# Analysis of Fractional-Order Regularized Long-Wave Models via a Novel Transform

Authors: Nehad Ali Shah; Essam R. El-Zahar; Ali Akgül; Adnan Khan; Jeevan Kafle
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

A new integral transform method for regularized long-wave (RLW) models having fractional-order is presented in this study. Although analytical approaches are challenging to apply to such models, semianalytical or numerical techniques have received much attention in the literature. We propose a new technique combining integral transformation, the Elzaki transform (ET), and apply it to regularized long-wave equations in this study. The RLW equations describe ion-acoustic waves in plasma and shallow water waves in seas. The results obtained are extremely important and necessary for describing various physical phenomena. This work considers an up-to-date approach and fractional operators in this context to obtain satisfactory approximate solutions to the proposed problems. We first define the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD) and implement them for solving RLW equations. We can readily obtain numerical results that provide us with improved approximations after only a few iterations. The derived solutions were found to be in close contact with the exact solutions. Furthermore, the suggested procedure has attained the best level of accuracy. In fact, when compared to other analytical techniques for solving nonlinear fractional partial differential equations, the present method might be considered one of the finest.


---

## Body

## 1. Introduction

Fractional calculus (FC) is a model discipline of mathematics that focuses entirely on fractional-order derivatives and integration. Fractional derivatives and fractional integrations are noninteger-order derivatives and integration that can model various phenomena in engineering and science. FC began in 1695 when L’Hospital asked Leibniz, “What would be the physical meaning of fractional derivative?” This question inspired many great scientists in the eighteenth and nineteenth centuries to focus on fractional calculus, which has a wide range of applications in applied science and technology. Many researchers have demonstrated that fractional generalizations of integer-order models efficiently represent natural phenomena [1–5]. The classical derivatives are local. In contrast, the Caputo fractional derivative is nonlocal, i.e., we can study changes in the neighborhood of a point using classical derivatives. Still, we may analyze changes in an interval using Caputo fractional derivatives. Because of this quality, the Caputo fractional derivative can be used to model a wider range of physical phenomena, including solid mechanics [6, 7], diffusion procedures [8], continuum and statistical mechanics [9], electromagnetism [10], viscoelastic materials [11], fluid mechanics [12], propagation of spherical flames [13], viscoelastic materials [14], and so on [15–17].Fractional differential equations have been studied for decades due to their widespread application in science and engineering. Fractional partial differential equations are used to describe various phenomena in acoustics, electromagnetics, material science, viscoelasticity, electrochemistry, and plasma physics. Fractional differential equations have numerical solutions that are of great interest. For fractional differential equations, no method provides an accurate solution. Only series solution methods or linearization can generate approximate solutions [18–21]. Nonlinear phenomena can be found in various engineering and science domains, including chemical kinetics, nonlinear spectroscopy, solid state physics, fluid physics, computational biology, quantum mechanics, and thermodynamics. Different higher-order nonlinear partial differential equations (PDEs) define the idea of nonlinearity. Nonlinear models for all physical systems describe basic phenomena. The literature has presented integrative transform approaches for solving fractional differential equations. Elzaki transform (ET) is an integral transform [22] in this context. Several scholars [23–30] have looked at some essential solution approaches for real-world issues, as well as numerical simulations obtained using the novel integral transformation.In this article, three alternative fractional homogeneous RLW equations are studied; the RLW equations, according to some scientists, are the best equations than the classical Korteweg-de Vries (KdV) equation [31]. We use the Elzaki transform combined with the CFD and ABC operator [32] to solve three special RLW problems. The approximate solutions are then obtained, and the numerical simulations of the solutions are analyzed [33, 34] and provided the nonlinear RLW equations.
(1)DIδφψ,I−φψψIψ,I+φψψ,I+φψ,Iφψψ,I=0,having initial source
(2)φψ,0=3αsech2δζ,(3)DIδφψ,I−2φψψIψ,I+φψψ,I=0,having initial source
(4)φψ,0=e−ψ,(5)DIδφψ,I+φψψψψψ,I=0,having initial source
(6)φψ,0=sinψ.Equation (1) is known as a general regularized long-wave equation (GRLWE) having fractional-order, whereas Equations (3) and (5) represents fractional regularized long-wave equations (RLWEs).Magnetohydrodynamic waves in plasma, ion-acoustic waves in plasma, stress waves in compressed gas bubble mixes, rotating tube flow, and longitudinal dispersive waves in elastic rods are just some of the applications of the RLW equations. The RLW equations are suitable models for many significant physical structures in applied physics and engineering. They also work on a variety of liquid flow phenomena where diffusion is essential, such as in viscous or shock situations. It can be used to simulate any nonlinear wave diffusion problem, including dissipation. Depending on the problem modeling [35], this dissipation could result in heat conduction, viscosity, thermal radiation, chemical reaction, mass diffusion, or other sources. Many necessary ocean research and engineering phenomena, such as minor frequency shallow-water waves and long-waves, are defined by fractional RLW equations. Several experts in ocean shallow liquid waves are interested in nonlinear waves described using the RLW equations having fractional-order. The fractional RLW equations were used to represent nonlinear waves in the ocean. Indeed, the tsunami’s massive surface waves are defined by fractional RLW equations. Huge internal waves in the ocean’s interior caused by temperature differences that can destroy marine ships could be defined as fractional RLW equations in the existing, exceedingly complex framework.The article is given as follows: In Section2, some basic definitions are essential for the formulation of the problem. The method is described in Section 3, using a novel integral transformation. Section 4 presents the main results, numerical simulations, and graphical representations. Finally, Section 5 presents all of the research study’s significant findings.


## 2. Preliminaries

The fundamental concepts with and without a singular kernel of fractional derivatives, fractional integrals, and their Elzaki transform are presented in this section.Definition 1.
The Caputo fractional derivative (CFD) is defined as [1]
(7)CDIδμI=1Γm−δ∫0IμmηI−ηδ+1−mdη,m−1<δ<m,dmdImμI,δ=m.Definition 2.
The Atangana-Baleanu derivative having fractional-order in the Caputo manner (ABC) is defined as [36]
(8)ABCDIδμI=Nδ1−δ∫mIμ′ηEδ−δI−ηδ1−δdη,where μ∈H1α,β,β>α,δ∈0,1. A normalization function equal to 1 when δ=0 and δ=1 is represented by Nδ in Equation (8).Definition 3.
The ABC operator fractional integral is given by [36].
(9)IIδμI=1−δNδμI+δΓδNδ∫mIμηI−ηδ−1dη.Definition 4.
In setA, the exponential function Elzaki transform is defined as [37, 38]
(10)A=μI:∃G,p1,p2>0,μI<GeI/pj,ifI∈−1j×0,∞.
G is a finite number for a specific function in the set, but p1, p2 can be finite or infinite.Definition 5.
The Elzaki transformation of a functionμI is given by [38].
(11)EμIϖ=U~ϖ=ϖ∫0∞e−I/ϖμIdI,where I≥0,p1≤ϖ≤p2.Theorem 6 (Elzaki transformation convolution theorem, [39]).
The following equality holds:(12)Eμ∗v=1ϖEμEv,where Elzaki transform is represented by E..Definition 7.
The Elzaki transform of the CFD operatorDIδμI is given by [40].
(13)EDCIδμIϖ=ϖ−δU~ϖ−∑k=0m−1ϖ2−δ+kμk0,where m−1<δ<m.Theorem 8.
The ABC fractional derivativeDIδμI Elzaki transform is defined as [32].
(14)EDABCIδμIϖ=Nδϖδϖδ+1−δU~ϖϖ−ϖμ0,where EμIϖ=U~ϖ.Proof.
From Definition2, we have
(15)EDABCIδμIϖ=ENδ1−δ∫0Iμ′ηEδ−δI−ηδ1−δdηϖ.
Then, taking into account the Elzaki transform’s definition and convolution, we get(16)EABCDIδμIϖ=ENδ1−δ∫0Iμ′ηEδ−δI−ηδ1−δdη=Nδ1−δ1ϖEμ′ηEEδ−δIδ1−δdη=Nδ1−δU~ϖϖ−ϖμ0∫0∞e−1/ϖEδ−δIδ1−δdI=Nδϖδϖδ+1−δU~ϖϖ−ϖμ0.


## 3. Description of the Technique via a New Integral Transform

The essential technique that was employed in this research will be presented in this section of the study. We use the following fractional nonlinear PDE general form to study this methodology:(17)DIδφψ,I+Lφψ,I+Nφψ,I=θψ,I,ψ,I∈0,1×0,T,κ−1<δ<κ,having initial source
(18)∂zφ∂Izψ,0=μzψ,z=0,1,⋯,κ−1,and the boundary sources
(19)φ0,I=γ0I,φ1,I=γ1I,I≥0,where known functions are μz,θ,γ0, and γ1. In Equation (17), DIδφψ,I represents the Caputo or ABC fractional derivatives, L. and N. denote the linear and nonlinear terms. The recursive steps for handling the specified problems are described (1)-(2), (3)-(4), and (5)-(6). We investigate Eφψ,Iϖ=ζ~ψ,ϖ for Equation (17) by taking the Elzaki transform with the aid of CFD in Equation (13) and ABC in Equation (14). The modified functions for the Caputo fractional derivative can then be obtained.
(20)ζ~ψ,ϖ=ϖδθ~ϖ,I−ELφψ,I+Nφψ,I+ϖ2φψ,0.We also get the modified functions for the ABC derivative, which are as follows.(21)ζ~ψ,ϖ=δϖδ+1−δNδθ~ψ,ϖ−ELφψ,I+Nφψ,I+ϖ2φψ,0,where Eθψ,I=θ~ψ,ϖ. We also get when we consider the Elzaki transforms of the boundary conditions
(22)Eγ0I=ζ~0,ϖ,Eγ1I=ζ~1,ϖ,ϖ≥0.The solution to Equations (17)–(19) is then obtained by using the perturbation method as
(23)ζ~ψ,ϖ=∑E=0∞XEζ~Eψ,ϖ,E=0,1,2,⋯.The nonlinear component in Equation (17) can be calculated as
(24)Nφψ,I=∑E=0∞XEΨEψ,I,and the parts ΨEψ,I are define in as
(25)ΨEφ0,φ1,⋯,φE=1E!∂E∂νEN∑i=0∞νiφiλ=0,E=0,1,2,⋯.By putting Equations (23) and (24) into Equation (20), we obtain the components of the Caputo operator’s solution:
(26)∑E=0∞XEζ~ψ,ϖ=−XϖδEL∑E=0∞XEφEψ,I+∑E=0∞XEΨEψ,I+ϖδθ~ψ,ϖ+ϖ2φψ,0.and by putting Equations (23) and (24) into Equation (21), we have the recursive relation that provides the Atangana-Baleanu operator’s solution:
(27)∑E=0∞XEζ~ψ,ϖ=−Xδϖδ+1−δNδEL∑E=0∞XEφEψ,I+∑E=0∞XEΨEψ,I+δϖδ+1−δNδθ~ψ,ϖ+ϖ2φψ,0.Thus, on solving Equations (26) and (27) with respect to X, the given Caputo homotopies are identified:
(28)X0:ζ~0ψ,ϖ=ϖδθ~ψ,ϖ+ϖ2φψ,0,X1:ζ~1ψ,ϖ=−ϖδELφ0ψ,I+Ψ0ψ,I,X2:ζ~2ψ,ϖ=−ϖδELφ1ψ,I+Ψ1ψ,I,⋮Xn+1:ζ~n+1ψ,ϖ=−ϖδELφnψ,I+Ψnψ,I.Furthermore, we determine the ABC homotopies as follows:(29)X0:ζ~0ψ,ϖ=δϖδ+1−δNδθ~ψ,ϖ+ϖ2φψ,0,X1:ζ~1ψ,ϖ=−δϖδ+1−δNδELφ0ψ,I+Ψ0ψ,I,X2:ζ~2ψ,ϖ=−δϖδ+1−δNδELφ1ψ,I+Ψ1ψ,I,⋮Xn+1:ζ~n+1ψ,ϖ=−δϖδ+1−δNδELφnψ,I+Ψnψ,I.WhenX⟶1, we can assume that Equations (28) and (29) represent the approximate solution to Equations (26) and (27); thus, the result is determined by
(30)Δnψ,ϖ=∑σ=0nζ~σψ,ϖ.We get the approximate solution of Equation (17), by taking the inverse ET to Equation (30).
(31)φψ,ϖ≅φnψ,I=E−1Δnψ,ϖ.


## 4. Applications

In this part, we will examine the problems in Equations (1)–(6) by means of Elzaki transform. First, we implement the Elzaki transform technique with the aid of Caputo derivative to solve problem (1) having initial source (2). By taking the Elzaki transform, we get
(32)ζ~ψ,ϖ=ϖδEφψψIψ,I−φψψ,I−φψ,Iφψψ,I+ϖ2φψ,0.We use the Elzaki perturbation transform approach to solve Equation (32) and obtain
(33)∑E=0∞XEζ~Eψ,ϖ=XϖδE∑E=0∞XEφEψ,IψψI−∑E=0∞XEφEψ,Iψ−XϖδE∑E=0∞XEΨEψ,I+ϖ2φψ,0.We now have by taking the Elzaki inverse transform to Equation (33),
(34)∑E=0∞XEφEψ,ϖ=XE−1ϖδE∑E=0∞XEφEψ,IψψI−∑E=0∞XEφEψ,Iψ−XE−1ϖδE∑E=0∞XEΨEψ,I+E−1ϖ2φψ,0.TheΨE. values in Equation (34) are functions that indicate the nonlinear terms assumed in Equation (26) and are analyzed as follows:
(35)Ψ0φ=φ0φ0ψ,Ψ1φ=φ0φ1ψ+φ1φ0ψ,Ψ2φ=φ0φ2ψ+φ1φ1ψ+φ2φ0ψ,⋮Then, by examining the associated powers ofX, we obtain the terms of the Caputo operator solution:
(36)X0:ζ~0ψ,I=E−1ϖ23αsech2δψ=3αsech2δψ,X1:ζ~1ψ,I=E−1ϖδELφ0ψ,I−E−1ϖδEΨ0ψ,I=3αδ1+6αδ+cosh2δψsech4δψtanhδψIδΓδ+1,X2:ζ~2ψ,I=E−1ϖδELφ1ψ,I−E−1ϖδEΨ1ψ,I=−332αδ2−8−96α−576α2+3−3−16α+144α2cosh2δψ+48αcosh4δψ+cosh6δψsech8δψI2δΓ2δ+1,⋮As a result, the approximate solution to the problem is(37)φψ,I=3αsech2δψ+3αδ1+6αδ+cosh2δψsech4δψtanhδψIδΓδ+1−332αδ2−8−96α−576α2+3−3−16α+144α2cosh2δψ+48αcosh4δψ+cosh6δψsech8δψI2δΓ2δ+1,+⋯,providing the problem’s integer-order δ=1 solution, φψ,I=3αsech2δψ−1+αI.On the other hand, we use the Elzaki transform in combination with the Atangana-Baleanu operator to solve the problem. First, we use the Elzaki transform to solve the problem:(38)ζ~ψ,ϖ=δϖδ+1−δNδEφψψIψ,I−φψψ,I−φψ,Iφψψ,I+ϖ2φψ,0.To Equation (38), we use the Elzaki perturbation transform approach and get
(39)∑E=0∞XEζ~Eψ,ϖ=Xδϖδ+1−δNδE∑E=0∞XEφEψ,IψψI−∑E=0∞XEφEψ,Iψ−Xδϖδ+1−δNδE∑E=0∞XEΨEψ,I+ϖ2φψ,0.By taking the inverse ET of Equation (39), we get
(40)∑E=0∞XEφEψ,I=XE−1δϖδ+1−δNδE∑E=0∞XEφEψ,IψψI−∑E=0∞XEφEψ,Iψ−XE−1δϖδ+1−δNδE∑E=0∞XEΨEψ,I+E−1ϖ2φψ,0.TheΨE. terms in Equation (40) are nonlinear polynomials that were described in Equation (25). We derive the following results by repeating the methods for nonlinear polynomials:
(41)X0:φ0ψ,I=E−1ϖ23αsech2δψ=3αsech2δψ,X1:φ1ψ,I=E−1δϖδ+1−δNδEφ0ψ,I−E−1δϖδ+1−δNδEΨ0ψ,I=−3αδ1+6αδ+cosh2δψsech4δψtanhδψNδδIδΓδ+1+1−δ,(42)X2:φ2ψ,I=E−1δϖδ+1−δNδEφ1ψ,I−E−1δϖδ+1−δNδEΨ1ψ,I=−−3/32αδ2−8−96α−576α2+3−3−16α+144α2cosh2δψ+48αcosh4δψ+cosh6δψsech8δψNδ2δIδ2Γ2δ+1+2δ1−δIδΓδ+1+1−δ2,⋮As a result, based on the ABC operator, the approximate solution is as follows:(43)φψ,I=∑σ=0nφσψ,I=3αsech2δψ−3αδ1+6αδ+cosh2δψsech4δψtanhδψNδδIδΓδ+1+1−δ−−3/32αδ2−8−96α−576α2+3−3−16α+144α2cosh2δψ+48αcosh4δψ+cosh6δψsech8δψNδ22δIδ2Γ2δ+1+2δ1−δIδΓδ+1+1−δ2+⋯,providing the problem’s integer-order δ=1 solution, φψ,I=3αsech2δψ−1+αI.Secondly, we implement the Elzaki transform technique with the aid of Caputo derivative to solve problem (3) having initial source (4). By taking the Elzaki transform, we get(44)ζ~ψ,ϖ=ϖδE2φψψIψ,I−φψψ,I+ϖ2φψ,0.We use the Elzaki perturbation transform approach to solve Equation (44) and obtain
(45)∑E=0∞XEζ~Eψ,ϖ=XϖδE∑E=0∞XEφEψ,IψψI−∑E=0∞XEφEψ,Iψ+ϖ2φψ,0.We now have by taking the Elzaki inverse transform to Equation (45),
(46)∑E=0∞XEζ~Eψ,I=XE−1ϖδE∑E=0∞XEφEψ,I+E−1ϖ2φψ,0.Then, by examining the associated powers ofX, we obtain the terms of the Caputo operator solution:
(47)X0:φ0ψ,I=E−1ϖ2e−ψ=e−ψ,X1:φ1ψ,I=E−1ϖδELφ0ψ,I,=e−ψIδΓδ+1,X2:φ1ψ,I=E−1ϖδELφ1ψ,I,=e−ψI2δΓ2δ+1,⋮As a result, the approximate solution to the problem is(48)φψ,I=e−ψ+e−ψIδΓδ+1+e−ψI2δΓ2δ+1+⋯,providing the problem’s integer-order δ=1 solution φψ,I=eI−ψ.On the other hand, we use the Elzaki transform in combination with the Atangana-Baleanu operator to solve the problem. First, we use the Elzaki transform to solve the problem:(49)ζ~ψ,ϖ=δϖδ+1−δNδEφψψIψ,I−φψψ,I+ϖ2φψ,0.To Equation (49), we use the Elzaki perturbation transform approach and get
(50)∑E=0∞XEζ~Eψ,ϖ=δϖδ+1−δNδE∑E=0∞XEφEψ,IψψI−∑E=0∞XEφEψ,Iψ+ϖ2φψ,0.By taking the inverse ET of the last equation, we get(51)∑E=0∞XEζ~Eψ,I=XE−1δϖδ+1−δNδE∑E=0∞XEφEψ,I+E−1ϖ2φψ,0.Thus, on comparing both sides(52)X0:φ0ψ,I=E−1ϖ2e−ψ=e−ψ,X1:φ1ψ,I=E−1δϖδ+1−δNδEφ0ψψI−φ0ψ=e−ψNδIδΓδ+1−δ,X2:φ2ψ,I=E−1δϖδ+1−δNδEφ1ψψI−φ1ψ=e−ψNδ2δ2I2δΓ2δ+1+1−δδI2δΓδ+1+1−δ2As a result, based on the ABC operator, the approximate solution is as follows:(53)φψ,I=e−ψ+e−ψNδIδΓδ+1−δ+e−ψNδ2δ2I2δΓ2δ+1+1−δδI2δΓδ+1+1−δ2+⋯,providing the problem’s integer-order δ=1 solution, φψ,I=eI−ψ.Finally, we use the Elzaki transform approach with the aid of Caputo and ABC derivative operators to solve the problem in Equations (5)–(6). To Equations (5)–(6), we first apply the Elzaki transform with the aid of Caputo derivative:
(54)ζ~ψ,ϖ=ϖδEφψψψψψ,I+ϖ2φψ,0.We use the Elzaki perturbation transform approach to solve Equation (54) and obtain
(55)∑E=0∞XEζ~Eψ,ϖ=XϖδE∑E=0∞XEφEψ,ϖψψψψ+ϖ2φψ,0.We now have by taking the Elzaki inverse transform to Equation (55)
(56)∑E=0∞XEφEψ,I=XE−1ϖδE∑E=0∞XEφEψ,I+E−1ϖ2φψ,0.Then, by examining thsse associated powers ofX, we obtain the terms of the Caputo operator solution:
(57)X0:φ0ψ,I=E−1ϖ2sinψ=sinψ,X1:φ1ψ,I=E−1ϖδELφ0ψ,I,=−sinψIδΓδ+1,X2:φ1ψ,I=E−1ϖδELφ1ψ,I,=sinψI2δΓ2δ+1,⋮As a result, the approximate solution to the problem is(58)φψ,I=sinψ−sinψIδΓδ+1+sinψI2δΓ2δ+1+⋯,providing the problems integer-order δ=1 solution φψ,I=sinψe−I.On the other hand, we use the Elzaki transform in combination with the Atangana-Baleanu operator to solve the problem. First, we use the Elzaki transform to solve the problem:(59)ζ~ψ,ϖ=δϖδ+1−δNδEφψψψψψ,I+ϖ2φψ,0.To Equation (59), we use the Elzaki perturbation transform approach and get
(60)∑E=0∞XEζ~Eψ,ϖ=δϖδ+1−δNδE∑E=0∞XEφEψ,Iψψψψ+ϖ2φψ,0.By taking the inverse ET of the last equation, we get(61)∑E=0∞XEφEψ,I=XE−1δϖδ+1−δNδE∑E=0∞XEφEψ,I+E−1ϖ2φψ,0.Thus, on comparing both sides(62)X0:φ0ψ,I=E−1ϖ2sinψ=sinψ,X1:φ1ψ,I=E−1δϖδ+1−δNδEφ0ψψψψ=sinψNδIδΓδ+1−δ,X2:φ2ψ,I=E−1δϖδ+1−δNδEφ1ψψψψ=sinψNδ2δ2I2δΓ2δ+1+1−δδI2δΓδ+1+1−δ2.As a result, based on the ABC operator, the approximate solution is as follows:(63)φψ,I=sinψ+sinψNδIδΓδ+1−δ+sinψNδ2δ2I2δΓ2δ+1+1−δδI2δΓδ+1+1−δ2+⋯,providing the problems integer-order δ=1 solution, φψ,I=sinψexp−I.


## 5. Results and Discussion

Figures1(a) and 1(b) demonstrates the comparison between approximate solution and exact solution, while Figures 1(c) and 1(d) shows the 3D and 2D behavior of proposed methods results at different fractional-orders of the problem given by Equation (1). Figure 1 indicates that approximate solution obtained by the suggested techniques is more close to exact solution. We have shown the exact and approximate solutions in Figures 2(a) and 2(b), and the results to the problem given by Equation (3) with respect to various values of fractional parameter in Caputo and Atangana-Baleanu manner can be seen in Figures 2(c) and 2(d). Figures 3(a) and 3(b) represents the comparison between proposed method solution and exact solution, whereas Figures 3(c) and 3(d) shows the behavior of the proposed methods as various fractional-orders of the problem given by Equation (5). On the other hand, in Tables 1–3 we presented the absolute error analysis of RLW equation obtained with the help of proposed method at various values of ψ and I. It is observed from tables that proposed method solution are in good contact with the exact solution and have high level of precisions between results and shows absolute error between results.Figure 1
Example 1 solution graph (a) exact solution, (b) analytical solution atδ=1, (c) analytical solution at various fractional-orders of δ, and (d) I=0.5.
(a)(b)(c)(d)Figure 2
Example 2 solution graph (a) exact solution, (b) analytical solution atλ=1, (c) analytical solution at various fractional-orders of δ, and (d) I=0.5.
(a)(b)(c)(d)Figure 3
Example 3 solution graph (a) exact solution, (b) analytical solution atδ=1, (c) analytical solution at various fractional-orders of δ, and (d) I=0.5.
(a)(b)(c)(d)Table 1
Comparison at different fractional-order ofδ on the basis of error for example 1.
Iψδ=0.4δ=0.6δ=0.8δ=1ETMCFDδ=1ETMABC10.23.2544637000E-052.2120827000E-051.1245160000E-064.6950000000E-094.6950000000E-090.45.6252371000E-053.8122367000E-051.9326610000E-064.5160000000E-094.5160000000E-090.66.6977543000E-054.5190884000E-052.2817165000E-062.8920000000E-092.8920000000E-090.86.6088137000E-054.4352512000E-052.2282931000E-069.6700000000E-109.6700000000E-1015.7917905000E-053.8643903000E-051.9309485000E-064.3500000000E-104.3500000000E-1020.23.2724604000E-052.2243772000E-051.1309350000E-061.4503000000E-081.4503000000E-080.45.6554664000E-053.8325285000E-051.9428321000E-061.1962000000E-081.1962000000E-080.66.7332527000E-054.5426381000E-052.2932956000E-066.3710000000E-096.3710000000E-090.86.6435085000E-054.4580264000E-052.2393475000E-068.5500000000E-108.5500000000E-1015.8219810000E-053.8840174000E-051.9403804000E-062.7360000000E-092.7360000000E-0930.23.2889983000E-052.2359041000E-051.1372439000E-062.9430000000E-082.9430000000E-080.45.6826239000E-053.8509365000E-051.9522428000E-062.2347000000E-082.2347000000E-080.66.7648218000E-054.5636779000E-052.3036960000E-061.0447000000E-081.0447000000E-080.86.6741663000E-054.4781765000E-052.2490874000E-063.3000000000E-103.3000000000E-1015.8485407000E-053.9012637000E-051.9485796000E-066.9030000000E-096.9030000000E-0940.23.3047770000E-052.2471143000E-051.1436601000E-064.9484000000E-084.9484000000E-080.45.7079185000E-053.8682371000E-051.9612665000E-063.5686000000E-083.5686000000E-080.66.7939008000E-054.5831283000E-052.3133582000E-061.5130000000E-081.5130000000E-080.86.7022070000E-054.4966048000E-052.2579430000E-062.5830000000E-092.5830000000E-0915.8727131000E-053.9169160000E-051.9559187000E-061.2932000000E-081.2932000000E-0850.23.3201253000E-052.2582205000E-051.1502870000E-067.4676000000E-087.4676000000E-080.45.7319182000E-053.8847969000E-051.9700805000E-065.1991000000E-085.1991000000E-080.66.8211657000E-054.6014233000E-052.3224915000E-062.0433000000E-082.0433000000E-080.86.7282969000E-054.5137364000E-052.2661180000E-065.8960000000E-095.8960000000E-0915.8950823000E-053.9313444000E-051.9625736000E-062.0821000000E-082.0821000000E-08Table 2
Comparison at different fractional-order ofδ on the basis of error for example 2.
Iψδ=0.4δ=0.6δ=0.8δ=1ETMCFDδ=1ETMABC10.22.4586042500E-021.6390008900E-028.1946786000E-034.0000000000E-094.0000000000E-090.42.0129349000E-021.3419004200E-026.7092353000E-033.4000000000E-093.4000000000E-090.61.6480517100E-021.0986551500E-025.4930573000E-032.7000000000E-092.7000000000E-090.81.3493106100E-028.9950275000E-034.4973349000E-032.3000000000E-092.3000000000E-0911.1047220900E-027.3645057000E-033.6821064000E-031.9000000000E-091.9000000000E-0920.22.4605791800E-021.6402716400E-028.2008084000E-031.6300000000E-081.6300000000E-080.42.0145518400E-021.3429408200E-026.7142540000E-031.3500000000E-081.3500000000E-080.61.6493755500E-021.0995069600E-025.4971663000E-031.1000000000E-081.1000000000E-080.81.3503944800E-029.0020016000E-034.5006991000E-039.0000000000E-099.0000000000E-0911.1056095000E-027.3702156000E-033.6848608000E-037.3000000000E-097.3000000000E-0930.22.4623948800E-021.6414429900E-028.2064699000E-033.6900000000E-083.6900000000E-080.42.0160384100E-021.3438998600E-026.7188893000E-033.0200000000E-083.0200000000E-080.61.6505926500E-021.1002921400E-025.5009613000E-032.4700000000E-082.4700000000E-080.81.3513909600E-029.0084302000E-034.5038062000E-032.0200000000E-082.0200000000E-0811.1064253400E-027.3754788000E-033.6874046000E-031.6600000000E-081.6600000000E-0840.22.4641082400E-021.6425501600E-028.2118269000E-036.5500000000E-086.5500000000E-080.42.0174411900E-021.3448063300E-026.7232751000E-035.3700000000E-085.3700000000E-080.61.6517411400E-021.1010343000E-025.5045521000E-034.3900000000E-084.3900000000E-080.81.3523312700E-029.0145064000E-034.5067461000E-033.6000000000E-083.6000000000E-0811.1071952000E-027.3804536000E-033.6898117000E-032.9400000000E-082.9400000000E-0850.22.4657459500E-021.6436096000E-028.2169554000E-031.0230000000E-071.0230000000E-070.42.0187820300E-021.3456737200E-026.7274740000E-038.3900000000E-088.3900000000E-080.61.6528389400E-021.1017444700E-025.5079899000E-036.8600000000E-086.8600000000E-080.81.3532300600E-029.0203207000E-034.5095607000E-035.6200000000E-085.6200000000E-0811.1079310700E-027.3852140000E-033.6921161000E-034.6000000000E-084.6000000000E-08Table 3
Comparison at different fractional-order ofδ on the basis of error for example 3.
Iψδ=0.4δ=0.6δ=0.8δ=1ETMCFDδ=1ETMABC10.25.9659347000E-033.9771240000E-031.9884840000E-031.0000000000E-091.0000000000E-090.41.1694026400E-027.7956926000E-033.8976932000E-031.9000000000E-091.9000000000E-090.61.6955914200E-021.1303471500E-025.6515139000E-032.8000000000E-092.8000000000E-090.82.1541823200E-021.4360616700E-027.1800265000E-033.6000000000E-093.6000000000E-0912.5268927700E-021.6845249400E-028.4222940000E-034.2000000000E-094.2000000000E-0920.25.9707330000E-033.9802135000E-031.9899773000E-034.0000000000E-094.0000000000E-090.41.1703431600E-027.8017484000E-033.9006205000E-037.8000000000E-097.8000000000E-090.61.6969551500E-021.1312252300E-025.6557583000E-031.1300000000E-081.1300000000E-080.82.1559148800E-021.4371772300E-027.1854188000E-031.4300000000E-081.4300000000E-0812.5289250900E-021.6858335100E-028.4286193000E-031.6800000000E-081.6800000000E-0830.25.9751488000E-033.9830657000E-031.9913610000E-038.9000000000E-098.9000000000E-090.41.1712087200E-027.8073393000E-033.9033328000E-031.7500000000E-081.7500000000E-080.61.6982101800E-021.1320358900E-025.6596910000E-032.5400000000E-082.5400000000E-080.82.1575093500E-021.4382071500E-027.1904152000E-033.2300000000E-083.2300000000E-0812.5307954300E-021.6870416200E-028.4344802000E-033.7900000000E-083.7900000000E-0840.25.9793203000E-033.9857663000E-031.9926749000E-031.5900000000E-081.5900000000E-080.41.1720263900E-027.8126327000E-033.9059080000E-033.1100000000E-083.1100000000E-080.61.6993957600E-021.1328034100E-025.6634250000E-034.5200000000E-084.5200000000E-080.82.1590155800E-021.4391822500E-027.1951591000E-035.7400000000E-085.7400000000E-0812.5325622600E-021.6881854300E-028.4400448000E-036.7300000000E-086.7300000000E-0850.25.9833121000E-033.9883550000E-031.9939372000E-032.4900000000E-082.4900000000E-080.41.1728088500E-027.8177068000E-033.9083824000E-034.8700000000E-084.8700000000E-080.61.7005302900E-021.1335391300E-025.6670127000E-037.0500000000E-087.0500000000E-080.82.1604569600E-021.4401169600E-027.1997171000E-038.9600000000E-088.9600000000E-0812.5342530300E-021.6892818700E-028.4453916000E-031.0520000000E-071.0520000000E-07


## 6. Conclusion

The approximate solutions of some particular regularized long-wave equations of fractional-order are determined in this paper using a new integral transform method known as the Elzaki transformation. To begin, we consider the Elzaki transform of the fractional Atangana-Baleanu operator and used it to solve the proposed problems. The used scheme’s trustworthiness and efficiency are based on its capacity to provide an appropriate convergence zone for the solution. The excellent accuracy of the findings and the simplicity of the solution approach confirm suggested method supremacy over other numerical methods. Also, we have shown how the Caputo and Atangana-Baleanu fractional operators differ when it comes to finding approximate solutions to the illustrative examples. To ensure the validity of the suggested technique, we showed the results in graphs and tables. The representations of graphs and tables demonstrate that the results obtained by suggested scheme are very accurate. In addition, the behavior of fractional-order results is discussed which confirm that the solution gets closer as the fractional-order tends toward integer-order. Finally, the approximation solution strategy employed is highly efficient and applicable to a wide range of nonlinear equations defining real systems.



---

*Source: 2754507-2022-06-06.xml*