ=xββ−(√−2α1(m2−1))tαα.
Result 14: If we substitute q=−12(1+m2), r=12(m2−1)2,c=14 into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=1,α=β21(1+2m2)9γ1(m2+1),a0=−β13γ1,a1=0,b1=−β13γ1√2(m2+1),c1=β13√−1γ1(m2+1),γ1<0} (4.48)
Substituting (4.48) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)=−β13γ1[1−m√2(m2+1)sn(ξ,m)], (4.49)
where ξ=xββ−(β13√−1γ1(m2+1))tαα. If m→1 , then we have the same dark soliton solution (4.26).
Result 15: If we substitute q=−12(1+m2),r=−12(m2−1)2,c=−14 into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=2,α1=−3β21(m2−1)64γ1,a0=−3β18γ1,a1=0,b1=−3β18γ1,c1=β14√−3γ1,γ1<0} (4.50)
Substituting (4.50) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)={−3β18γ1[1∓ns(ξ,m)]}12, β1>0,γ1<0. (4.51)
where ξ=xββ−(β14√−3γ1)tαα. If m→1, then we have the singular soliton solution:
u(x,t)={−3β18γ1[1∓coth(xββ−β14√−3γ1 tαα)]}12, (4.52)
while if m→0, then we have the periodic solutions.
u(x,t)={−3β18γ1[1∓cosec(xββ−β14√−3γ1 tαα)]}12. (4.53)
Result 16: If we substitute q=(m2−6m+1), r=−2, c=4m (m−1)2 into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=1,β1=0,a0=0,a1=0,b1=√2α1γ1(2m2−12m+1),c1=√−α1(2m2−12m+1),α1(2m2−12m+1)<0,γ1<0} (4.54)
Substituting (4.54) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)=√2α1γ1(2m2−12m+1)[cs(ξ,m)dn(ξ,m)(msn2(ξ,m)+1msn2(ξ,m)−1)], (4.55)
wher ξ=xββ−(√−α1(2m2−12m+1))tαα. If m→0 , then we have the same periodic solution (4.11), while if m→1, then we have the solitary wave solutions:
u(x,t)=−√−2α19γ1[coth(xββ−√α13tαα)+tanh(xββ−√α13tαα)]. (4.56)
Result 17: If we substitute q=(m2+6m+1), r=−2, c=−4m(m+1)2 into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=2,a0=−3β18γ1,a1=−3β18γ1(m+1),b1=0,α1=−3β21(m2−18m+5)256mγ1,c1=β18√−3mγ1,γ1<0} (4.57)
Substituting (4.57) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)={−3β18γ1[1+(m+1)sn(ξ,m)msn2(ξ,m)+1]}12, β1>0,γ1<0 (4.58)
where ξ=xββ−(β18√−3mγ1)tαα. If m→1 , then we have the solitary wave solution:
u(x,t)={−3β18γ1[1+2tanh(xββ−β18√−3γ1 tαα)1+tanh2(xββ−β18√−3γ1 tαα)]}12. (4.59)
Result 18: If we substitute q=−12(1+m2), r=12(m2−1), c=14(m2−1) into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=2,a0=−3β18γ1,b1=−3β18γ1,a1=0,α1=−3β21(m2−1)64γ1,c1=β14√−3γ1,γ1<0} (4.60)
Substituting (4.60) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)={−3β18γ1[1∓dc(ξ,m)]}12, β1>0,γ1<0 (4.61)
where ξ=xββ−(β14√−3γ1)tαα. If m→0 , then we have the same periodic solutions (4.17) and (4.41) respectively.
Result 19: If we substitute q=12(2−m2),r=−12m4,c=−14 into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=2,a0=−3β18γ1,a1=3β18γ1m2(−1+√1−m2),c1=β12m2√3m2+6√1−m2−6γ1 α1=−3β2116γ1m4(((10−m2)√1−m2+m4+6m2−10)(m2−2+2√1−m2)(−1+√1−m2)2),b1=0,(3m2+6√1−m2−6)γ1>0, } (4.62)
Substituting (4.62) into (4.4), (4.6),we have the Jacobi elliptic solutions:
u(x,t)={−3β18γ1[1−(−1+√1−m2m2)(dn(ξ,m)±1sn(ξ,m))]}12,β1γ1<0 (4.63)
where ξ=xββ−(β12m2√3m2+6√1−m2−6γ1)tαα. If m→1 , then we have the solitary wave solutions:
u(x,t)={−3β18γ1[1+cosech(xββ−β12√−3γ1tαα)±coth(xββ−β12√−3γ1tαα)]}12. (4.64)
Result 20: If we substitute q=12(2m2−1), r=−12, c=−14 into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=1,a0=0,b1=0,β1=0,a1=√−α1γ1(2m2+1),c1=√2α1(2m2+1),α1>0,γ1<0 } (4.65)
Substituting (4.65) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)=√−α1γ1(2m2+1)(sn(ξ,m)1±cn(ξ,m)), α1>0,γ1<0 (4.66)
where ξ=xββ−(√2α1(2m2+1))tαα. If m→1 , then we have the solitary wave solution:
u(x,t)=√−α13γ1[tanh(xββ−√2α13tαα)1±sech(xββ−√2α13tαα))], (4.67)
while if m→0 , then we have the periodic solution:
u(x,t)=√−α1γ1[tan(xββ−√2α1tαα)sec(xββ−√2α1tαα)±1]. (4.68)
Result 21: If we substitute q=−12(1+m2), r=−12, c=−14(m2−1)2 into the algebraic equations (4.7) and use the Maple, we have
{m=m,n=1,a0=−β13γ1,a1=0,b1=−β13γ1√2(1+m2),c1=β13√−1γ1(1+m2),α1=β21(1+2m2)9γ1(1+m2),γ1<0,} (4.69)
Substituting (4.69) into (4.4), (4.6), we have the Jacobi elliptic solutions:
u(x,t)=−β13γ1[1±√2(1+m2)ns(ξ,m)], (4.70)
where ξ=xββ−(β13√−1γ1(1+m2))tαα. If m→1 , then we have the singular soliton solution:
u(x,t)=−β13γ1[1±coth(xββ−β13√−12γ1tαα)], (4.71)
while if m→0 , then we have the periodic solution:
u(x,t)=−β13γ1[1±√2cosec(xββ−β13√−1γ1tαα)]. (4.72)
In this section, we present some graphs of the solitons and other solutions of Eq.(1.1). Let us now examine Figures (1 - 12) as it illustrates some of our solutions obtained in this article. To this aim, we select some special values of the parameters obtained for example, in some of the solutions of (4.10), (4.12), (4.22), (4.23), (4.28), (4.37), (4.41), (4.56), (4.58), (4.64), (4.67) and (4.72) of the conformable space-time fractional fourth-order Pochhammer-Chree equation (1.1). For more convenience the graphical representations of these solutions are shown in the following figures.
From the above Figures, one can see that the obtained solutions possess the Jacobi elliptic solutions, the conformable fractional solitary wave solution, the Jacobi elliptic conformable fractional solution and the solitary wave solutions . Also, these Figures expressing the behaviour of these solutions which give some perspective readers how the behaviour solutions are produced.
We have derived many Jacobi elliptic function solutions, the solitary wave solutions, singular solitary wave solutions and the trigonometric function solutions of the conformable space-time fractional fourth-order Pochhammer-Chree equation (1.1) using the unified sub-equation method combined with the conformable space-time fractional derivatives described in Sec. 3. On comparing our results in this article with that obtained in42–50 using different methods, we conclude that the Jacobi elliptic solutions obtained in our article are new, while some solitary wave solutions, singular solitary wave solutions and the trigonometric function solutions obtained in our article are equivalent to that obtained in42–50. From these discussions, we conclude that the proposed method of Sec.3, is direct, concise and effective powerful mathematical tools for obtaining the exact solutions of other nonlinear evolution equations. Finally, our results in this article have been checked using the Maple by putting them back into the original equation (1.1).57
None.
The author declares that there is no conflict of interest.
©2018 Zayed, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.