# Fractional-View Analysis of Jaulent-Miodek Equation via Novel Analytical Techniques

Authors: Ahmad Haji Zadeh; Kavikumar Jacob; Nehad Ali Shah; Jae Dong Chung
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we analyze the analytical result of fractional-order Jaulent-Miodek equations with the help of two novel methods, namely,ρ-Laplace decomposition method and ρ-Laplace variational iteration method. The achieved results are shown in a series form, which is rapidly converging. The approximate simulations were performed in absolute error to ensure that the suggested methods are accurate and reliable. The achieved results are graphically presented to confirm the validity and accuracy of the techniques. The study results reveal that the ρ-Laplace decomposition method is computationally very effective and accurate compared to ρ-Laplace variational iteration method to analyze the nonlinear system of fractional-order Jaulent-Miodek equations.


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## Body

## 1. Introduction

Jaulent and Miodek developed the Jaulent-Miodek equation in 1979 as an extension to energy-dependent potentials [1, 2]. The JM equation was developed in many related fields of physics, optics [3], condensed matter physics [4], plasma physics [5], and including fluid mechanics [6]. The purpose of this research is to investigate the approximate results for arbitrary order of the following anomalous and difficult physical model,
(1)∂ϑμ∂Iϑ+∂3μ∂ε3+32ω∂3ω∂ε3+92∂ω∂ε∂2ω∂ε2−6μ∂μ∂ε−6μω∂ω∂ε−32∂μ∂εω2=0,∂ϑμ∂Iϑ+∂3ω∂ε3−6∂μ∂εω−6μ∂ω∂ε−152∂ω∂εω2=0.The nonlinear system JM equation has been analyzed in past years by several researchers through different methods like homotopy perturbation transform method [7], optimal homotopy asymptotic method and Hermite wavelets [8], invariant subspace technique [9], and others [10–12].Systems of nonlinear partial differential equation [13–17] appear in a variety of scientific physical models. The classical Jaulent-Miodek equations have been the subject of extensive investigation in recent years. Several techniques such as suitable algebraic technique [18], Adomian decomposition technique [19], tanh-sech technique [20], homotopy perturbation technique [21], Exp-function technique [22], and homotopy analysis technique [23] had been applied for solving of the system of Jaulent-Miodek equations. But the complete evaluation of the nonlinear fractional-order combined with the Jaulent-Miodek equation is only an initiation by the available information.Abdeljawad and Jarad [24] recently developed the Laplace transform of the fractional-order Caputo derivatives. We proposed a new iterative technique based on the ρ-Laplace transformation to study fractional-order ordinary and partial differential equations with fractional-order Caputo derivatives. This novel proposed method is used to solve numerous fractional-order differential equations, including linear and nonlinear diffusion equations, fractional-order Fokker-Planck equations, and Zakharov-Kuznetsov equations [25–27]. Adomian, an American mathematician, introduced the Adomian decomposition method. It concentrates on obtaining series-like solutions and decomposing the nonlinear operator into a sequence, with the terms currently obtained using Adomian polynomials [28–31]. The ρ-Laplace transform is used to modify this procedure, resulting in the ρ-Laplace decomposition method. This method is applied to the nonhomogeneous fractional differential equations [32–34]. He [35, 36] was the first to introduce the variational iteration method (VIM), which was successfully applied to autonomous ordinary differential equations in [37], nonlinear polycrystal line solids in [38], and other fields. The variational iteration method is modified with the help of ρ-Laplace transform, known as ρ-Laplace variational iteration method. Different types of ODEs and PDEs have been analyzed with the aid of the variational iteration method. For instance, this method is investigated for solving fractional differential equations in [37]. In [38], this method is used to solve nonlinear oscillator equations. Compared to the Adomians decomposition method, variational iteration transforms approach solving without the need to compute Adomian’s polynomials. The [39] mesh point procedures provide an analytical solution, whereas this scheme provides a fast response to the equation [40–42]. This technique can also be used to approximate accurate results.This article has applied theρ-Laplace decomposition method and ρ-Laplace variational iteration method to solve the fractional coupled Jaulent-Miodek equation with the Caputo fractional derivative operator. The analytical results attained via the ρ-Laplace decomposition method were compared with exact results and those derived by using the ρ-Laplace variational iteration method in case of fractional order.


## 2. Basic Preliminaries

### 2.1. Definition

The fractional-order generalized integralϑ of a continuous function f:0,+∞⟶R is given as [24]
(2)Iϑ,ρgε=1Γϑ∫0εερ−sρρϑ−1gsdss1−ρ,the gamma function is defined by Γ, ρ>0, ε>0, and 0<ϑ<1.


### 2.2. Definition

The fractional-order generalized derivative ofϑ of a continuous function g:0,+∞⟶R is defined as [24]
(3)Dϑ,ρgε=I1−ϑ,ρgε=1Γ1−ϑddε∫0εερ−sρρ−ϑ−1gsdss1−ρ.where expressed the gamma function Γ, ρ>0, ε>0, and 0<ϑ<1.


### 2.3. Definition

The fractional-order Caputo derivativeϑ of a continuous function g:0,+∞⟶R is defined as [24]
(4)Dϑ,ρgε=1Γ1−ϑddε∫0εερ−sρρ−ϑ−1ϑngsdss1−ρ.where ρ>0, ε>0, ϑ=ε1−ϑd/dε and 0<ϑ<1.


### 2.4. Definition

Theρ-Laplace transform of a continuous function g:0,+∞⟶R is given as [24]
(5)Lρgεs=∫0∞e−sερ/ρgεdεε1−ρ.The fractional-order Caputo generalizedρ-Laplace transformation derivative of a continuous function g is given by [24]
(6)LρDϑ,ρgεs=sϑLρgε−∑k=0n−1sϑ−k−1Iϑ,ρϑng0.


### 2.5. Definition

The generalized Mittag-Leffler function is defined by(7)Eϑ,ρz=∑k=0∞zϑΓϑk+β,where ϑ>0, β>0, and Eϑz=Eϑ,1z.


## 2.1. Definition

The fractional-order generalized integralϑ of a continuous function f:0,+∞⟶R is given as [24]
(2)Iϑ,ρgε=1Γϑ∫0εερ−sρρϑ−1gsdss1−ρ,the gamma function is defined by Γ, ρ>0, ε>0, and 0<ϑ<1.


## 2.2. Definition

The fractional-order generalized derivative ofϑ of a continuous function g:0,+∞⟶R is defined as [24]
(3)Dϑ,ρgε=I1−ϑ,ρgε=1Γ1−ϑddε∫0εερ−sρρ−ϑ−1gsdss1−ρ.where expressed the gamma function Γ, ρ>0, ε>0, and 0<ϑ<1.


## 2.3. Definition

The fractional-order Caputo derivativeϑ of a continuous function g:0,+∞⟶R is defined as [24]
(4)Dϑ,ρgε=1Γ1−ϑddε∫0εερ−sρρ−ϑ−1ϑngsdss1−ρ.where ρ>0, ε>0, ϑ=ε1−ϑd/dε and 0<ϑ<1.


## 2.4. Definition

Theρ-Laplace transform of a continuous function g:0,+∞⟶R is given as [24]
(5)Lρgεs=∫0∞e−sερ/ρgεdεε1−ρ.The fractional-order Caputo generalizedρ-Laplace transformation derivative of a continuous function g is given by [24]
(6)LρDϑ,ρgεs=sϑLρgε−∑k=0n−1sϑ−k−1Iϑ,ρϑng0.


## 2.5. Definition

The generalized Mittag-Leffler function is defined by(7)Eϑ,ρz=∑k=0∞zϑΓϑk+β,where ϑ>0, β>0, and Eϑz=Eϑ,1z.


## 3. The Methodology ofρ-Laplace Decomposition Method

In this section, theρ-Laplace decomposition method solution system for fractional partial differential fractional equations.
(8)DIϑμε,I+M¯1μ,ω+N1μ,ω−P1ε,I=0,DIϑωε,I+G¯2μ,ω+N2μ,ω−P2ε,I=0,κ−1<ϑ≤κ,with initial conditions
(9)με,0=g1ε,ωε,0=g2ε.where isDIϑ=∂ϑ/∂Iϑ the Caputo fractional derivative of order ϑ, M¯1, G¯2, and N1, N2 are linear and nonlinear functions, respectively, and P1 and P2 are source operators.Theρ-Laplace transformation is applied to Eq. (8), we have
(10)LDIϑμε,I+LM¯1μ,ω+N1μ,ω−P1ε,I=0,LDIϑωε,I+LG¯2μ,ω+N2μ,ω−P2ε,I=0.Using theρ-Laplace transform differentiation property, we get
(11)Lμε,I=1sϑ∑J=0κ−1sϑ−J−1∂Jμε,I∂kII=0+1sϑLP1ε,I−1sϑLM¯1μ,ω+N1μ,ω,Lωε,I=1sϑ∑J=0κ−1sϑ−J−1∂Jωε,I∂kII=0+1sϑLP2ε,I−1sϑLG¯2μ,ω+N2μ,ω,ρ-LDM defines the result of infinite series με,I and ωε,I,
(12)με,I=∑κ=0∞μκε,I,ωε,I=∑κ=0∞ωκε,I.Adomian polynomial decomposition of nonlinear terms ofN1 and N2 is described as
(13)N1μ,ω=∑κ=0∞Aκ,N2μ,ω=∑κ=0∞Bκ,All forms of nonlinearity the Adomian polynomials can be represented as(14)Aκ=1κ!∂κ∂λκN1∑J=0∞λJμJ,∑J=0∞λJωJλ=0,Bκ=1κ!∂κ∂λκN2∑J=0∞λJμJ,∑J=0∞λJωJλ=0.Putting equations (12) and (14) into equation (11) gives
(15)L∑κ=0∞μκε,I=1sϑ∑J=0κ−1sϑ−J−1∂Jμε,I∂kII=0+1sϑLP1ε,I−1sϑLM¯1∑κ=0∞μκ,∑κ=0∞ωm+∑m=0∞Aκ,(16)L∑κ=0∞ωκε,I=1sϑ∑J=0κ−1sϑ−J−1∂Jωε,I∂kII=0+1sϑLP2ε,I−1sϑLG¯2∑κ=0∞μκ,∑κ=0∞ωκ+∑κ=0∞Bκ.Using the inverseρ-Laplace transform of equation (15), we get
(17)∑κ=0∞μκε,I=L−11sϑ∑J=0κ−1sϑ−k−1∂Jμε,I∂JII=0+1sϑLP1ε,I−1sϑLM¯1∑κ=0∞μκ,∑κ=0∞ωκ+∑κ=0∞Aκ,(18)∑κ=0∞ωκε,I=L−11sϑ∑J=0κ−1sϑ−J−1∂kωε,I∂JII=0+1sϑLP2ε,I−1sϑLG¯2∑κ=0∞μκ,∑κ=0∞ωκ+∑κ=0∞Bκ,we define the following terms,
(19)μ0ε,I=L−11sϑ∑J=0κ−1sϑ−J−1∂Jμε,I∂JII=0+1sϑLP1ε,I,ω0ε,I=L−11sϑ∑J=0κ−1sϑ−J−1∂Jωε,I∂JII=0+1sϑLP2ε,I,(20)μ1ε,I=−L−11sϑLM¯1μ0,ω0+A0,ω1ε,I=−L−11sϑLG¯2μ0,ω0+B0,the general forκ≥1 is given by
(21)μκ+1ε,I=−L−11sϑLM¯1μκ,ωκ+Aκ,ωκ+1ε,I=−L−11sϑLG¯2μκ,ωκ+Bκ.


## 4. The Producer ofρ-LVITM

In this section, explain theρ-Laplace variational iteration method of the solution of FPDEs.
(22)DIϑμε,I+M¯1μ,ω+N1μ,ω−P1ε,I=0,DIϑωε,I+G¯2μ,ω+N2μ,ω−P2ε,I=0,m−1<ϑ≤m,with initial conditions
(23)με,0=g1ε,ωε,0=g2ε.where is DIϑ=∂ϑ/∂Iϑ the Caputo fractional derivative of order ϑ, M¯1, G¯2 and N1, N2 are linear and nonlinear functions, respectively, and P1,P2 are source operators.Theρ-Laplace transformation is applied to Eq. (22),
(24)LDIϑμε,I+LM¯1μ,ω+N1μ,ω−P1ε,I=0,LDIϑωε,I+LG¯2μ,ω+N2μ,ω−P2ε,I=0.Using the differentiation propertyρ-Laplace transform, we get
(25)Lμε,I−∑κ=0J−1sϑ−J−1∂Jμε,I∂JII=0=−LM¯1μ,ω+N1μ,ω−P1ε,I,Lωε,I−∑κ=0J−1sϑ−J−1∂Jωε,I∂JII=0=−LG¯2μ,ω+N2μ,ω−P2ε,I.The procedure iteration method is defined as(26)Lμκ+1ε,I=Lμκε,I+λssϑμκε,I−∑J=0κ−1sϑ−J−1∂Jμε,I∂JII=0−LP1ε,I−LM¯1μ,ω+N1μ,ω,(27)Lωκ+1ε,I=Lωκε,I+λssϑωκε,I−∑J=0κ−1sϑ−J−1∂Jωε,I∂JII=0−LP2ε,I−LG¯2μ,ω+N2μ,ω.A Lagrange multiplier as(28)λs=−1sϑ,the inverse ρ-Laplace transformation L−1, the iteration method Eq. (26) can be given as
(29)μκ+1ε,I=μκε,I−L−11sϑ∑J=0κ−1sϑ−J−1∂Jμε,I∂JII=0−LP1ε,I−LM¯1μ,ω+N1μ,ω,ωκ+1ε,I=ωκε,I−L−11sϑ∑J=0κ−1sϑ−J−1∂Jωε,I∂JII=0−LP2ε,I−LG¯2μ,ω+N2μ,ω,the initial iteration can be find as(30)μ0ε,I=L−11sϑ∑J=0κ−1sϑ−J−1∂Jμε,I∂JII=0,ω0ε,I=L−11sϑ∑J=0κ−1sϑ−J−1∂Jωε,I∂JII=0.


## 5. Implementation of Techniques

### 5.1. Problem

Consider the fractional-order non-linear Jaulent-Miodek equation is given as(31)∂ϑμ∂Iϑ+∂3μ∂ε3+32ω∂3ω∂ε3+92∂ω∂ε∂2ω∂ε2−6μ∂μ∂ε−6μω∂ω∂ε−32∂μ∂εω2=0,(32)∂ϑμ∂Iϑ+∂3ω∂ε3−6∂μ∂εω−6μ∂ω∂ε−152∂ω∂εω2=0.With initial conditions(33)με,0=18λ21−4sech2λε2,(34)ωε,0=λsechλε2.Takingρ-Laplace transform of (31),
(35)sϑLμε,I−sϑ−1με,0=−L∂3μ∂ε3+32ω∂3ω∂ε3+92∂ω∂ε∂2ω∂ε2−6μ∂μ∂ε−6μω∂ω∂ε−32∂μ∂εω2,(36)sϑLωε,I−sϑ−1ωε,0=−L∂3ω∂ε3−6∂μ∂εω−6μ∂ω∂ε−152∂ω∂εω2.Using inverseρ-Laplace transformation, we get
(37)με,I=L−1με,0s−1sϑL∂3μ∂ε3+32ω∂3ω∂ε3+92∂ω∂ε∂2ω∂ε2−6μ∂μ∂ε−6μω∂ω∂ε−32∂μ∂εω2,(38)ωε,I=L−1ωε,0s−1sϑL∂3ω∂ε3−6∂μ∂εω−6μ∂ω∂ε−152∂ω∂εω2.Applying Adomian procedure, we have(39)μ0ε,I=L−1με,0s=L−11/8λ21−4sech2λε/2s,μ0ε,I=18λ21−4sech2λε2,(40)ω0ε,I=L−1ωε,0s=L−1λsechλε/2s,ω0ε,I=λsechλε2.(41)∑κ=0∞μκ+1ε,I=−L−11sϑL∑κ=0∞∂3μκ∂ε3+32∑κ=0∞Aκ+92∑κ=0∞Bκ−6∑κ=0∞Cκ−6∑κ=0∞Dκ−32∑κ=0∞Eκ,κ=0,1,2,⋯,(42)∑κ=0∞ωκ+1ε,I=−L−11sϑL∑κ=0∞∂3ωκ∂ε3−6∑κ=0∞Fκ−6∑κ=0∞Gκ−152∑κ=0∞Hκ,κ=0,1,2,⋯.The nonlinear terms find with the help of Adomain polynomialsForκ=1(43)μ1ε,I=−2λ5csch3λεsech4λε/2Iρ/ρϑΓϑ+1,ω1ε,I=λ4csch2λεsech3λε/2Iρ/ρϑΓϑ+1,for κ=2(44)μ2ε,I=−λ8−2+coshλεsech4λε/2Iρ/ρ2ϑ16Γ2ϑ+1,ω2ε,I=λ7−3+coshλεsech3λε/2Iρ/ρ2ϑ32Γ2ϑ+1,for κ=2(45)μ3ε,I=−11024Γϑ+12Γ3ϑ+1λ112−165+28coshλε+cosh2λεΓϑ+12+343−20coshλε+cosh2λεΓ2ϑ+1sech6λε2tanhλε2Iρρ3ϑ,(46)ω3ε,I=1512Γϑ+12Γ3ϑ+1λ10147−92coshλε+cosh2λεΓϑ+12+12−7+3coshλεΓ2ϑ+1sech5λε2tanhλε2Iρρ3ϑ.Theρ-LDM solution of Example 1 is
(47)με,I=μ0ε,I+μ1ε,I+μ2ε,I+μ3ε,I+⋯,ωε,I=ω0ε,I+ω1ε,I+ω2ε,I+ω3ε,I+⋯,(48)με,I=18λ21−4sech2λε2−2λ5csch3λεsech4λε/2Iρ/ρϑΓϑ+1−λ8−2+coshλεsech4λε/2Iρ/ρ2ϑ16Γ2ϑ+1−11024Γϑ+12Γ3ϑ+1λ112−165+28coshλε+cosh2λεΓϑ+12+343−20coshλε+cosh2λεΓ2ϑ+1sech6λε2tanhλε2Iρρ3ϑ−⋯.(49)ωε,I=λsechλε2+λ4csch2λεsech3λε/2Iρ/ρϑΓϑ+1+λ7−3+coshλεsech3λε/2Iρ/ρ2ϑ32Γ2ϑ+1+1512Γϑ+12Γ3ϑ+1λ10147−92coshλε+cosh2λεΓϑ+12+12−7+3coshλεΓ2ϑ+1sech5λε2tanhλε2Iρρ3ϑ+⋯.The approximate solution byρ-LVIM.Apply the iteration technique, we have(50)∑κ=0∞μκ+1ε,I=μκε,I−L−11sϑL∑κ=0∞∂3μκ∂ε3+32∑κ=0∞ωκ∂3ωκ∂ε3+92∑κ=0∞∂ωκ∂ε∂2ωκ∂ε2−6∑κ=0∞μκ∂μκ∂ε−6∑κ=0∞μκωκ∂ωκ∂ε−32∑κ=0∞∂μκ∂εωκ2,κ=0,1,2,⋯,(51)∑κ=0∞ωκ+1ε,I=ωκε,I−L−11sϑL∑κ=0∞∂3ωκ∂ε3−6∑κ=0∞∂μκ∂εωκ−6∑κ=0∞μκ∂ωκ∂ε−152∑κ=0∞∂ωκ∂εωκ2,where
(52)μ0ε,I=18λ21−4sech2λε2,ω0ε,I=λsechλε2.Forκ=0,1,2,⋯(53)μ1ε,I=μ0ε,I−L−11sϑL∂3μ0∂ε3+32ω0∂3ω0∂ε3+92∂ω0∂ε∂2ω0∂ε2−6μ0∂μ0∂ε−6μ0ω0∂ω0∂ε−32∂μ0∂εω02,(54)ω1ε,I=ω0ε,I−L−11sϑL∂3ω0∂ε3−6∂μ0∂εω0−6μ0∂ω0∂ε−152∂ω0∂εω02,(55)μ1ε,I=18λ21−4sech2λε2−2λ5csch3λεsech4λε/2Iρ/ρϑΓϑ+1,(56)ω1ε,I=λsechλε2+λ4csch2λεsech3λε/2Iρ/ρϑΓϑ+1,(57)μ2ε,I=μ1ε,I−L−11sϑL∂3μ1∂ε3+32ω1∂3ω1∂ε3+92∂ω1∂ε∂2ω1∂ε2−6μ1∂μ1∂ε−6μ1ω1∂ω1∂ε−32∂μ1∂εω12,(58)ω2ε,I=ω1ε,I−L−11sϑL∂3ω1∂ε3−6∂μ1∂εω1−6μ1∂ω1∂ε−152∂ω1∂εω12,(59)μ2ε,I=18λ21−4sech2λε2−2λ5csch3λεsech4λε/2Iρ/ρϑΓϑ+1−λ8−2+coshλεsech4λε/2Iρ/ρ2ϑ16Γ2ϑ+1,(60)ω2ε,I=λsechλε2+λ4csch2λεsech3λε/2Iρ/ρϑΓϑ+1+λ7−3+coshλεsech3λε/2Iρ/ρ2ϑ32Γ2ϑ+1,(61)μ3ε,I=μ2ε,I−L−11sϑL∂3μ2∂ε3+32ω2∂3ω2∂ε3+92∂ω2∂ε∂2ω2∂ε2−6μ2∂μ2∂ε−6μ2ω2∂ω2∂ε−32∂μ2∂εω22,(62)ω3ε,I=ω2ε,I−L−11sϑL∂3ω2∂ε3−6∂μ2∂εω2−6μ2∂ω2∂ε−152∂ω2∂εω22,(63)μ3ε,I=18λ21−4sech2λε2−2λ5csch3λεsech4λε/2Iρ/ρϑΓϑ+1−λ8−2+coshλεsech4λε/2Iρ/ρ2ϑ16Γ2ϑ+1−11024Γϑ+12Γ3ϑ+1λ112−165+28coshλεcosh2λεΓϑ+12++343−20coshλε+cosh2λεΓ2ϑ+1sech6λε2tanhλε2Iρρ3ϑ.(64)ω3ε,I=λsechλε2+λ4csch2λεsech3λε/2Iρ/ρϑΓϑ+1+λ7−3+coshλεsech3λε/2Iρ/ρ2ϑ32Γ2ϑ+1+1512Γϑ+12Γ3ϑ+1λ10147−92coshλεΓϑ+12+cosh2λε+12−7+3coshλεΓ2ϑ+1sech5λε2tanhλε2Iρρ3ϑ.(65)με,I=18λ21−4sech2λε2−2λ5csch3λεsech4λε/2Iρ/ρϑΓϑ+1−λ8−2+coshλεsech4λε/2Iρ/ρ2ϑ16Γ2ϑ+1−11024Γϑ+12Γ3ϑ+1λ112−165+28coshλε+cosh2λεΓϑ+12+343−20coshλε+cosh2λεΓ2ϑ+1sech6λε2tanhλε2Iρρ3ϑ−⋯.(66)ωε,I=λsechλε2+λ4csch2λεsech3λε/2Iρ/ρϑΓϑ+1+λ7−3+coshλεsech3λε/2Iρ/ρ2ϑ32Γ2ϑ+1+1512Γϑ+12Γ3ϑ+1λ10147−92coshλε+cosh2λεΓϑ+12+12−7+3coshλεΓ2ϑ+1sech5λε2tanhλε2Iρρ3ϑ+⋯.The exact solution of equation (8) at ϑ=1,
(67)με,I=18λ21−4sech2λ2ε+12λ2I,ωε,I=λsechλ2ε+12λ2I.In Figure1, the first figure shows the ρ-Laplace decomposition method and ρ-Laplace variational transform method and exact solutions graphs of the given problem at με,I and ωε,I. From the given graphs, it can be observed that both the exact, ρ-Laplace decomposition method, and ρ-Laplace variational transform method solution are in strong agreement with each other. In Figure 2, the exact and approximate solution of με,I at ϑ=1. Figure 3, different fractional-order of ϑ=0.8 and 0.6 of the model. Similarly, in Figure 4, the exact and approximate solution of ωε,I at ϑ=1. Figure 5, different fractional-order of ϑ=0.8 and 0.6 of the model. It is investigated that results of fractional-order problem are convergent to an integer-order result as fractional-order analysis to integer-order. The same phenomenon of convergence of fractional-order solutions towards integral-order solutions is observed.Figure 1
Exact and approximate solution ofμε,I and ωε,I at ϑ=1.Figure 2
Exact and approximate solution ofμε,I at ϑ=1.Figure 3
The different fractional-order ofϑ=0.8 and 0.6 of the model.Figure 4
Exact and approximate solution ofωε,I at ϑ=1.Figure 5
The different fractional-order ofϑ=0.8 and 0.6 of the model.


## 5.1. Problem

Consider the fractional-order non-linear Jaulent-Miodek equation is given as(31)∂ϑμ∂Iϑ+∂3μ∂ε3+32ω∂3ω∂ε3+92∂ω∂ε∂2ω∂ε2−6μ∂μ∂ε−6μω∂ω∂ε−32∂μ∂εω2=0,(32)∂ϑμ∂Iϑ+∂3ω∂ε3−6∂μ∂εω−6μ∂ω∂ε−152∂ω∂εω2=0.With initial conditions(33)με,0=18λ21−4sech2λε2,(34)ωε,0=λsechλε2.Takingρ-Laplace transform of (31),
(35)sϑLμε,I−sϑ−1με,0=−L∂3μ∂ε3+32ω∂3ω∂ε3+92∂ω∂ε∂2ω∂ε2−6μ∂μ∂ε−6μω∂ω∂ε−32∂μ∂εω2,(36)sϑLωε,I−sϑ−1ωε,0=−L∂3ω∂ε3−6∂μ∂εω−6μ∂ω∂ε−152∂ω∂εω2.Using inverseρ-Laplace transformation, we get
(37)με,I=L−1με,0s−1sϑL∂3μ∂ε3+32ω∂3ω∂ε3+92∂ω∂ε∂2ω∂ε2−6μ∂μ∂ε−6μω∂ω∂ε−32∂μ∂εω2,(38)ωε,I=L−1ωε,0s−1sϑL∂3ω∂ε3−6∂μ∂εω−6μ∂ω∂ε−152∂ω∂εω2.Applying Adomian procedure, we have(39)μ0ε,I=L−1με,0s=L−11/8λ21−4sech2λε/2s,μ0ε,I=18λ21−4sech2λε2,(40)ω0ε,I=L−1ωε,0s=L−1λsechλε/2s,ω0ε,I=λsechλε2.(41)∑κ=0∞μκ+1ε,I=−L−11sϑL∑κ=0∞∂3μκ∂ε3+32∑κ=0∞Aκ+92∑κ=0∞Bκ−6∑κ=0∞Cκ−6∑κ=0∞Dκ−32∑κ=0∞Eκ,κ=0,1,2,⋯,(42)∑κ=0∞ωκ+1ε,I=−L−11sϑL∑κ=0∞∂3ωκ∂ε3−6∑κ=0∞Fκ−6∑κ=0∞Gκ−152∑κ=0∞Hκ,κ=0,1,2,⋯.The nonlinear terms find with the help of Adomain polynomialsForκ=1(43)μ1ε,I=−2λ5csch3λεsech4λε/2Iρ/ρϑΓϑ+1,ω1ε,I=λ4csch2λεsech3λε/2Iρ/ρϑΓϑ+1,for κ=2(44)μ2ε,I=−λ8−2+coshλεsech4λε/2Iρ/ρ2ϑ16Γ2ϑ+1,ω2ε,I=λ7−3+coshλεsech3λε/2Iρ/ρ2ϑ32Γ2ϑ+1,for κ=2(45)μ3ε,I=−11024Γϑ+12Γ3ϑ+1λ112−165+28coshλε+cosh2λεΓϑ+12+343−20coshλε+cosh2λεΓ2ϑ+1sech6λε2tanhλε2Iρρ3ϑ,(46)ω3ε,I=1512Γϑ+12Γ3ϑ+1λ10147−92coshλε+cosh2λεΓϑ+12+12−7+3coshλεΓ2ϑ+1sech5λε2tanhλε2Iρρ3ϑ.Theρ-LDM solution of Example 1 is
(47)με,I=μ0ε,I+μ1ε,I+μ2ε,I+μ3ε,I+⋯,ωε,I=ω0ε,I+ω1ε,I+ω2ε,I+ω3ε,I+⋯,(48)με,I=18λ21−4sech2λε2−2λ5csch3λεsech4λε/2Iρ/ρϑΓϑ+1−λ8−2+coshλεsech4λε/2Iρ/ρ2ϑ16Γ2ϑ+1−11024Γϑ+12Γ3ϑ+1λ112−165+28coshλε+cosh2λεΓϑ+12+343−20coshλε+cosh2λεΓ2ϑ+1sech6λε2tanhλε2Iρρ3ϑ−⋯.(49)ωε,I=λsechλε2+λ4csch2λεsech3λε/2Iρ/ρϑΓϑ+1+λ7−3+coshλεsech3λε/2Iρ/ρ2ϑ32Γ2ϑ+1+1512Γϑ+12Γ3ϑ+1λ10147−92coshλε+cosh2λεΓϑ+12+12−7+3coshλεΓ2ϑ+1sech5λε2tanhλε2Iρρ3ϑ+⋯.The approximate solution byρ-LVIM.Apply the iteration technique, we have(50)∑κ=0∞μκ+1ε,I=μκε,I−L−11sϑL∑κ=0∞∂3μκ∂ε3+32∑κ=0∞ωκ∂3ωκ∂ε3+92∑κ=0∞∂ωκ∂ε∂2ωκ∂ε2−6∑κ=0∞μκ∂μκ∂ε−6∑κ=0∞μκωκ∂ωκ∂ε−32∑κ=0∞∂μκ∂εωκ2,κ=0,1,2,⋯,(51)∑κ=0∞ωκ+1ε,I=ωκε,I−L−11sϑL∑κ=0∞∂3ωκ∂ε3−6∑κ=0∞∂μκ∂εωκ−6∑κ=0∞μκ∂ωκ∂ε−152∑κ=0∞∂ωκ∂εωκ2,where
(52)μ0ε,I=18λ21−4sech2λε2,ω0ε,I=λsechλε2.Forκ=0,1,2,⋯(53)μ1ε,I=μ0ε,I−L−11sϑL∂3μ0∂ε3+32ω0∂3ω0∂ε3+92∂ω0∂ε∂2ω0∂ε2−6μ0∂μ0∂ε−6μ0ω0∂ω0∂ε−32∂μ0∂εω02,(54)ω1ε,I=ω0ε,I−L−11sϑL∂3ω0∂ε3−6∂μ0∂εω0−6μ0∂ω0∂ε−152∂ω0∂εω02,(55)μ1ε,I=18λ21−4sech2λε2−2λ5csch3λεsech4λε/2Iρ/ρϑΓϑ+1,(56)ω1ε,I=λsechλε2+λ4csch2λεsech3λε/2Iρ/ρϑΓϑ+1,(57)μ2ε,I=μ1ε,I−L−11sϑL∂3μ1∂ε3+32ω1∂3ω1∂ε3+92∂ω1∂ε∂2ω1∂ε2−6μ1∂μ1∂ε−6μ1ω1∂ω1∂ε−32∂μ1∂εω12,(58)ω2ε,I=ω1ε,I−L−11sϑL∂3ω1∂ε3−6∂μ1∂εω1−6μ1∂ω1∂ε−152∂ω1∂εω12,(59)μ2ε,I=18λ21−4sech2λε2−2λ5csch3λεsech4λε/2Iρ/ρϑΓϑ+1−λ8−2+coshλεsech4λε/2Iρ/ρ2ϑ16Γ2ϑ+1,(60)ω2ε,I=λsechλε2+λ4csch2λεsech3λε/2Iρ/ρϑΓϑ+1+λ7−3+coshλεsech3λε/2Iρ/ρ2ϑ32Γ2ϑ+1,(61)μ3ε,I=μ2ε,I−L−11sϑL∂3μ2∂ε3+32ω2∂3ω2∂ε3+92∂ω2∂ε∂2ω2∂ε2−6μ2∂μ2∂ε−6μ2ω2∂ω2∂ε−32∂μ2∂εω22,(62)ω3ε,I=ω2ε,I−L−11sϑL∂3ω2∂ε3−6∂μ2∂εω2−6μ2∂ω2∂ε−152∂ω2∂εω22,(63)μ3ε,I=18λ21−4sech2λε2−2λ5csch3λεsech4λε/2Iρ/ρϑΓϑ+1−λ8−2+coshλεsech4λε/2Iρ/ρ2ϑ16Γ2ϑ+1−11024Γϑ+12Γ3ϑ+1λ112−165+28coshλεcosh2λεΓϑ+12++343−20coshλε+cosh2λεΓ2ϑ+1sech6λε2tanhλε2Iρρ3ϑ.(64)ω3ε,I=λsechλε2+λ4csch2λεsech3λε/2Iρ/ρϑΓϑ+1+λ7−3+coshλεsech3λε/2Iρ/ρ2ϑ32Γ2ϑ+1+1512Γϑ+12Γ3ϑ+1λ10147−92coshλεΓϑ+12+cosh2λε+12−7+3coshλεΓ2ϑ+1sech5λε2tanhλε2Iρρ3ϑ.(65)με,I=18λ21−4sech2λε2−2λ5csch3λεsech4λε/2Iρ/ρϑΓϑ+1−λ8−2+coshλεsech4λε/2Iρ/ρ2ϑ16Γ2ϑ+1−11024Γϑ+12Γ3ϑ+1λ112−165+28coshλε+cosh2λεΓϑ+12+343−20coshλε+cosh2λεΓ2ϑ+1sech6λε2tanhλε2Iρρ3ϑ−⋯.(66)ωε,I=λsechλε2+λ4csch2λεsech3λε/2Iρ/ρϑΓϑ+1+λ7−3+coshλεsech3λε/2Iρ/ρ2ϑ32Γ2ϑ+1+1512Γϑ+12Γ3ϑ+1λ10147−92coshλε+cosh2λεΓϑ+12+12−7+3coshλεΓ2ϑ+1sech5λε2tanhλε2Iρρ3ϑ+⋯.The exact solution of equation (8) at ϑ=1,
(67)με,I=18λ21−4sech2λ2ε+12λ2I,ωε,I=λsechλ2ε+12λ2I.In Figure1, the first figure shows the ρ-Laplace decomposition method and ρ-Laplace variational transform method and exact solutions graphs of the given problem at με,I and ωε,I. From the given graphs, it can be observed that both the exact, ρ-Laplace decomposition method, and ρ-Laplace variational transform method solution are in strong agreement with each other. In Figure 2, the exact and approximate solution of με,I at ϑ=1. Figure 3, different fractional-order of ϑ=0.8 and 0.6 of the model. Similarly, in Figure 4, the exact and approximate solution of ωε,I at ϑ=1. Figure 5, different fractional-order of ϑ=0.8 and 0.6 of the model. It is investigated that results of fractional-order problem are convergent to an integer-order result as fractional-order analysis to integer-order. The same phenomenon of convergence of fractional-order solutions towards integral-order solutions is observed.Figure 1
Exact and approximate solution ofμε,I and ωε,I at ϑ=1.Figure 2
Exact and approximate solution ofμε,I at ϑ=1.Figure 3
The different fractional-order ofϑ=0.8 and 0.6 of the model.Figure 4
Exact and approximate solution ofωε,I at ϑ=1.Figure 5
The different fractional-order ofϑ=0.8 and 0.6 of the model.


## 6. Conclusion

In this paper, two analytical methods are applied to solve fractional-order nonlinear Jaulent-Miodek equation. The analytical results of the equations are calculated to show the reliability and validity of the suggested techniques. Figures of the results are plotted to show the closed contact between the achieved and exact solutions. Moreover, the proposed methods provide easily computable components for the series-form solutions. It has been discovered that the results obtained in series form have a better rate of convergence to the exact solutions. Finally, the proposed strategies are found to be a sophisticated strategy for solving various fractional-order nonlinear partial differential equations.



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