# Pure Traveling Wave Solutions for Three Nonlinear Fractional Models

Authors: Qinjun Li; Danyal Soybaş; Onur Alp Ilhan; Gurpreet Singh; Jalil Manafian
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

Three nonlinear fractional models, videlicet, the space-time fractional1+1 Boussinesq equation, 2+1-dimensional breaking soliton equations, and SRLW equation, are the important mathematical approaches to elucidate the gravitational water wave mechanics, the fractional quantum mechanics, the theoretical Huygens’ principle, the movement of turbulent flows, the ion osculate waves in plasma physics, the wave of leading fluid flow, etc. This paper is devoted to studying the dynamics of the traveling wave with fractional conformable nonlinear evaluation equations (NLEEs) arising in nonlinear wave mechanics. By utilizing the oncoming exp−Θq-expansion technique, a series of novel exact solutions in terms of rational, periodic, and hyperbolic functions for the fractional cases are derived. These types of long-wave propagation phenomena played a dynamic role to interpret the water waves as well as mathematical physics. Here, the form of the accomplished solutions containing the hyperbolic, rational, and trigonometric functions is obtained. It is demonstrated that our proposed method is further efficient, general, succinct, powerful, and straightforward and can be asserted to install the new exact solutions of different kinds of fractional equations in engineering and nonlinear dynamics.


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## Body

## 1. Introduction

The solutions of fractional partial differential equations (FPDEs) are often of interest and are applied in practical life. One of the main purposes of mathematical physics is to determine the exact solutions. Some scholars had perused the plenty of physical wave equations. Therefore, several analytical methods were systematically developed and applied to achieve exact and approximate solutions of fractional ordinary and partial differential equations with applications in various fields of sciences like fluid flow, mechanics, biology, nonlinear optics, substance energy, system identification, and geooptical filaments which are expressed in fractional forms [1–11].In the past few decades, a lot of studies have been executed to find the new and further exact traveling wave solution of space-time fractional PDEs by many research. With the collaboration of potential symbolic computer programming software, they have been appointed for researching appropriate solution to the nonlinear space-time fractional PDEs by executing powerful techniques, for example, the tan-ϕ/2 expansion method, the sin-Gordon expansion method, the G′/G expansion method, and the advanced exponential expansion method [12–25].This work mainly investigates three nonlinear fractional models by utilizing the oncomingexp−Θq-expansion method [17, 25]. Recently, Bashar and Roshid [17] and Rahhman et al. [25] have studied this technique to some fractional and nonfractional NLEEs. They found that this introduced method provides some simple general form of traveling wave solutions. Rahhman et al. [25] did not give any fruitful discussion about fractional NLEEs with this proposed method. The important idea of this method is too explicit: the exact solutions of NLEEs satisfy the nonlinear ODE, Θ′q+θexpΘq+δexp−Θq, where θ and δ are real parameters.The space-time fractional Boussinesq equation with theρ-derivative [26, 27] is presented as follows:
(1)Dt2αΨ+bDx2αΨ+βDx2αΨ2+γDx4αΨ=0,t>0,0<α≤1,where Ψx,t is the vertical deflection. In Ref. [28], authors constructed an analytical solution for both linear and nonlinear time-fractional Boussinesq equations by an iterative method. Also, Hemeda [29] studied the fractional Boussinesq-like equation via a new iterative method. Authors of [30] investigated nonlinear two-point boundary value problem to the fractional Boussinesq-like equation. Furthermore, the space-time fractional breaking soliton equations [27, 31] are taken in the following form:
(2)Dt2αΨ+aDx2y3αΨ+4aΨDxαΩ+4aΩDxαΨ=0,t>0,0<α≤1,DyαΨ−DxαΩ=0.Meng and Feng [32] used an auxiliary equation method to the space-time fractional 2+1-dimensional breaking soliton equation. Authors of [27, 33, 34] discussed the space-time fractional SRLW equation (STFSRLWE) in the following case:
(3)Dt2αΨ+Dx2αΨ+ΨDtαDxαΨ+DtαΨDxαΨ+Dt2αDx2αΨ=0,t>0,0<α≤1.The functional variable method, exp-function method, and G′/G-expansion method to the fractional SRLW equation in the sense of the modified Riemann-Liouville derivative were utilized in Ref. [35]. The interested readers can see more works in Refs. ([36–50]). Inc and coworkers presented the new soliton structures to some time-fractional nonlinear differential equations with a conformable derivative via the Ricatti–Bernoulli sub-ODE method [51]. Salahshour and colleagues worked on the truncated M-fractional derivative as a novel and effective derivative under interval uncertainty and investigated the existence and uniqueness conditions of the solution [52]. Authors of [53] studied a coupled nonlinear Maccari’s system which describes the motion of isolated waves localized in a small part of space. Authors of [54] employed the exponential function method for the combined KdV-mKdV equation.One technique, videlicet, the oncomingexp−Θq-expansion technique, was employed by many researchers for solving the number of nonlinear PDEs or fractional PDEs. For both methods, the interested readers can refer for the first technique to Refs. ([55–59]).The pattern of this article is summarized as follows. In Sections2 and 3, the properties and the detail of technique are given, which are to be utilized for getting the exact solutions of the fractional Boussinesq, breaking soliton, and SRLW equations along with numerical simulation and details of graph in Sections 4–7. Finally, some conclusions are given in the end.


## 2. Analysis ofμ-Derivative

Definition 1 (see [60]).
Definition ofμ-derivative: let λ:0;1→R; then, the μ-derivative of λ of order μ is defined as
(4)Dtμλt=limε→0λt+εt+1/Γμ1−μ−λtε,μ∈0,1,t>0.The features and novel theorems will be utilized as follows:The proofs of the aboveμ-derivative properties are obviously given in [60].Theorem 2.
Letμ∈0,1;λ,ϑ be μ-differentiable at point t; therefore, we get
(1)
Dtμaλt+bϑt=aDtαλt+bDtαϑt,fora,b∈R(2)
Dtμc=0,forc∈R(3)
Dtμλtϑt=λtDtμϑt+ϑtDtμλt(4)
Dtμλt/ϑt=λtDtαϑt−ϑtDtμλt/ϑ2t(5)
Dtμλt=t+1/Γμ1−μdλt/dtWe have the following features as follows:
(5)Dtμsint=limε→0sint+εt+1/Γμ1−μ−sintε=t+1Γμ1−αcost,Dtμcost=limε→0cost+εt+1/Γμ1−μ−costε=−t+1Γμ1−μsint,Dtμexpt=limε→0expt+εt+1/Γμ1−μ−exptε=t+1Γμ1−αexpt.Theorem 3.
Letλ:0;1→R; be a function such that λ is differentiable and also μ-differentiable. Also, let λ be a differentiable function defined in the range of ϑ. Then, we get
(6)Dtμλoϑt=t+1Γμ1−μϑ′tλ′ϑt,where prime denotes the classical derivatives with respect to t.


## 3. The Oncomingexp−Θq-Expansion Technique

This method was summarized and improved for achieving the analytic solutions of NLPDEs.Step 1.
Assume that a nonlinear partial differential equation is given in the general form as follows:(7)Fu,ux,uy,ut,uxx,utt,⋯=0.After simple algebraic operations, this equation is transformed into an ordinary differential equation (ODE) with the below transformation:
(8)ux,y,t=Uq,q=x+ky+ωt,as well as into nonlinear ODE:
(9)SU,U′,ωU′,kU′,U′′,ω2U′′,⋯=0.Step 2.
Then, assume that the searched wave solutions of equation (9) have the following representation:
(10)Uq=∑j=0χπjYjq+∑j=0χτjY−jq,where Yq=exp−Θq and πj0≤j≤χ,τj0≤j≤χ are constants to be determined, such that πχ,τχ≠0, and Θ=Θq is the solution of the following first-order differential equation:
(11)Θ′=δY−1q+Yq+θ.If we try to find the solution of (11), then we obtain special solutions that vary according to the state of the coefficients:
Solution 1 (hyperbolic function solution). Ifδ≠0 and θ2−4δ>0, then we achieve
(12)Θq=ln−θ2−4δ2δtanhθ2−4δ2q+Σ−θ2δ,where Σ is the integral constant.
Solution 2 (trigonometric function solution). Ifδ≠0 and θ2−4δ<0, afterward we achieve
(13)Θq=ln−θ2+4δ2δtan−θ2+4δ2q+Σ−θ2δ.
Solution 3. Ifδ=0, θ≠0, and θ2−4δ>0, afterward we achieve
(14)Θq=−lnθexpθq+Σ−1.
Solution 4. Ifδ≠0, θ≠0, and θ2−4δ=0, afterward we achieve
(15)Θq=ln−2θq+Σ+4θ2q+Σ.
Solution 5. Ifδ=0, θ=0, and θ2−4δ=0, afterward we achieve
(16)Θq=lnq+Σ,where πj0≤j≤χ, τj0≤j≤χ, θ, and δ are also the constants to be explored later. As usual, for determining χ, the highest-order derivative should be balanced with the highest-order nonlinear terms in equation (10). However, the positive integer χ can be determined in this way.Step 3.
Following these operations, according to them value obtained above, let (11) be substituted into equation (10). Therefore, we obtain a set of algebraic equations that contains Ysqs=0,1,2,⋯. Then, setting each coefficient of exp−Θqs to zero, we can get a set of overdetermined equations for π0,π1,τ1,⋯,πχ,τχ, θ, and δ. Since the obtained algebraic equation system will be difficult to solve manually, symbolic computation such as Maple can be used at this stage. Assume that the estimation of the constants can be gotten by fathoming the mathematical conditions in step 2. Substituting the estimations of the constants together with the arrangements of equation (11), we will acquire new and far reaching precise traveling wave arrangements of the nonlinear development equation (7).
TheΣ has the following features as follows:
(17)Uq≃ξYχ,U′q≃ξYχ−1Y′=ξYχ+1,U′′q≃ξYχ+2,Uq2≃ξ2Y2χ,where ξ=πχ. Balancing U′′ with U2 yields
(18)χ+2=2χ⇒χ=2.


## 4. The First Equation

By utilizing the following transformation:(19)q=kx+1Γμμ−ωt+1Γμμ,then, equation (1) transformed to
(20)ω2Ψ′′+bk2Ψ′′+βk2Ψ2′′+γk4Ψ′′′′=0,where Ψ′=dΨ/dq, and by integrating equation (20) twice with respect to q, it can be seen as
(21)ω2+bk2Ψ+βk2Ψ2+γk4Ψ′′=0.The balance number will be obtained χ=2 by using the balance principle. Then, the exact solution is given as
(22)Ψq=π0+π1Yq+π2Y2q+τ1Yq+τ2Y2q.Firstly, we substitute the expressions of Ψq in (22) into (21) and collect all terms with the same order of Yq. Then, by equating the coefficient of each polynomial to zero, we obtain a set of algebraic equations as follows:
(23)6γk4τ2+βk2τ22=0,10γk4θτ2+2γk4τ1+2βk2τ1τ2=0,4γk4θ2τ2+8δγk4τ2+3γk4θτ1+2βk2π0τ2+βk2τ12+bk2τ2+ω2τ2=0,6δγk4θτ2+γk4θ2τ1+2δγk4τ1+2βk2π0τ1+2βk2π1τ2+bk2τ1+ω2τ1=0,2δ2γk4τ2+δγk4θτ1+γk4θπ1+2γk4π2+βk2π02+2βk2π1τ1+2βk2π2τ2+bk2π0+ω2π0=0,γk4θ2π1+2δγk4π1+6γk4θπ2+2βk2π0π1+2βk2π2τ1+bk2π1+ω2π1=0,3δγk4θπ1+4γk4θ2π2+8δγk4π2+2βk2π0π2+βk2π12+bk2π2+ω2π2=0,2δ2γk4π1+10δγk4θπ2+2βk2π1π2=0,6δ2γk4π2+βk2π22=0.First:(24)ω=kγk2θ2−4δγk2−b,π0=−k2γθ2+2δβ,τ1=−6k2γθβ,τ2=−6k2γβ.According to Family I, (22) becomes
(25)Ψ1q=−k2γθ2+2δβ−6k2γθ/β−θ2−4δ/2δtanhθ2−4δ/2q+Σ−θ/2δ−6k2γ/β−θ2−4δ/2δtanhθ2−4δ/2q+Σ−θ/2δ2,where
(26)q=kx+1Γμμ−kγk2θ2−4δγk2−bt+1Γμμ.According to Family II, (22) becomes
(27)Ψ2q=−k2γθ2+2δβ−6k2γθ/β−θ2+4δ/2δtan−θ2+4δ/2q+Σ−θ/2δ−6k2γ/β−θ2+4δ/2δtan−θ2+4δ/2q+Σ−θ/2δ2,where
(28)q=kx+1Γμμ−kγk2θ2−4δγk2−bt+1Γμμ.According to Family III, (22) can be written as
(29)Ψ3q=−k2γθ2β−6k2γθβθexpθq+Σ−1−6k2γβθexpθq+Σ−12,where
(30)q=kx+1Γμμ−kγk2θ2−bt+1Γμμ.According to Family IV, (22) becomes
(31)Ψ4q=−k2γθ2+2δβ−6k2γθβ−θ2q+E2θq+Σ+4−6k2γβ−θ2q+E2θq+Σ+42,where
(32)q=kx+1Γμμ−k−bt+1Γμμ.Second:(33)ω=k−γk2θ2+4δγk2−b,π0=−6k2γδβ,τ1=−6k2γθβ,τ2=−6k2γβ.According to Family I, (22) can be written as
(34)Ψ5q=−6k2γδβ−6k2γθ/β−θ2−4δ/2δtanhθ2−4δ/2q+Σ−θ/2δ−6k2γ/β−θ2−4δ/2δtanhθ2−4δ/2q+Σ−θ/2δ2,where
(35)q=kx+1Γμμ−k−γk2θ2+4δγk2−bt+1Γμμ.According to Family II, (22) can be written as
(36)Ψ6q=−6k2γδβ−6k2γθ/β−θ2+4δ/2δtan−θ2+4δ/2q+Σ−θ/2δ−6k2γ/βθ2−4δ/2δtan−θ2+4δ/2q+Σ−θ/2δ2,where
(37)q=kx+1Γμμ−k−γk2θ2+4δγk2−bt+1Γμμ.According to Family III, (22) can be written as
(38)Ψ7q=−6k2γθβθexpθq+Σ−1−6k2γβθexpθq+Σ−12,where
(39)q=kx+1Γμμ−k−γk2θ2−bt+1Γμμ.According to Family IV, (22) becomes
(40)Ψ8q=−6k2γδβ−6k2γθβ−θ2q+Σ2θq+E+4−6k2γβ−θ2q+Σ2θq+Σ+42,where
(41)q=kx+1Γμμ−k−bt+1Γμμ.Third:(42)ω=kγk2θ2−4δγk2−b,π0=−k2γθ2+2δβ,π1=−6k2γθβ,π2=−6k2γβ.According to Family I, (22) becomes
(43)Ψ9q=−k2γθ2+2δβ−6k2γθβ−θ2−4δ2δtanhθ2−4δ2q+Σ−θ2δ−6k2γβ−θ2−4δ2δtanhθ2−4δ2q+Σ−θ2δ2,where
(44)q=kx+1Γμμ−kγk2θ2−4δγk2−bt+1Γμμ.According to Family II, (22) can be written as
(45)Ψ10q=−k2γθ2+2δβ−6k2γθβ−θ2+4δ2δtan−θ2+4δ2q+Σ−θ2δ−6k2γβ−θ2+4δ2δtan−θ2+4δ2q+Σ−θ2δ2,where
(46)q=kx+1Γμμ−kγk2θ2−4δγk2−bt+1Γμμ.According to Family III, (22) can be written as
(47)Ψ11q=−k2γθ2β−6k2γθβexpθq+Σ−1θ−6k2γβexpθq+Σ−1θ2,where
(48)q=kx+1Γαα−kγk2θ2−bt+1Γαα.According to Family IV, (22) can be written as
(49)Ψ12q=−k2γθ2+2δβ−6k2γθβ−2θq+Σ+4θ2q+E−6k2γβ−2θq+Σ+4θ2q+E2,where
(50)q=kx+1Γμμ−k−bt+1Γμμ.Fourth:(51)ω=k−γk2θ2+4δγk2−b,π0=−6k2γδβ,π1=−6k2γθβ,π2=−6k2γβ.According to Family I, (22) becomes
(52)Ψ13q=−6k2γδβ−6k2γθβ−θ2−4δ2δtanhθ2−4δ2q+Σ−θ2δ−6k2γβ−θ2−4δ2δtanhθ2−4δ2q+Σ−θ2δ2,where
(53)q=kx+1Γμμ−k−γk2θ2+4δγk2−bt+1Γμμ.According to Family II, (22) can be written as
(54)Ψ14q=−6k2γδβ−6k2γθβ−θ2+4δ2δtan−θ2+4δ2q+Σ−θ2δ−6k2γβθ2−4δ2δtan−θ2+4δ2q+Σ−θ2δ2,where
(55)q=kx+1Γμμ−k−γk2θ2+4δγk2−bt+1Γμμ.According to Family III, (22) becomes
(56)Ψ15q=−6k2γθβexpθq+Σ−1θ−6k2γβexpθq+Σ−1θ2,where
(57)q=kx+1Γαα−k−γk2θ2−bt+1Γαα.According to Family IV, (22) can be written as
(58)Ψ16q=−6k2γδβ−6k2γθβ−2θq+Σ+4θ2q+E−6k2γβ−2θq+E+4θ2q+Σ2,where
(59)q=kx+1Γμμ−k−bt+1Γμμ.Fifth:(60)k=16−6γβπ2δγ,ω=162δγβπ23bδ+8βπ2δ2γ,π0=2π2δ,τ2=π2δ2.According to Family I, (22) becomes
(61)Ψ17q=2π2δ−π2δ−δcoth2−4δ2q+Σ−π2δ2−δδtanh2−4δ2q+Σ,where
(62)q=16−6γβπ2δγx+1Γμμ−162δγβπ23bδ+8βπ2δ2γt+1Γμμ.According to Family II, (22) becomes
(63)Ψ18q=2π2δ+π22δ4δcot24δ2q+Σ+π2δ24δ2δtan24δ2q+Σ,where
(64)q=16−6γβπ2δγx+1Γμμ−162δγβπ23bδ+8βπ2δ2γt+1Γμμ.


## 5. The Second Equation

By utilizing the following transformation:(65)q=kx+1Γμμ+cy+1Γμμ−ωt+1Γμμ,then, equation (1) transformed to
(66)−cΨ′+ak2ωΨ′′′+4akΨΩ′+4akΨ′Ω=0,(67)ωΨ′−kΩ′=0,where Ψ′=dΨ/dq and Ω′=dΩ/dq. By putting Ω=ω/kΨ into equation (66), it can be seen as
(68)−cΨ′+ak2ΨΨ′′′+8aωΨΨ′=0.The balance number will be obtained χ=2 by using the balance principle. Then, the exact solution is given as
(69)Ψq=π0+π1Yq+π2Y2q+τ1Yq+τ2Y2q.Firstly, we substitute the expressions of Ψq in (69) into (68) and collect all terms with the same order of Yq. Then, by equating the coefficient of each polynomial to zero, we obtain a set of algebraic equations as follows:
(70)6ak2ωτ2+4aωτ22=0,10ak2ωθτ2+2ak2ωτ1+8aωτ1τ2=0,4ak2ωθ2τ2+8aδk2ωτ2+3ak2ωθτ1+8aωπ0τ2+4aωτ12−cτ2=0,6aδk2ωθτ2+ak2ωθ2τ1+2aδk2ωτ1+8aωπ0τ1+8aωπ1τ2−cτ1=0,2aδ2k2ωτ2+aδk2ωθτ1+ak2ωθπ1+2ak2ωπ2+4aωπ02+8aωπ1τ1+8aωπ2τ2−cπ0=0,ak2ωθ2π1+2aδk2ωπ1+6ak2ωθπ2+8aωπ0π1+8aωπ2τ1−cπ1=0,3aδk2ωθπ1+4ak2ωθ2π2+8aδk2ωπ2+8aωπ0π2+4aωπ12−cπ2=0,2aδ2k2ωπ1+10aδk2ωθπ2+8aωπ1π2=0,6aδ2k2ωπ2+4aωπ22=0.First:(71)c=−ak2θ2−4aδk2ω,π0=−14k2θ2−12δk2,τ1=−32θk2,τ2=−32k2.According to Family I, (69) can be written as
(72)Ψ1q=−14k2θ2−12δk2−3/2θk2−θ2−4δ/2δtanhθ2−4δ/2q+Σ−θ/2δ−3/2k2−θ2−4δ/2δtanhθ2−4δ/2q+Σ−θ/2δ2,where
(73)q=kx+1Γμμ−ak2θ2−4aδk2ωy+1Γμμ−ωt+1Γμμ.According to Family II, (69) becomes
(74)Ψ2q=−14k2θ2−12δk2−3/2θk2θ2−4δ/2δtan−θ2+4δ/2q+Σ−θ/2δ−3/2k2θ2−4δ/2δtan−θ2+4δ/2q+Σ−θ/2δ2,where
(75)q=kx+1Γμμ−ak2θ2−4aδk2ωy+1Γμμ−ωt+1Γμμ.According to Family III, (69) becomes
(76)Ψ3q=−14k2θ2−3/2θk2expθq+Σ−1/θ−3/2k2expθq+Σ−1/θ2,where
(77)q=kx+1Γμμ−ak2θ2ωy+1Γμμ−ωt+1Γμμ.Second:(78)c=−−ak2θ2+4aδk2ω,π0=−32δk2,τ1=−32θk2,τ2=−32k2.According to Family I, (69) can be seen as
(79)Ψ4q=−32δk2−3/2θk2−θ2−4δ/2δtanhθ2−4δ/2q+Σ−θ/2δ−3/2k2−θ2−4δ/2δtanhθ2−4δ/2q+Σ−θ/2δ2,where
(80)q=kx+1Γμμ−−ak2θ2+4aδk2ωy+1Γμμ−ωt+1Γμμ.According to Family II, (69) becomes
(81)Ψ5q=−32δk2−3/2θk2θ2−4δ/2δtan−θ2+4δ/2q+Σ−θ/2δ−3/2k2θ2−4δ/2δtan−θ2+4δ/2q+Σ−θ/2δ2,where
(82)q=kx+1Γμμ−−ak2θ2+4aδk2ωy+1Γμμ−ωt+1Γμμ.According to Family III, (69) can be written as
(83)Ψ6q=−3/2θk2expθq+Σ−1/θ−3/2k2expθq+Σ−1/θ2,where
(84)q=kx+1Γμμ+ak2θ2ωy+1Γμμ−ωt+1Γμμ.Third:(85)c=−ak2θ2−4aδk2ω,π0=−14k2θ2−12δk2,π1=−32θk2,π2=−32k2.According to Family I, (69) becomes
(86)Ψ7q=−14k2θ2−12δk2−32θk2−θ2−4δ2δtanhθ2−4δ2q+Σ−θ2δ−32k2−θ2−4δ2δtanhθ2−4δ2q+Σ−θ2δ2,where
(87)q=kx+1Γμμ−ak2θ2−4aδk2ωy+1Γμμ−ωt+1Γμμ.According to Family II, (69) can be written as
(88)Ψ8q=−14k2θ2−12δk2−32θk2θ2−4δ2δtan−θ2+4δ2q+Σ−θ2δ−32k2θ2−4δ2δtan−θ2+4δ2q+Σ−θ2δ2,where
(89)q=kx+1Γμμ−ak2θ2−4aδk2ωy+1Γμμ−ωt+1Γμμ.According to Family III, (69) can be written as
(90)Ψ9q=−14k2θ2−3/2θk2θ/expθq+Σ−1−3/2k2θ/expθq+Σ−12,where
(91)q=kx+1Γμμ−ak2θ2ωy+1Γμμ−ωt+1Γμμ.Fourth:(92)c=−−ak2θ2+4aδk2ω,π0=−32δk2,π1=−32θk2,π2=−32k2.According to Family I, (69) can be seen as
(93)Ψ10q=−32δk2−32θk2−θ2−4δ2δtanhθ2−4δ2q+Σ−θ2δ−32k2−θ2−4δ2δtanhθ2−4δ2q+Σ−θ2δ2,where
(94)q=kx+1Γμμ−−ak2θ2+4aδk2ωy+1Γμμ−ωt+1Γμμ.According to Family II, (69) can be written as
(95)Ψ11q=−32δk2−32θk2θ2−4δ2δtan−θ2+4δ2q+Σ−θ2δ−32k2θ2−4δ2δtan−θ2+4δ2q+Σ−θ2δ2,where
(96)q=kx+1Γαα−−ak2θ2+4aδk2ωy+1Γμμ−ωt+1Γμμ.According to Family III, (69) becomes
(97)Ψ12q=−3/2θk2θ/expθq+Σ−1−3/2k2θ/expθq+Σ−12,where
(98)q=kx+1Γμμ+ak2θ2ωy+1Γμμ−ωt+1Γμμ.Fifth:(99)k=14−aδωcaδω,π0=316caω,π2=3cδ32aω,τ2=3c32aδω.According to Family I, (22) can be written as
(100)Ψ13q=316caω−3cδ32aωδ−δcoth2−4δ2q+Σ−3c32aδω−δδtanh2−4δ2q+Σ,where
(101)q=14−aδωcaδωx+1Γαα−cy+1Γαα−ωt+1Γαα.According to Family II, (22) becomes
(102)Ψ14q=316caω+3cδ32aωδδcot24δ2q+Σ+3c32aδωδδtan24δ2q+Σ,where
(103)q=14−aδωcaδωx+1Γμμ−cy+1Γμμ−ωt+1Γμμ.Sixth:(104)k=14aδωcaδω,π0=116caω,π2=−3cδ32aω,τ2=−3c32aδω.According to Family I, (22) can be written as
(105)Ψ15q=116caω+3cδ32aωδ−δcoth2−4δ2q+Σ+3c32aδω−δδtanh2−4δ2q+Σ,where
(106)q=14aδωcaδωx+1Γαα−cy+1Γαα−ωt+1Γαα.According to Family II, (22) can be seen as
(107)Ψ16q=116caω−3cδ32aωδδcot24δ2q+Σ−3c32aδωδδtan24δ2q+Σ,where
(108)q=14aδωcaδωx+1Γμμ−cy+1Γμμ−ωt+1Γμμ.


## 6. The Third Equation

By utilizing the following transformation:(109)η=kx+1Γμμ−ωt+1Γμμ,then, equation (1) transformed to
(110)2ω2+k2Ψ+kωΨ2+2k2ω2Ψ′′=0,where Ψ′=dΨ/dq, and by integrating equation (110) twice with respect to q, it can be seen as
(111)ω2+bk2Ψ+βk2Ψ2+γk4Ψ′′=0.The balance number will be obtained χ=2 by using the balance principle. Then, the exact solution is given as
(112)Ψq=π0+π1Yq+π2Y2q+τ1Yq+τ2Y2q.Firstly, we substitute the expressions of Ψq in (112) into (111) and collect all terms with the same order of Yq. Then, by equating the coefficient of each polynomial to zero, we obtain a set of algebraic equations as follows:
(113)12k2ω2τ2+kωτ22=0,20k2ω2θτ2+4k2ω2τ1+2kωτ1τ2=0,8k2ω2θ2τ2+16δk2ω2τ2+6k2ω2θτ1+2kωπ0τ2+kωτ12+2k2τ2+2ω2τ2=0,12δk2ω2θτ2+2k2ω2θ2τ1+4δk2ω2τ1+2kωπ0τ1+2kωπ1τ2+2k2τ1+2ω2τ1=0,4δ2k2ω2τ2+2δk2ω2θτ1+2k2ω2θπ1+4k2ω2π2+kωπ02+2kωπ1τ1+2kωπ2τ2+2k2π0+2ω2π0=0,2k2ω2θ2π1+4δk2ω2π1+12k2ω2θπ2+2kωπ0π1+2kωπ2τ1+2k2π1+2ω2π1=0,6δk2ω2θπ1+8k2ω2θ2π2+16δk2ω2π2+2kωπ0π2+kωπ12+2k2π2+2ω2π2=0,4δ2k2ω2π1+20δk2ω2θπ2+2kωπ1π2=0,12δ2k2ω2π2+kωπ22=0.First:(114)δ=116k2+ω2k2ω2,θ=0,π0=−32k2+ω2kω,π2=−3k2+ω2264k3ω3,τ2=−12kω.According to Family I, (112) can be written as
(115)Ψ1q=−32k2+ω2kω+3k2+ω2264k3ω3δcoth2−4δ2q+Σ+12kωδtanh2−4δ2q+Σ,where
(116)q=kx+1Γαα−ωt+1Γαα.According to Family II, (112) becomes
(117)Ψ2q=−32k2+ω2kω−3k2+ω2264k3ω3δcot24δ2q+Σ+−12kωδtan24δ2q+Σ,where
(118)δ=116k2+ω2k2ω2,q=kx+1Γμμ−ωt+1Γμμ.Second:(119)δ=−116k2+ω2k2ω2,θ=0,π0=−12k2+ω2kω,π2=−3k2+ω2264k3ω3,τ2=−12kω.According to Family I, (112) becomes
(120)Ψ3q=−12k2+ω2kω+3k2+ω2264k3ω3δcoth2−4δ2q+Σ+12kωδtanh2−4δ2q+Σ,where
(121)q=kx+1Γαα−ωt+1Γαα.According to Family II, (112) can be written as
(122)Ψ4q=−12k2+ω2kω−3k2+ω2264k3ω3δcot24δ2q+Σ+−12kωδtan24δ2q+Σ,where
(123)δ=−116k2+ω2k2ω2,q=kx+1Γμμ−ωt+1Γμμ.Third:(124)δ=14k2ω2θ2+k2+ω2k2ω2,θ=0,π0=−3k2ω2θ2+k2+ω2kω,π1=−3k2ω2θ2+k2+ω2θkω,π2=−34k2ω2θ2+k2+ω22k3ω3.According to Family II, (112) can be seen as
(125)Ψ5q=−3k2ω2θ2+k2+ω2kω−3k2ω2θ2+k2+ω2θ/kω−θ2+4δ/2δtan−θ2+4δ/2q+Σ−θ/2δ−3/4k2ω2θ2+k2+ω22/k3ω3−θ2+4δ/2δtan−θ2+4δ/2q+Σ−θ/2δ2,where
(126)θ2−4δ=−k2+ω2k2ω2<0,q=kx+1Γμμ−ωt+1Γμμ.Fourth:(127)δ=14k2ω2θ2−k2−ω2k2ω2,θ=0,π0=−3k2ω2θ2−k2−ω2kω,π1=−3k2ω2θ2−k2−ω2θkω,π2=−34k2ω2θ2−k2−ω22k3ω3.According to Family I, (112) becomes
(128)Ψ6q=−3k2ω2θ2−k2−ω2kω−3k2ω2θ2−k2−ω2θ/kω−θ2−4δ/2δtanhθ2−4δ/2q+Σ−θ/2δ−3/4k2ω2θ2−k2−ω22/k3ω3−θ2−4δ/2δtanhθ2−4δ/2q+Σ−θ/2δ2,where
(129)θ2−4δ=k2+ω2k2ω2>0,q=kx+1Γμμ−ωt+1Γμμ.Fifth:(130)δ=−144k2+144ω2−τ12576k2ω2,θ=−112τ1kω,π0=14848k2+48ω2−τ12kω,τ2=−12kω.According to Family I, (112) can be written as
(131)Ψ7q=14848k2+48ω2−τ12kω+τ1−θ2−4δ2δtanhθ2−4δ2q+Σ−θ2δ−12kω−θ2−4δ2δtanhθ2−4δ2q+Σ−θ2δ2,where
(132)θ2δ=24τ1kω144k2+144ω2−τ12,θ2−4δ=k2+ω2k2ω2>0,q=kx+1Γαα−ωt+1Γαα.According to Family III, (112) becomes
(133)Ψ8q=14848k2+48ω2−τ12kω+τ1θexpθq+Σ−1−12kωθexpθq+Σ−12,where
(134)τ1=12k2+ω2,θ=−k2+ω2kω,q=kx+1Γμμ−ωt+1Γμμ.Sixth:(135)δ=−144k2+144ω2−τ12576k2ω2,θ=−112τ1kω,π0=14848k2+48ω2−τ12kω,τ2=−12kω.According to Family II, (112) can be written as
(136)Ψ9q=−14848k2+48ω2−+τ12kω+τ1−θ2+4δ2δtan−θ2+4δ2q+Σ−θ2δ−12kω−θ2+4δ2δtan−θ2+4δ2q+Σ.−θ2δ2,where
(137)θ2δ=−24τ1kω144k2+144ω2+τ12,θ2−4δ=−k2+ω2k2ω2<0,q=kx+1Γμμ−ωt+1Γμμ.


## 7. Results and Discussion

In this section, we will discuss the physical explanation of the found exact solutions to three nonlinear fractional models. We expose the graphical representation of these solutions and accomplish the different kinds of solution. To sketch plots, we have utilized the Maple software package. Every exact solutions are offered in a Maple 3D plot view for proper understanding. Some appropriate values are given in Figures1–6 to analyze the dynamic properties briefly. These figures are dependent on the family conditions which are of importance physically. It has been investigated that all figures include a 3D plot with three fractional-order μ=0.5, μ=0.75, and μ=1 designed for three nonlinear fractional models involving the μ-derivative.Figure 1
The different types of graphs to bright solution (25) for the parameters θ=3, δ=2, k=−2, γ=1, b=−10, and β=2 with providing amounts of (a) μ=0.5, (b) μ=0.75, and (c) μ=1 within the interval 0≤x≤20,0≤t≤10.
(a)(b)(c)Figure 2
The different types of graphs to periodic solution (27) for the parameters θ=2, δ=3, k=−2, γ=−1, b=1, and β=2 with providing amounts of (a) μ=0.5, (b) μ=0.75, and (c) μ=1 within the interval 0≤x≤20,0≤t≤10.
(a)(b)(c)Figure 3
The different types of graphs to kink solution (72) for the parameters θ=3, δ=2, d=1, k=−2, a=1, b=−10, ω=2, and y=1 with providing amounts of (a) μ=0.5, (b) μ=0.75, and (c) μ=1 within the interval 0≤x,t≤10.
(a)(b)(c)Figure 4
The different types of graphs to periodic solution (74) for the parameters θ=2, δ=3, d=1, k=−2, a=1, b=−10, ω=2, and y=1 with providing amounts of (a) μ=0.5, (b) μ=0.75, and (c) μ=1 within the interval 0≤x,t≤10.
(a)(b)(c)Figure 5
The different types of graphs to bright solution (115) for the parameters δ=−2, d=1, k=2, and ω=2 with providing amounts of (a) μ=0.5, (b) μ=0.75, and (c) μ=1 within the interval 0≤x,t≤10.
(a)(b)(c)Figure 6
The different types of graphs to periodic solution (117) for the parameters δ=−2, d=1, k=2, and ω=2 with providing amounts of (a) μ=0.5, (b) μ=0.75, and (c) μ=1 within the interval 0≤x,t≤10.
(a)(b)(c)


## 8. Conclusion

In this study, the efficient and significant solutions to three nonlinear fractional models were established which include kink solution, periodic wave solution, singular kink, single soliton, and other types of soliton found by preferring different free parameters. These applied parameters have important conjugation, such as choosing various inputs of free parameters from an individual solution; known solutions must be found identically. It is important to notice that the new type solution of the space-time fractional1+1-dimensional Boussinesq equation, 2+1-dimensional breaking soliton equations, and SRLW equations has not been exposed by the oncoming exp−Ωq-expansion technique in the previous literature. So we claim that in this current study, the obtained solutions are unique and thus could be more effective in the study of space-time fractional nonlinear physical phenomena. All calculations in this paper have been made quickly with the aid of Maple. The development of offered methods may allow the mitigating Internet bottleneck with quadratic-cubic nonlinearity to be used in more general configurations. The solutions are all verified by putting them back into the original equations with the aid of the Maple symbolic computation package 18.



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*Source: 6680874-2021-04-10.xml*