# Novel Analysis of Fractional-Order Fifth-Order Korteweg–de Vries Equations

Authors: Ahmed B. Khoshaim; Muhammad Naeem; Ali Akgul; Nejib Ghanmi; Shamsullah Zaland
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, theρ-homotopy perturbation transformation method was applied to analysis of fifth-order nonlinear fractional Korteweg–de Vries (KdV) equations. This technique is the mixture form of the ρ-Laplace transformation with the homotopy perturbation method. The purpose of this study is to demonstrate the validity and efficiency of this method. Furthermore, it is demonstrated that the fractional and integer-order solutions close in on the exact result. The suggested technique was effectively utilized and was accurate and simple to use for a number of related engineering and science models.


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## Body

## 1. Introduction

A number of researchers have recently become interested in fractional calculus, which was first developed during Newton’s period. Within the fractional calculus structure, many interesting and significant steps have been discovered within the last thirty decades. A fractional derivative was invented as a result of the complexity of a heterogeneous phenomenon. The fractional derivative operators, by incorporating diffusion methods, are capable of capturing the attitudes of multidimensional media [1–4]. The use of differential equations of any scale has proved useful in showing a number of problems more quickly and accurately. Increasingly, scholars turned to generalized calculus to convey their viewpoints while analyzing complex phenomena in the context of mathematical methods using software [5–10].Nonlinear impacts occur in several implemented scientific fields, such as fluid, mathematical biology, nonlinear image sensors, quantum field theory, kinetics, thermodynamics, and fluid dynamics. It is based on nonlinear partial differential equations of various degrees of complexity to model these processes. Partial differential equations are generally applied in the description of physical processes [11–15]. Most of the essential physical systems do not exhibit linear behavior. There is no way to determine the exact result of such nonlinear phenomena. Only techniques that are appropriate for solving nonlinear equations can be used to investigate this phenomenon [16–22].In 1895, Korteweg and de Vries proposed a KdV equation to design Russell’s soliton phenomenon, such as small and huge water waves. Solitons are steady solitary waves, which mean that these solitary waves are a particle. KdV equations are applied in different applied fields such as quantam mechanics, fluid dynamics, optics, and plasma physics. Fifth-order KdV form equations were utilized to analyze many nonlinear phenomena in particle physics [23–25]. It plays a vital role in the distribution of waves [26]. In their analysis, the KdV form equation has dispersive terms of the third and fifth-order relevant to the magnetoacoustic wave problem in cold plasma free collision plasma and dispersive terms appear near-critical angle propagation [27]. Plasma is a dynamic, quasineutral, and electrically conductive fluid. It consists of neutral particles, electrons, and ions. It consists of magnetic and electric areas due to the electrically conducting behavior of plasma. The mixture of particles and areas supports plasma waves of various forms. A magnetic lock is a less longitudinal ion dispersion. The magnetoacoustic wave behaves as an ion-acoustic wave in the low magnetic field range, while in the low-temperature capacity, it acts as an Alfven wave [28, 29].The general model for the analysis of magnetic properties-acoustic waves in plasma and shallow water waves with surface tension is equated with the fifth order of KdV. Recent study reveals that the solutions to this equation for travelling waves do not vanish at infinity [30, 31]. Consider the well-known three types of the fifth-order KdV equations as follows [32, 33]:(1)DτγV+Vζ+V2V2ζ+VζV2ζ−20V2V3ζ+V5ζ=0,0<β≤1,with initial condition Vζ,0=1/ζ,(2)DτβV+VVζ−VV3ζ+V5ζ=0,0<β≤1,with initial condition Vζ,0=eζ, and(3)DτβV+VVζ+V3ζ−V5ζ=0,0<β≤1,with initial condition Vζ,0=105/169sech4ζ−ϕ/213.(1) and (2) are called fifth-order KdV equations and (3) is called the Kawahara equation. Analytic techniques for these mathematical model are particularly difficult to come across due to their severe nonlinearity. Several researchers have employed various analytical and computational strategies to the solution of linear and nonlinear KdV equations throughout the last decade, such as the multisymplectic method [34], variational iteration method [33], He’s homotopy perturbation method [35], and Exp-function method [36].Recently, Fahd and Abdeljawad [37] developed the Laplace transform of the generalized fractional Caputo derivatives. We established a novel methodology with ρ-Laplace transform for solving fractional differential equations with a generalized fractional Caputo derivative. The homotopy perturbation method is merged with the Laplace transform method to create a highly effective method for handling nonlinear terms which is known as the homotopy perturbation transformation technique. This technique can provide the result in quick convergent series. Ghorbani pioneered the use of He’s polynomials in nonlinear terms [38–40]. Later on, many scholars utilized the homotopy perturbation transformation method for linear and nonlinear differential equations such as heat-like equations [41], Navier–Stokes equations [42], hyperbolic equation and Fisher’s equation [43], and gas dynamic equation [44].


## 2. Basic Definitions

### 2.1. Definition

The fractional generalized integral of orderβ of a continuous function (CF) g:0,+∞⟶R is defined as [37](4)Iβ,ρgζ=1Γβ∫0ζζρ−sρρβ−1gsdss1−ρ,ρ>0,ζ>0,0<β<1.


### 2.2. Definition

The orderβ fractional generalized derivative of a CF g:0,+∞⟶R is given as [37](5)Dβ,ρgζ=I1−β,ρgζ=1Γ1−βddζ∫0ζζρ−sρρ−β−1gsdss1−ρ,ρ>0,ζ>0and0<β<1.


### 2.3. Definition

The Caputo derivative of fractional-orderβ of a CF g:0,+∞⟶R is defined as [37](6)Dβ,ρgζ=1Γ1−βddζ∫0ζζρ−sρρ−β−1βngsdss1−ρ,where ρ>0, ζ>0, β=ζ1−βd/dζ, and 0<β<1.


### 2.4. Definition

Theρ-Laplace transform of a CF g:0,+∞⟶R is defined as [37](7)Lρgζs=∫0∞e−sζρ/ρgζdζζ1−ρ.The fractional generalized Caputo derivative ofρ-Laplace transformation of a CF g is given by [37](8)LρDβ,ρgζs=sβLρgζ−∑k=0n−1sβ−k−1Iβ,ρβng0.


### 2.5. Definition

The generalized Mittag-Leffler function is defined by(9)Eβ,ρz=∑k=0∞zβΓβk+γ,where β>0, γ>0, and Eβz=Eβ,1z.


## 2.1. Definition

The fractional generalized integral of orderβ of a continuous function (CF) g:0,+∞⟶R is defined as [37](4)Iβ,ρgζ=1Γβ∫0ζζρ−sρρβ−1gsdss1−ρ,ρ>0,ζ>0,0<β<1.


## 2.2. Definition

The orderβ fractional generalized derivative of a CF g:0,+∞⟶R is given as [37](5)Dβ,ρgζ=I1−β,ρgζ=1Γ1−βddζ∫0ζζρ−sρρ−β−1gsdss1−ρ,ρ>0,ζ>0and0<β<1.


## 2.3. Definition

The Caputo derivative of fractional-orderβ of a CF g:0,+∞⟶R is defined as [37](6)Dβ,ρgζ=1Γ1−βddζ∫0ζζρ−sρρ−β−1βngsdss1−ρ,where ρ>0, ζ>0, β=ζ1−βd/dζ, and 0<β<1.


## 2.4. Definition

Theρ-Laplace transform of a CF g:0,+∞⟶R is defined as [37](7)Lρgζs=∫0∞e−sζρ/ρgζdζζ1−ρ.The fractional generalized Caputo derivative ofρ-Laplace transformation of a CF g is given by [37](8)LρDβ,ρgζs=sβLρgζ−∑k=0n−1sβ−k−1Iβ,ρβng0.


## 2.5. Definition

The generalized Mittag-Leffler function is defined by(9)Eβ,ρz=∑k=0∞zβΓβk+γ,where β>0, γ>0, and Eβz=Eβ,1z.


## 3. The Rod Map of the Proposed Method

Consider the general partial differential equation given as(10)DτγVζ,τ+MVζ,τ+NVζ,τ=hζ,τ,τ>0,0<γ≤1,Vζ,0=gζ,ν∈ℜ.Applyingρ-Laplace transformation of (10), we get(11)LρDτγVζ,τ+MVζ,τ+NVζ,τ=Lρhζ,τ,τ>0,0<γ≤1,μζ,τ=1sgζ+1sβLρhζ,τ−1sβLρMVζ,τ+NVζ,τ.Now, applying the inverseρ-Laplace transform, we get(12)Vζ,τ=Fζ,τ−Lρ−11sβLρMVζ,τ+NVζ,τ,where(13)Fζ,τ=Lρ−11sgζ+1sβLρhζ,τ=gν+Lρ−11sβLρhζ,τ.Now, the perturbation procedure in terms of power series with parameterp is presented as(14)Vζ,τ=∑κ=0∞pκVκζ,τ,where p is the perturbation parameter and p∈0,1.The nonlinear term can be defined as(15)NVζ,τ=∑κ=0∞pκHκVκ,where Hn are He’s polynomials in terms of V0,V1,V2,…,Vn and can be calculated as(16)HnV0,V1,…,Vn=1γn+1DpκN∑κ=0∞pκVκp=0,where Dpκ=∂κ/∂pκ.Substituting (15) and (16) in (12), we get(17)∑κ=0∞pκVκζ,τ=Fζ,τ−p×Lρ−11sβLρM∑κ=0∞pκVκζ,τ+∑κ=0∞pκHκVκ.The coefficients comparison on both sides ofp, we have(18)p0:V0ζ,τ=Fζ,τ,p1:V1ζ,τ=Lρ−11sβLρMV0ζ,τ+H0V,p2:V2ζ,τ=Lρ−11sβLρMV1ζ,τ+H1V,⋮pκ:Vκζ,τ=Lρ−11sβLρMVκ−1ζ,τ+Hκ−1V,κ>0,κ∈N.TheVκζ,τ component can be determined easily which quickly leads us to the convergent series. We can get p⟶1:(19)Vζ,τ=limM⟶∞∑κ=1MVκζ,τ.


## 4. Numerical Implementations

Example 1.
Consider the fifth-order nonlinear KdV equation(20)DτβV+Vζ+V2V2ζ−VζV2ζ−20V2V3ζ+V5ζ=0,0<β≤1,with the IC(21)Vζ,τ=1ζ.
Applying theρ-Laplace transform on (20), we get(22)LρVζ,τ=1sζ−1sβLρVζ+V2V2ζ+VζV2ζ−20V2V3ζ+V5ζ.
Next, using the inverse ofρ-Laplace transform of (22),(23)Vζ,τ=1ζ−Lρ−11sβLρVζ+V2V2ζ+VζV2ζ−20V2V3ζ+V5ζ.
Now, we apply HPM(24)∑n=0∞pnVnζ,τ=1ζ−pLρ−11sβLρ∑n=0∞pnHnV+∑n=0∞pnVnζ,τζ+∑n=0∞pnVnζ,τ5ζ,where Hnx represents the nonlinear function of He’s polynomial. For the first few components, we present He’s polynomials(25)H0V=V02V02ζ+V0ζV02ζ−20V02V03ζ,H1V=V02V12ζ+2V0V1Vo2ζ+VoζV12ζ+Vo2ζV1ζ−20V02V13ζ−40V0V1Vo3ζ,H2V=V02V22ζ+2V0V1V12ζ+2V0V2Vo2ζ+V12Vo2ζ+VoζV22ζ+V1ζV12ζ+Vo2ζV2ζ−20V02V23ζ−40V0V1V13ζ−40V0V2Vo3ζ−20V12Vo3ζ,H3V=V02V32ζ+2V0V1V22ζ+2V0V2V12ζ+2V0V3Vo2ζ+V12V12ζ+2V1V2Vo2ζ+VoζV32ζ+V1ζV22ζ+V12ζV2ζ+V02ζV3ζ−20V02V33ζ−40V0V1V23ζ−40V0V2V13ζ−40V0V3Vo3ζ−20V12V13ζ−40V1V2Vo3ζ,H4V=V02V2ζ+2V0V1V32ζ+2V0V2V22ζ+2V0V3V12ζ+2V0V4Vo2ζ+V12V22ζ+2V1V2V12ζ+2V1V3Vo2ζ+V22Vo2ζ+VoζV42ζ+V1ζV32ζ+V22ζV2ζ+V12ζV3ζ+Vo2ζV4ζ−20V02V43ζ−40V0V1V33ζ−40V0V2V23ζ−40V0V3V13ζ−40V0V4Vo3ζ−20V12V23ζ−40V1V2V13ζ−40V1V3Vo3ζ−20V22Vo2ζ,⋮
Comparing theP-like coefficients, we have(26)P0:V0ζ,τ=1ζ,P1:V1ζ,τ=−Lρ−11sβLρH0V+V0ζ+V05ζ=τρ/ρβζ2Γβ+1,P2:V2ζ,τ=−Lρ−11sβLρH1V+V1ζ+V15ζ=τρ/ρ2βζ3Γ2β+1,P3:V3ζ,τ=−Lρ−11sβLρH2V+V2ζ+V25ζ=τρ/ρ3βζ4Γ3β+1,P4:V4ζ,τ=−Lρ−11sβLρH3V+V3ζ+V35ζ=τρ/ρ4βζ5Γ4β+1,P5:V5ζ,τ=−Lρ−11sβLρH4V+V4ζ+V45ζ=τρ/ρ5βζ6Γ5β+1.⋮
The analytical solution ofVζ,τ is defined as(27)Vζ,τ=∑i=0∞Vζ,τi=1ζ+τρ/ρβζ2Γβ+1+τρ/ρ2βζ3Γ2β+1+τρ/ρ3βζ4Γ3β+1+τρ/ρ4βζ5Γ4β+1+τρ/ρ5βζ6Γ5β+1+⋯.
Then, putβ=1 in (27):(28)Vζ,τ=∑i=0∞Viζ,τ=1ζ+τζ2+τ2ζ3+τ3ζ4+⋯.
The exact result isVζ,τ=1/ζ−τ.
In Figure1, the three-dimensional figures of ρ-HPTM and exact results in graphs (a) and (b) respectively at β=1 and the close contact of the exact and ρ-HPTM solutions are investigated. In Figure 2, represent that various fractional order of ρ-HPTM results at β=1,0.8,0.6,0.4. The nonclassical results are investigated to be converge to an integer-order result of the given problem.Figure 1
Graph of (a) exact and (b) analytic solutions ofβ=1 of Example 1.
(a)(b)Figure 2
Figure of various fractional orders of Example1.Example 2.
Consider the fifth-order nonlinear fraction KdV equation(29)DτβV+VVζ−VV3ζ+V5ζ=0,0<β≤1,with the IC(30)Vζ,τ=eζ.
Applying theρ-Laplace transform on (29), we get(31)LρVζ,τ=1seζ+1sβLρVV3ζ−VVζ−V5ζ.
Next, using the inverse ofρ-Laplace transform of (31),(32)Vζ,τ=eζ+Lρ−11sβLρVV3ζ−VVζ−V5ζ.
Now, we apply HPM(33)∑n=0∞pnVnζ,τ=eζ+pLρ−11sβLρ∑n=0∞pnHnV−∑n=0∞pnVnζ,τ5ζ,where Hnx represents the nonlinear term of He’s polynomial. For the first few components, we present He’s polynomials(34)H0V=VoV03ζ−V0V0ζ,H1V=V1V03ζ+V0V13ζ−V1V0ζ−V0V1ζ,H2V=V2V03ζ+V1V13ζ+V0V23ζ−V2V0ζ−V1V1ζ−V0V2ζ,H3V=V3V03ζ+V2V13ζ+V1V23ζ+V0V33ζ−V3V0ζ−V2V1ζ−V1V2ζ−V0V3ζ,H4V=V4V03ζ+V3V13ζ+V2V23ζ+V1V33ζ+V0V43ζ−V4V0ζ−V3V1ζ−V2V2ζ−V1V3ζ−V0V4ζ,⋮
Comparing theP-like coefficients, we have(35)p0:V0ζ,τ=eζ,p1:V1ζ,τ=Lρ−11sβLρH0V−V05ζ=−τρ/ρβΓβ+1eζ,p2:V2ζ,τ=Lρ−11sβLρH1V−V15ζ=τρ/ρ2βΓ2β+1eζ,p3:V3ζ,τ=Lρ−11sβLρH2V−V25ζ=−τρ/ρ3βΓ3β+1eζ,p4:V4ζ,τ=Lρ−11sβLρH3V−V35ζ=τρ/ρ4βΓ4β+1eζ,p5:V5ζ,τ=Lρ−11sβLρH4V−V45ζ=−τρ/ρ5βΓ5β+1eζ.⋮
Therefore, the analytic solution ofVζ,τ is defined as(36)Vζ,τ=∑i=0∞Viζ,τ=eζ1−τρ/ρβΓβ+1+τρ/ρ2βΓ2β+1−τρ/ρ3βΓ3β+1+τρ/ρ4βΓ4β+1−τρ/ρ5βΓ5β+1+⋯.
Then,β=1 for (36), and we get(37)Vζ,τ=∑i=0∞Viζ,τ=eζ1−τ+τ22!−τ33!+τ44!−τ55!+⋯.
The exact solution isVζ,τ=eζ−τ.
In Figure3, the three-dimensional figures of ρ-HPTM and exact results in graphs (a) and (b) respectively at β=1 and the close contact of the exact and ρ-HPTM solutions are investigated. In Figure 4, represent that various fractional order of ρ-HPTM results at β=1,0.8,0.6,0.4. The nonclassical results are investigated to be converge to an integer-order result of the given problem.Figure 3
Graph of (a) exact and (b) analytic solutions ofβ=1 of Example 2.
(a)(b)Figure 4
Figure of (a) and (b) at various fractional-order of Example2.
(a)(b)Example 3.
Consider nonlinear fractional-order Kawahara equation(38)DτβV+VVζ+V3ζ−V5ζ=0,0<β≤1,with the IC(39)Vζ,τ=105169sech4ζ−ϕ213.
Applying theρ-Laplace transform on (38), we get(40)LρVζ,τ=1s105169sech4ζ−ϕ213+1sβLρV5ζ−V3ζ−VVζ.
Next, using the inverse ofρ-Laplace transform of (40),(41)Vζ,τ=105169sech4ζ−ϕ213+Lρ−11sβLρV5ζ−V3ζ−VVζ.
Now, we apply HPM(42)∑n=0∞pnVnζ,τ=105169sech4ζ−ϕ213+pLρ−11sβLρ∑n=0∞pnVnζ,τ5ζ−∑n=0∞pnVnζ,τ3ζ−∑n=0∞pnHnV,where HnV represent the nonlinear terms of He’s polynomial. For the first few components, we present He’s polynomials(43)H0V=V0V0ζ,H1V=V0V1ζ+V1V0ζ,H2V=V0V2ζ+V1V1ζ+V2V0ζ,H3V=V0V3ζ+V1V2ζ+V2V1ζ+V3V0ζ,H4V=V0V4ζ+V1V3ζ+V2V2ζ+V3V1ζ+V4V0ζ,⋮
Comparing the P-like coefficients, we get(44)p0:V0ζ,τ=105169sech4ζ−ϕ213,p1:V1ζ,τ=Lρ−11sβLρV05ζ−V05ζ−H0V=−10037713sech4ζ−ϕ213tanhζ−ϕ213τρ/ρβΓβ+1,p2:V2ζ,τ=Lρ−11sβLρV15ζ−V13ζ−H1V=−2168710×10713sech6ζ−ϕ213−3+2coshζ−ϕ213τρ/ρ2βΓ2β+1,p3:V3ζ,τ=Lρ−11sβLρV25ζ−V23ζ−H2V=−46196210×10713sech7ζ−ϕ213×−13sinhζ−ϕ213+2sinh3ζ−ϕ2`13τρ/ρ3βΓ3β+1,p4:V4ζ,τ=Lρ−11sβLρV35ζ−V33ζ−H3V=−378485410×10713sech8ζ−ϕ213×−49scoshζ−ϕ213+4cosh2ζ−ϕ2`13+52τρ/ρ4βΓ4β+1,p5:V5ζ,τ=Lρ−11sβLρV45ζ−V43ζ−H4V=−3.22496310×10713sech9ζ−ϕ213×171sinh3ζ−ϕ213−8sinh5ζ−ϕ2`13−661sinh5ζ−ϕ2sqrt13τρ/ρ5βΓ5β+1.⋮
The analytic solutionVζ,τ is achieved as(45)Vζ,τ=∑i=0∞Viζ,τ,Vζ,τ=105169sech4ζ−ϕ213−10037713sech4ζ−ϕ213tanhζ−ϕ213τρ/ρβΓβ+1−2168710×10713sech6ζ−ϕ213−3+2coshζ−ϕ213τρ/ρ2βΓ2β+1−46196210×10713sech7ζ−ϕ213×−13sinhζ−ϕ213+2sinh3ζ−ϕ2`13τρ/ρ3βΓ3β+1−378485410×10713sech8ζ−ϕ213×−49scoshζ−ϕ213+4cosh2ζ−ϕ2`13+52τρ/ρ4βΓ4β+1−3.22496310×10713sech9ζ−ϕ213×171sinh3ζ−ϕ213−8sinh5ζ−ϕ2`13−661sinh5ζ−ϕ213τρ/ρ5βΓ5β+1+⋯.
The exact solution isVζ,τ=105/169sech41/213ζ+36τ/169−ϕ.
In Figure5, the three-dimensional figures of ρ-HPTM and exact results in graphs (a) and (b) respectively at β=1 and the close contact of the exact and ρ-HPTM solutions are investigated. In Figure 6, represent that various fractional order of ρ-HPTM results at β=1,0.8,0.6,0.4. The nonclassical results are investigated to be converge to an integer-order result of the given problem.Figure 5
Graph of (a) exact and (b) analytical results ofβ=1 of Example 3.
(a)(b)Figure 6
Figure of (a) at various fractional-order ofβ and (b) error graph of Example 3.
(a)(b)


## 5. Conclusions

This paper determined the fractional-order Kawahara and fifth-order KdV equations, applying theρ-homotopy perturbation transform method. The present method is used to describe the results for specific examples. The ρ-HPTM result is highly congruent with the precise solution of the suggested problems. Additionally, the proposed method estimated the results of the cases using fractional-order derivatives. The graphical examination of the resulting fractional-order results proved their convergence to integer-order outcomes. Additionally, the ρ-HPTM technique is straightforward, simple, and computationally efficient; the suggested method can be adapted to solve additional fractional-order partial differential equations.



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