# Numerical Investigation of Fractional-Order Kawahara and Modified Kawahara Equations by a Semianalytical Method

Authors: Weaam Alhejaili; Sharifah E. Alhazmi; Rashid Nawaz; Aatif Ali; Joshua Kiddy Kwasi Asamoah; Laiq Zada
Journal: Journal of Nanomaterials
 (2022)

---

## Abstract

In this work, the optimal homotopy asymptotic method (OHAM) has been used to find approximate solutions to the nonlinear fractional-order Kawahara and modified Kawahara equations. The method convergence is controlled by a flexible function known as the auxiliary function. The values of the unknown arbitrary constants in the auxiliary function are computed using the Caputo derivative fractional-order and the well-known approach of least squares. Fractional-order derivatives are taken in the Caputo sense with numerical values in the closed interval0,1. The suggested method is directly applied to fractional-order Kawahara and modified Kawahara equations, with no need for small or large parameter assumptions. The numerical results obtained by the proposed method are compared to the new iterative method (NIM). Results reveal that the proposed method converges faster to the exact solution than other methods in the literature.


---

## Body

## 1. Introduction

Fractional computation was established as an important subject of mathematics in 1695. Fractional calculus ideas have recently been successfully expanded to numerous sectors, and academics have increasingly realized that fractional calculus may well reflect many nonlocal occurrences in the fields of natural science and architecture. Rheology, liquid flow, dispersion diffusion transport, dynamic cycles in self-compatible and porous materials, viscoelasticity, and optics are some of the key areas of fractional calculation today. Very few researchers have drawn on the successful use of fractional systems in these fields to examine their mathematical approximation methods, since diagnostic frameworks are usually difficult to obtain. A variety of real-world problems can be modeled using fractional-order differential equations. These equations have many applications in fluid mechanics, electromagnetic theory, electric grids, diffuse transport, groundwater problems, biological sciences, etc. [1–9]. The exact solution for nonlinear problems is very hard to obtain, and an alternative way is to find the approximate solution. Some familiar approximation methods are used in the series of papers [9–21], etc. Similarly, we extend the well-known optimal homotopy asymptotic method (OHAM) to fractional-order Kawahara and modified Kawahara equations.The proposed approach was presented by Marinca and Herisanu and applied to resolve nonlinear differential equations in the literature series [22–26]. Recently, Sarwar et al. extended the idea of OHAM fractional-order partial differential equations and used them for different problems having fractional-order derivatives [27, 28]. Nawaz et al. applied the suggested approach to the fractional-order Zakharov-Kuznetsov equations [29]. Likewise, Zada et al. applied the proposed approach to various fractional PDEs in the series of articles [30, 31]. In this article, the application of OHAM is extended to the modified Kawahara and Kawahara equations together with initial conditions:
(1)∂αϒξ,τ∂τα+ϒξ,τ∂ϒξ,τ∂ξ+ϒξ,τ−∂5ϒξ,τ∂ξ5=0,0<α≤1,ϒξ,0=105169sech4ξ213,(2)∂αϒξ,τ∂τα+ϒ2ξ,τ∂ϒξ,τ∂ξ+ρ∂3ϒξ,τ∂ξ3+μ∂5ϒξ,τ∂ξ5=0,0<α≤1,ϒξ,0=3ρ−10μsech2kξ,k=12−ρ2μ.Here,ρ and μ are constants. Equations (1) and (2) have become the subject of active and wide research topics in recent times [32–34].


## 2. Preliminaries

Definition 1.
The Riemann-Liouville fractional integral operator of an orderα≥0 of a function G∈Cμ,μ≥−1 is presented by
(3)IaαGξ=1Γα∫aξξ−μα−1Gμdμ,α>0,ξ>0,Ia0Gξ=Gξ.Definition 2.
The fractional derivativeGξ according to Caputo is presented by
(4)DaαGξ=Iam−αDmGξ=1Γm−a∫aξξ−μm−α−1Gmμdμ,m−1<α≤m,m∈N,ξ>0,G∈C−1m.Definition 3.
Ifm−1<α≤m,m∈N, and G∈Cμm,μ≥−1, then DaαIaαGξ=Gξ and DaαIaαGξ=Gξ−∑k=0m−1Gkξ−a/k!,ξ>0.The properties of operatorIα are found in [3, 11]. We introduce the subsequent.ForG∈Cμm,α,β>0,μ≥−1,andγ≥−1,
(1)
IaαGξ exist for almost every ξ∈a,b(2)
IaαIaβGξ=Iaα+βGξ(3)
IaαJaβGξ=IaβJaαGξ(4)
Iaαξ−aγ=Γγ+1/Γα+γ+1ξ−aα+γ


## 3. OHAM Methodology to Fractional-Order PDEs [27, 28]

To extend the basic theory of OHAM for fractional-order PDEs, we assume that the subsequent general fractional differential system(5)∂αϒξ,τ∂τα=Aϒξ,τ+Fξ,τα>0,with initial condition
(6)D0α−κϒξ,0=hκr,κ=0,1,2,⋯,n−1,D0α−nϒξ,0=0,n=α,D0κϒξ,0=gκr,κ=0,1,2,⋯,n−1,D0nϒξ,0=0,n=α.In the above equation,∂α/∂τα represents the Caputo fractional derivative operator, A stands for the differential operator, and ϒξ,τ represents an unknown function. Fξ,τ is a function that serves as an analytical function.The homotopy using OHAM for equation (5) is ϕξ,τ;p:Ω×0,1⟶R which is satisfied:
(7)1−p∂αϕξ,τ∂τα−Fξ,τ−Hξ,p∂αϕξ,τ∂τα−Aϕξ,τ+Fξ,τ=0.Hence,p∈0,1 which is an embedding parameter and Hξ,τ shows the auxiliary function such that
(8)Hξ,τ≠0forp≠0andHξ,0=0.Remark 4.
The approximate solutionϕξ,τ approaches to the closed solution when the numerical values of p varies for 0 to 1 in the closed interval 0,1. The convergence of the OHAM purely depends on the auxiliary function.The auxiliary functionHξ,p is set out below:
(9)Hξ,p=pk1ξ,Ci+p2k2ξ,Ci+p3k3ξ,Ci+⋯+pmkmξ,Ci.In equation (9), Ci,i=1,2,⋯, convergence control parameters kiξ,i=1,2,⋯, is a function of ξ.By extendingϕξ,τ;p,Ci in Taylor’s series about p, one can obtain
(10)ϕξ,τ;Ci=ϒ0ξ,τ+∑k=1mϒkξ,τ;Cipk,i=1,2,3,⋯.Puttingp=1, in the above equation, we have
(11)ϒξ,τ;Ci=ϒ0ξ,τ+∑k=1∞ϒkξ,τ;Ci,i=1,2,3,⋯.By substituting equation (10) in equation (7) and equating the coefficient of the same powers of p, we acquire the series of problems:
(12)p0:∂αϒ0ξ,τ∂τα−F=0,p1:∂αϒ1ξ,τ,C1∂τα−1+C1∂αϒ0ξ,τ∂τα+1+C1F+C1Aϒ0ξ,τ=0,p2:∂αϒ2ξ,τ,C1,C2∂τα−1+C1∂αϒ1ξ,τ,C1∂τα−C2∂αϒ0ξ,τ∂τα+C1Aϒ1ξ,τ,C1+C2F+Aϒ0ξ,τ=0⋯.The above problems contain fractional-order derivatives. So, we apply the inverse of the operatorIα on both sides of the above problems:
(13)ϒ0ξ,τ=IαF,ϒ1ξ,τ;C1=Iα1+C1∂αϒ0ξ,τ∂τα−1+C1F−C1Aϒ0ξ,τ,ϒ2ξ,τ;C1,C2=Iα1+C1∂αϒ1ξ,τ;C1∂τα+C2∂αϒ0ξ,τ∂τα−C1Aϒ1ξ,τ;C1−C2F+Aϒ0ξ,τ⋯.By using these solutions in equation (11), we obtain the approximate solution:
(14)ϒ~ξ,τ;Ci=ϒ0ξ,τ+ϒ1ξ,τ;C1+ϒ2ξ,τ;C1,C2+⋯.The residualRξ,τ;Ci is acquired by using equation (14) into equation (5).C1,C2,⋯ can be found by using either the Ritz method, the least squared method, the collocation method, or Galerkin’s method. The least-square approach is used here. Here, we introduce the functional
(15)χCi=∫0t∫ΩR2ξ,τ;Cidξdτ,then calculate the optimal values for auxiliary constants Ci by solving the following equation system:
(16)∂χ∂C1=∂χ∂C2=⋯=∂χ∂Cm=0.


### 3.1. Convergence Theorem

If the series (11) converge to ϒξ,τ, where ϒkξ,τ∈LR+ is generated by the zero-order system and the K-order deformation, then ϒξ,τ is the exact solution of (5).Proof.
The following series(17)∑k=1∞ϒi,kξ,τ;C1,C2,⋯,Ck
converges and is presented by(18)ψiξ,τ=∑k=1∞ϒi,kξ,τ;C1,C2,⋯,Ck,which satisfies the following:
(19)limk⟶∞ϒi,kξ,τ;C1,C2,⋯,Ck=0.
Indeed, the subsequent equation is fulfilled:(20)ϒi,1ξ,τ;C1+∑k=2nϒi,kξ,τ;C⟶k−∑k=2nϒi,k−1ξ,τ;C⟶k−1=ϒi,2ξ,τ;C⟶2−ϒi,1ξ,τ;C1+⋯+ϒi,nξ,τ;C⟶n−ϒi,n−1ξ,τ;C⟶n−1=ϒi,nξ,τ;C⟶n.
Now, we have(21)Li,1ϒi,1ξ,τ;C1+∑k=2∞L1ϒi,kξ,τ;C⟶k−∑k=2∞Liϒi,k−1ξ,τ;C⟶k−1=Liϒi,1ξ,τ;C1+∑k=2∞Liϒi,kξ,τ;C⟶k−∑k=2∞Liϒi,k−1ξ,τ;C⟶k−1=0,which satisfies
(22)Li,1ϒi,1ξ,τ;C1+Li∑k=2∞ϒi,kξ,τ;C⟶k−Li∑k=2∞ϒi,k−1ξ,τ;C⟶k−1=∑k=2∞CmLiϒi,k−mξ,τ;C⟶k−m+Ni,k−mϒi,k−1ξ,τ;Ck−1+giξ,τ=0.
Now, ifCm,m=1,2,3,⋯, is correctly selected, then the equation leading to
(23)Liϒiξ,τ+A=0is the exact solution.


## 3.1. Convergence Theorem

If the series (11) converge to ϒξ,τ, where ϒkξ,τ∈LR+ is generated by the zero-order system and the K-order deformation, then ϒξ,τ is the exact solution of (5).Proof.
The following series(17)∑k=1∞ϒi,kξ,τ;C1,C2,⋯,Ck
converges and is presented by(18)ψiξ,τ=∑k=1∞ϒi,kξ,τ;C1,C2,⋯,Ck,which satisfies the following:
(19)limk⟶∞ϒi,kξ,τ;C1,C2,⋯,Ck=0.
Indeed, the subsequent equation is fulfilled:(20)ϒi,1ξ,τ;C1+∑k=2nϒi,kξ,τ;C⟶k−∑k=2nϒi,k−1ξ,τ;C⟶k−1=ϒi,2ξ,τ;C⟶2−ϒi,1ξ,τ;C1+⋯+ϒi,nξ,τ;C⟶n−ϒi,n−1ξ,τ;C⟶n−1=ϒi,nξ,τ;C⟶n.
Now, we have(21)Li,1ϒi,1ξ,τ;C1+∑k=2∞L1ϒi,kξ,τ;C⟶k−∑k=2∞Liϒi,k−1ξ,τ;C⟶k−1=Liϒi,1ξ,τ;C1+∑k=2∞Liϒi,kξ,τ;C⟶k−∑k=2∞Liϒi,k−1ξ,τ;C⟶k−1=0,which satisfies
(22)Li,1ϒi,1ξ,τ;C1+Li∑k=2∞ϒi,kξ,τ;C⟶k−Li∑k=2∞ϒi,k−1ξ,τ;C⟶k−1=∑k=2∞CmLiϒi,k−mξ,τ;C⟶k−m+Ni,k−mϒi,k−1ξ,τ;Ck−1+giξ,τ=0.
Now, ifCm,m=1,2,3,⋯, is correctly selected, then the equation leading to
(23)Liϒiξ,τ+A=0is the exact solution.


## 4. Main Results

We test our adopted procedure OHAM for finding the approximate solution of the fractional-order Kawahara equation. For most of the computational work, we used MathType and Mathematica 10.


### 4.1. Numerical Solution of Fractional Kawahara Equation

First, we assume that the time-fractional Kawahara equation is given in [35]:
(24)∂αϒξ,τ∂τα+ϒξ,τ∂ϒξ,τ∂ξ+∂3ϒξ,τ∂ξ3−∂5ϒξ,τ∂ξ5=0,0<α≤1.Subject to I.C.,(25)ϒξ,0=105169sech4ξ213.Forα=1, an exact solution for equation (24) is found by [35] as
(26)ϒξ,τ=105169sech41213ξ−36τ169.Recall the OHAM preparation given in Section3, we obtain the subsequent problems:Zero-order problem:(27)∂αϒ0ξ,τ∂τα=0,ϒ0ξ,τ=105169sech4ξ213.First-order problem:(28)∂αϒ1ξ,τ∂τα=∂αϒ0ξ,τ∂τα+C1∂αϒ0ξ,τ∂τα−C1ϒ0∂ϒ0ξ,τ∂ξ+C1∂3ϒ0ξ,τ∂ξ3−C1∂5ϒ0ξ,τ∂ξ5.Second-order problem:(29)∂αϒ2ξ,τ∂τα=C2∂αϒ0ξ,τ∂τα+∂αϒ0ξ,τ∂τα+C1∂αϒ1ξ,τ∂τα+C2ϒ0ξ,τ∂ϒ0ξ,τ∂xξ+C1ϒ0ξ,τ∂ϒ0ξ,τ∂ξ+C1ϒ0ξ,τ∂ϒ1ξ,τ∂ξ+C2∂3ϒ0ξ,τ∂ξ3+C1∂3ϒ1ξ,τ∂ξ3−C2∂5ϒ1ξ,τ∂ξ5.Apply the inverse operatorIα, the solution of the above problems is given as follows:
(30)ϒ0ξ,τ=105169sech4ξ213,ϒ1ξ,τ,C1=−7560C1ταsech4ξ/213tanξ/2132856113Γ1+α,ϒ2ξ,τ,C1,C2=162748517Γ1+α1890e−ξ/13τα−16913C1+C12+C2−1+e2ξ/13+923−2αC121−3eξ/13+e2ξ/13πτα13Γ1+αsech6ξ213.The second-order OHAM solution is presented as follows:(31)ϒ~ξ,τ,Ci=ϒ0ξ,τ+ϒ1ξ,τ,C1+ϒ2ξ,τ,C1,C2.Forα=1, second-order OHAM solution for Kawahara equation is
(32)ϒ~ξ,τ=105sech4ξ/213371293+72τ−9C12τ−4+5sech2ξ/213−16913C12+C1+C2tanhξ/21362748517.Forα=2/3, second-order OHAM solution for Kawahara equation is
(33)ϒ~ξ,τ=105sech4ξ/213371293+72τ2/3−18C12τ2/3−4+5sech2ξ/213/Γ7/3−16913C12+C2tanhξ/213/Γ5/362748517.


### 4.2. Numerical Solution of Fractional Modified Kawahara Equation

Assume the following time-fractional modified Kawahara system presented by(34)∂αϒξ,τ∂τα+ϒ2ξ,τ∂ϒξ,τ∂ξ+p∂3ϒξ,τ∂ξ3+q∂5ϒξ,τ∂ξ5=0,0<α≤1,with I.C.,
(35)ϒξ,0=3ρ−10μsech2kξ,k=12−ρ2μ.Whenα=1, the exact solution is given by [35] as
(36)ϒξ,τ=3ρ−10μsech2kξ−ψt,ψ=25μ−4ρ225μ.Following the OHAM procedure, we have the following.Zero-order problem:(37)∂αϒ0ξ,τ∂τα=0,ϒ0ξ,0=3ρ−10μsech2kξ,k=12−ρ2μ.First-order problem:(38)∂αϒ1ξ,τ∂τα=∂αϒ0ξ,τ∂τα+C1∂αϒ0ξ,τ∂τα−C1ϒ0∂ϒ0ξ,τ∂ξ+C1∂3ϒ0ξ,τ∂ξ3−C1∂5ϒ0ξ,τ∂ξ5.Apply the inverse operatorIα, the solution of the above problems is given as follows:
(39)ϒ0ξ,τ=3ρ−10μsech212−ρ2μξ,ϒ1ξ,τ,C1=3C1ρ7/2ταsech5ρξ/25μ−59sinρξ/25μ+sinρξ/25μtanξ/213500−μμ3/22αΓα.The 1st-order OHAM solution is given by the following expression:
(40)ϒ~ξ,τ,Ci=ϒ0ξ,τ+ϒ1ξ,τ,C1.


## 4.1. Numerical Solution of Fractional Kawahara Equation

First, we assume that the time-fractional Kawahara equation is given in [35]:
(24)∂αϒξ,τ∂τα+ϒξ,τ∂ϒξ,τ∂ξ+∂3ϒξ,τ∂ξ3−∂5ϒξ,τ∂ξ5=0,0<α≤1.Subject to I.C.,(25)ϒξ,0=105169sech4ξ213.Forα=1, an exact solution for equation (24) is found by [35] as
(26)ϒξ,τ=105169sech41213ξ−36τ169.Recall the OHAM preparation given in Section3, we obtain the subsequent problems:Zero-order problem:(27)∂αϒ0ξ,τ∂τα=0,ϒ0ξ,τ=105169sech4ξ213.First-order problem:(28)∂αϒ1ξ,τ∂τα=∂αϒ0ξ,τ∂τα+C1∂αϒ0ξ,τ∂τα−C1ϒ0∂ϒ0ξ,τ∂ξ+C1∂3ϒ0ξ,τ∂ξ3−C1∂5ϒ0ξ,τ∂ξ5.Second-order problem:(29)∂αϒ2ξ,τ∂τα=C2∂αϒ0ξ,τ∂τα+∂αϒ0ξ,τ∂τα+C1∂αϒ1ξ,τ∂τα+C2ϒ0ξ,τ∂ϒ0ξ,τ∂xξ+C1ϒ0ξ,τ∂ϒ0ξ,τ∂ξ+C1ϒ0ξ,τ∂ϒ1ξ,τ∂ξ+C2∂3ϒ0ξ,τ∂ξ3+C1∂3ϒ1ξ,τ∂ξ3−C2∂5ϒ1ξ,τ∂ξ5.Apply the inverse operatorIα, the solution of the above problems is given as follows:
(30)ϒ0ξ,τ=105169sech4ξ213,ϒ1ξ,τ,C1=−7560C1ταsech4ξ/213tanξ/2132856113Γ1+α,ϒ2ξ,τ,C1,C2=162748517Γ1+α1890e−ξ/13τα−16913C1+C12+C2−1+e2ξ/13+923−2αC121−3eξ/13+e2ξ/13πτα13Γ1+αsech6ξ213.The second-order OHAM solution is presented as follows:(31)ϒ~ξ,τ,Ci=ϒ0ξ,τ+ϒ1ξ,τ,C1+ϒ2ξ,τ,C1,C2.Forα=1, second-order OHAM solution for Kawahara equation is
(32)ϒ~ξ,τ=105sech4ξ/213371293+72τ−9C12τ−4+5sech2ξ/213−16913C12+C1+C2tanhξ/21362748517.Forα=2/3, second-order OHAM solution for Kawahara equation is
(33)ϒ~ξ,τ=105sech4ξ/213371293+72τ2/3−18C12τ2/3−4+5sech2ξ/213/Γ7/3−16913C12+C2tanhξ/213/Γ5/362748517.


## 4.2. Numerical Solution of Fractional Modified Kawahara Equation

Assume the following time-fractional modified Kawahara system presented by(34)∂αϒξ,τ∂τα+ϒ2ξ,τ∂ϒξ,τ∂ξ+p∂3ϒξ,τ∂ξ3+q∂5ϒξ,τ∂ξ5=0,0<α≤1,with I.C.,
(35)ϒξ,0=3ρ−10μsech2kξ,k=12−ρ2μ.Whenα=1, the exact solution is given by [35] as
(36)ϒξ,τ=3ρ−10μsech2kξ−ψt,ψ=25μ−4ρ225μ.Following the OHAM procedure, we have the following.Zero-order problem:(37)∂αϒ0ξ,τ∂τα=0,ϒ0ξ,0=3ρ−10μsech2kξ,k=12−ρ2μ.First-order problem:(38)∂αϒ1ξ,τ∂τα=∂αϒ0ξ,τ∂τα+C1∂αϒ0ξ,τ∂τα−C1ϒ0∂ϒ0ξ,τ∂ξ+C1∂3ϒ0ξ,τ∂ξ3−C1∂5ϒ0ξ,τ∂ξ5.Apply the inverse operatorIα, the solution of the above problems is given as follows:
(39)ϒ0ξ,τ=3ρ−10μsech212−ρ2μξ,ϒ1ξ,τ,C1=3C1ρ7/2ταsech5ρξ/25μ−59sinρξ/25μ+sinρξ/25μtanξ/213500−μμ3/22αΓα.The 1st-order OHAM solution is given by the following expression:
(40)ϒ~ξ,τ,Ci=ϒ0ξ,τ+ϒ1ξ,τ,C1.


## 5. Results and Discussion

We implemented OHAM to provide approximate numerical solutions to fractional and modified Kawahara equations. Numerical values are tabulated for the auxiliary constants in Tables1 and 2 for Kawahara and modified Kawahara equations at various values of α. Table 3 gives the estimation of the second-order OHAM solution and the third-order NIM solution for the Kawahara fractional equation. Table 4 compares the absolute errors of the second-order OHAM solution for various α values. Table 4 presents the values of the first-order OHAM solution and the third-order NIM solution for the various values of α. Table 5 compares first-order OHAM solution with third-order NIM solution for time-fractional modified Kawahara equations for different values of α.Table 1
Numerical values ofC1, C2 for time-fractional Kawahara equation for several values of α.
αC1C21.0‐0.99999830317063540.000002.41750578770330623‐0.99996099663193420.000030849221963805824Table 2
Numerical values ofC1 for time-fractional modified Kawahara equations for several values of α.
αC11.0‐0.464723497961125423‐0.9999609966319342Table 3
Comparison of second-order OHAM solution with third-order NIM solution for time-fractional Kawahara equation for different values ofα.
ξτOHAMα=2/3OHAMα=1Exactα=1-50.020.2528770.2539850.2539850.040.2520170.2536250.2536250.060.2512980.2532650.2532650.080.2506570.2529050.2529050.10.2500690.2525460.25254600.020.6212920.6213010.6213010.040.6212770.62130.62130.060.6212590.6212980.6212980.080.6212390.6212950.6212950.10.6212170.6212910.62129150.020.2558210.2547070.2547070.040.2566910.2550680.2550680.060.2574220.2554290.2554290.080.2580760.2557910.2557910.10.2586780.2561530.256153Table 4
Comparison absolute errors of 2nd-order OHAM solution with 3rd-order NIM solution for time-fractional Kawahara equations for different values of α.
ξτResidualα=0.5NIM [35]α=1OHAMα=1-50.02−1.31504×10−62.27500×10−108.04633×10−100.04−3.18365×10−61.81933×10−91.20722×10−90.06−5.04947×10−66.13795×10−98.08547×10−100.08−6.91298×10−61.45438×10−87.88584×10−100.1−8.77447×10−62.83953×10−83.97934×10−900.02−8.08613×10−81.76636×10−131.29552×10−120.04−2.14003×10−72.82618×10−123.06177×10−120.06−3.84142×10−71.43068×10−111.05937×10−120.08−5.84492×10−74.52161×10−112.1665×10−110.1−8.11039×10−71.10391×10−107.35924×10−1150.021.33389×10−62.27670×10−108.05429×10−100.043.22731×10−61.82204×10−91.20837×10−90.065.12378×10−66.15166×10−98.03524×10−100.087.02284×10−61.45871×10−88.16471×10−100.18.92419×10−62.85011×10−84.061×10−9Table 5
Comparison of 1st-order OHAM solution with 3rd-order NIM solution for time-fractional modified Kawahara equations For different values of α.
ξτExact solutionOHAM solutionAbsolute error NIM for [35]α=1Absolute error OHAM forα=1-50.029.474889415×10−49.474984314×10−49.48992×10−99.48992×10−90.049.474794138×10−49.474984314×10−41.90176×10−81.90176×10−80.069.474698483×10−49.474984314×10−42.8583×10−82.8583×10−80.089.474602454×10−49.474984314×10−43.81862×10−83.81862×10−80.19.474506042×10−49.474984314×10−44.78271×10−84.78271×10−800.029.486832790×10−49.486832980×10−41.89737×10−111.89737×10−110.049.486832221×10−49.486832980×10−47.58947×10−117.58947×10−110.069.486831272×10−49.486832980×10−41.70763×10−101.70763×10−100.089.486829944×10−49.486832980×10−43.03579×10−103.03579×10−100.19.486828237×10−49.486832980×10−44.74342×10−104.74342×10−1050.029.475078835×10−49.474984314×10−49.45216×10−99.45216×10−90.049.475172979×10−49.474984314×10−41.88666×10−81.88666×10−80.069.475266744×10−49.474984314×10−42.82432×10−82.82432×10−80.089.475360132×10−49.474984314×10−43.75821×10−83.75821×10−80.19.475453144×10−49.474984314×10−44.68832×10−84.68832×10−8Figures1–3 depict the 3D surfaces obtained by second-order OHAM as well as the accurate solutions to fractional Kawahara equation at α=0.5 and 1. Figure 4 shows the residual for α=0.5, whereas Figure 5 shows the 2D surface of the second-order OHAM solution for various values of α. Figures 6–8 show the 3D plots for the first-order OHAM solution and exact solution for the fractional modified Kawahara equation at α=0.5 and 1. Figure 9 depicts a two-dimensional graph of the first-order OHAM solution for different values of α. The residual for α=0.5 is shown in Figure 10.Figure 1
3D surface obtained by OHAM solution for fractional Kawahara equation atα=0.5.Figure 2
3D surface obtained by OHAM solution for fractional Kawahara equation atα=1.Figure 3
3D surface obtained by exact solution for fractional Kawahara equation atα=1.Figure 4
The curves show the comparison between exact solution and approximate solution for different values ofα at τ=0.5 for Kawahara equation.Figure 5
Residual obtained by OHAM forα=0.5 at τ=0.5, for Kawahara equation.Figure 6
3D plot of OHAM solution for fractional modified Kawahara equation atα=0.5.Figure 7
3D surface obtained by OHAM solution for fractional modified Kawahara equation atα=1.Figure 8
3D surface obtained by exact solution for fractional modified Kawahara equation atα=1.Figure 9
The curves show the comparison between exact solution and approximate solution for different values ofα at τ=0.5 for modified Kawahara equation.Figure 10
Residual obtained by OHAM forα=0.5 at τ=0.5, for modified Kawahara equation.The results obtained by the second-order OHAM solution for the Kawahara fractional equation agree with both the closed and the NIM solution. Similarly, for fractional modified Kawahara equation, the results achieved by the first-order OHAM solutions are exactly the same as for the third-order NIM solutions.


## 6. Conclusions

We observe that OHAM converges rapidly towards the closed solution with a lower sequence of approximation of fractional orders of the Kawahara equations and modified Kawahara equations based on the calculated results. The results achieved with the proposed approach are highly encouraging compared to the new iterative method (NIM). This proposed approach is capable of providing the greatest accuracy within the lowest approximation sequence. This approach does not require choices between small and large parameter assumptions in problems. The results are analyzed and explained with the help of graphs by considering different values of parameters. Results reveal that as the value of fractional-order derivatives approaches to 1, the approximate solution converges to the exact solution. The convergence of this approach is independent of initial assumptions. The precision of the proposed approach can be improved by assuming high approximations, and therefore, it may be highly attractive for researchers to use our approach to solve fractional-order systems emerging in the science of technology.



---

*Source: 1985572-2022-02-03.xml*