# Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative

Authors: Majid Bagheri; Ali Khani
Journal: Advances in Mathematical Physics
 (2020)

---

## Abstract

The present work is related to solving the fractional generalized Korteweg-de Vries (gKdV) equation in fractional time derivative form of orderα. Some exact solutions of the fractional-order gKdV equation are attained by employing the new powerful expansion approach by using the beta-fractional derivative which is used to get many solitary wave solutions by changing various parameters. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of nonlinearity. Some of the nonlinear equations arise in fluid dynamics and nonlinear phenomena.


---

## Body

## 1. Introduction

The differential equation of fractional order is the new form of classical integer order differential equations. Different types of differential equations of both ordinary differential equations (ODEs) and partial differential equations (PDEs) in various fields of science like fluid mechanics and biological systems are expressed in fractional forms [1]. There is no any particular method for accessing the exact type of solutions of fractional PDEs, but some approximate solutions are determined by using the Adomian decomposition approach, the homotopy perturbation approach, and the homotopy analysis approach [2–4]. The analytical method and its various forms are well known for determining soliton wave-type solutions of the nonlinear PDEs. With chronology, some investigators have utilized the new analytical approach on fractional-type nonlinear PDEs for obtaining the solitary solutions. This work is related to the fractional order generalized Korteweg-de Vries (gKdV) equation [5]; for this, the fractional form of the gKdV equation [5] is taken as(1)utα+Fuux+uxxx=0,0≤α<1,0<d<4,Fu=λud+μu2d,λ,μ∈R.Equation (1) is a fractional form of the classical generalized Korteweg-de Vries equation which exists by changing the 1st-order time derivatives by fractional derivatives.In the past two decades, the fractional calculus theory gained great attention and popularity in various fields of science and engineering due to its demonstrated applications. These contributions to the fields of sciences and engineering are based on mathematical analysis. They cover widely known classical fields such as Abel’s integral equation and viscoelasticity. They also include the analysis of feedback amplifiers, the fractional-order Chua-Hartley systems, electrode-electrolyte interface models, fractional-order models of neurons, electric conductance of biological systems, generalized voltage dividers, fitting of experimental data, capacitor theory, and the fields of special functions [6–9].Several robust methods have been used to solve the FDEs, the fractional differential equations and dynamic systems containing fractional derivatives. Some of the most important methods are Adomian’s decomposition method [10–12], the exp-function method [13], He’s variational iteration method [14, 15], the fractional subequation method [16], the first integral method [17], the homotopy analysis method [18], the (G′/G)-expansion method [19], the homotopy perturbation method [20, 21], the spectral methods [22], and the transform methods [23]. In [24], the authors presented two methods, which are the exp(−ϕξ)-expansion method and the Kudryashov method. In [25], the authors demonstrated three methods, which are the csch function method, the tanh-coth method, and the modified simple equation method. In [26–29], the authors introduced the semi-inverse variational principle method, the extended Kudryashov method, the modified simple equation method, and the expanded trail equation method, respectively. Moreover, the fractional differential equations have been studied by powerful authors, and their applications were introduced in sciences and engineering branches [30–32].One of the well-known equations is the ZK equation, first obtained as a description of weakly nonlinear ion-acoustic modes in a strongly magnetized plasma; it is of particular interest as it is the simplest equation that admits cylindrical and spherical solitary wave solutions in addition to the planar KdV soliton solutions [33]. Another powerful analytical method is called the exp-function method (EFM), which was first presented by He [34]. EFM has successfully been applied to many situations. For example, He and Wu [34] solved the nonlinear wave equations via EFM. Abdou [35] solved generalized solitonary and periodic solutions for nonlinear partial differential equations by EFM. For further information, refer to vigorous references in ([35–40]). In the following years, this proposed method was improved by many researchers. Yang developed general fractional derivatives, along with theory, methods, and applications, to some nonlinear fractional differential equations [41]. Recently, there has been a new fractal nonlinear Burgers’ equation arising from the acoustic signal propagation studied by Yang [42]. Also, Yang et al. investigated fundamental solutions of anomalous equations implemented with the decay exponential kernel [43]. A new integral transform operator for solving the heat-diffusion problem has been utilized by Yang [44]. In [45], Liu et al. probed the group analysis to the time fractional nonlinear wave equation and found many exact solutions. Moreover, Liu et al. worked on the time-fractional nonlinear diffusion equation [46]. Same authors proposed the fractional symmetry group method for the time fractional nonlinear heat conduction equation which usually appears in mathematics, physics, integrable systems, fluid mechanics, and nonlinear areas [47]. Furthermore, in the last decade, many real world problems have been explained using fractional partial differential equations which can be linked to the following valuable researches: utilizing the q-homotopy analysis transform method and the fractional natural decomposition method for the fractional Benney-Lin equation [48]; applying the quasiaffine biframelets to the fractional differential equations [49]; analyzing the moderate epidemiological model to describe computer viruses with an arbitrary order derivative [50]; solving the fractional Kaup-Kupershmidt equation with the Atangana-Beleanu derivative and the Caputo-Fabrizio derivative [51]; employing the Riemann-Liouville integral, the Atangana-Baleanu integral operator, and the nonlinear Telegraph equation [52]; solving singular fractional integrodifferential equations with applications to hematopoietic stem cell modeling [53]; applying the truncatedM-fractional derivative to fractional differential equations [54]; applying them+1/G′-expansion method to the study of the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation [55], the study of unreported cases of 2019-nCOV epidemic outbreaks [56–58], and the study of the conformable (2+1)-dimensional Ablowitz-Kaup-Newell-Segur equation [59]; and employing a reliable hybrid numerical method for a time-dependent vibration model [60]. To make this paper more self-contained, a general trigonometric, hyperbolic, and exponential function of the gKdV equation is constructed with the help of an expansion approach, which is crucial in obtaining the lump solution of equation (1).The pattern of this article is summarized as follows: Section2 gives the details of the initial definitions; Section 3 gives an introduction of the direct truncation method which is utilized for getting the exact solutions of the gKdV equation in Section 4; Section 5 gives the numerical simulation and graph details; and Section 6 gives some conclusions at the end.


## 2. Initial Definitions

Definition 1.
Definition ofβ-derivative: suppose φ:0;1→R, then the β derivative of φ of order α is defined as(2)Dtαφt=limε→0φt+εt+1/Γα1−α−φtε,α∈0,1,t>0.The following properties and new theorems will be used.Theorem 2.
Supposeα∈0,1; let φ,ω be α-differentiable at point t. Therefore, we have(3)Dtαaφt+bωt=aDtαφt+bDtαωt,fora,b∈R,Dtαc=0,forc∈R,Dtαφtωt=φtDtαωt+ωtDtαφt,Dtαφtωt=φtDtαωt−ωtDtαφtω2t,Dtαφt=t+1Γα1−αdφtdt.Theorem 3 (see [61–63]).
Supposeφ:0;1→R is a function such that φ is differentiable and also α-differentiable. Assume ω is a differentiable function defined in the range of φ. Therefore, we have(4)Dtαφoωt=t+1Γα1−αω′tχ′ωt,where a prime denotes the classical derivatives with respect to t.


## 3. Methodology

In this section, we give a description of the direct truncation method and introduce it to a partial differential equation.For a given partial differential equation(5)Pu,ux,uxx,⋯,Dtαu,Dxαu,Dxxαu,⋯=0,0<α≤1.Using a transformation as(6)ux,t=uϕ,η=kx+lαt+1Γαα,where k and l are constants to be determined later. We can rewrite equation (5) in the following nonlinear ODE:(7)Qu,ku′,k2u′′,⋯,lu′,⋯=0,where the prime denotes the derivative with respect to ϕ. If possible, we integrate equation (7) term by term one or more times. This yields constants of integration. For simplicity, the integration constants can be set to zero. Suppose g has the following truncation form:(8)gϕ=∑j=0τajξϕjζϕτ,in which ξϕ and ζϕ are introduced in the following forms:(9)ξϕ=p1Fχϕ+q1Gχϕ+r1,ζϕ=p2Fχϕ+q2Gχϕ+r2,uξ=gϕ=∑j=0τajp1Fχϕ+q1Gχϕ+r1jp2Fχϕ+q2Gχϕ+r2τ,where aj, p1, q1, r1, p2, q2, and r2 are constants to be determined, χϕ is given, and eitherFandGare functions determined by an ordinary differential system or F and G are functions given by direct ansatz such that their derivations are combinations of F and G, and χϕ is determined by an ordinary differential equation:(10)dχϕdϕ=Hχϕ=LFχϕ+MGχϕ+N,in which the function H is also given by a direct ansatz according to the context, and the exponent τ is determined by utilizing the homogeneous balance method in equation (5). The value τ is determined by equalizing the maximum order nonlinear term and the maximum order partial derivative term appearing in (7). Ifτis rational, then the appropriate transformations can be applied to conquer these hurdles. Substituting (9) and (10) into (8) leads to a polynomial in Fϕ and Gϕ; then, we set the coefficients of FiϕGjϕ and the constant term to zero to get a system of algebraic equations on the unknown parameters in H together with the unknown numbers aj, p1, q1, r1, p2, q2, and r2 for j=0,1,⋯,τ. By solving the system, one can get aj, p1, q1, r1, p2, q2, and r2 and the unknown parameters in H; then, by solving equation (10) to getχϕ, the solutions of equation (5) can be obtained.


## 4. Traveling Wave Solution Fractional Order Form of Generalized KdV Equation

The given section deals with the application of a new powerful expansion technique by determining the traveling wave form solutions of the fractional order generalized KdV equation:(11)utα+Fuux+uxxx=0,Fu=λud+μu2d,λ,μ≠0,by applying the aforementioned method. By using the fractional beta complex transform η=kx+l/αt+1/Γαα, equation (11) is reduced to the following ODE:(12)lu′+λud+μu2dku′+k3u‴=0.Integrating equation (12) once and setting the constant of integration equal to zero results in(13)lu+kλd+1ud+1+kμd+1u2d+1+k3u″=0.Balancingu″ and u2d+1 by employing the homogenous principle, we get(14)M+2=2d+1M,⇒M=1d.To get a closed form solution, we use the transformation(15)uη=vη1/d.Substituting (15) into equation (13), we get(16)lv2+kλd+1v3+kμ2d+1v4+k3d1−ddv′2+vv″=0.Balancingvv″ and v4, we get(17)2M+2=4M,⇒M=1.Case I.
Then, the exact solution will be as(18)vη=e2χϕa1p1+eχϕa1q1+a1r1+a0p2e2χϕ+q2eχϕ+r2.
Inserting (18) into equation (16), we obtain the below relation(19)d2d+12d+1p2e2χϕ+q2eχϕ+r24−1∑n=011Cnexpnχϕ=0,where Cn0≤n≤11 are polynomial statements in terms of a0, a1, p1, p2, q1, q2, r1, and r2. Hence, solving the resulting system Cn=00≤n≤11 simultaneously, we acquire the following sets of parameters of solutions.Set I.
(20)L=0,M=0,N=12−μd+12d+1λdμd+1d+2k,k=k,l=λ22d+1kμd3+5d2+8d+4,a0=−r2λ2d+1d+2μ,a1=0,p1=p1,p2=p2,q1=q1,q2=0,r1=r1,r2=r2.
We, therefore, gained the following generalized solitary solution:(21)u1ϕ=−r2λ2d+1d+2μp2e2χϕ+r21/d,χϕ=12−μd+12d+1λdμd+1d+2kϕ+C,in which(22)ϕ=kx+λ22d+1kμαd3+5d2+8d+4t+1Γαα.Set II.
(23)L=0,M=0,N=−μd+12d+1λdμd+1d+2k,k=k,l=λ22d+1kμd3+5d2+8d+4,a0=−r2λ2d+1d+2μ,a1=0,p1=p1,p2=0,q1=q1,q2=q2,r1=r1,r2=r2.
We, therefore, gained the following generalized solitary solution:(24)u2ϕ=−r2λ2d+1d+2μq2eχϕ+r21/d,χϕ=−μd+12d+1λdμd+1d+2kϕ+C,in which(25)ϕ=kx+λ22d+1kμαd3+5d2+8d+4t+1Γαα.Set III.
(26)L=0,M=M,N=N,k=−μd+12d+1λdμd+1d+2N,l=λ32d+1−μd+12d+1dμ2d+1d3+5d2+8d+4d+2N,a0=−Mq2−Np2λ2d+1r1Nd+2μp1,a1=Mq2−Np2λ2d+1r1Nd+2μp1,p1=p1,p2=p2,q1=0,q2=q2,r1=r1,r2=0.
We, therefore, gained the following generalized solitary solution:(27)u3ϕ=Mq2−Np2λ2d+1eχϕNd+2μeχϕp2+q21/d,χϕ=Nϕ+C+lnN1−MexpNϕ+C,in which(28)ϕ=−μd+12d+1λdμd+1d+2Nx+λ32d+1−μd+12d+1dμ2αd+1d3+5d2+8d+4d+2Nt+1Γαα.Set IV.
(29)L=0,M=M,N=12−μd+12d+1λdμd+1d+2k,k=k,l=λ22d+1kμd3+5d2+8d+4,r1=r1,r2=r2,a0=4M2d3k2μr2+20M2d2k2μr2+32M2dk2μr2+16M2k2μr2+2d3λ2p2+d2λ2p2r1d+2μd2λp1,p1=p1,p2=p2,q1=0,a1=−4M2d3k2μr2+20M2d2k2μr2+32M2dk2μr2+16M2k2μr2+2d3λ2p2+d2λ2p2d+2μd2λp1,q2=4r2Md+2kμd+1−μd+12d+1λd.
We, therefore, gained the following generalized solitary solution:(30)u4ϕ=−124M2k2μr2d+1d+22+d2λ2p22d+1−μd+12d+1e2χϕμ2d+1dd+21/2−μd+12d+1λdp2e2χϕ+r2/μd+1+2r2Mkeχϕd+2,in which(31)χϕ=Nϕ+C+lnN1−MexpNϕ+C,ϕ=kx+λ22d+1kαμd3+5d2+8d+4t+1Γαα.Set V.
(32)L=0,M=M,N=N,k=k,l=−N2k3d2,a0=2Md2r2−Nd2q2+6Mdr2−3Ndq2+4Mr2−2Nq2Nk2r1q1d2λ,a1=−2Md2r2−Nd2q2+6Mdr2−3Ndq2+4Mr2−2Nq2Nk2q1d2λ,p1=Mq1N,q1=q1,q2=q2,r1=r1,r2=r2,p2=4Mk2μr2d+1d+22Mr2−Nq2+q22N2d3k2μ+5N2d2k2μ+8N2dk2μ+4N2k2μ+2d3λ2+d2λ24d2λ2r22d+1.
We, therefore, gained the following generalized solitary solution:(33)u5ϕ=−4d+2d+12Mr2−Nq2k2λ2d+1r2eχϕMeχϕ+Ne2χϕ4M2d3k2μr22−4MNd3k2μq2r2+N2d3k2μq22+20M2d2k2μr22−20MNd2k2μq2r2+5N2d2k2μq22+32M2dk2μr22−32MNdk2μq2r2+8N2dk2μq22+16M2k2μr22−16MNk2μq2r2+4N2k2μq22+2d3λ2q22+d2λ2q22+4q2eχϕd2λ2r22d+1+4d2λ2r222d+11/d,in which(34)χϕ=Nϕ+C+lnN1−MexpNϕ+C,ϕ=kx−N2k3αd2t+1Γαα.Set VI.
(35)L=0,M=−2d+1dλp1q1k−μd+12d+1d+2,N=−μd+12d+1λdμd+1d+2k,k=k,l=kλ22d+1d+1μd+22,a0=λq22d+1r1μq1d+2,a1=−λq22d+1μq1d+2,p1=p1,p2=p2,q1=q1,q2=q2,r1=r1,r2=0.
We, therefore, gained the following generalized solitary solution:(36)u6ϕ=−λq22d+1eχϕp1+q1μq1d+2eχϕp2+q21/d,in which(37)χϕ=Nϕ+C+lnN1−MexpNϕ+C,ϕ=kx+kλ22d+1αd+1μd+22t+1Γαα.Set VII.
(38)L=0,M=M,N=N,k=−μd+12d+1λdμd+1d+2N,l=λ32d+1−μd+12d+1dμ2d+1d3+5d2+8d+4d+2N,a0=−Mq1r2+Np1r2−Nq1q2λ2d+1r1Nd+2μq12,a1=Mq1r2+Np1r2−Nq1q2λ2d+1Nd+2μq12,p1=p1,p2=−p1p1r2−q1q2q12,q1=q1,q2=q2,r1=r1,r2=r2.
We, therefore, gained the following generalized solitary solution:(39)u7ϕ=−Mq1r2+Np1r2−Nq1q2λ2d+1eχϕeχϕp1r2−q2eχϕq1−q1r2Nd+2μ1/d,in which(40)χϕ=Nϕ+C+lnN1−MexpNϕ+C,ϕ=−μd+12d+1λdμd+1d+2Nx+λ32d+1−μd+12d+1dαμ2d+1d3+5d2+8d+4d+2Nt+1Γαα.Set VIII.
(41)L=0,M=0,N=−μd+12d+1λdμd+1d+2k,k=k,l=kλ22d+1d+1μd+22,a0=−dμa1r1+2dλr2+2μa1r1+λr2d+2μ,a1=a1,p1=0,p2=0,q1=0,q2=q2,r1=r1,r2=r2.
We, therefore, gained the following generalized solitary solution:(42)u8ϕ=−r2λ2d+1d+2μq2eχϕ+r21/d,in which(43)χϕ=Nϕ+C,ϕ=kx+kλ22d+1αd+1μd+22t+1Γαα.Set IX.
(44)L=0,M=0,N=−μd+12d+1λdμd+1d+2k,k=k,l=kλ22d+1d+1μd+22,a0=−r2λ2d+1d+2μ,a1=0,p1=p1,p2=0,q1=q1,q2=q2,r1=r1,r2=r2.
We, therefore, gained the following generalized solitary solution:(45)u9ϕ=a1q1eχϕμd+2a1q1eχϕμ−2dλr2−λr2d+2μr21/d,in which(46)χϕ=Nϕ+C,ϕ=kx+kλ22d+1αd+1μd+22t+1Γαα.Set X.
(47)L=0,M=−μ2d2+3d+1a1q1d2d2+3d+1r2k,N=dλ2d2+3d+1d+2−μ2d2+3d+1kd+1,k=k,l=kλ22d+1d+1μd+22,a0=−dμa1r1+2dλr2+2μa1r1+λr2d+2μ,a1=a1,p1=0,p2=0,q1=q1,q2=q2,r1=r1,r2=r2.
We, therefore, gained the following generalized solitary solution:(48)u10ϕ=a1q1eχϕμd+2a1q1eχϕμ−2dλr2−λr2d+2μq2eχϕ+r21/d,in which(49)χϕ=Nϕ+C+lnN1−MexpNϕ+C,ϕ=kx+kλ22d+1αd+1μd+22t+1Γαα.Set XI.
(50)L=0,M=0,N=−μd+12d+1λdμd+1d+2k,k=k,l=kλ22d+1d+1μd+22,a0=−dμa1r1+2dλr2+2μa1r1+λr2d+2μ,a1=a1,p1=0,p2=p2,q1=q1,q2=−d2μ2a12q12+4dμ2a12q12+4d2λ2p2r2+4μ2a12q12+4dλ2p2r2+λ2p2r2μ2d2+5d+2a1q1λ,r1=r1,r2=r2.
We, therefore, gained the following generalized solitary solution:(51)u11ϕ=λq1a12d+12eχϕdλp2−dμa1q1+eχϕλp2−2μa1q11/d,in which(52)χϕ=Nϕ+C,ϕ=kx+kλ22d+1αd+1μd+22t+1Γαα.Set XII.
(53)L=0,M=−14q2dλ2d+1r2k−μd+12d+1d+2,N=12−μd+12d+1μd+1,k=k,l=kλ22d+1d+1μd+22,a0=−dμa1r1+2dλr2+2μa1r1+λr2d+2μ,a1=a1,p1=−142d+1q22λd+2a1r2μ,p2=p2,q1=−q2λ2d+1d+2a1μ,q2=q2,r1=r1,r2=r2.
We, therefore, gained the following generalized solitary solution:(54)u12ϕ=−14λ2q22e2χϕd+q22e2χϕ+8q2eχϕr2d+4q2eχϕr2+8dr22+4r22p2e2χϕ+q2eχϕ+r2d+2r2μ1/d,in which(55)χϕ=Nϕ+C+lnN1−MexpNϕ+C,ϕ=kx+kλ22d+1αd+1μd+22t+1Γαα.Case II.
Then the exact solution will be as(56)vη=sinhχϕa1p1+coshχϕa1q1+a1r1+a0p2sinhχϕ+q2coshχϕ+r2.
Inserting (56) into equation (16), we obtain(57)d2d+12d+1p2sinhχϕ+q2coshχϕ+r24−1∑i+j=6Cijsinhiχϕcoshjχϕ=0,where Ciji+j=6,0≤i,j≤6 are polynomial statements in terms of a0, a1, p1, p2, q1, q2, r1, and r2. Hence, solving the resulting system Cij=0i+j=6,0≤i,j≤6 simultaneously, we acquire the following sets of parameters of solutions.Set I.
(58)L=0,M=0,N=12−μd+12d+1λdμd+1d+2k,k=k,l=λ22d+1kμd3+5d2+8d+4,a0=−r2λ2d+1d+2μ,a1=0,p1=p1,p2=q2,q1=q1,q2=q2,r1=r1,r2=r2.
We, therefore, gained the following generalized solitary solution:(59)u1ϕ=−λ2d+1r2μd+2q2sinhχϕ+q2coshχϕ+r21/d,χϕ=12−μd+12d+1λdμd+1d+2kϕ+C,in which(60)ϕ=kx+λ22d+1kμαd3+5d2+8d+4t+1Γαα.Set II.
(61)L=0,M=0,N=−μd+12d+1λdμd+1d+2k,k=k,l=λ22d+1kμd3+5d2+8d+4,a0=−r2λ2d+1d+2μ,a1=0,p1=p1,p2=−q2,q1=q1,q2=q2,r1=r1,r2=r2.
We, therefore, gained the following generalized solitary solution:(62)u2ϕ=λ2d+1r2μd+2q2sinhχϕ−q2coshχϕ−r21/d,χϕ=−μd+12d+1λdμd+1d+2kϕ+C,in which(63)ϕ=kx+λ22d+1kμαd3+5d2+8d+4t+1Γαα.Set III.
(64)L=0,M=0,N=12−μd+12d+1λdμd+1d+2k,k=k,l=λ22d+1kμd3+5d2+8d+4,a0=12λ2dp2+2dq2+p2+q2r1q1d+2μ,a1=−12λ2dp2+2dq2+p2+q2q1d+2μ,p1=q1,p2=p2,q1=q1,q2=q2,r1=r1,r2=0.
We, therefore, gained the following generalized solitary solution:(65)u3ϕ=−12λ2dp2+2dq2+p2+q2sinhχϕ+coshχϕμd+2p2sinhχϕ+q2coshχϕ1/d,χϕ=12−μd+12d+1λdμd+1d+2kϕ+C,in which(66)ϕ=kx+λ22d+1kμαd3+5d2+8d+4t+1Γαα.Set IV.
(67)L=0,M=0,N=12−μd+12d+1λdμd+1d+2k,k=k,l=λ22d+1kμd3+5d2+8d+4,a0=12λ2dp2−2dq2+p2−q2r1q1d+2μ,a1=−12λ2dp2−2dq2+p2−q2q1d+2μ,p1=−q1,p2=p2,q1=q1,q2=q2,r1=r1,r2=0.
We, therefore, gained the following generalized solitary solution:(68)u4ϕ=−12λ2dp2−2dq2+p2−q2sinhχϕ−coshχϕμd+2p2sinhχϕ+q2coshχϕ1/d,χϕ=12−μd+12d+1λdμd+1d+2kϕ+C,in which(69)ϕ=kx+λ22d+1kμαd3+5d2+8d+4t+1Γαα.Set V.
(70)L=0,M=0,N=12−μd+12d+1λdμd+1d+2k,k=k,l=λ22d+1kμd3+5d2+8d+4,a0=−12q2λ2d+1r1μp1d+2,a1=12q2λ2d+1μp1d+2,p1=p1,p2=0,q1=−p1,q2=q2,r1=r1,r2=0.
We, therefore, gained the following generalized solitary solution:(71)u5ϕ=122d+1λsinhχϕ−coshχϕμd+2coshχϕ1/d,χϕ=12−μd+12d+1λdμd+1d+2kϕ+C,in which(72)ϕ=kx+λ22d+1kμαd3+5d2+8d+4t+1Γαα.Set VI.
(73)L=0,M=0,N=12−μd+12d+1λdμd+1d+2k,k=k,l=λ22d+1kμd3+5d2+8d+4,a0=12q2λ2d+1r1μp1d+2,a1=−12q2λ2d+1μp1d+2,p1=p1,p2=0,q1=p1,q2=q2,r1=r1,r2=0.
We, therefore, gained the following generalized solitary solution:(74)u6ϕ=−122d+1λsinhχϕ+coshχϕμd+2coshχϕ1/d,χϕ=12−μd+12d+1λdμd+1d+2kϕ+C,in which(75)ϕ=kx+λ22d+1kμαd3+5d2+8d+4t+1Γαα.Set VII.
(76)L=0,M=0,N=12−μd+12d+1λdμd+1d+2k,k=k,l=λ22d+1kμd3+5d2+8d+4,a0=12q2λ2d+1r1μp1,a1=−12q2λ2d+1μp1,p1=p1,p2=d+1q2,q1=p1,q2=q2,r1=r1,r2=0.
We, therefore, gained the following generalized solitary solution:(77)u7ϕ=−122d+1λsinhχϕ+coshχϕμsinhχϕd+sinhχϕ+coshχϕ1/d,χϕ=12−μd+12d+1λdμd+1d+2kϕ+C,in which(78)ϕ=kx+λ22d+1kμαd3+5d2+8d+4t+1Γαα.Set VIII.
(79)L=0,M=0,N=12−μd+12d+1λdμd+1d+2k,k=k,l=λ22d+1kμd3+5d2+8d+4,a0=−12q2λ2d+1r1μp1,a1=12q2λ2d+1μp1,p1=p1,p2=−d+1q2,q1=−p1,q2=q2r1=r1,r2=0.
We, therefore, gained the following generalized solitary solution:(80)u8ϕ=−122d+1λsinhχϕ−coshχϕμsinhχϕd+sinhχϕ−coshχϕ1/d,χϕ=12−μd+12d+1λdμd+1d+2kϕ+Cin which(81)ϕ=kx+λ22d+1kμαd3+5d2+8d+4t+1Γαα.


## 5. Graphical Representation

The graphical description of derived soliton and other solutions have been expressed in the mentioned figures by allotting the different values of the parameters. The 3D plots for some solutions of the considered equation for four fractional order casesα=0.25, 0.5, 0.85, and 0.99 have been shown. When all obtained exact solutions for the fractional generalized Korteweg-de Vries equation are examined, the exact solution (32) is similar to the solution of Sahadevan and Bakkyaraj [64], the solution of Li et al. [65] (48), and the solution of Akbulut and Tascan [66] (74) in the literature. There are other obtained exact solutions that are not included in the literature, and it can be said that they are new exact solutions obtained by the new version trial equation method. The results are new, interesting, and have a great impact in the field of nonlinear sciences where the (2+1)-dimensional KdV equation will be used for the dynamics of nonlinear solitons and other solutions. Also, two- and three-dimensional graphics of the obtained solution functions are illustrated in Figures 1–6 which demonstrate suitable parametric choices.Figure 1
The 3D plot of (21) at d=0.2, μ=−1, p2=1.5, r2=2, λ=2.2, and k=3 when (a) α=0.25, (b) α=0.5, (c) α=0.85, and (d) α=0.99.
(a)(b)(c)(d)Figure 2
The 3D plot of (27) at d=0.2,μ=−1, p2=1.5, q2=2, λ=2.2, M=−3, N=2, and k=3 when (a) α=0.25, (b) α=0.5, (c) α=0.85, and (d) α=0.99.
(a)(b)(c)(d)Figure 3
The 3D plot of (45) at d=2, μ=−1, a1=1.5, q1=0.2, r2=2, λ=2.2, M=−3, N=2, and k=3 when (a) α=0.25, (b) α=0.5, (c) α=0.85, and (d) α=0.99.
(a)(b)(c)(d)Figure 4
The 3D plot of (59) at d=0.2, μ=−1, q2=1.5, r2=2, λ=2.2, M=3, N=2, and k=3 when (a) α=0.25, (b) α=0.5, (c) α=0.85, and (d) α=0.99.
(a)(b)(c)(d)Figure 5
The 3D plot of (65) at d=0.2, μ=−1, p2=1.5, q2=2, λ=2.2, M=3, N=2, and k=3 when (a) α=0.25, (b) α=0.5, (c) α=0.85, and (d) α=0.99.
(a)(b)(c)(d)Figure 6
The 3D plot of (80) at d=0.2, μ=−1, r2=1.5, q2=2, λ=2.2, M=3, N=2, and k=3, when (a) α=0.25, (b) α=0.5, (c) α=0.85, and (d) α=0.99.
(a)(b)(c)(d)


## 6. Conclusion

We used the direct truncation method to find the explicit solution of the fractional generalized Korteweg-de Vries equation, comparing it with the known results in the literature, and we were able to get some novel solitary wave solutions, including bright, dark, kink, and periodic solitons. Numerical simulation has been performed by utilizing the Maple software. The solution of each PDE is always utilized for understanding the system and various phenomena described by it. The new analytical expansion method is helpful for obtaining the solutions in the form of hyperbolic and ergometric forms which are exact and helpful in understanding its fractional forms. It is not hard to see that the general solution is an algebraically localized wave decayed in all space directions and existing at all times. Finally, a transformation is used to draw soliton solutions of equation (1) using the Maple software. The 3D plot for some solutions of the considered equation for four fractional order cases α=0.25, 0.5, 0.85, and 0.99 are shown in Figures 1–6. Finally, we show some graphs to explain these solutions. So, this gives the efficient applications of the new analytical expansion for the fractional PDEs. In future works with the generalized expansion function based on combined functions, we next aim at the fractional generalized KdV equation or nonlinear wave equations, which possess more terms and higher nonlinearity than the standard equations.



---

*Source: 8819183-2020-11-04.xml*