# An Extension of the DoubleG′/G,1/G-Expansion Method for Conformable Fractional Differential Equations

Authors: Altaf A. Al-Shawba; Farah A. Abdullah; Amirah Azmi; M. Ali Akbar
Journal: Complexity
 (2020)

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## Abstract

The phenomena, molecular path in a liquid or a gas, fluctuating price stoke, fission and fusion, quantum field theory, relativistic wave motion, etc., are modeled through the nonlinear time fractional clannish random Walker’s parabolic (CRWP) equation, nonlinear time fractional SharmaTassoOlver (STO) equation, and the nonlinear space-time fractional KleinGordon equation. The fractional derivative is described in the sense of conformable derivative. From there, theG′/G,1/G-expansion method is found to be ensuing, effective, and capable to provide functional solutions to nonlinear models concerning physical and engineering problems. In this study, an extension of the G′/G,1/G-expansion method has been introduced. This enhancement establishes broad-ranging and adequate fresh solutions. In addition, some existing solutions attainable in the literature also confirm the validity of the suggested extension. We believe that the extension might be added to the literature as a reliable and efficient technique to examine a wide variety of nonlinear fractional systems with parameters including solitary and periodic wave solutions to nonlinear FDEs.


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## Body

## 1. Introduction

The subject FDEs can be considered as a generalization of the typical ordinary differential equations (ODEs). The advantages of the FDEs become apparent for us to understand real world problems. In fact, the FDEs have attracted significant attention to the past couple of decades due to their relatively much effectiveness than ODEs. There are several definitions of fractional derivatives, as for instance, the RiemannLiouville derivative, the Caputo derivative, and the conformal fractional derivative [1–3]. Recently, Khalil et al. [4] proposed a compatible definition of fractional derivative called the conformable fractional derivative. Therefore, several properties related to this new definition have been studied. Jarad et al. [5] used conformable type derivatives defined in [6] to generate new type of generalized fractional derivatives with memory effect. In [7, 8], the authors used conformable derivatives and integrals to formulate new generalized Liapunov-type inequality. The exact solutions of nonlinear FDEs play fundamental role in describing different qualitative and quantitative features of nonlinear complex physical phenomena. For this reason, researchers have proposed different methods to obtain exact solutions to nonlinear FDEs such as exp−φξ-expansion method [9–12], the G′/G-expansion method [13–19], auxiliary equation method [20, 21], G′/G,1/G-expansion method [22–27] the trial equation method [28, 29], fractional subequation method [30, 31], modified simple equation method [32], generalized Kudrayshov method [33, 34], and others [35–38].In this study, we introduce an extension of theG′/G,1/G-expansion method for analyzing nonlinear FDEs in mathematical physics, engineering, and applied mathematics. To demonstrate the reliability and advantages of the suggested extension, the time-fractional CRWP equation, the time fractional STO equation, and the space-time fractional KleinGordon equation are examined and further broad-ranging and new families of exact wave solutions are established. This new extension can be applied to further nonlinear FDEs which can be done in forthcoming work.


## 2. Conformable Fractional Derivative and Its Important Properties

In the last years, Khalil et al. [4] introduced a simple, interesting, and compatible with typical definition of derivative named conformable fractional derivative, which can rectify the deficiencies of the other definitions. One can also find several useful studies related to this new definition in [5–8]. In [39], the geometrical and physical interpretations of this definition are investigated and the potential applications in science and engineering are pointer out. The conformable fractional derivative of a function g of order β is defined as(1)Tβgt=limx⟶0gt+xt1−β−gtx,where g:0,∞⟶R,t>0 and β∈0,1. Some important properties of above definition are given below:(i)
Tβag+bf=aTβg+bTβf,∀a,b∈R(ii)
Tβtμ=μtμ−β,∀μ∈R(iii)
Tβλ=0, for each constant function gt=λ(iv)
Tβgοft=t1−βf′tg′ft


## 3. Methodology

In this section, we will suggest an extension of theG′/G,1/G-expansion method to ascertain the analytic solutions to nonlinear FDEs. We begin with considering the second-order linear ODE:(2)G″ξ+λGξ−μ=0.We choose(3)ϕ=G′G,ψ=1G′.Thus, from equation (2) and (3), it is found(4)1/ϕ′=λ1ϕ2−μψ1ϕ+1,ψ′=λψ1ϕ−μψ2.From the different general solutions of the linear ODE (2), we attain the following:Case 1: whenλ<0, the hyperbolic function solution is(5)Gξ=A1sinh−λξ+A2cosh−λξ+μλ,and thus we obtain(6)ψ′=−λλ2σ+μ2λ1ϕ2−2μψ1ϕ+1,whereσ=A12−A22 and A1,A2 are arbitrary constants.Case 2: whenλ>0, the trigonometric function solution is(7)Gξ=A1sinλξ+A2cosλξ+μλ,and corresponding relation is(8)ψ2=λλ2σ−μ2λ1/ϕ2−2μψ1/ϕ+1,whereσ=A12+A22 and A1 and A2 are arbitrary constants.Case 3: whenλ=0, the rational function solution is(9)Gξ=μ2ξ2+A1ξ+A2,  and thus it is found(10)ψ2=1A12−2μA2−2μψ1ϕ+1,where A1 and A2 are arbitrary constants.Let us consider a general nonlinear FDE:(11)Pu,Dtαu,Dxαu,Dyαu,DtαDtαu,=0,0<α<1,where Dtαu,Dxαu and Dyαu are the conformable fractional derivatives of the wave function u with respect to spatial variables xandy and the temporal variable t, and P is a polynomial of u=ux,y,t,… and its various partial derivatives.The main steps of the new extension of theG′/G,1/G-expansion method to seek exact solutions of nonlinear FDEs are as follows:Step 1: we estimate the new form of the fractional wave variable:(12)ux,y,t=Uξ,ξ=kxαα+lyαα±vtαα,wherekandl are nonzero constants and v is the wave velocity to be determined later. The complex wave transformation (12) translates equation (11) into an ODE as follows:(13)PU,U′,U″,…=0.Step 2: suppose that the general solution of nonlinear ODE (13) can be expressed by a polynomial in 1/ϕ and ψ as(14)Uξ=∑i=0Nai1ϕi+∑i=0Nbi1ϕiψ,whereG=Gξ satisfies the auxiliary linear ODE (2) and ai and bi are arbitrary constants to be determined later, and balancing the highest order derivative with the nonlinear terms in equation (13), we can find the positive integer N.Step 3: inserting solution (14) into equation (13) and utilizing (4) and (6) (here case 1 is selected as an example), the left-hand side of equation (13) can be converted into a polynomial in 1/ϕ and ψ, where the degree of ψ is not more than one. Equating all coefficients of this polynomial to zero yield a set of algebraic equations for ai,bi,k,l,v,A1,A2,λλ<0, and μ.Step 4: the solution of the algebraic equations found in the step 3 can be found with the aid of Maple software package. Making use of the values ofai,bi,k,l,v,A1,A2,λ, and μ into (14), we might determine exact solutions expressed by hyperbolic functions of equation (13).Step 5: similar to step 3 and step 4, substituting (14) into equation (13) and utilizing equation (4) and (6) (or equation (4) and (10)), we can get the exact solutions of equation (13) expressed by the trigonometric functions (or expressed by the rational functions).


## 4. Determination of Solutions

In this paragraph, we will search out solutions to three conformable FDEs as appertain of the new extension of theG′/G,1/G-expansion method.


### 4.1. Time Fractional CRWP Equation

First, we consider the time fractional CRWP equation [22, 40]:(15)Dtαu−ux+2uux+uxx=0,0<α≤1.The ensuing wave transformation is(16)ux,t=Uξ,ξ=kx−ctαα,where k is the wave number and c is the velocity; reducing the equation (15) into the subsequent ODE, we get(17)−c+kU′+2kUU′+k2U″=0.Integrating (17) once, we find(18)−c+kU+kU2+k2U′+ξ0=0,where ξ0 is a constant of integration. It is clear that the homogeneous balance between U2 and U′ present in equation (18) gives N=1. Therefore, the shape of the exact solution of equation (18) is(19)Uξ=a0+a11ϕ+b1ψ.There are three cases to be considered:Case 1: whenλ<0, substituting solution (19) into equation (18) and utilizing (4) and (6), equation (18) can be transmuted to a polynomial in 1/ϕ and ψ. Equalizing its coefficients to zero yields a set of algebraic equations in a0, a1, b1, k, c, σ, λ, μ, and ξ0:(20)1ϕ2:b1k2μλ2λ2σ+μ2−b12kλ2λ2σ+μ2+a12k+a1k2λ=0,1ϕψ:2b12kλμλ2σ+μ2−2b1k2μ2λλ2σ+μ2−a1k2μ+b1k2λ+2a1b1k=0,1ϕ:a1k−a1c+2a0a1k=0,ψ:b1k+2a0b1k−b1c=0,ψ0:b1k2μλλ2σ+μ2−b12kλλ2σ+μ2+a02k−a0c+a0k+a1k2+ξ0=0.Resolving these algebraic equations by Maple software package, we attain the following values:(21)a0=c+k2k,a1=−kλ,b1=μk,ξ0=4k4λ+c2+2ck+k24k,(22)a0=c+k2k,a1=−12kλ,b1=12kμ∓λ−σ,ξ0=k4λ+c2+2ck+k24k,wherek and c are free constants.Inserting the values of the parameters assembled in (21) into solution (19) along with (6) and (16), the following wide-ranging hyperbolic function solution is ascertained:(23)u1x,t=c+k2k+k−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+μκ−λ1A1cosh−λξ+A2sinh−λξ.In particular, if we setA1=0, A2≠0, and μ=0 (or A2=0, A1≠0 and μ=0) into the solution (23), we obtain the successive singular kink and kink soliton solutions to the CRWP equation (15), respectively:(24)u11=c+k2k+k−λcoth−λξ,(25)u12=c+k2k+k−λtanh−λξ,whereξ=kx−ctα/α.Furthermore, by means of the parameter values gathered in (22), we derive(26)u2x,t=c+k2k+12k−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+κ2−λμ∓λ−σA1cosh−λξ+A2sinh−λξ.SinceA1 and A2 are integral constants, one might select their values spontaneously. Thus, if we select A1=0, A2>0, and μ=0 (or A2=0, A1>0, and μ=0) into solution (26), we gain the following solitary wave solutions to the CRWP equation, respectively:(27)u21=c+k2k+12k−λcoth−λξ±csch−λξ,(28)u22=c+k2k+12k−λtanh−λξ±isech−λξ,whereξ=kx−ctα/α.Case 2: whenλ>0, embedding solution (19) int1/ϕo (18) and putting in use (4) and (8), equation (18) becomes polynomial in and ψ. Computing the action similar to case 1 and after resolving the algebraic equations, we ascertain the following values:(29)a0=c+k2k,a1=−kλ,b1=μk,ξ0=4k4λ+c2+2ck+k24k,(30)a0=c+k2k,a1=−12kλ,b1=12kμ±λσ,ξ0=k4λ+c2+2ck+k24k,wherek and c are arbitrary constants.Substituting the values scheduled in (29) into solution (19) along with (8) and (16), the succeeding trigonometric solution to the CRWP equation (15) is established:(31)u3x,t=c+k2k−kλA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+μkλ1A1cosλξ−A2sinλξ.SinceA1 and A2 are free constants, we may set A1=0,A2≠0, and μ=0 (or A2=0,A1≠0, and μ=0) into solution (31), we derive the under mentioned periodic wave solutions to the CRWP equation:(32)u31=c+k2k+kλcotλξ,(33)u32=c+k2k−kλtanλξ,whereξ=kx−ctα/α.By means of the values organized in (30), we extract the following solution:(34)u4x,t=c+k2k−12kλA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+k2λμ±λσA1cosλξ−A2sinλξ.Now, if we setA1=0,A2>0, and μ=0 (or A2=0,A1>0, and μ=0) into solution (34), we obtain other periodic wave solutions to the CRWP equation:(35)u41=c+k2k+12kλcotλξ∓icscλξ,(36)u42=c+k2k−12kλtanλξ∓secλξ,whereξ=kx−ctα/α.Case 3: whenλ=0, introducing (19) into equation (18) and executing (4) and (10), equation (18) becomes a polynomial in 1/ϕ and ψ. Vanishing the coefficients yields a set of algebraic equations, and solving this system the following results are obtained:(37)a0=c+k2k,a1=0,b1=μk,ξ0=c+k24k.Setting the values gathered in (37) into solution (19) along with (10) and (16), we carry out the next rational function solution to the CRWP equation:(38)u5=c+k2k+kμμξ+A1,where ξ=kx−ctα/α.The obtained solutions can be compared with the solutions accessible in the literature. We detect that, by settingc=−k2λ+2a0k−k and μ=−λ2/4, the obtained solutions (24) and (25) fully agree with the corresponding solutions (3.19) and (3.20) established in [22], while the other solutions are different.


### 4.2. General Time Fractional STO Equation

The conformable general time fractional STO equation [41–43] is(39)Dtαu+3βux2+3βu2ux+3βuuxx+βuxxx=0,t>0,0<α≤1,where β is nonzero constant. The wave transformation is as follows:(40)ux,t=Uξ,ξ=x+ωtαα,where ω is the velocity of the travelling wave; converting equation (39) into an ODE, we get(41)ωU′+3βU′2+3βU2U′+3βUU″+βU′″=0.Integrating equation (41) once, we find(42)ωU+3βUU′+βU3+βU′=0.BalancingU′ and U3 obtained from equation (42), we get N=1 Thus, the solution structure of equation (42) is same of solution (19). Three cases as described in Section 3 will be discussed further:Case 1: whenλ<0, placing the solution (19) into (42) and utilizing (4) and (6), equation (42) modifies to a polynomial in 1/ϕ and ψ. Setting each coefficient of the polynomial to zero yields a set of algebraic equations in a0, a1, b1, σ, λ, μ, β, and ω. From these equations with the help of Maple algebra software, we find the ensuing results of constants:(43)a0=±12−λ,a1=−12λ,b1=12μ∓λ−σ,ω=βλ,(44)a0=0,a1=−12λ,b1=12μ∓λ−σ,ω=14βλ,(45)a0=0,a1=−λ,b1=μ∓λ−σ,ω=βλ,(46)a0=±−λ,a1=−λ,b1=μ,ω=4βλ,(47)a0=0,a1=−λ,b1=μ,ω=βλ,(48)a0=0,a1=−2λ,b1=2μ,ω=4βλ.By means of parameter values sorted out in (43), along with (6) and (40), the following hyperbolic function solutions of the general time fractional STO equation are obtained:(49)u1x,t=±12−λ+12−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+12−λμ∓λ−σA1cosh−λξ+A2sinh−λξ.In solution (49), A1 and A2 are free parameters; therefore, we freely can choose our values. Thus, if we choose A1=0, A2>0, and μ=0 or A2=0, A1>0, and μ=0, from solution (49), we obtain the under mentioned soliton solutions to the general time fractional STO equation:(50)u11=±12−λ1±coth−λξ+csch−λξ,(51)u12=±12−λ1±tanh−λξ±12λsech−λξ,whereξ=x+βλtα/α.Furthermore, by means of the values scheduled in (44), we establish(52)u2x,t=12−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+12−λμ∓λ−σA1cosh−λξ+A2sinh−λξ.Now, if we useA1=0, A2>0, and μ=0 or A2=0, A1>0, and μ=0, in (52), we attain the next soliton solutions to the general time fractional STO equation:(53)u21=12−λcoth−λξ±csch−λξ,(54)u22=12−λtanh−λξ±12λsech−λξ,whereξ=x+1/4βλtα/α.Moreover, by means of the values organized in (45), we carry out(55)u3x,t=−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+1−λμ∓λ−σA1cosh−λξ+A2sinh−λξ.If we considerA1=0,A2≠0, and μ=0 or A2=0,A1≠0, and μ=0, from solution (55), we derive the solitary wave solutions given underneath:(56)u31=−λcoth−λξ±csch−λξ,(57)u32=−λtanh−λξ±λsech−λξ,whereξ=x+βλtα/α.Similarly, making use of the values accumulated in (46), we determine(58)u4x,t=±−λ+−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+μ−λ1A1cosh−λξ+A2sinh−λξ.ForA1=0, A2≠0, and μ=0 or A2=0, A1≠0, and μ=0, the solution (58) generates the kink and singular kink solutions given underneath:(59)u41=±−λ1±coth−λξ,(60)u42=±−λ1±tanh−λξ,whereξ=x+4βλtα/α.Likewise, with the help of the parametric values amassed in (47), we achieve(61)u5x,t=−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+μ−λ1A1cosh−λξ+A2sinh−λξ.ChoosingA1=0, A2≠0, and μ=0 or A2=0, A1≠0, and μ=0, solution (61) turns into(62)u51=−λcoth−λξ,(63)u52=−λtanh−λξ,whereξ=x+βλtα/α.For the values gathered in (48), we found(64)u6x,t=2−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+2μ−λ1A1cosh−λξ+A2sinh−λξ.In particular, if we setA1=0,A2≠0, and μ=0 (or A2=0,A1≠0, and μ=0), the solution (64) reduces to(65)u61=2−λcoth−λξ,(66)u62=2−λtanh−λξ,whereξ=x+4βλtα/α.Case 2: whenλ>0, we established by completing the parallel course of algorithms to case 1 and the following values for the constants:(67)a0=±12iλ,a1=−12λ,b1=12μ±λσ,ω=βλ,(68)a0=0,a1=−12λ,b1=12μ±λσ,ω=14βλ,(69)a0=0,a1=−λ,b1=μ±λσ,ω=βλ,(70)a0=±iλ,a1=λ,b1=μ,ω=4βλ,(71)a0=0,a1=−λ,b1=μ,ω=βλ,(72)a0=0,a1=−2λ,b1=2μ,ω=4βλ.Substituting the sets of values ((67)–(72)) into solution (19) along with (8) and (40), we get the trigonometric function solutions for equation (39) as follows:By means of (67), we obtain(73)u7x,t=±12iλ−12λA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+12λμ±λσA1cosλξ−A2sinλξ.If we considerA1=0,A2>0, and μ=0 (or A2=0,A1>0, and μ=0) in (73), the periodic wave solutions of equation (39) are found as follows:(74)u71=±12λi±cotλξ−cscλξ,(75)u72=±12λi∓tanλξ+secλξ,where ξ=x+βλtα/α.Through the values of the parameters gathered in (68), we derive(76)u8x,t=−12λA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+12λμ±λσA1cosλξ−A2sinλξ.SinceA1 and A2 are subjective constants, one may pick their values freely. Thus, if we pick A1=0, A2≠0, and μ=0 (or A2=0, A1≠0, and μ=0), from (76), we found the periodic wave solutions of equation (39) as follows:(77)u81=12λcotλξ∓cscλξ,(78)u82=−12λtanλξ∓secλξ,where ξ=x+1/4βλtα/α.By virtue of the values scheduled in (69), we obtain(79)u9x,t=−λA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+1λμ±λσA1cosλξ−A2sinλξ.If we setA1=0, A2≠0, and μ=0 (or A2=0, A1≠0, and μ=0) into solution (79), we ascertain the following periodic wave solutions to the general time fractional STO equation:(80)u91=λcotλξ∓cscλξ,(81)u92=−λtanλξ∓secλξ,where ξ=x+βλtα/α.For the values accumulated in (70), we establish(82)u10x,t=±iλ−λA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+μλ1A1cosλξ−A2sinλξ.SettingA1=0, A2≠0, and μ=0 (or A2=0, A1≠0, and μ=0), the solution (82) transformed to(83)u101=±λi±cotλξ,(84)u102=±λi∓tanλξ,where ξ=x+4βλtα/α.Utilizing the values of the parameters arranged in (71), we obtain(85)u11x,t=−λA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+μλ1A1cosλξ−A2sinλξ.Now settingA1=0, A2≠0, and μ=0 (or A2=0 and μ=0), the solution (85) becomes(86)u111=λcotλξ,(87)u112=−λtanλξ,where ξ=x+βλtα/α.Embedding the parametric values compiled in (72), we find out(88)u12x,t=−2λA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+2μλ1A1cosλξ−A2sinλξ.Suppose thatA1=0,A2≠0, and μ=0 (or A2=0,A1≠0, and μ=0), then the solution (88) developed into(89)u121=2λcotλξ,(90)u122=−2λtanλξ,where ξ=x+4βλtα/α.From the obtained broad-ranging solutions, it is observed that setting definite values of the associated parameters, we manage to determine some particular solutions which coincide with those accessible in the literature and some fresh solutions are established. It is seen that, by puttingβ=d, the obtained solutions (53), (54), (77), and (78) completely agree with the solutions (4.8), (4.7), (4.11), and (4.10), respectively, found in [43]. In addition to these solutions, we found many more solutions that were not found in any other studies.


### 4.3. Space-Time Fractional KleinGordon Equation

In this subsection, we extract the closed form solutions to the space-time fractional KleinGordon equation. Let us consider the KleinGordon equation with space-time fractional order [44, 45]:(91)DtαDtαu−DxαDxαu−βu−γu3=0,t>0,0<α≤1,where βandγ are nonzero constants. We apply the following transformation for reducing equation (91) to an ODE:(92)ux,t=Uξ,ξ=kxαα−ctαα,where kandc are nonzero constants. Thus, the space-time fractional KleinGordon equation turns out as follows:(93)k2−c2U″+βU+γU3=0.BalancingU″ and U3 in equation (93), we found N=1. On account of this, the structure of the solution of equation (93) is identical to the shape of the solution of equation (19) and therefore has not been repeated. There are three cases should be discussed as described in Section 3:Case 1: whenλ<0, plugging in (19) into equation (93) and utilizing (4) and (6), equation (93) will be converted into a polynomial in 1/ϕ and ψ. Equalizing the coefficients of this polynomial to zero yields a set of algebraic equations for a0, a1, b1, c, k, σ, λ, μ, β, and γ. Resolving these equations with the assistance of computer algebra, like Maple software package, we found the following values for the constants:(94)a0=0,a1=±λβγ,b1=±μβλγ,c=±2λk2+β2λ,(95)a0=0,a1=0,b1=±2λβσγ,c=±λk2−βλ,(96)a0=0,a1=±λβγ,b1=±βλγμ±λ−σ,c=∓λk2+2βλ,wherek, β, γ, and μ are free parameters.Inserting the sets of constant values from (94) to (96) into solution (19) along with (6) and (92), the following hyperbolic function solutions to the space-time fractional KleinGordon are obtained:By means of the values assembled in (94), we attain(97)u1x,t=±−βγA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1∓μλ−βγ1A1cosh−λξ+A2sinh−λξ,whereξ=kxα/α∓2λk2+β/2λtα/α.With the help of the values arranged in (95), we carry out(98)u2x,t=±−2βσγ1A1cosh−λξ+A2sinh−λξ,whereξ=kxα/α∓λk2−β/λtα/α.On the other hand, by means of (96), we derive(99)u3x,t=±−βγA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1∓1λ−βγμ±λ−σA1cosh−λξ+A2sinh−λξ,whereξ=kxα/α±λk2+2β/λtα/α.Case 2: whenλ>0, executing the analogous steps as case 1, a system of algebraic equations can be found and after resolving this system of equations, we obtain three sets of values of the constants as follows:(100)a0=0,a1=∓λβγ,b1=±μβλγ,c=±2λk2+β2λ,(101)a0=0,a1=0,b1=±−2λβσγ,c=±λk2−βλ,(102)a0=0,a1=±λβγ,b1=∓βλγμ∓λσ,c=±λk2+2βλ,where k, β, γ, and μ are arbitrary constants.By the use of the constant values scheduled in (100)–(102) into solution (19) alongside with (8) and (92) function solutions to the space-time, the following trigonometric function solutions to the space-time fractional KleinGordon equation are obtained.For the values arranged in (100), we obtain the general solution:(103)u4x,t=∓βγA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1±μλβγ1A1cosλξ−A2sinλξ,where ξ=kxα/α∓2λk2+β/2λtα/α.Similarly, for the values organized in (101), we accomplish(104)u5x,t=±−2βσγ1A1cosλξ−A2sinλξ,where ξ=kxα/α∓λk2−β/λtα/α.And, for the values laid out in (102), we ascertain(105)u6x,t=±βγA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1∓1λβγμ∓λσA1cosλξ−A2sinλξ,where ξ=kxα/α∓λk2+2β/λtα/α.In this section, we have established the general solutions (97)–(99) and (103)–(105) to the space-time fractional KleinGordon equation from where scores of periodic solitary wave solutions can be extracted selecting special values for parameters. But, for simplicity, particular solutions are omitted.The obtained results can be compared with the exact solutions accessible in the literature. In [45], the exact solutions of the space-time fractional KleinGordon equation are established by using the G′/G,1/G-expansion method. It is seen that the solutions established in this study are different than the solutions found in [45].


## 4.1. Time Fractional CRWP Equation

First, we consider the time fractional CRWP equation [22, 40]:(15)Dtαu−ux+2uux+uxx=0,0<α≤1.The ensuing wave transformation is(16)ux,t=Uξ,ξ=kx−ctαα,where k is the wave number and c is the velocity; reducing the equation (15) into the subsequent ODE, we get(17)−c+kU′+2kUU′+k2U″=0.Integrating (17) once, we find(18)−c+kU+kU2+k2U′+ξ0=0,where ξ0 is a constant of integration. It is clear that the homogeneous balance between U2 and U′ present in equation (18) gives N=1. Therefore, the shape of the exact solution of equation (18) is(19)Uξ=a0+a11ϕ+b1ψ.There are three cases to be considered:Case 1: whenλ<0, substituting solution (19) into equation (18) and utilizing (4) and (6), equation (18) can be transmuted to a polynomial in 1/ϕ and ψ. Equalizing its coefficients to zero yields a set of algebraic equations in a0, a1, b1, k, c, σ, λ, μ, and ξ0:(20)1ϕ2:b1k2μλ2λ2σ+μ2−b12kλ2λ2σ+μ2+a12k+a1k2λ=0,1ϕψ:2b12kλμλ2σ+μ2−2b1k2μ2λλ2σ+μ2−a1k2μ+b1k2λ+2a1b1k=0,1ϕ:a1k−a1c+2a0a1k=0,ψ:b1k+2a0b1k−b1c=0,ψ0:b1k2μλλ2σ+μ2−b12kλλ2σ+μ2+a02k−a0c+a0k+a1k2+ξ0=0.Resolving these algebraic equations by Maple software package, we attain the following values:(21)a0=c+k2k,a1=−kλ,b1=μk,ξ0=4k4λ+c2+2ck+k24k,(22)a0=c+k2k,a1=−12kλ,b1=12kμ∓λ−σ,ξ0=k4λ+c2+2ck+k24k,wherek and c are free constants.Inserting the values of the parameters assembled in (21) into solution (19) along with (6) and (16), the following wide-ranging hyperbolic function solution is ascertained:(23)u1x,t=c+k2k+k−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+μκ−λ1A1cosh−λξ+A2sinh−λξ.In particular, if we setA1=0, A2≠0, and μ=0 (or A2=0, A1≠0 and μ=0) into the solution (23), we obtain the successive singular kink and kink soliton solutions to the CRWP equation (15), respectively:(24)u11=c+k2k+k−λcoth−λξ,(25)u12=c+k2k+k−λtanh−λξ,whereξ=kx−ctα/α.Furthermore, by means of the parameter values gathered in (22), we derive(26)u2x,t=c+k2k+12k−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+κ2−λμ∓λ−σA1cosh−λξ+A2sinh−λξ.SinceA1 and A2 are integral constants, one might select their values spontaneously. Thus, if we select A1=0, A2>0, and μ=0 (or A2=0, A1>0, and μ=0) into solution (26), we gain the following solitary wave solutions to the CRWP equation, respectively:(27)u21=c+k2k+12k−λcoth−λξ±csch−λξ,(28)u22=c+k2k+12k−λtanh−λξ±isech−λξ,whereξ=kx−ctα/α.Case 2: whenλ>0, embedding solution (19) int1/ϕo (18) and putting in use (4) and (8), equation (18) becomes polynomial in and ψ. Computing the action similar to case 1 and after resolving the algebraic equations, we ascertain the following values:(29)a0=c+k2k,a1=−kλ,b1=μk,ξ0=4k4λ+c2+2ck+k24k,(30)a0=c+k2k,a1=−12kλ,b1=12kμ±λσ,ξ0=k4λ+c2+2ck+k24k,wherek and c are arbitrary constants.Substituting the values scheduled in (29) into solution (19) along with (8) and (16), the succeeding trigonometric solution to the CRWP equation (15) is established:(31)u3x,t=c+k2k−kλA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+μkλ1A1cosλξ−A2sinλξ.SinceA1 and A2 are free constants, we may set A1=0,A2≠0, and μ=0 (or A2=0,A1≠0, and μ=0) into solution (31), we derive the under mentioned periodic wave solutions to the CRWP equation:(32)u31=c+k2k+kλcotλξ,(33)u32=c+k2k−kλtanλξ,whereξ=kx−ctα/α.By means of the values organized in (30), we extract the following solution:(34)u4x,t=c+k2k−12kλA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+k2λμ±λσA1cosλξ−A2sinλξ.Now, if we setA1=0,A2>0, and μ=0 (or A2=0,A1>0, and μ=0) into solution (34), we obtain other periodic wave solutions to the CRWP equation:(35)u41=c+k2k+12kλcotλξ∓icscλξ,(36)u42=c+k2k−12kλtanλξ∓secλξ,whereξ=kx−ctα/α.Case 3: whenλ=0, introducing (19) into equation (18) and executing (4) and (10), equation (18) becomes a polynomial in 1/ϕ and ψ. Vanishing the coefficients yields a set of algebraic equations, and solving this system the following results are obtained:(37)a0=c+k2k,a1=0,b1=μk,ξ0=c+k24k.Setting the values gathered in (37) into solution (19) along with (10) and (16), we carry out the next rational function solution to the CRWP equation:(38)u5=c+k2k+kμμξ+A1,where ξ=kx−ctα/α.The obtained solutions can be compared with the solutions accessible in the literature. We detect that, by settingc=−k2λ+2a0k−k and μ=−λ2/4, the obtained solutions (24) and (25) fully agree with the corresponding solutions (3.19) and (3.20) established in [22], while the other solutions are different.


## 4.2. General Time Fractional STO Equation

The conformable general time fractional STO equation [41–43] is(39)Dtαu+3βux2+3βu2ux+3βuuxx+βuxxx=0,t>0,0<α≤1,where β is nonzero constant. The wave transformation is as follows:(40)ux,t=Uξ,ξ=x+ωtαα,where ω is the velocity of the travelling wave; converting equation (39) into an ODE, we get(41)ωU′+3βU′2+3βU2U′+3βUU″+βU′″=0.Integrating equation (41) once, we find(42)ωU+3βUU′+βU3+βU′=0.BalancingU′ and U3 obtained from equation (42), we get N=1 Thus, the solution structure of equation (42) is same of solution (19). Three cases as described in Section 3 will be discussed further:Case 1: whenλ<0, placing the solution (19) into (42) and utilizing (4) and (6), equation (42) modifies to a polynomial in 1/ϕ and ψ. Setting each coefficient of the polynomial to zero yields a set of algebraic equations in a0, a1, b1, σ, λ, μ, β, and ω. From these equations with the help of Maple algebra software, we find the ensuing results of constants:(43)a0=±12−λ,a1=−12λ,b1=12μ∓λ−σ,ω=βλ,(44)a0=0,a1=−12λ,b1=12μ∓λ−σ,ω=14βλ,(45)a0=0,a1=−λ,b1=μ∓λ−σ,ω=βλ,(46)a0=±−λ,a1=−λ,b1=μ,ω=4βλ,(47)a0=0,a1=−λ,b1=μ,ω=βλ,(48)a0=0,a1=−2λ,b1=2μ,ω=4βλ.By means of parameter values sorted out in (43), along with (6) and (40), the following hyperbolic function solutions of the general time fractional STO equation are obtained:(49)u1x,t=±12−λ+12−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+12−λμ∓λ−σA1cosh−λξ+A2sinh−λξ.In solution (49), A1 and A2 are free parameters; therefore, we freely can choose our values. Thus, if we choose A1=0, A2>0, and μ=0 or A2=0, A1>0, and μ=0, from solution (49), we obtain the under mentioned soliton solutions to the general time fractional STO equation:(50)u11=±12−λ1±coth−λξ+csch−λξ,(51)u12=±12−λ1±tanh−λξ±12λsech−λξ,whereξ=x+βλtα/α.Furthermore, by means of the values scheduled in (44), we establish(52)u2x,t=12−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+12−λμ∓λ−σA1cosh−λξ+A2sinh−λξ.Now, if we useA1=0, A2>0, and μ=0 or A2=0, A1>0, and μ=0, in (52), we attain the next soliton solutions to the general time fractional STO equation:(53)u21=12−λcoth−λξ±csch−λξ,(54)u22=12−λtanh−λξ±12λsech−λξ,whereξ=x+1/4βλtα/α.Moreover, by means of the values organized in (45), we carry out(55)u3x,t=−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+1−λμ∓λ−σA1cosh−λξ+A2sinh−λξ.If we considerA1=0,A2≠0, and μ=0 or A2=0,A1≠0, and μ=0, from solution (55), we derive the solitary wave solutions given underneath:(56)u31=−λcoth−λξ±csch−λξ,(57)u32=−λtanh−λξ±λsech−λξ,whereξ=x+βλtα/α.Similarly, making use of the values accumulated in (46), we determine(58)u4x,t=±−λ+−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+μ−λ1A1cosh−λξ+A2sinh−λξ.ForA1=0, A2≠0, and μ=0 or A2=0, A1≠0, and μ=0, the solution (58) generates the kink and singular kink solutions given underneath:(59)u41=±−λ1±coth−λξ,(60)u42=±−λ1±tanh−λξ,whereξ=x+4βλtα/α.Likewise, with the help of the parametric values amassed in (47), we achieve(61)u5x,t=−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+μ−λ1A1cosh−λξ+A2sinh−λξ.ChoosingA1=0, A2≠0, and μ=0 or A2=0, A1≠0, and μ=0, solution (61) turns into(62)u51=−λcoth−λξ,(63)u52=−λtanh−λξ,whereξ=x+βλtα/α.For the values gathered in (48), we found(64)u6x,t=2−λA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1+2μ−λ1A1cosh−λξ+A2sinh−λξ.In particular, if we setA1=0,A2≠0, and μ=0 (or A2=0,A1≠0, and μ=0), the solution (64) reduces to(65)u61=2−λcoth−λξ,(66)u62=2−λtanh−λξ,whereξ=x+4βλtα/α.Case 2: whenλ>0, we established by completing the parallel course of algorithms to case 1 and the following values for the constants:(67)a0=±12iλ,a1=−12λ,b1=12μ±λσ,ω=βλ,(68)a0=0,a1=−12λ,b1=12μ±λσ,ω=14βλ,(69)a0=0,a1=−λ,b1=μ±λσ,ω=βλ,(70)a0=±iλ,a1=λ,b1=μ,ω=4βλ,(71)a0=0,a1=−λ,b1=μ,ω=βλ,(72)a0=0,a1=−2λ,b1=2μ,ω=4βλ.Substituting the sets of values ((67)–(72)) into solution (19) along with (8) and (40), we get the trigonometric function solutions for equation (39) as follows:By means of (67), we obtain(73)u7x,t=±12iλ−12λA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+12λμ±λσA1cosλξ−A2sinλξ.If we considerA1=0,A2>0, and μ=0 (or A2=0,A1>0, and μ=0) in (73), the periodic wave solutions of equation (39) are found as follows:(74)u71=±12λi±cotλξ−cscλξ,(75)u72=±12λi∓tanλξ+secλξ,where ξ=x+βλtα/α.Through the values of the parameters gathered in (68), we derive(76)u8x,t=−12λA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+12λμ±λσA1cosλξ−A2sinλξ.SinceA1 and A2 are subjective constants, one may pick their values freely. Thus, if we pick A1=0, A2≠0, and μ=0 (or A2=0, A1≠0, and μ=0), from (76), we found the periodic wave solutions of equation (39) as follows:(77)u81=12λcotλξ∓cscλξ,(78)u82=−12λtanλξ∓secλξ,where ξ=x+1/4βλtα/α.By virtue of the values scheduled in (69), we obtain(79)u9x,t=−λA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+1λμ±λσA1cosλξ−A2sinλξ.If we setA1=0, A2≠0, and μ=0 (or A2=0, A1≠0, and μ=0) into solution (79), we ascertain the following periodic wave solutions to the general time fractional STO equation:(80)u91=λcotλξ∓cscλξ,(81)u92=−λtanλξ∓secλξ,where ξ=x+βλtα/α.For the values accumulated in (70), we establish(82)u10x,t=±iλ−λA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+μλ1A1cosλξ−A2sinλξ.SettingA1=0, A2≠0, and μ=0 (or A2=0, A1≠0, and μ=0), the solution (82) transformed to(83)u101=±λi±cotλξ,(84)u102=±λi∓tanλξ,where ξ=x+4βλtα/α.Utilizing the values of the parameters arranged in (71), we obtain(85)u11x,t=−λA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+μλ1A1cosλξ−A2sinλξ.Now settingA1=0, A2≠0, and μ=0 (or A2=0 and μ=0), the solution (85) becomes(86)u111=λcotλξ,(87)u112=−λtanλξ,where ξ=x+βλtα/α.Embedding the parametric values compiled in (72), we find out(88)u12x,t=−2λA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1+2μλ1A1cosλξ−A2sinλξ.Suppose thatA1=0,A2≠0, and μ=0 (or A2=0,A1≠0, and μ=0), then the solution (88) developed into(89)u121=2λcotλξ,(90)u122=−2λtanλξ,where ξ=x+4βλtα/α.From the obtained broad-ranging solutions, it is observed that setting definite values of the associated parameters, we manage to determine some particular solutions which coincide with those accessible in the literature and some fresh solutions are established. It is seen that, by puttingβ=d, the obtained solutions (53), (54), (77), and (78) completely agree with the solutions (4.8), (4.7), (4.11), and (4.10), respectively, found in [43]. In addition to these solutions, we found many more solutions that were not found in any other studies.


## 4.3. Space-Time Fractional KleinGordon Equation

In this subsection, we extract the closed form solutions to the space-time fractional KleinGordon equation. Let us consider the KleinGordon equation with space-time fractional order [44, 45]:(91)DtαDtαu−DxαDxαu−βu−γu3=0,t>0,0<α≤1,where βandγ are nonzero constants. We apply the following transformation for reducing equation (91) to an ODE:(92)ux,t=Uξ,ξ=kxαα−ctαα,where kandc are nonzero constants. Thus, the space-time fractional KleinGordon equation turns out as follows:(93)k2−c2U″+βU+γU3=0.BalancingU″ and U3 in equation (93), we found N=1. On account of this, the structure of the solution of equation (93) is identical to the shape of the solution of equation (19) and therefore has not been repeated. There are three cases should be discussed as described in Section 3:Case 1: whenλ<0, plugging in (19) into equation (93) and utilizing (4) and (6), equation (93) will be converted into a polynomial in 1/ϕ and ψ. Equalizing the coefficients of this polynomial to zero yields a set of algebraic equations for a0, a1, b1, c, k, σ, λ, μ, β, and γ. Resolving these equations with the assistance of computer algebra, like Maple software package, we found the following values for the constants:(94)a0=0,a1=±λβγ,b1=±μβλγ,c=±2λk2+β2λ,(95)a0=0,a1=0,b1=±2λβσγ,c=±λk2−βλ,(96)a0=0,a1=±λβγ,b1=±βλγμ±λ−σ,c=∓λk2+2βλ,wherek, β, γ, and μ are free parameters.Inserting the sets of constant values from (94) to (96) into solution (19) along with (6) and (92), the following hyperbolic function solutions to the space-time fractional KleinGordon are obtained:By means of the values assembled in (94), we attain(97)u1x,t=±−βγA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1∓μλ−βγ1A1cosh−λξ+A2sinh−λξ,whereξ=kxα/α∓2λk2+β/2λtα/α.With the help of the values arranged in (95), we carry out(98)u2x,t=±−2βσγ1A1cosh−λξ+A2sinh−λξ,whereξ=kxα/α∓λk2−β/λtα/α.On the other hand, by means of (96), we derive(99)u3x,t=±−βγA1cosh−λξ+A2sinh−λξA1sinh−λξ+A2cosh−λξ+μ/λ−1∓1λ−βγμ±λ−σA1cosh−λξ+A2sinh−λξ,whereξ=kxα/α±λk2+2β/λtα/α.Case 2: whenλ>0, executing the analogous steps as case 1, a system of algebraic equations can be found and after resolving this system of equations, we obtain three sets of values of the constants as follows:(100)a0=0,a1=∓λβγ,b1=±μβλγ,c=±2λk2+β2λ,(101)a0=0,a1=0,b1=±−2λβσγ,c=±λk2−βλ,(102)a0=0,a1=±λβγ,b1=∓βλγμ∓λσ,c=±λk2+2βλ,where k, β, γ, and μ are arbitrary constants.By the use of the constant values scheduled in (100)–(102) into solution (19) alongside with (8) and (92) function solutions to the space-time, the following trigonometric function solutions to the space-time fractional KleinGordon equation are obtained.For the values arranged in (100), we obtain the general solution:(103)u4x,t=∓βγA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1±μλβγ1A1cosλξ−A2sinλξ,where ξ=kxα/α∓2λk2+β/2λtα/α.Similarly, for the values organized in (101), we accomplish(104)u5x,t=±−2βσγ1A1cosλξ−A2sinλξ,where ξ=kxα/α∓λk2−β/λtα/α.And, for the values laid out in (102), we ascertain(105)u6x,t=±βγA1cosλξ−A2sinλξA1sinλξ+A2cosλξ+μ/λ−1∓1λβγμ∓λσA1cosλξ−A2sinλξ,where ξ=kxα/α∓λk2+2β/λtα/α.In this section, we have established the general solutions (97)–(99) and (103)–(105) to the space-time fractional KleinGordon equation from where scores of periodic solitary wave solutions can be extracted selecting special values for parameters. But, for simplicity, particular solutions are omitted.The obtained results can be compared with the exact solutions accessible in the literature. In [45], the exact solutions of the space-time fractional KleinGordon equation are established by using the G′/G,1/G-expansion method. It is seen that the solutions established in this study are different than the solutions found in [45].


## 5. Physical Explanations

In this section, we put forth the physical explanation and the 2D and 3D graphical representation of the solutions obtained for the time fractional CRWP equation, the general time fractional STO equation, and the space-time fractional KleinGordon equation as follows:Solutions (25), (28), (51), (54), (57), (60), (63), (66), and (97) are the kink-shape soliton. Kink waves are travelling waves which arise from one asymptotic position to another. Figure 1 shows the shape of the kink solution (63). Other figures are omitted for convenience.Figure 1
Kink-shape wave of (63) when λ=−1, β=−1, α=0.5, −50≤x≤50, and 0≤t≤50.
(a)(b)Moreover, solutions (24), (27), (50), (53), (56), (59), (62), (65), and (99) are singular kink solitons. Figure 2 shows the shape of the singular kink solution (24). The residual figures are left for simplicity.Figure 2
Singular kink-shape wave of (24) when c=1, k=−1, λ=−1, α=0.5, −20≤x≤20, and 0.1≤t≤20.
(a)(b)On the other hand, solutions (32), (33), (35), (36), (74), (75), (77), (78), (80), (81), (83), (84), (86), (87), (89), 90), and (103)–(105) are the exact periodic wave solutions. Periodic solutions are travelling wave solutions that are periodic like sinx−t. Figure 3 shows the singular periodic solution of (104). The remaining graphs are left for minimalism.Figure 3
Singular periodic wave shape of (104) when β=1, k=1, γ=−1, λ=1, α=0.5, −20≤x≤20, and 0≤t≤20.
(a)(b)Solution (98) is the soliton solution. Figure 4 shows the shape of the exact soliton solution of (98) of the space-time fractional KleinGordon equation.Figure 4
Soliton wave shape of (98) when β=1, k=1, γ=−1, λ=−1, α=0.95, −10≤x≤10, and 0≤t≤7.
(a)(b)


## 6. Conclusion

In this article, we have introduced an extension of theG′/G,1/G-expansion method to look into nonlinear fractional differential equations in the sense of conformable derivative. Taking the advantage of this extension, the time fractional CRWP equation, the general time fractional STO equation, and the space-time fractional Klein-Gordon equation have been investigated. Scores of broad-ranging exact solutions have successfully been found as a linear combination of hyperbolic, trigonometric, and rational function associated with free parameters. For definite values of these parameters, some known periodic, kink, and solitary wave solutions accessible in the literature are derived from the general solutions and some fresh solutions are originated. This study shows that the proposed extension is quite efficient, useful, direct, and easily computable with the aid of Maple software package and practically well suited to be used in finding analytical exact solutions to many other nonlinear FDEs, and this is our scheme in the future.



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