# A Comparative Analysis of the Fractional-Order Coupled Korteweg–De Vries Equations with the Mittag–Leffler Law

Authors: Noufe H. Aljahdaly; Ali Akgül; Rasool Shah; Ibrahim Mahariq; Jeevan Kafle
Journal: Journal of Mathematics
 (2022)

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## Abstract

This article applies efficient methods, namely, modified decomposition method and new iterative transformation method, to analyze a nonlinear system of Korteweg–de Vries equations with the Atangana–Baleanu fractional derivative. The nonlinear fractional coupled systems investigated in this current analysis are the system of Korteweg–de Vries and the modified system of Korteweg–de Vries equations applied as a model in nonlinear physical phenomena arising in chemistry, biology, physics, and applied sciences. Approximate analytical results are represented in the form of a series with straightforward components, and some aspects showed an appropriate dependence on the values of the fractional-order derivatives. The convergence and uniqueness analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. The series result achieved applying this technique is proved to be accurate and reliable with minimal calculations. The numerical simulations for obtained solutions are discussed for different values of the fractional order.


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## Body

## 1. Introduction

Many researchers have been working on various aspects of fractional derivatives in recent years. Caputo and Fabrizio modified the existing Caputo derivative to develop the Caputo–Fabrizio fractional derivative [1–5] based on a nonsingular kernel. Because of its advantages, numerous researchers utilized this operator to investigate various types of fractional-order partial differential equations [6–9]. To address this issue, Atangana and Baleanu proposed a new fractional operator called the Atangana–Baleanu derivative, which combines Caputo and Riemann–Liouville derivatives. Because of the existence of the Mittag–Leffler kernel, which is a generalization of the exponential kernel, this new Atangana–Baleanu derivative has a long memory. Moreover, the Atangana–Baleanu operator outperforms other operators, and different scientific models have been successfully solved. Many advances have been made in fractional calculus over the last few years by borrowing ideas from classical calculus, but it does not remain easy. Scholars have the main concern to obtain a numerical solution; for this, numerous efficient methodologies have been constructed for fractional differential equations, such as the Adomian decomposition transform method [10], variational iteration transform method [11, 12], optimal homotopy asymptotic method [13], and homotopy perturbation method [14, 15].Korteweg and de Vries introduced the Korteweg–de Vries equation in 1895 to model shallow water waves in a canal [16]. The suggested system Korteweg–de Vries equations play a crucial role in diverse engineering and applied sciences such as plasma physics, water waves, hydrodynamics, and theory of the quantum field. The Korteweg–de Vries equations are usually investigated in the analysis of nonlinear dispersive waves [17]. They define the interactions among two long waves with various dispersion relations. Many researchers have been interested in these schemes, and a lot of works have been done. For example, Ghoreishi et al. applied the homotopy analysis method to achieve numerical results of a modified system of Korteweg–de Vries equations [18]. Kaya and Inan in [19] achieved traveling wave results of the system of Korteweg–de Vries and modified system of Korteweg–de Vries equations. The fractional-order system of Korteweg–de Vries equations is defined as follows:(1)∂γU∂ℑγ=−ρ∂3U∂φ3−6ρU∂U∂φ+6V∂V∂φ,∂γV∂ℑγ=−ρ∂3V∂φ3−3ϑU∂V∂φ,ℑ>0,0<γ≤1,where γ is the fractional-order derivative of Uφ,ℑ and Vφ,ℑ, ϑ, and ρ are constants, respectively. The functions Uφ,ℑ and Vφ,ℑ are considered as important functions of time and space, disappearing for ℑ and φ, respectively. The other method eliminates to the conventional coupled Korteweg–de Vries equations since ρ=ϑ=1 is implemented.A classic model in this hierarchy is the modified coupled Korteweg–de Vries system. The following nonlinear partial differential equations govern this model [20]:(2)∂γU∂ℑγ=12∂3U∂ℑ3−3U2∂U∂φ+32W∂2V∂φ2+3∂V∂φ∂W∂φ+32V∂2W∂φ2+3yx∂U∂φ+3zx∂V∂φ+3zy∂W∂φ,∂γV∂ℑγ=−∂3V∂φ3−3∂U∂φ∂V∂φ−3V∂2U∂φ2−3V2∂W∂φ+6zy∂U∂φ+3U2∂V∂φ,∂γW∂ℑγ=−∂3W∂φ3−3∂U∂φ∂W∂φ−3W∂2U∂φ2−3W2∂V∂φ+6zx∂U∂φ+3U2∂W∂φ,ℑ>0,0<γ≤1.The modified Korteweg–de Vries equation in its standard type is simplified by the modified couple Korteweg–de Vries equation (2), with V=W=0. Korteweg–de Vries models are a source of nonevolution equations with a wide range of implementations in science and engineering. The Korteweg–de Vries models, for instance, generate ion-acoustic result in fluid mechanics [21, 22]. Long waves characterise geophysical fluid dynamics in shallow and deep oceans [23, 24]. Various studies have suggested numerous systems to overcome the fractional-order Korteweg–de Vries equation employing various methodologies, such as the differential transform method [25], Adomian decomposition method [26], natural decomposition method [27], homotopy analysis method [28], Elzaki projected differential transform method [29], variational iteration method [30], new iterative method [31], modified tanh technique [32], and Lie symmetry analysis [33]. Analogously, same solutions for (2) have been suggested by Inc and Cavlak [34], Fan [35], Lin et al. [36], Inc et al. [37], and Ghoreishi et al. [18].Daftardar-Gejji and Jafari [38] proposed an innovative iterative method of solving functional equations with approximation solutions. The new iterative approach is constructed on the justification of disappearing the nonlinear functions is identified as the iterative transformation technique [39]. This procedure is quick and accurate, and it avoids the utilization of complicated integrals, unconditioned matrix, and infinite series forms. This technique does not require any expressive parameters for the model. Numerous researchers have analyzed new iterative transformation methods to solve partial differential equations, such as the Fornberg–Whitham equation [40], KdV equation [31], and Klein–Gordon equation [41].The Adomian decomposition method was firstly introduced by Adomian in 1980 and implemented by several investigators. In recent decades, numerous researchers have investigated the solutions of integral and differential equations by different techniques with the mixed Laplace transform. The Adomian decomposition method was modified with many integral transformations, such as Laplace,ρ-Laplace, Elzaki, Aboodh, and Mohand. Modification of Laplace Adomian decomposition method for solving nonlinear Volterra integral and integro-differential equations based on Newton Raphson formula [42] for solving nonlinear integrodifferential and Volterra integral equations based on the Newton–Raphson method, discrete Adomian decomposition technique [43] applied for investigating the fractional-order Navier–Stokes model, Laplace–Adomian decomposition method [44] study of implicit-impulsive differential equations involving Caputo-Fabrizio fractional derivative.


## 2. Basic Definitions

Definition 1.
The fractional-order Caputo derivative is defined by(3)DLCℑγfℑ=1n−γ∫0ℑℑ−kn−γ−1fnkdk,wheren<γ≤n+1.Definition 2.
The Laplace transformation connected with fractional Caputo derivativeDLCℑγfℑ is expressed by(4)LDLCℑγfℑs=1sn−γsnLfx,ℑs−sn−1fx,0−⋯−fn−1x,0.Definition 3.
In the Caputo sense, the Atangana–Baleanu derivative is defined as(5)DABCℑγfℑ=Aγ1−γ∫aℑf′kEγ−γ1−γ1−kγdk,where Aγ is a normalization function such that A0=A1=1,f∈H1a,b,b>a,γ∈0,1, and Eγ represents the Mittag–Leffler function.Definition 4.
The Atangana–Baleanu derivative in the Riemann–Liouville sense is defined as(6)DABCℑγfℑ=Aγ1−γddℑ∫aℑfkEγ−γ1−γ1−kγdk.Definition 5.
The Laplace transform connected with the Atangana–Baleanu operator is defined as(7)DABℑγfℑs=AγsγLfℑs−sγ−1f01−γsγ+γ/1−γ.Definition 6.
Consider0<γ<1, and f is a function of γ; then, the fractional-order integral operator of γ is given as(8)IABCℑγfℑ=1−γAγfℑ+γAγΓγ∫aℑfkℑ−kγ−1dk.


## 3. The General Implementation of the Modified Decomposition Method

Suppose the nonlinear fractional partial differential equations(9)DℑγUφ,ℑ+LUφ,ℑ+NUφ,ℑ=ℋφ,ℑ,ℑ>0,0<γ≤1,with the condition(10)Uφ,0=Gφ,where Dℑγ=∂γUφ,ℑ/∂ℑγ show the fractional-order Caputo derivative operator with 0<γ≤1, while ℒ is linear, N are nonlinear functions, and ℋφ,ℑ defines the source term.Applying the Laplace transformation to (9), we get(11)LDℑγUφ,ℑ+LUφ,ℑ+NUφ,ℑ=Lℋφ,ℑ.Taking the Laplace transformation differentiation, we find(12)vγvγ1−γ+γUv,ω=∑κ=0j−11vγ−κ−1Uκ0+LLUφ,ℑ+NUφ,ℑ+Lℋφ,ℑ.The inverse Laplace transformation of (12) gives(13)Uφ,ℑ=L−1∑κ=0j−11vγ−κ−1Uκ0+vγ1−γ+γvγLℋφ,ℑ−L−1vγ1−γ+γvγLLUφ,ℑ+NUφ,ℑ.The adomain decomposition method series form solution is defined as(14)Uφ,ℑ=∑j=0∞Ujφ,ℑ.Thus, the nonlinear functionNφ,ℑ can be calculated by the Adomian polynomials defined as(15)NUφ,ℑ=∑j=0∞A˜jU0,U1,…,j=0,1,…,where(16)A˜jU0,U1,…=1j!djdλjN∑J=0∞λJUJλ=0,j>0.Putting (14) and (15) into (13), we have(17)∑j=0∞Ujφ,ℑ=Gφ+G˜φ−L−1vγ1−γ+γvγLLUφ,ℑ+∑j=0∞A˜j.Lastly, the iterative methodology for (17) is achieved as(18)U0φ,ℑ=Gφ+G˜φ,j=0,Uj+1φ,ℑ=−L−1vγ1−γ+γvγLLUφ,ℑ+∑j=0∞A˜j,j≥1.


## 4. The General Discussion of the New Iterative Transformation Method

Let us assume the following general fractional partial differential equation(19)DℑγUφ,ℑ+LUφ,ℑ+NUφ,ℑ=ℋφ,ℑ,ℑ>0,j−1<γ≤j,j∈ℕ,with the condition(20)Uκφ,0=Gκφ,κ=0,1,2,…,j−1,where ℒ and N are linear and nonlinear terms and ℋφ,ℑ shows the source term.Using the Laplace transformation to (19), we get(21)LDℑγUφ,ℑ+LUφ,ℑ+NUφ,ℑ=Lℋφ,ℑ.Taking the Laplace transformation differentiation property, we get(22)vγvγ1−γ+γUv,ω=∑κ=0j−11vγ−κ−1Uκ0+LLUφ,ℑ+NUφ,ℑ+Lℋφ,ℑ.The inverse Laplace transformation of (22) gives(23)Uφ,ℑ=L−1∑κ=0j−11vγ−κ−1Uκ0+vγ1−γ+γvγLℋφ,ℑ−L−1vγ1−γ+γvγLLUφ,ℑ+NUφ,ℑ.From the iterative connection, we achieve(24)Uφ,ℑ=∑j=0∞Ujφ,ℑ.Also, the linear operator isℒ; therefore,(25)ℒ∑j=0∞Ujφ,ℑ=∑j=0∞ℒUjφ,ℑ,and N defines the nonlinear term as in [38].(26)N∑j=0∞Ujφ,ℑ=NU0φ,ℑ+∑j=0∞N∑k=0∞Uκφ,ℑ−N∑k=1∞Uκφ,ℑ=NU0+∑k=1∞Dj,where Dj=N∑κ=0jUκ−N∑κ=0j−1Uκ.By putting (24), (25), and (??) into (23), we obtain(27)∑j=0∞Ujφ,ℑL−1∑κ=0j−11vγ−κ−1Uκ0+vγ1−γ+γvγLℋφ,ℑ−L−1vγ1−γ+γvγLℒ∑κ=0∞Uκφ,ℑ+NU0+∑κ=1jDj.As a result, we determine the next iteration(28)U0φ,ℑ=L−1∑κ=0j−11vγ−κ−1Uκ0+vγ1−γ+γvγLℋφ,ℑ,U1φ,ℑ=−L−1vγ1−γ+γvγLℒU0φ,ℑ+NU0φ,ℑ,⋮Uj+1φ,ℑ=−L−1vγ1−γ+γvγLℒUjφ,ℑ+Dj,m≥1.Finally, (19) and (20) yield the j-term result in the series form, defined as(29)Uφ,ℑ≈U0φ,ℑ+U1φ,ℑ+U2φ,ℑ+⋯+Ujφ,ℑ,j∈ℕ.


## 5. Uniqueness and Existence Solutions for the Modified Decomposition Method

Theorem 1 (uniqueness theorem).
The unique result of equation (9) provide space whenever 0<ε<1, where ε=L˘1+L˘2+L˘31−γ+γℑγ/Γγ+1.Proof.
Assume thatJ=CI,. represents all continuous mappings on the Banach space, defined on I=0,T having the norm .. For this, we introduce a mapping W:M↦M, and we have(30)Un+1φ,ℑ=Uφ,ℑ+L−1vγ1−γ+γvγLℒUnφ,ℑ+ℛUnφ,ℑ+NUnφ,ℑ,n≥0,when ℒUφ,ℑ≡∂3Uφ,ℑ/∂φ2 and ℛUφ,ℑ≡∂Uφ,ℑ/∂φ. Suppose that ℒUφ,ℑ and ℳUφ,ℑ are also Lipschitzian with ℛz−ℛU˘<L˘1U−U˘ and LU−ℒU˘<L˘2U−U˘ where L˘1 and L˘2 are Lipschitz constants, respectively, and U,U˘ are various values of the mapping.(31)WU−WU˘=maxℑ∈IL−1vγ1−γ+γvγLℒUφ,ℑ+ℛUφ,ℑ+NUφ,ℑ−L−1vγ1−γ+γvγLℒU˘φ,ℑ+ℛU˘φ,ℑ+NU˘φ,ℑ≤maxℑ∈IL−1vγ1−γ+γvγLℒUφ,ℑ−ℒU˘φ,ℑ+L−1vγ1−γ+γvγLℛUφ,ℑ−ℛU˘φ,ℑ+L−1vγ1−γ+γvγLNUφ,ℑ−NU˘φ,ℑ≤maxℑ∈IL˘1L−1vγ1−γ+γvγLUφ,ℑ−U˘φ,ℑ+L˘2L−1vγ1−γ+γvγLUφ,ℑ−U˘φ,ℑ+L˘3L−1vγ1−γ+γvγLUφ,ℑ−U˘φ,ℑ≤maxℑ∈IL˘1+L˘2+L˘3L−1vγ1−γ+γvγLUφ,ℑ−U˘φ,ℑ≤L˘1+L˘2+L˘3L−1vγ1−γ+γvγLUφ,ℑ−U˘φ,ℑ=L˘1+L˘2+L˘31−γ+γℑγΓγ+1Uφ,ℑ−U˘φ,ℑ.
The mapping is a contraction under the assumption0<ε<1. As a result of the Banach contraction fixed point theorem, there is a unique solution to (9). As a result, the proof is complete.Theorem 2 (convergence analysis).
The solution general form of (9) will be convergent.Proof.
SupposeS^n is the nth partial sum; that is, W^n=∑j=0nUjφ,ℑ. Firstly, we define that W^n is a Banach space Cauchy sequence in M. Using into consideration of Adomian polynomials, we achieve(32)R¯W^n=H˘n+∑p=0n−1H˘p,N¯W^n=H˘n+∑c=0n−1H˘c.
Now,(33)W^n−W^q=maxℑ∈IW^n−W^q=maxℑ∈I∑j=q+1nU˘φ,ℑ,j=1,2,3,…≤maxℑ∈IL−1vγ1−γ+γvγL∑j=q+1nℒUn−1φ,ℑ+L−1vγ1−γ+γvγL∑j=q+1nℛUn−1φ,ℑ+L−1vγ1−γ+γvγL∑j=q+1nH˘n−1φ,ℑ=maxℑ∈IL−1vγ1−γ+γvγL∑j=qn−1ℒUnφ,ℑ+L−1vγ1−γ+γvγL∑j=qn−1ℛUnφ,ℑ+L−1vγ1−γ+γvγL∑j=qn−1H˘nφ,ℑ≤maxℑ∈IL−1vγ1−γ+γvγL∑j=qn−1ℒW^n−1−ℒW^q−1+L−1vγ1−γ+γvγL∑j=qn−1ℛW^n−1−ℛW^q−1+L−1vγ1−γ+γvγL∑j=qn−1NW^n−1−NW^q−1≤maxℑ∈IL−1vγ1−γ+γvγLℒW^n−1−ℒW^q−1+L−1vγ1−γ+γvγLℛW^n−1−ℛW^q−1+L−1vγ1−γ+γvγLNW^n−1−NW^q−1≤L˘1maxℑ∈IL−1vγ1−γ+γvγLW^n−1−W^q−1+L˘2maxℑ∈Ivγ1−γ+γvγLW^n−1−W^q−1+L˘3maxℑ∈Ivγ1−γ+γvγLW^n−1−W^q−1=L˘1+L˘2+L˘31−γ+γℑγΓγ+1W^n−1−W^q−1.
Considern=q+1; then,(34)W^q+1−W^q≤εW^q−W^q−1≤ε2W^q−1−W^q−2≤⋯≤εqW^1−W^0,where L˘1+L˘2+L˘3ℑγ−1/γ!. Similarly, we have the triangular inequality(35)W^n−W^q≤W^q+1−W^q+W^q+2−W^q+1+⋯+W^n−W^n−1≤εq+εq+1+⋯+εn−1W^1−W^0≤εq1−εn−qεU1,and since 0<ε<1, we get 1−εn−q<1; then,(36)W^n−W^q≤εq1−εmaxℑ∈IU1.
However,U1<∞ (since Uφ,ℑ is bounded). Thus, as q↦∞, W^n−W^q↦0. Hence, W^1 is a Cauchy sequence in K. As a solution, the series ∑n=0∞Un converges, and this completes the proof.Theorem 3 (error estimate).
The maximum absolute truncation error of series solution (9) to (??) is computed as(37)maxℑ∈IUφ,ℑ∑n=1qUnφ,ℑ≤εq1−εmaxℑ∈IU1.


## 6. Numerical Results

This section describes several test examples by applying two novel techniques, modified decomposition technique and new iterative transformation technique, via the Atangana–Baleanu derivative operator. Also, the stability and convergence of the technique are discussed.Example 1 (see [31]).
Consider the fractional-order nonlinear system of Korteweg–de Vries equation (1) with ϑ=ρ=1, with the initial conditions (38)Uφ,0=ϱ2sech2δ2+ϱφ2,Vφ,0=ρ2ϱ2sech2δ2+ϱφ2.Case I: first, we apply the modified decomposition technique for Example1.Applying the Laplace transform to (1), we get(39)vγvγ1−γ+γUφ,v−∑κ=0j−11vγ−κ−1Uκ0=L−ρ∂3U∂φ3−6ρU∂U∂φ+6V∂V∂φ,vγvγ1−γ+γVφ,v−∑κ=0j−11vγ−κ−1Vκ0=L−ρ∂3V∂φ3−3ρU∂V∂φ.In view of (38) and analytical method procedure as follows:(40)Uφ,v=1vU0φ,0+vγ1−γ+γvγL−ρ∂3U∂φ3−6ρU∂U∂φ+6V∂V∂φ,Vφ,v=1vV0φ,0+vγ1−γ+γvγL−ρ∂3V∂φ3−3ρU∂V∂φ.Using the inverse Laplace transformation, we get(41)Uφ,ℑ=L−11vUφ,0+L−1vγ1−γ+γvγL−ρ∂3U∂φ3−6ρU∂U∂φ+6V∂V∂φ,Vφ,ℑ=L−11vVφ,0+L−1vγ1−γ+γvγL−ρ∂3V∂φ3−3ρU∂V∂φ.By morality of the modified decomposition technique, we get(42)U0φ,ℑ=L−1ωvUφ,0=L−11vϱ2sech2δ2+ϱφ2=ϱ2sech2δ2+ϱφ2,V0φ,ℑ=L−1ωvVφ,0=ρ2ϱ2sech2δ2+ϱφ2,∑j=0∞Uj+1φ,ℑ=L−1vγ1−γ+γvγL−ρ∑j=0∞Uφφφj−6ρ∑j=0∞Aj+6∑j=0∞ℬj,∑j=0∞Vj+1φ,ℑ=L−1vγ1−γ+γvγL−ρ∑j=0∞Vφφφj−3ρ∑j=0∞Cj,j=0,1,2,….The Adomian polynomials’ some terms are defined as follows:(43)A0UUφ=U0U0φ,A1UUφ=U0U1φ+U1U0φ,A2UUφ=U1U2φ+U1U1φ+U2U0φ,ℬ0VVφ=V0V0φ,ℬ1VVφ=V0V1φ+V1V0φ,ℬ2VVφ=V1V2φ+V1V1φ+V2V0φ,C0UVφ=U0V0φ,C1UVφ=U0V1φ+U1V0φ,C2UVφ=U1V2φ+U1V1φ+U2V0φ.Forj=0,1,2,3,…,(44)U1φ,ℑ=L−1vγ1−γ+γvγL−ρUφφφ0−6ρA0+6ℬ0=L−1ωγ+2vγ+2ϱ5ρtanhδ2+ϱφ2sech2δ2+ϱφ2=ϱ5ρtanhδ2+ϱφ2sech2δ2+ϱφ21−γ+γℑγΓγ+1V1φ,ℑ=L−1vγ1−γ+γvγL−ρVφφφ0−3ρC0=ϱ5ρ3/22tanhδ2+ϱφ2sech2δ2+ϱφ21−γ+γℑγΓγ+1U2φ,ℑ=L−1vγ1−γ+γvγL−ρUφφφ1−6ρA1+6ℬ1=L−1ω2γ+2v2γ+2ϱ8ρ222cosh2ρ2+ϱφ2−3sech4ρ2+ϱφ2=ϱ8ρ222cosh2ρ2+ϱφ2−3sech4ρ2+ϱφ21−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1V2φ,ℑ=L−1vγ1−γ+γvγL−ρVφφφ1−3ρC1=ϱ5ρ5/2222cosh2ρ2+ϱφ2−3sech4ρ2+ϱφ21−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1⋮.The modified decomposition technique result for Example1 is shown as(45)Uφ,ℑ=U0φ,ℑ+U1φ,ℑ+U2φ,ℑ+U3φ,ℑ+⋯=ϱ2sech2δ2+ϱφ2+ϱ5ρtanhδ2+ϱφ2sech2δ2+ϱφ21−γ+γℑγΓγ+1+ϱ8ρ222cosh2ρ2+ϱφ2−3sech4ρ2+ϱφ21−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1+⋯.Similarly, we get(46)Vφ,ℑ=ρ2ϱ2sech2δ2+ϱφ2+ϱ5ρ3/22tanhδ2+ϱφ2sech2δ2+ϱφ21−γ+γℑγΓγ+1+ϱ5ρ5/2222cosh2ρ2+ϱφ2−3sech4ρ2+ϱφ21−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1+⋯.By puttingγ=1, we achieve the exact result of the system of Korteweg–de Vries equation (1):(47)Uφ,ℑ=ϱ2sech2δ2+ϱφ2−ρϱ3ℑ2,Vφ,ℑ=ρ2ϱ2sech2δ2+ϱφ2−ρϱ3ℑ2.Case II: now, we apply the new iterative transformation technique on Example1.
Using the suggested analytical method, we have(48)U0φ,ℑ=L−11vUφ,0=L−11vϱ2sech2δ2+ϱφ2=ϱ2sech2δ2+ϱφ2V0φ,ℑ=L−11vVφ,0=ρ2ϱ2ech2δ2+ϱφ2U1φ,ℑ=L−1vγ1−γ+γvγL−ρ∂3U0∂φ3−6ρU0∂U0∂φ+6V0∂V0∂φ=L−1vγ1−γ+γvγϱ5ρtanhδ2+ϱφ2sech2δ2+ϱφ2=ϱ5ρtanhδ2+ϱφ2sech2δ2+ϱφ21−γ+γℑγΓγ+1V1φ,ℑ=L−1vγ1−γ+γvγL−ρ∂3V0∂φ3−3ρU0∂V0∂φ=ϱ5ρ3/22tanhδ2+ϱφ2sech2δ2+ϱφ21−γ+γℑγΓγ+1U2φ,ℑ=L−1vγ1−γ+γvγL−ρ∂3U1∂φ3−6ρU1∂U1∂φ+6V1∂V1∂φ=L−1ω2γ+2v2γ+2ϱ8ρ222cosh2ρ2+ϱφ2−3sech4ρ2+ϱφ2=ϱ8ρ222cosh2ρ2+ϱφ2−3sech4ρ2+ϱφ21−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1V2φ,ℑ=L−1vγ1−γ+γvγL−ρ∂3V1∂φ3−3ρU1∂V1∂φ=ϱ5ρ5/2222cosh2ρ2+ϱφ2−3sech4ρ2+ϱφ21−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1⋮Unφ,ℑ=L−1vγ1−γ+γvγL−ρ∂3Uj−1∂φ3−6ρUj−1∂Uj−1∂φ+6Vj−1∂Vj−1∂φVjφ,ℑ=L−1vγ1−γ+γvγL−ρ∂3Vj−1∂φ3−3ρUj−1∂Vj−1∂φ.
The series of solutions for Example1 is expressed as(49)Uφ,ℑ=U0φ,ℑ+U1φ,ℑ+U2φ,ℑ+U3φ,ℑ+⋯Ujφ,ℑ,Vφ,ℑ=V0φ,ℑ+V1φ,ℑ+V2φ,ℑ+V3φ,ℑ+⋯Vjφ,ℑ.
Consequently, we have(50)Uφ,ℑ=ϱ2sech2δ2+ϱφ2+ϱ5ρtanhδ2+ϱφ2sech2δ2+ϱφ21−γ+γℑγΓγ+1+ϱ8ρ222cosh2ρ2+ϱφ2−3sech4ρ2+ϱφ21−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1+⋯,Vφ,ℑ=ρ2ϱ2sech2δ2+ϱφ2+ϱ5ρ3/22tanhδ2+ϱφ2sech2δ2+ϱφ21−γ+γℑγΓγ+1+ϱ5ρ5/2222cosh2ρ2+ϱφ2−3sech4ρ2+ϱφ21−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1+⋯.
By puttingγ=1, we get the exact result of the system of Korteweg–de Vries equation (1):(51)Uφ,ℑ=ϱ2sech2δ2+ϱφ2−ρϱ3ℑ2,Vφ,ℑ=ρ2ϱ2sech2δ2+ϱφ2−ρϱ3ℑ2.
In Figures1 and 2, the actual and analytical solutions of Uφ,ℑ and Vφ,ℑ are proved at δ=2, ρ=0.5, and ϱ=1. In Figures 3 and 4, the surface and two-dimensional figure for Uφ,ℑ and Vφ,ℑ for numerous fractional orders are described which demonstrate that the modified decomposition technique and new iterative transformation technique obtained series form solutions are in close contact with the analytical and the exact results. This comparison shows a strong connection among the modified decomposition method and actual solutions. Consequently, the modified decomposition technique and new iterative transformation technique are accurate innovative techniques which need less calculation time and are very simple and more flexible than the homotopy analysis technique and homotopy perturbation technique.Figure 1
The actual and analytical (MDM/NITM) result figure atUφ,ℑ of Example 1 for ϱ=1, ρ=0.5, and δ=2.Figure 2
The actual and analytical (MDM/NITM) result figure atVφ,ℑ of Example 1 for ϱ=1, ρ=0.5, and δ=2.Figure 3
Analytical investigation of figureUφ,ℑ for Example 1 for different fractional orders γ=1.0,0.8,0.6,0.4, ρ=0.5, ϱ=1, and δ=2.Figure 4
Analytical investigation of figureVφ,ℑ for Example 1 for different fractional orders γ=1.0,0.8,0.6,0.4, ρ=0.5, ϱ=1, and δ=2.Example 2 (see [31]).
Consider the fractional-order nonlinear system of Korteweg–de Vries equation given as(52)∂γU∂ℑγ=−∂V∂φ−12∂U2∂φ,∂γV∂ℑγ=−∂U∂φ−∂3U∂φ3−∂zy∂φ,ℑ>0,0<γ≤1,with the conditions(53)Uφ,0=ρtanhϱ2+ρφ2+1,Vφ,0=ρ22sech2ϱ2+ρφ2−1.Case I: first, we apply the modified decomposition technique for Example2.Applying the Laplace transform to (52), we find(54)vγvγ1−γ+γUφ,v−∑κ=0j−11vγ−κ−1Uκ0=L−∂V∂φ−12∂U2∂φ,vγvγ1−γ+γVφ,v−∑κ=0j−11vγ−κ−1Vκ0=L−∂U∂φ−∂3U∂φ3−∂zy∂φ.In view of (29) and straightforward approximate achieve(55)Uφ,v=1vU0φ,0+vγ1−γ+γvγL−∂V∂φ−12∂U2∂φ,Vφ,v=1vV0φ,0+vγ1−γ+γvγL−∂U∂φ−∂3U∂φ3−∂zy∂φ.Using the inverse Laplace transformation, we get(56)Uφ,ℑ=L−11vUφ,0+L−1vγ1−γ+γvγL−∂V∂φ−12∂U2∂φ,Vφ,ℑ=L−11vVφ,0+L−1vγ1−γ+γvγL−∂U∂φ−∂3U∂φ3−∂zy∂φ.By the consequence of the modified decomposition technique, we get(57)U0φ,ℑ=L−11vUφ,0=L−11vρtanhϱ2+ρφ2+1=ρtanhϱ2+ρφ2+1,V0φ,ℑ=L−11vVφ,0=ρ22sech2ϱ2+ρφ2−1.It follows that(58)∑j=0∞Uj+1φ,ℑ=L−1vγ1−γ+γvγL−ρ∑j=0∞Vφj−12∑j=0∞Dj,∑j=0∞Vj+1φ,ℑ=L−1vγ1−γ+γvγL−∑j=0∞Uφj−∑j=0∞Vφφφj−∑j=0∞UVφj,j=0,1,2,….The Adomian polynomials’ some terms are expressed as(59)D0U2=U02,D1U2=2U0U1,D2U2=2U0U2+U12.Forj=0,1,2,…,(60)U1φ,ℑ=L−1vγ1−γ+γvγL−ρVφ0−12D0=−ρ22L−1ωγ+2vγ+2sech2ϱ2+ρφ2=−ρ22sech2ϱ2+ρφ21−γ+γℑγΓγ+1V1φ,ℑ=L−1vγ1−γ+γvγL−Uφ0−Vφφφ0−zyφ0=ρ32sinhϱ2+ρφ2sech3ϱ2+ρφ21−γ+γℑγΓγ+1U2φ,ℑ=L−1vγ1−γ+γvγL−ρVφ1−12D1=L−1−ρ54sech2ϱ2+ρφ2+3ρ54ω2γ+2v2γ+2sinh2ϱ2+ρφ2sech4ϱ2+ρφ2+ρ74L−1Γ2γ+1Γ2γ+1ω3γ+2v3γ+2sinhϱ2+ρφ2sech5ϱ2+ρφ2=−ρ54sech2ϱ2+ρφ2+3ρ54sinh2ϱ2+ρφ2sech4ϱ2+ρφ21−γ+γℑγΓγ+1+21−γγℑγΓγ+1+ρ74sinhϱ2+ρφ2sech5ϱ2+ρφ2Γ2γ+1ℑ3γΓ2γ+1Γ3γ+1V2φ,ℑ=L−1vγ1−γ+γvγ−Uφ1−Vφφφ1−zyφ1=ρ642cosh2ϱ2+ρφ2−3sech4ϱ2+ρφ21−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1⋮.The modified decomposition technique result for Example2 is represented as(61)Uφ,ℑ=U0φ,ℑ+U1φ,ℑ+U2φ,ℑ+⋯,=ρtanhϱ2+ρφ2+1−ρ22sech2ϱ2+ρφ21−γ+γℑγΓγ+1+−ρ54sech2ϱ2+ρφ2+3ρ54sinh2ϱ2+ρφ2sech4ϱ2+ρφ21−γ+γℑγΓγ+1+21−γγℑγΓγ+1+ρ74sinhϱ2+ρφ2sech5ϱ2+ρφ2Γ2γ+1ℑ3γΓ2γ+1Γ3γ+1+⋯.Consequently, we get(62)Vφ,ℑ=−1+ρ22sech2ϱ2+ρφ2+ρ32sinhϱ2+ρφ2sech3ϱ2+ρφ21−γ+γℑγΓγ+1+ρ642cosh2ϱ2+ρφ2−3sech4ϱ2+ρφ21−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1+⋯.By puttingγ=1, we achieve the exact result of the system of Korteweg–de Vries equation:(63)Uφ,ℑ=ρtanhϱ2+ρφ2−ρ2ℑ2+1,Vφ,ℑ=ρ22sech2ϱ2+ρφ2−ρ2ℑ2−1.Case II: now, we implement the new iterative transformation technique on Example2.
By using the suggested analytical technique, we get(64)U0φ,ℑ=L−11vUφ,0=L−11vρtanhϱ2+ρφ2+1=ρtanhϱ2+ρφ2+1V0φ,ℑ=L−11vVφ,0=ρ22sech2ϱ2+ρφ2−1U1φ,ℑ=L−1vγ1−γ+γvγL−∂V0∂φ−12∂U02∂φ=−ρ22L−1ωγ+2vγ+2sech2ϱ2+ρφ2=−ρ22sech2ϱ2+ρφ21−γ+γℑγΓγ+1V1φ,ℑ=L−1vγ1−γ+γvγL−∂U0∂φ−∂3U0∂φ3−∂U0V0∂φ=ρ32sinhϱ2+ρφ2sech3ϱ2+ρφ21−γ+γℑγΓγ+1U2φ,ℑ=L−1vγ1−γ+γvγL−∂V1∂φ−12∂U12∂φ=L−1−ρ54sech2ϱ2+ρφ2+3ρ54ω2γ+2v2γ+2sinh2ϱ2+ρφ2sech4ϱ2+ρφ2+ρ74L−1Γ2γ+1Γ2γ+1ω3γ+2v3γ+2sinhϱ2+ρφ2sech5ϱ2+ρφ2=−ρ54sech2ϱ2+ρφ2+3ρ54sinh2ϱ2+ρφ2sech4ϱ2+ρφ21−γ+γℑγΓγ+1+21−γγℑγΓγ+1+ρ74sinhϱ2+ρφ2sech5ϱ2+ρφ2Γ2γ+1ℑ3γΓ2γ+1Γ3γ+1V2φ,ℑ=L−1vγ1−γ+γvγ−∂U1∂φ−∂3U1∂φ3−∂U1V1∂φ=ρ642cosh2ϱ2+ρφ2−3sech4ϱ2+ρφ21−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1⋮Ujφ,ℑ=L−1vγ1−γ+γvγL−∂Vj−1∂φ−12∂Uj−12∂φVjφ,ℑ=L−1vγ1−γ+γvγ−∂Uj−1∂φ−∂3Uj−1∂φ3−∂Uj−1Vj−1∂φ.
The series of results for Example2 is expressed as(65)Uφ,ℑ=U0φ,ℑ+U1φ,ℑ+U2φ,ℑ+⋯Ujφ,ℑ,Vφ,ℑ=V0φ,ℑ+V1φ,ℑ+V2φ,ℑ+⋯Vjφ,ℑ.
Consequently, we have(66)Uφ,ℑ=ρtanhϱ2+ρφ2+1−ρ22sech2ϱ2+ρφ21−γ+γℑγΓγ+1+−ρ54sech2ϱ2+ρφ2+3ρ54sinh2ϱ2+ρφ2sech4ϱ2+ρφ21−γ+γℑγΓγ+1+21−γγℑγΓγ+1+ρ74sinhϱ2+ρφ2sech5ϱ2+ρφ2Γ2γ+1ℑ3γΓ2γ+1Γ3γ+1+⋯Vφ,ℑ=−1+ρ22sech2ϱ2+ρφ2+ρ32sinhϱ2+ρφ2sech3ϱ2+ρφ21−γ+γℑγΓγ+1ρ642cosh2ϱ2+ρφ2−3sech4ϱ2+ρφ21−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1+⋯.
By puttingγ=1, we obtain the actual result of the system of Korteweg–de Vries equation (??):(67)Uφ,ℑ=ρtanhϱ2+ρφ2−ρ2ℑ2+1,Vφ,ℑ=ρ22sech2ϱ2+ρφ2−ρ2ℑ2−1.
In Figures5 and 6, the actual and analytical solutions of Uφ,ℑ and Vφ,ℑ are proved at δ=2, ρ=0.5, and ϱ=1. In Figures 7 and 8, the surface and two-dimensional figure for Uφ,ℑ and Vφ,ℑ for numerous fractional orders are described which demonstrate that the modified decomposition technique and new iterative transformation technique approximated obtained results are in close contact with the analytical and the exact results. This comparison shows a strong connection among the modified decomposition method and actual solutions. Consequently, the modified decomposition technique and new iterative transformation technique are accurate innovative techniques which need less calculation time and are very simple and more flexible than the homotopy analysis technique and homotopy perturbation technique.Figure 5
The analytical and exact (MDM/NITM) solution plot atUφ,ℑ of Example 2 for ρ=0.5, ϱ=1, and δ=2.Figure 6
The analytical and exact (MDM/NITM) solution plot atVφ,ℑ of Example 2 for ρ=0.5, ϱ=1, and δ=2.Figure 7
Mathematical analysis of the plot ofVφ,ℑ for Example 2 for different fractional orders γ=1.0,0.8,0.6,0.4, ρ=0.5, ϱ=1, and δ=2.Figure 8
Mathematical analysis of the plot ofVφ,ℑ for Example 2 for different fractional orders γ=1.0,0.8,0.6,0.4ρ=0.5, ϱ=1, and δ=2.Example 3.
(see [31]). Consider the fractional-order nonlinear system of modified Korteweg–de Vries equations given as (2) with the conditions(68)Uφ,0=2+tanhφ2,Vφ,0=2−tanhφ4,Wφ,0=2−tanhφ.Case I: first, we apply the modified decomposition technique for Example3.Using the Laplace transformation to (2), we have(69)vγvγ1−γ+γUφ,v−∑κ=0j−11vγ−κ−1Uκ0=L12∂3U∂ℑ3−3U2∂U∂φ+32W∂2V∂φ2+3∂V∂φ∂W∂φ+32V∂2W∂φ2+3yx∂U∂φ+3zx∂V∂φ+3zy∂W∂φ,vγvγ1−γ+γVφ,v−∑κ=0j−11vγ−κ−1Vκ0=L−∂3V∂φ3−3∂U∂φ∂V∂φ−3V∂2U∂φ2−3V2∂W∂φ+6zy∂U∂φ+3U2∂V∂φ,vγvγ1−γ+γWφ,v−∑κ=0j−11vγ−κ−1Wκ0=L−∂3W∂φ3−3∂U∂φ∂W∂φ−3W∂2U∂φ2−3V2∂W∂φ+6zy∂U∂φ+3U2∂V∂φ.In view of (68) and straightforward calculations,(70)Uφ,v=1vU0φ,0+vγ1−γ+γvγL12∂3U∂ℑ3−3U2∂U∂φ+32W∂2V∂φ2+3∂V∂φ∂W∂φ+32V∂2W∂φ2+3yx∂U∂φ+3zx∂V∂φ+3zy∂W∂φ,Vφ,v=1vV0φ,0+vγ1−γ+γvγL−∂3V∂φ3−3∂U∂φ∂V∂φ−3V∂2U∂φ2−3V2∂W∂φ+6zy∂U∂φ+3U2∂V∂φ,Wφ,v=1vW0φ,0+vγ1−γ+γvγL−∂3W∂φ3−3∂U∂φ∂W∂φ−3W∂2U∂φ2−3W2∂V∂φ+6zx∂U∂φ+3U2∂W∂φ.Applying the Laplace transform, we have(71)Uφ,ℑ=L−11vUφ,0+L−1vγ1−γ+γvγL12∂3U∂ℑ3−3U2∂U∂φ+32W∂2V∂φ2+3∂V∂φ∂W∂φ+32V∂2W∂φ2+3yx∂U∂φ+3zx∂V∂φ+3zy∂W∂φ,Vφ,ℑ=L−11vVφ,0+L−1vγ1−γ+γvγL−∂3V∂φ3−3∂U∂φ∂V∂φ−3V∂2U∂φ2−3V2∂W∂φ+6zy∂U∂φ+3U2∂V∂φ,Wφ,ℑ=L−11vWφ,0+L−1vγ1−γ+γvγL−∂3W∂φ3−3∂U∂φ∂W∂φ−3W∂2U∂φ2−3W2∂V∂φ+6zx∂U∂φ+3U2∂W∂φ.By the consequence of the modified decomposition technique, we get(72)U0φ,ℑ=L−11vUφ,0=12L−11v2+tanhφ=122+tanhφ,V0φ,ℑ=L−11vVφ,0=142−tanhφ,W0φ,ℑ=L−11vWφ,0=2−tanhφ.It follows that(73)∑j=0∞Uj+1φ,ℑ=L−1vγ1−γ+γvγL12∑j=0∞Uφφφj−3∑j=0∞Ej+32∑j=0∞ℱj+3∑j=0∞Gj+32∑j=0∞ℋj+3∑j=0∞Ij+3∑j=0∞Jj+3∑j=0∞Kj,j=0,1,2,…,∑j=0∞Vj+1φ,ℑ=L−1vγ1−γ+γvγL−∑j=0∞Vφφφj−3∑j=0∞ℳj−3∑j=0∞Nj−3∑j=0∞Oj+6∑j=0∞xj+3∑j=0∞Qj,∑j=0∞Wj+1φ,ℑ=L−1vγ1−γ+γvγL−∑j=0∞Wφφφj−3∑j=0∞ℛj−3∑j=0∞Wj−3∑j=0∞Tj+6∑j=0∞Xj+3∑j=0∞Yj.The Adomian polynomials’ some terms are defined as(74)EJU2Uφ=U02U0φ,forJ=0,2U0U1U0φ+U02U1φ,forJ=1,2U0U2+U12U0φ+2U0U1U1φ+U02U2φ,forJ=2,ℱJWVφφ=W0V0φφ,forJ=0,W1V0φφ+W0V1φφ,forJ=1,W2V0φφ+W1V1φφ+W0V2φφ,forJ=2,GJVφWφ=V0φW0φ,forJ=0,V0φW1φ+V1φW0φ,forJ=1,V2φW0φ+V1φW1φ+V0φW2φ,forJ=2,ℋJVφWφφ=V0φW0φφ,forJ=0,V0φW1φφ+V1φW0φφ,forJ=1,V2φW0φφ+V1φW1φφ+V0φW2φφ,forJ=2,IJVzUφ=yx0U0φ,forJ=0,yx0U1φ+yx1U0φ,forJ=1,yx0U2φ+yx1U1φ+yx2U0φ,forJ=2,JJUzVφ=zx0U0φ,forJ=0,zx0U1φ+zx1U0φ,forJ=1,zx0U2φ+zx1U1φ+zx2U0φ,forJ=2,KJzyWφ=zy0U0φ,forJ=0,zy0U1φ+zy1U0φ,forJ=1,zy0U2φ+zy1U1φ+zy2U0φ,forJ=2,ℳJUφVφ=U0φV0φ,forJ=0,U0φV1φ+UφV0φ,forJ=1,U2φV0φ+U1φV1φ+U0φV1φ,forJ=2,NJVUφφ=V0U0φφ,forJ=0,V0U1φφ+V1U0φφ,forJ=1,V2U0φφ+V1U1φφ+V0U2φφ,forJ=2,OJV2Wφ=V02W0φ,forJ=0,2V0V1W0φ+V02W1φ,forJ=1,2V0V2+V12W0φ+2V0V1W1φ+V02W2φ,forJ=2,xJzyWφ=zy0U0φ,forJ=0,zy0U1φ+zy0U1φ,forJ=1,zy0U2φ+zy1U1φ+zy2U0φ,forJ=2,QJU2Vφ=U02V0φ,forJ=0,2U0U1V0φ+U02V1φ,forJ=1,2U0U2+U12V0φ+2U0U1V1φ+U02V2φforJ=2,ℛ0UφVφ=U0φV0φ,forJ=0,U0φV1φ+U1φV0φ,forJ=1,U2φV0φ+U1φV1φ+U0φV2φ,forJ=2,SJWUφφ=W0U0φφ,forJ=0,W0U1φφ+W1U0φφ,forJ=1,W2U0φφ+W1U1φφ+W0U2φφ,forJ=2,TJW2Vφ=W02V0φ,forJ=0,2W0W1V0φ+W02V1φ,forJ=1,2W0W2+W12V0φ+2W0W1V1φ+W02V2φ,forJ=2,XJUW1Uφ=zx0U0φ,forJ=0,zx0U1φ+zx1U0φ,forJ=1,zx2U0φ+zx1U1φ+zx2U0φ,forJ=2,YJU2Wφ=U02W0φ,forJ=0,2U0U1W0φ+U02W1φ,forJ=1,2U0U2+U12W0φ+2U0U1W1φ+U02W2φ,forJ=2.Forj=0,1,2,3,…,(75)U1φ,ℑ=L−1vγ1−γ+γvγL12Uφφφ0−3E0+32ℱ0+3G0+32ℋ0+3I0+3J0+3K0=112sech2φ1−γ+γℑγΓγ+1V1φ,ℑ=L−1vγ1−γ+γvγLVφφφ0−3ℳ0−3N0−3O0+6x0+3Q0=−118sech2φ1−γ+γℑγΓγ+1W1φ,ℑ=L−1vγ1−γ+γvγLWφφφ0−3ℛ0−3S^0−3T0+6X0+3Y0=−112sech2φ1−γ+γℑγΓγ+1U2φ,ℑ=L−1vγ1−γ+γvγL12Uφφφ1−3E1+32ℱ1+3G1+32ℋ1+3I1+3J1+3K1=−1218tanhφsech2φ1−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1V2φ,ℑ=L−1vγ1−γ+γvγLVφφφ1−3ℳ1−3N1−3O1+6x1+3Q1=1218tanhφsech2φ1−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1W2φ,ℑ=L−1vγ1−γ+γvγLWφφφ1−3ℛ1−3S^1−3T1+6X1+3Y1=2428anhφsech2φ1−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1U3φ,ℑ=L−1vγ1−γ+γvγL12Uφφφ2−3E2+32ℱ2+3G2+32ℋ2+3I2+3J2+3K2=133148sech4φcosh2φ−21−γ3+γ1−γ1+γ+2γ2ℑγΓγ+1+3γ21−γℑ2γΓ2γ+1+γ3Γ2γ+1ℑ3γΓ3γ+1V3φ,ℑ=L−1vγ1−γ+γvγLVφφφ2−3ℳ2−3N2−3O2+6x2+3Q2=266296sech4φcosh2φ−21−γ3+γ1−γ1+γ+2γ2ℑγΓγ+1+3γ21−γℑ2γΓ2γ+1+γ3Γ2γ+1ℑ3γΓ3γ+1W3φ,ℑ=L−1vγ1−γ+γvγLWφφφ2−3ℛ2−3S^2−3T2+6X2+3Y2=−266248sech4φcosh2φ−21−γ3+γ1−γ1+γ+2γ2ℑγΓγ+1+3γ21−γℑ2γΓ2γ+1+γ3Γ2γ+1ℑ3γΓ3γ+1⋮.The modified decomposition technique result for Example3 is given as(76)Uφ,ℑ=U0φ,ℑ+U1φ,ℑ+U2φ,ℑ+U3φ,ℑ…,=122+tanhφ+112sech2φ1−γ+γℑγΓγ+1−1218tanhφsech2φ1−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1+133148sech4φcosh2φ−21−γ3+γ1−γ1+γ+2γ2ℑγΓγ+1+3γ21−γℑ2γΓ2γ+1+γ3Γ2γ+1ℑ3γΓ3γ+1+⋯.Consequently, we get(77)Vφ,ℑ=142−tanhφ−118sech2φ1−γ+γℑγΓγ+1+1218tanhφsech2φ1−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1−133148sech4φcosh2φ−21−γ3+γ1−γ1+γ+2γ2ℑγΓγ+1+3γ21−γℑ2γΓ2γ+1+γ3Γ2γ+1ℑ3γΓ3γ+1+⋯Wφ,ℑ=2−tanhφ−112sech2φ1−γ+γℑγΓγ+1+1218tanhφsech2φ1−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1−266296sech4φcosh2φ−21−γ3+γ1−γ1+γ+2γ2ℑγΓγ+1+3γ21−γℑ2γΓ2γ+1+γ3Γ2γ+1ℑ3γΓ3γ+1+⋯.By puttingγ=1, we obtain the exact result of the system of Korteweg–de Vries equation (2):(78)Uφ,ℑ=122+tanhφ−11ℑ2,Vφ,ℑ=142−tanhφ−11ℑ2,Wφ,ℑ=2−tanhφ−11ℑ2.Case II: now, we apply the new iterative transformation technique for Example3.
By using the suggested analytical method, we get(79)U0φ,ℑ=122+tanhφV0φ,ℑ=142−tanhφW0φ,ℑ=2−tanhφU1φ,ℑ=L−1vγ1−γ+γvγL12∂3U0∂ℑ3−3U02∂U0∂φ+32W0∂2V0∂φ2+3∂V0∂φ∂W0∂φ+32V0∂2W0∂φ2+3V0W0∂U0∂φ+3U0W0∂V0∂φ+3U0V0∂W0∂φ=112sech2φL−1ωγ+2vγ+2=112sech2φ1−γ+γℑγΓγ+1V1φ,ℑ=L−1vγ1−γ+γvγL−∂3V0∂φ3−3∂U0∂φ∂V0∂φ−3V0∂2U0∂φ2−3V02∂W0∂φ+6U0V0∂U0∂φ+3U02∂V0∂φ=−118sech2φ1−γ+γℑγΓγ+1W1φ,ℑ=L−1vγ1−γ+γvγL−∂3W0∂φ3−3∂U0∂φ∂W0∂φ−3W0∂2U0∂φ2−3W02∂V0∂φ+6U0W0∂U0∂φ+3U02∂W0∂φ=−112sech2φ1−γ+γℑγΓγ+1U2φ,ℑ=L−1vγ1−γ+γvγL12∂3U1∂ℑ3−3U12∂U1∂φ+32W1∂2V1∂φ2+3∂V1∂φ∂W1∂φ+32V1∂2W1∂φ2+3V1W1∂U1∂φ+3U1W1∂V1∂φ+3U1V1∂W1∂φ=−1218tanhφsech2φ1−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1V2φ,ℑ=L−1vγ1−γ+γvγL−∂3V1∂φ3−3∂U1∂φ∂V1∂φ−3V1∂2U1∂φ2−3V12∂W1∂φ+6U1V1∂U1∂φ+3U12∂V1∂φ=1218tanhφsech2φ1−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1W2φ,ℑ=L−1vγ1−γ+γvγL−∂3W1∂φ3−3∂U1∂φ∂W1∂φ−3W1∂2U1∂φ2−3W12∂V1∂φ+6U1W1∂U1∂φ+3U12∂W1∂φ=2428tanhφsech2φ1−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1U3φ,ℑ=L−1vγ1−γ+γvγL12∂3U2∂ℑ3−3U22∂U2∂φ+32W2∂2V2∂φ2+3∂V2∂φ∂W2∂φ+32V2∂2W2∂φ2+3V2W2∂U2∂φ+3U2W2∂V2∂φ+3U2V2∂W2∂φ=133148sech4φcosh2φ−21−γ3+γ1−γ1+γ+2γ2ℑγΓγ+1+3γ21−γℑ2γΓ2γ+1+γ3Γ2γ+1ℑ3γΓ3γ+1V3φ,ℑ=L−1vγ1−γ+γvγL−∂3V2∂φ3−3∂U2∂φ∂V2∂φ−3V2∂2U2∂φ2−3V22∂W2∂φ+6U2V2∂U2∂φ+3U22∂V2∂φ=266296sech4φcosh2φ−21−γ3+γ1−γ1+γ+2γ2ℑγΓγ+1+3γ21−γℑ2γΓ2γ+1+γ3Γ2γ+1ℑ3γΓ3γ+1W3φ,ℑ=L−1vγ1−γ+γvγL−∂3W2∂φ3−3∂U2∂φ∂W2∂φ−3W2∂2U2∂φ2−3W22∂V2∂φ+6U2W2∂U2∂φ+3U22∂W2∂φ=−266248sech4φcosh2φ−21−γ3+γ1−γ1+γ+2γ2ℑγΓγ+1+3γ21−γℑ2γΓ2γ+1+γ3Γ2γ+1ℑ3γΓ3γ+1⋮Ujφ,ℑ=L−1vγ1−γ+γvγL12∂3Uj−1∂ℑ3−3Uj−12∂Uj−1∂φ+32Wj−1∂2Vj−1∂φ2+3∂Vj−1∂φ∂Wj−1∂φ+32Vj−1∂2Wj−1∂φ2+3Vj−1Wj−1∂Uj−1∂φ+3Uj−1Wj−1∂Vj−1∂φ+3Uj−1Vj−1∂Wj−1∂φVjφ,ℑ=L−1vγ1−γ+γvγL−∂3Vj−1∂φ3−3∂Uj−1∂φ∂Vj−1∂φ−3Vj−1∂2Uj−1∂φ2−3Vj−12∂Wj−1∂φ+6Uj−1Vj−1∂Uj−1∂φ+3Uj−12∂Vj−1∂φWjφ,ℑ=L−1vγ1−γ+γvγL−∂3Wj−1∂φ3−3∂Uj−1∂φ∂Wj−1∂φ−3Wj−1∂2Uj−1∂φ2−3Wj−12∂Vj−1∂φ+6Uj−1Wj−1∂Uj−1∂φ+3Uj−12∂Wj−1∂φ.
The series of results for Example3 is given as(80)Uφ,ℑ=U0φ,ℑ+U1φ,ℑ+U2φ,ℑ+U3φ,ℑ+⋯Ujφ,ℑ=122+tanhφ+112sech2φ1−γ+γℑγΓγ+1−1218tanhφsech2φ1−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1+133148sech4φcosh2φ−21−γ3+γ1−γ1+γ+2γ2ℑγΓγ+1+3γ21−γℑ2γΓ2γ+1+γ3Γ2γ+1ℑ3γΓ3γ+1+⋯.
Consequently, we get(81)Vφ,ℑ=142−tanhφ−118sech2φ1−γ+γℑγΓγ+1+1218tanhφsech2φ1−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1−133148sech4φcosh2φ−21−γ3+γ1−γ1+γ+2γ2ℑγΓγ+1+3γ21−γℑ2γΓ2γ+1+γ3Γ2γ+1ℑ3γΓ3γ+1+⋯,Wφ,ℑ=2−tanhφ−112sech2φ1−γ+γℑγΓγ+1+1214tanhφsech2φ1−γ2+γ2ℑ2γΓ2γ+1+21−γγℑγΓγ+1−266248sech4φcosh2φ−21−γ3+γ1−γ1+γ+2γ2ℑγΓγ+1+3γ21−γℑ2γΓ2γ+1+γ3Γ2γ+1ℑ3γΓ3γ+1+⋯.
By puttingγ=1, we obtain the exact result of modified couple Korteweg–de Vries equation (2):(82)Uφ,ℑ=122+tanhφ−11ℑ2,Vφ,ℑ=142−tanhφ−11ℑ2,Wφ,ℑ=2−tanhφ−11ℑ2.
In Figures9–11, the actual and analytical solutions of Uφ,ℑ,Vφ,ℑ, and Wφ,ℑ are proved at δ=2, ρ=0.5, and ϱ=1. In Figures 12–14, the surface and two-dimensional figure for Uφ,ℑ,Vφ,ℑ, and Wφ,ℑ for numerous fractional orders are described which demonstrate that the modified decomposition technique and new iterative transformation technique approximated obtained results are in close contact with the analytical and the exact results. This comparison shows a strong connection among the modified decomposition method and actual solutions. Consequently, the modified decomposition technique and new iterative transformation technique are accurate innovative techniques which need less calculation time and are very simple and more flexible than the homotopy analysis technique and homotopy perturbation technique.Figure 9
The analytical and exact result plot atUφ,ℑ of Example 3 for ρ=0.5, ϱ=1, and δ=2.Figure 10
The analytical and exact result plot atVφ,ℑ of Example 3 for ρ=0.5, ϱ=1, and δ=2.Figure 11
The analytical and exact result plot atWφ,ℑ of Example 3 for ρ=0.5, ϱ=1, and δ=2.Figure 12
The three and two dimensional different fractional order of Example 3 with respect toU (φ, ℑ).Figure 13
The three and two dimensional different fractional order of Example 3 with respect toV (φ, ℑ).Figure 14
The three and two dimensional different fractional order of Example 3 with respect toW (φ, ℑ).


## 7. Conclusion

In this article, we have considered the nonlinear fractional-order Korteweg–de Vries equations in the sense of the Atangana–Baleanu derivative which is able to perform more extensive analysis due to the nonsingular kernel in its structure. The mathematical solutions are obtained with the help of the modified decomposition method and new iterative transformation method associated with the Atangana–Baleanu derivative. The present analysis illuminates the effectiveness of the considered derivative operator. We can conclude from the analytical results that these are very reliable, simple, and powerful methods for finding approximate results of many fractional physical models which arise in applied sciences. In this approach, we do not need the Lagrange multiplier, correction functional, and stationary conditions or to calculate heavy integrals because the results established are noise free, which overcomes the shortcomings of existing methods. It is remarkable that the projected approaches are well-organized analytical methods for finding approximate analytical solutions to complex nonlinear partial differential equations. Finally, we conclude that this scheme, in future, will be taken into account in order to cope with other complex nonlinear fractional-order systems of equations.



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