Research article

New analytical modelling of fractional generalized Kuramoto-Sivashinky equation via Atangana-Baleanu operator and $\mathbb{J} $-transform method

  • Metonou Richard ,
  • Weidong Zhao ,
  • Shehu Maitama , *
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  • School of Mathematics, Shandong University, Jinan, Shandong, China
* E-mail addresses: metonourichard@yahoo.fr (M. Richard), (W. Zhao), (S. Maitama)

Received date: 2022-04-26

  Revised date: 2022-06-09

  Accepted date: 2022-06-10

  Online published: 2022-06-16

Abstract

In this paper, we propose a new analytical modelling of the well-known fractional generalized Kuramoto-Sivashinky equation (FGKSE) using fractional operator with non-singular kernel and the homotopy analysis transform method via $\mathbb{J} $-transform method. Also, using fixed-point theorem, we prove the existence and uniqueness of our proposed solution to the fractional Kuramoto-Sivashinky equation. To further validate the efficiency of the suggested technique, we proved the convergence analysis of the method and provide the error estimate. The obtained solutions of the FGKSE, describing turbulence processes in the field of ocean engineering are analytically and numerically compared to show the behaviors of many parameters of the present model.

Cite this article

Metonou Richard , Weidong Zhao , Shehu Maitama . New analytical modelling of fractional generalized Kuramoto-Sivashinky equation via Atangana-Baleanu operator and $\mathbb{J} $-transform method[J]. Journal of Ocean Engineering and Science, 2024 , 9(1) : 66 -88 . DOI: 10.1016/j.joes.2022.06.025

1. Introduction

The seminal work of Leibniz and the L’Hôpital contributed immensely to the advancement of the basic idea of fractional calculus [1], [2], [3]. The beauty of fractional calculus is its ability to accurately capture the exact behavior of many complex fractional models in science, engineering, and finance [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. The concept of fractional calculus is believed to be a generalization of integer order calculus [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. For the most recent related works on fractional calculus, the readers may to refer to [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. The previous work of Kuramoto and Sivashinsky [36], [37], [38], [39] on the fourth-order nonlinear partial differential equation which provided the general idea on the GKSE also known as the Generalized Kuramato-Sivashinky equation exhibited a chaotic behavior with non-zero constants ( α , β , γ R +) of the form
W t + W W x + α 2 W x 2 + β 3 W x 3 + γ 4 W x 4 = 0 .
The GKSE was used to model important applications in science and engineering such as the chemical reaction dynamics, the flows in pipes and at interfaces, the flame propagation and reaction-diffusion systems. In the field of ocean engineering, the GKSE arises naturally in various applications such as the viscous flow problems, the magneto hydrodynamics, the weather forecasting and the climate modelling, the turbulence in micro-tides, the offshore industry, and so on. The GKSE provides solution to the most of the none-linear differential equations that exhibit chaotic behavior and is considered as a prototype to the larger class of the well-known Burger’s equations. If α = γ = 1 and β = 0, the generalized Kuramoto-Sivashinky Eq. (1) reduces to Kuramoto-Sivashinky equation (KSE) [40]. Shah et al. solved Eq. (1) using the Laplace transformation and the variational iteration method [41]. In 2019, Taneco-Hernandez et al. obtained the analytical solutions of the KSE via the homotopy perturbation transform method [42]. In [43], the modified Kudrayshov method was applied to Eq. (1) when α = β = γ = 1. Kurulay et al. used the well-known homotopy analysis method to analyzed the analytical solutions of the GKSE [44]. Using the B-spline function, the numerical solutions of Eq. (1) when α = γ = 1 and β = 0 [45] was illustrated. In 2018, Rosa et al. derive the classical symmetries and the low-order conservation laws of the GKSE [46]. Veeresha and Prakasha applied the q-homotopy analysis transform method to obtained the analytical solutions of the KSE [47]. Xu and Shu utilized the local discontinuous Galerkin techniques to analyzed the numerical solution of the KSE [48]. In 2011, Porshokouhi and Ghanbari solved the KSE via the variational iteration technique [49]. Khater and Temsah applied the Chebyshev spectral collocation techniques on KSE [50]. In [51], Ye et al. obtained the numerical solutions of the KSE via Lattice Boltzmann model. In 2015, Sahoo and Ray obtained a new exact solutions of the KSE [52]. For more numerical solutions of the GKSE, the readers are refereed to [53], [54], [55], [56], [57].
The aim of this work is to proposed an analytical technique called the homotopy analysis J-transform method (HAJTM) via the Caputo (C) [58], [59], the Caputo-Fabrizio (CF) [60], and the Atangana-Baleanu (AB) fractional derivatives [61] for solving the non-integer generalized Kuramato-Sivashinky equation. The HAJTM is a combination of the well-known homotopy analysis method which was first proposed by Liao [62], [63] and the J-transform method which is a modification of the well-known Sumudu integral transform and the natural transform method [64]. The J-transform was first introduced by Maitama and Zhao in 2020, and was successfully utilized to solved problems which cannot be solved by the natural transform and the Sumudu integral transform [65]. The J-transform converges to Laplace transform when the variable u = 1. In [66], the J-transform was applied to solved differential equations with variable coefficient which cannot be solve using the well-known Laplace transform. The advantages and disadvantages of the proposed HAJTM are as follows:
•The HAJTM can be applied to analyzed the solution of linear or nonlinear fractional models in ocean engineering without any restrictive assumptions.
•The proposed method gives a series solutions which converges rapidly within few iterations.
•The most important aspect of the suggested method is the existence of the non-zero convergence control parameter which is used to adjust and control the convergence of the series solutions.
•The HAJTM is used to investigate the analytical and numerical solutions of the fractional generalized Kuramato-Sivashinky equation which naturally arises in oceanic engineering.
•The HAJTM can only be applied to fractional models with boundary or initial conditions.
The work is organized as follows: In Section 2, definitions pertaining J-transform, fractional derivative with non-singular kernel, and some important theorems used in this paper are presented. In Section 3, the fractional generalized Kuramato-Sivashinky equation is examined via the Atangana-Baleanu fractional derivative. In Section 4, an iterative method via the Caputo, the Caputo-Fabrizio, and the Atangana-Baleanu fractional derivatives is given. In Section 5, the applications of the HAJTM to fractional generalized Kuramato-Sivashinky equation via the three fractional derivatives are illustrated. In Section 6, which is the last part of this paper, the concluding remarks are presented.

2. Preliminaries

In this section, we present the basic definition of the fractional derivative with non-singular kernel known as the Atangana-Baleanu fractional derivative. Moreover, the J-transform and some of its useful theorems used in this paper are also presented.
Definition 1 [61] Let W H 1 ( a , b ) , a < b , ξ [ 0 , 1 ] and not necessarily differentiable, then the Atangana-Baleanu fractional derivative in Riemann-Liouville sense is defined as
A B R b D t ξ ( W ( t ) ) = M ( ξ ) 1 ξ d d t b t W ( τ ) E ξ [ ξ ( t τ ) ξ 1 ξ ] d τ ,
where M ( ξ ) is a normalization function with property M ( 0 ) = M ( 1 ) = 1 .
Definition 2 [61] Let W H 1 ( a , b ) , a < b , ξ [ 0 , 1 ], then the Atangana-Baleanu fractional derivative in Caputo sense is defined as
A B C b D t ξ ( W ( t ) ) = M ( ξ ) 1 ξ b t W ( τ ) E ξ [ ξ ( t τ ) ξ 1 ξ ] d τ ,
where M ( ξ ) is a normalization function having the property M ( 0 ) = M ( 1 ) = 1 .
Definition 3 [14] The associative fractional integral related to the AB fractional derivative is defined as
A B a I t ξ ( W ( t ) ) = 1 ξ M ( ξ ) W ( t ) + ξ M ( ξ ) Γ ( ξ ) a t W ( τ ) ( t τ ) ξ 1 d τ ,
where M ( ξ ) satisfies M ( 0 ) = M ( 1 ) = 1 .
Definition 4 [59] The Mittag-Leffler function for one parameter is defined by the series expansion
E ξ ( t ) = k = 0 t k Γ ( ξ k + 1 ) , R ( ξ ) > 0 , ξ , t C .
Definition 5 [64], [66] The J-transform of the function W ( t ) is defined over the set of functions
A = { W ( t ) : C , ζ 1 , ζ 2 > 0 , s u c h t h a t | W ( t ) | < C exp ( | t | ζ i ) f o r t ( 1 ) i × [ 0 , ) } ,as
J [ W ( t ) ] ( s , u ) = W ( s , u ) = u 0 exp ( s u t ) W ( t ) d t , s > 0 , u > 0 .
Theorem 1 [64, 66], Convolution theorem for the J -transform is given by
J [ ( V * W ) ( t ) ] = 1 u V ( s , u ) W ( s , u ) .
Theorem 2 The J -transform of Caputo fractional derivative is defined as
J ( C b D t ξ W ( t ) ) = ( s u ) ξ J [ W ( t ) ] k = 0 n 1 s ξ ( k + 1 ) u ξ ( k + 2 ) W ( k ) ( 0 ) ,
where n 1 ξ n .
Proof Thanks to Caputo fractional derivative [59] which help us to get
C D t ξ W ( t ) = 1 Γ ( n ξ ) 0 t ( t τ ) n ξ 1 n W ( τ ) τ n d τ = 1 Γ ( n ξ ) W ( n ) ( t ) t n ξ 1 ,where
Γ ( . ) is the Euler-Gamma function.
Using the J-transform Definition and its Convolution Theorem [64], we get
J [ C D t ξ W ( t ) ] = 1 Γ ( n ξ ) J [ t n ξ 1 W ( n ) ( t ) ] = ( s u ) ξ J [ W ( t ) ] k = 0 n 1 s ξ ( k + 1 ) u ξ ( k + 2 ) W ( k ) ( 0 ) .
The proof ends. □
Theorem 3 Let W H 1 ( a , d ) , b > a , ξ [ 0 , 1 ] , then the J -transform of the Caputo-Fabrizio fractional derivative is defined as
J ( C F b D t ξ ( W ( t ) ) ) = u M ( ξ ) s + ξ ( u s ) [ s u J [ W ( t ) ] u W ( 0 ) ] ,
where M ( ξ ) satisfies M ( 0 ) = M ( 1 ) = 1 .
Proof From the definition of J-transform [64] and the CF [60], we get
J ( C F 0 D t ξ ( W ( t ) ) ) ( s , u ) = J ( M ( ξ ) 1 ξ b t W ( τ ) exp [ ξ ( t τ ) 1 ξ ] d τ ) ( s , u ) .
Applying the convolution property of J-transform, gives
J [ C F 0 D t ξ ( W ( t ) ) ] = M ( ξ ) 1 ξ 1 u J [ W ( t ) ] * J [ exp [ ξ t 1 ξ ] ] = M ( ξ ) 1 ξ 1 u ( s u J [ W ( t ) ] u W ( 0 ) ) ( u 0 exp ( s t u ) exp [ ξ t 1 ξ ] ) = u M ( ξ ) s + ξ ( u s ) [ s u J [ W ( t ) ] u W ( 0 ) ] .
This completes the proof. □
Theorem 4 The J -transform of the Atangana-Baleanu fractional derivative is given by
J ( A B C b D t ξ ( W ( t ) ) ) = u M ( ξ ) s ξ + ξ ( u ξ s ξ ) [ s u J [ W ( t ) ] u W ( 0 ) ] ,
where M ( ξ ) satisfies M ( 0 ) = M ( 1 ) = 1 .
Proof Using the definition of J-transform [64] and the AB [61], we get
J ( A B C 0 D t ξ ( W ( t ) ) ) ( s , u ) = J ( M ( ξ ) 1 ξ b t W ( τ ) E ξ [ ξ ( t τ ) ξ 1 ξ ] d τ ) ( s , u ) .
Applying the convolution property of J-transform, we get
J [ A B C 0 D t ξ ( W ( t ) ) ] = M ( ξ ) 1 ξ 1 u J [ W ( t ) ] * J [ exp [ ξ t ξ 1 ξ ] ] = M ( ξ ) 1 ξ 1 u ( s u J [ W ( t ) ] u W ( 0 ) ) ( u 0 exp ( s t u ) E ξ [ ξ t ξ 1 ξ ] ) = u M ( ξ ) s ξ + ξ ( u ξ s ξ ) [ s u J [ W ( t ) ] u W ( 0 ) ] .
This completes the proof. □
In the following section, we present the new fractional generalized Kuramoto-Sivashinky equation using the AB fractional derivative.

3. Modelling fractional generalized Kuramoto-Sivashinky equation with Atangana-Baleanu fractional derivative

In this section, we study the fractional model of generalized Kuramoto-Sivashinky equation via the AB fractional operator
A B C D t ( ξ ) W ( x , t ) + W W x + α W x x + β W x x x + γ W x x x x = 0 ,
where α , β , γ R +
Applying the AB fractional derivative operator on Eq. (12), we get
W ( x , t ) W ( x , 0 ) = ( 1 ξ ) M ( ξ ) ( W W x α W x x β W x x x γ W x x x x ) + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 ( W W x α W x x β W x x x γ W x x x x ) d τ .
For clarity, we adopt the following simple notation
Ψ ( x , t , W ) = W W x α W x x β W x x x γ W x x x x .
Then Eq. (13) becomes
W ( x , t ) W ( x , 0 ) = ( 1 ξ ) M ( ξ ) Ψ ( x , t , W ) + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 Ψ ( x , τ , W ) d τ .
In Eq. (15), the kernel Ψ ( x , t , W ) satisfies the Lipschitz condition provided the function W ( x , t ) is has an upper bound. Thus, if the function Ψ ( x , t , W ) has an upper bound then
Ψ ( x , t , W ) Ψ ( x , t , Y ) = ( W W x Y Y x ) α ( W x x Y x x ) β ( W x x x Y x x x ) γ ( W x x x x Y x x x x ) .
With the help of triangular inequality of the norm we arrive at
Ψ ( x , t , W ) Ψ ( x , t , Y ) 1 2 x ( W 2 Y 2 ) + α W x x Y x x + β W x x x Y x x x + γ W x x x x Y x x x x ω 2 W 2 Y 2 + α W Y + β W Y + γ W Y ω ( μ 1 + μ 2 ) 2 W Y + α W Y + β W Y + γ W Y = ( ω ( μ 1 + μ 2 ) 2 + α ν + β λ + γ ζ ) W Y ,
where μ 1 , μ 2 , ω , ζ , λ , ν R +.
In Eq. (17), the functions W ( x , t ) and Y ( x , t ) are bounded. This implies W ( x , t ) R 1 and Y ( x , t ) R 2, where R 1 , R 2 are constants. Thus, setting Λ = ω ( μ 1 + μ 2 ) 2 + α ν + β λ + γ ζ, we get
Ψ ( x , t , W ) Ψ ( x , t , Y ) Λ ( W ( x , t ) Y ( x , t ) .
Hence, the kernel Ψ ( x , t , W ) satisfies the Lipschitz condition.
Theorem 5 Let the function W ( x , t ) be bounded, then the operator
T ( W ( x , t ) ) = W ( x , 0 ) + ( 1 ξ ) M ( ξ ) Ψ ( x , t , W ) + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 Ψ ( x , τ , W ) d τ
satisfies the Lipschitz condition.
Proof Without loss of generality. Let assume the functions W ( x , t ) and Y ( x , t ) are bounded with initials W ( x , 0 ) = Y ( x , 0 ), then we get
T ( W ( x , t ) ) T ( Y ( x , t ) ) = ( 1 ξ ) M ( ξ ) ( Ψ ( x , t , W ) Ψ ( x , t , Y ) + + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 ( Ψ ( x , τ , W ) Ψ ( x , τ , Y ) ) d τ ( 1 ξ ) M ( ξ ) Ψ ( x , t , W ) Ψ ( x , t , Y ) + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 ( Ψ ( x , τ , W ) Ψ ( x , τ , Y ) ) d τ ( ( 1 ξ ) M ( ξ ) λ + 1 M ( ξ ) Γ ( ξ ) B t 0 ξ ) ( W ( x , t ) Y ( x , t ) Λ ( W ( x , t ) Y ( x , t ) ,
where Λ = ( 1 ξ ) M ( ξ ) λ + 1 M ( ξ ) Γ ( ξ ) B t 0 ξ.
This completes the proof. □
Theorem 6 If the function W ( x , t ) is bounded, then the operator T 1 denoted as
T 1 ( W ( x , t ) ) = W ( x , t ) W x ( x , t ) α W x x ( x , t ) β W x x x ( x , t ) γ W x x x x ( x , t )
satisfies the following result
| T 1 ( W ( x , t ) ) T 1 ( Y ( x , t ) ) , W ( x , t ) Y ( x , t ) | Λ ( W ( x , t ) Y ( x , t ) 2 ,
where . , . represent the inner product of function with the differentiation restricted in L 2.
Proof Let the functions W ( x , t ) be bounded, then we have
| T 1 ( W ( x , t ) ) T 1 ( Y ( x , t ) ) , W ( x , t ) Y ( x , t ) | = | ( W W x Y Y x ) α ( W x x Y x x ) β ( W x x x Y x x x ) γ ( W x x x x Y x x x x ) , W Y | | ( W W x Y Y x ) , W Y | + α | ( W x x Y x x ) , W Y | + β | W x x x Y x x x , W Y | + γ | ( W x x x x Y x x x x , W Y | ω ( μ 1 + μ 2 ) 2 W Y W Y + α ν W Y W Y + β λ ( W Y W Y + ζ ( W Y W Y ( ω ( μ 1 + μ 2 ) 2 + α ν + β λ + γ ζ ) ( W ( x , t ) Y ( x , t ) 2 Λ ( W ( x , t ) Y ( x , t ) 2 ,
where Λ = ω ( μ 1 + μ 2 ) 2 + α ν + β λ + γ ζ , μ 1 , μ 2 , ω , ζ , λ , ν R +
The proof is completed. □
Theorem 7 Suppose the function W ( x , t ) is bounded, then the operator T 1 defined in Eq. (21) satisfies
| T 1 ( W ( x , t ) ) T 1 ( Y ( x , t ) ) , ϑ | Λ ( W ( x , t ) Y ( x , t ) ϑ , 0 < ϑ < .
Proof Let 0 < ϑ < and the functions W ( x , t ) be bounded, then we have
| T 1 ( W ( x , t ) ) T 1 ( Y ( x , t ) ) , ϑ | = | ( W W x Y Y x ) α ( W x x Y x x ) β ( W x x x Y x x x ) γ ( W x x x x Y x x x x ) , ϑ | | ( W W x Y Y x ) , ϑ | + α | ( W x x Y x x ) , ϑ | + β | W x x x Y x x x , ϑ | + γ | ( W x x x x Y x x x x , ϑ | ω ( μ 1 + μ 2 ) 2 W ( x , t ) Y ( x , t ) ϑ + α ν W ( x , t ) Y ( x , t ) ϑ + β λ ( W ( x , t ) Y ( x , t ) ϑ + ζ ( W ( x , t ) Y ( x , t ) ϑ = ( ω ( μ 1 + μ 2 ) 2 + α ν + β λ + γ ζ ) ( W ( x , t ) Y ( x , t ) ϑ = Λ ( W ( x , t ) Y ( x , t ) ϑ ,
where Λ = ω ( μ 1 + μ 2 ) 2 + α ν + β λ + γ ζ , μ 1 , μ 2 , ω , ζ , λ , ν R +
This completes the proof. □

3.1. The existence and uniqueness analysis of the FGKSE

In this section we prove the existence and uniqueness of the fractional generalized Kuramoto-Sivashinky equation using the AB fractional derivative. Based on Eq. (15), we formulate the following iterative scheme
W m + 1 ( x , t ) = ( 1 ξ ) M ( ξ ) Ψ ( x , t , W m ) + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 Ψ ( x , τ , W m ) d τ
with initial
W 0 ( x , t ) = W ( x , 0 ) .
Moreover, the algebraic difference of the successive terms is given by
Θ m ( x , t ) = W m ( x , t ) W m 1 ( x , t ) = ( 1 ξ ) M ( ξ ) ( Ψ ( x , t , W m 1 ) Ψ ( x , t , W m 2 ) ) + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 ( Ψ ( x , τ , W m 1 ) Ψ ( x , τ , W m 2 ) ) d τ
At this stage, it is crucial to know that
W m ( x , t ) = j = 0 m Θ j ( x , t ) .
Then by virtue of Eq. (28), we deduce
Θ m ( x , t ) = W m ( x , t ) W m 1 ( x , t ) = ( 1 ξ ) M ( ξ ) ( Ψ ( x , t , W m 1 ) Ψ ( x , t , W m 2 ) ) + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 ( Ψ ( x , τ , W m 1 ) Ψ ( x , τ , W m 2 ) ) d τ .
Computing the triangular inequality of Eq. (30), yields
Θ m ( x , t ) ( 1 ξ ) M ( ξ ) Ψ ( x , t , W m 1 ) Ψ ( x , t , W m 2 ) + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 Ψ ( x , τ , W m 1 ) Ψ ( x , τ , W m 2 ) d τ .
Since the kernel Ψ ( x , t , W ) satisfies the Lipschiz condition, then
Θ m ( x , t ) ( 1 ξ ) M ( ξ ) B W m 1 ( x , t ) W m 2 ( x , t ) + ξ M ( ξ ) Γ ( ξ ) B 0 t ( t τ ) ξ 1 W m 1 ( x , τ ) W m 2 ( x , τ ) d τ .
or
Θ m ( x , t ) ( 1 ξ ) M ( ξ ) B Θ m 1 ( x , t ) + ξ M ( ξ ) Γ ( ξ ) B 0 t ( t τ ) ξ 1 Θ m 1 ( x , τ ) d τ .
From Eqs. (30) to (33), we obtain the following useful theorem
Theorem 8 The model of fractional generalized Kuramoto-Sivashinky equation given in Eq. (12) has solution which satisfies the following inequality
( 1 ξ ) M ( ξ ) B + 1 M ( ξ ) Γ ( ξ ) B t 0 ξ < 1 .
Proof Let the function W ( x , t ) be bounded. Besides, the kernel Ψ ( x , t , W ) satisfies Lipschitz condition. By virtue of Eq. (33) and the iteration scheme, we arrive at
Θ m ( x , t ) ( ( 1 ξ ) M ( ξ ) B + 1 M ( ξ ) Γ ( ξ ) B t ξ ) m W ( x , 0 ) .
Thus,
W m ( x , t ) = j = 0 m Θ j ( x , t ) ,
exist. The result of Eq. (36) is the solution of Eq. (12), to verify that we consider
W ( x , t ) W ( x , 0 ) = W m ( x , t ) D m ( x , t ) .
Then,
D m ( x , t ) = ( 1 ξ ) M ( ξ ) ( Ψ ( x , t , W m 1 ) Ψ ( x , t , W m 2 ) ) + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 ( Ψ ( x , τ , W m 1 ) Ψ ( x , τ , W m 2 ) ) d τ ( 1 ξ ) M ( ξ ) ( Ψ ( x , t , W m 1 ) Ψ ( x , t , W m 2 ) ) + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 ( Ψ ( x , τ , W m 1 ) Ψ ( x , τ , W m 2 ) ) d τ ( 1 ξ ) M ( ξ ) B W ( x , t ) W m 1 ( x , t ) + 1 M ( ξ ) Γ ( ξ ) B W ( x , t ) W m 1 ( x , t ) t ξ .
Besides, following the same fashion we recursively obtain
D m ( x , t ) ( ( 1 ξ ) M ( ξ ) + 1 M ( ξ ) Γ ( ξ ) ) m + 1 B m + 1 a .
At the moment, setting t = t 0, we get
D m ( x , t ) ( ( 1 ξ ) M ( ξ ) + 1 M ( ξ ) Γ ( ξ ) t 0 ξ ) m + 1 B m + 1 a .
Finally, computing the limit of Eq. (40) as m , yields
D m ( x , t ) 0 .
This completes the proof of the existence. □
Fig. 1. Comparison of the analytical and numerical solutions of the FGKS Eq. (77) using the Caputo fractional derivative for different x , t and ξ s.
Fig. 2. Comparison of the analytical and numerical solutions of the FGKS Eq. (78) using the Caputo-Fabrizio fractional derivative for different x , t and ξ s.
Fig. 3. Comparison of the analytical and numerical solutions of the FGKS Eq. (79) using the Atangana-Baleanu fractional derivative for different x , t and ξ s.
Fig. 4. Comparison of the analytical and numerical solutions of the FGKS Eq. (92) using the Caputo fractional derivative for different x , t and ξ s.
It now remain to show the uniqueness of solution for fractional generalized Kuramoto-Sivashinky equation with the Atangana-Baleanu fractional derivative.
Without loss of generality, let claims there exist another solution Y ( x , t ) for the model on Eq. (12), then their successive difference is given by
W ( x , t ) Y ( x , t ) = ( 1 ξ ) M ( ξ ) ( Ψ ( x , t , W ) Ψ ( x , t , Y ) + + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 ( Ψ ( x , τ , W ) Ψ ( x , τ , Y ) ) d τ .
Computing the norm on Eq. (42), we deduce
W ( x , t ) Y ( x , t ) = ( 1 ξ ) M ( ξ ) ( Ψ ( x , t , W ) Ψ ( x , t , Y ) + ξ M ( ξ ) Γ ( ξ ) 0 t ( t τ ) ξ 1 Ψ ( x , τ , W ) Ψ ( x , τ , Y ) d τ .
Recall that the kernel Ψ ( x , t , W ) already satisfies the Lipschitz condition, then
W ( x , t ) Y ( x , t ) ( 1 ξ ) M ( ξ ) B ( Ψ ( x , t , W ) Ψ ( x , t , Y ) + ξ M ( ξ ) Γ ( ξ ) B t ξ Ψ ( x , τ , W ) Ψ ( x , τ , Y ) d τ .
The inequality of Eq. (44) yields
W ( x , t ) Y ( x , t ) ( 1 ( 1 ξ ) M ( ξ ) B 1 M ( ξ ) Γ ( ξ ) B t ξ ) 0 .
As a direct consequence of inequality Eq. (45) above, we obtain the following theorem
Theorem 9 The fractional generalized Kuramoto-Sivashinky Eq. (12) possesses a unique solution if the following inequality is satisfied
W ( x , t ) Y ( x , t ) ( 1 ( 1 ξ ) M ( ξ ) B 1 M ( ξ ) Γ ( ξ ) B t ξ ) > 0 .
Proof Since the kernel Ψ ( x , t , W ) of inequality Eq. (14) satisfies Lipschitz condition, and the aforementioned requirement, then
W ( x , t ) Y ( x , t ) ( 1 ( 1 ξ ) M ( ξ ) B 1 M ( ξ ) Γ ( ξ ) B t ξ ) 0 .
Thus
W ( x , t ) Y ( x , t ) = 0 .
This implies
W ( x , t ) = Y ( x , t ) = 0 .
This completes the proof. □
In the next section, we introduce the new homotopy analysis J-transform method for solving nonlinear fractional generalized Kuramato-Sivashinky equation.

4. The homotopy analysis J-transform method via the Caputo, the Caputo-Fabrizio, and the Atangana-Baleanu fractional derivatives

Consider the following nonlinear fractional generalized Kuramoto-Sivashinky equations
C D t ( ξ ) W ( x , t ) + W W x + α 2 W x 2 + β 3 W x 3 + γ 4 W x 4 = F ( x , t )
C F C D t ( ξ ) W ( x , t ) + W W x + α 2 W x 2 + β 3 W x 3 + γ 4 W x 4 = F ( x , t )
A B C D t ( ξ ) W ( x , t ) + W W x + α 2 W x 2 + β 3 W x 3 + γ 4 W x 4 = F ( x , t ) .
Where C D t ( ξ ) W ( x , t ) , C F C D t ( ξ ) W ( x , t ) , A B C D t ( ξ ) W ( x , t ) is the Caputo, the Caputo-Fabrizio, and the Atangana-Baleanu fractional derivatives and α , β , γ R + .
Computing the J-transform of the Caputo, the Caputo-Fabrizio, and the Atangana-Baleanu fractional derivative on Eqs. (50) to (52), we arrive at
( s u ) ξ J [ W ( x , t ) ] s ξ 1 u ξ 2 W ( x , 0 ) + J [ W W x + α 2 W x 2 + β 3 W x 3 + γ 4 W x 4 F ( x , t ) ] = 0 ,
u M ( ξ ) s + ξ ( u s ) [ s u J [ W ( x , t ) ] ( s , u ) u W ( x , 0 ) ] + J [ W W x + α 2 W x 2 + β 3 W x 3 + γ 4 W x 4 F ( x , t ) ] = 0 ,
u M ( ξ ) s ξ + ξ ( u ξ s ξ ) [ s u J [ W ( x , t ) ] ( s , u ) u W ( x , 0 ) ] + J [ W W x + α 2 W x 2 + β 3 W x 3 + γ 4 W x 4 F ( x , t ) ] = 0 ,
respectively.
Upon simplifying Eqs. (53) to (55), we get
J [ W ( x , t ) ] u 2 s W ( x , 0 ) + Ψ ( ) J [ W W x + α 2 W x 2 + β 3 W x 3 + γ 4 W x 4 F ( x , t ) ] = 0 ,
where
Ψ ( C ) = u ξ s ξ , Ψ ( C F C ) = s + ξ ( u s ) M ( ξ ) s ,and
Ψ ( A B C ) = s ξ + ξ ( u ξ s ξ ) M ( ξ ) s ξ .
The required nonlinear operator is defined as
N * [ Θ ( x , t ; q ) ] = J [ Θ ( x , t ; q ) ] u 2 s Θ ( x , 0 ) + Ψ ( ) J [ Θ ( x , t ) Θ ( x , t ) x + α 2 Θ ( x , t ) x 2 + β 3 Θ ( x , t ) x 3 + γ 4 Θ ( x , t ) x 4 F ( x , t ) ] ,
where q [ 0 , 1 ] is a nonzero auxiliary parameter, Θ ( x , t ; q ) is a real-valued function of x , t , q, and Ψ ( ) is to be replaced by (C), (CFC), and (ABC) respectively. We defines 0th-order homotopy of the form
( 1 q ) J [ Θ ( x , t ; q ) W 0 ( x , t ) ] = q H ^ ( x , t ) N * [ Θ ( x , t ) ] ,
where J is the J-transform, q [ 0 , 1 ] is the embedding parameter, H ^ ( x , t ) 0 (auxiliary function), 0 (convergence control parameter), W 0 ( x , t ) is the initial guess of W ( x , t ), and Θ ( x , t ; q ) is the unknown function.
Interestingly, the HAJTM provide us with the great chance to select the auxiliary parameter and the initial guess. Setting q = 0 , 1, in Eq. (58) gives the following result
Θ ( x , t ; 0 ) = W 0 ( x , t ) , a n d Θ ( x , t ; 1 ) = W ( x , t ) .
When q rises from 0 to 1, the solution Θ ( x , t ; q ) moves from the initial guess to W 0 ( x , t ) to the solution W ( x , t ). Then applying the Taylor series expansion of Θ ( x , t ; q ) with respect to q we get
Θ ( x , t ; q ) = W 0 ( x , t ) + m = 1 + W m ( x , t ) q m ,
where
W m ( x , t ) = [ 1 Γ ( m + 1 ) m Θ ( x , t ; q ) q m ] q = 0 .
Besides, if 0, the auxiliary function H ^ ( x , t ) 0, the initial guess, are selected properly, Eq. (58) converges at q = 1, and
Θ ( x , t ) = W 0 ( x , t ) + m = 1 + W m ( x , t ) ,
is the solution of Eqs. (50) to (52). Moreover, based on Eq. (60), the governing equation can be deduced from the 0th-order deformation Eq. (58).
Define the vectors
W m = { W 0 ( x , t ) , W 1 ( x , t ) , W 2 ( x , t ) , , W n ( x , t ) } .
Dividing Eq. (58) by Γ ( m + 1 ) after m-times differentiation with respect to q, and setting q = 0 we get the Mth-order deformation equation
J [ W m ( x , t ) χ ^ m W m 1 ( x , t ) ] = H ^ ( x , t ) R ^ m ( W m 1 , x , t ) ,
where
R ^ m ( W m 1 , x , t ) = [ 1 Γ ( m ) ( m 1 ) N * [ ξ ( x , t ; q ) ] q ( m 1 ) ] q = 0 ,
and
χ ^ m = { 0 m 1 1 o t h e r w i s e .
Operating the inverse J-transform on both sides of Eq. (64) yields
W m ( x , t ) = χ ^ m W m 1 ( x , t ) + J 1 [ H ^ ( x , t ) R ^ m ( W m 1 , x , t ) ] .
Based on Eqs. (50) to (52), the R ^ m ( W m 1 ) is define as
R ^ m ( W m 1 ) = J [ W m 1 ( x , t ) ] u 2 s W ( x , 0 ) + Ψ ( ) J ( i = 0 m 1 W i W m 1 i x + α 2 W m 1 x 2 + β 3 W m 1 x 3 + γ 4 W m 1 x 4 ( 1 χ m ) F ( x , t ) ) .
Using Eq. (67), we solve W m ( x , t ) for m 1 , and get
W ( x , t ) = W 0 ( ) ( x , t ) + lim M m = 1 M W m ( x , t ) ,
where the superscript ( ) is to be substituted by the (C), the (CF), and the (ABC) respectively.
In the next theorem, we study the convergence analysis of the original problems Eqs. (50) to (52).
Theorem 10 Convergence analysis. As M , W ( x , t ) = lim M m = 0 M W m ( x , t ) < , which is computed via Eq. (67) with help of Eq. (68) . Then it must be closed solution of Eqs. (50) to (52).
Proof Considering
m = 0 W m ( x , t ) = W 0 ( x , t ) + m = 1 + W m ( x , t ) = C ( x , t ) ,
Since m = 0 M W m ( x , t ) < , Then lim M m = 1 M W m ( x , t ) = 0.
With the help of Eqs. (67) and (68), we arrive at
lim M [ H ^ ( x , t ) m = 1 M R m ( W m 1 , x , t ) ] = lim M [ m = 1 M J [ W m ( x , t ) χ m W m 1 ( x , t ) ] ] = J [ lim M m = 1 M W m ( x , t ) lim M m = 1 M χ m W m 1 ( x , t ) ] = J [ lim M m = 1 M W m ( x , t ) ] = 0 .Considering H ^ ( x , t ) 0, q 0 and the linearity property of Eq. (58), we have
lim M m = 1 M R m ( W m 1 , x , t ) = 0 .
With the help of Eq. (68), we finally get
lim M m = 1 M R m ( W m 1 , x , t ) = lim M m = 1 M ( K D t ( ξ ) W ( x , t ) + W W x + α 2 W x 2 + β 3 W x 3 + γ 4 W x 4 ( 1 χ m ) F ( x , t ) ) = K D t ( ξ ) lim M m = 1 M W m 1 ( x , t ) + lim M i = 0 m 1 W i W m 1 i x + α 2 x 2 lim M m = 1 M W m 1 ( x , t ) + β 3 x 3 lim M m = 1 M W m 1 ( x , t ) + γ 4 x 4 lim M m = 1 M W m 1 ( x , t ) ( 1 χ m ) F ( x , t ) = K D t ( ξ ) W m 1 ( x , t ) + W i W m 1 i x + α 2 W m 1 x 2 + β 3 W m 1 x 3 + γ 4 W m 1 x 4 ( 1 χ m ) F ( x , t ) = 0 .
Where K is to be replace by the Caputo, the Caputo-Fabrizio, and the Atangana-Baleanu fractional derivatives. Thus, Eq. (72) proved that C ( x , t ) is the closed solution of Eqs. (50), (51), and (52) respectively. □
Theorem 11 Let W n ( x , t ) and W ( x , t ) be in Banach space ( BS ) . Then the HAJTM series solutions i = 0 W n ( x , t ) defined by Eq. (67) converges to the solution of Eqs. (50) to (52) provided 0 < < 1.
Proof Considering the sequence of partial sum { W m ( x , t ) } m = 0 of the form
W 0 ( x , t ) = K 0 ( x , t ) W 1 ( x , t ) = K 0 ( x , t ) + K 1 ( x , t ) W 2 ( x , t ) = K 0 ( x , t ) + K 1 ( x , t ) + K 2 ( x , t ) W 3 ( x , t ) = K 0 ( x , t ) + K 1 ( x , t ) + K 2 ( x , t ) + K 3 ( x , t ) W m ( x , t ) = K 0 ( x , t ) + K 1 ( x , t ) + K 2 ( x , t ) + K 3 ( x , t ) + + K m ( x , t ) .We are interested to prove that W m ( x , t ) is a Cauchy sequence in BS. From the last hypothesis of the theorem, we have ( 0 , 1 ) then
W m + 1 ( x , t ) W m ( x , t ) = K m + 1 ( x , t ) K m ( x , t ) 2 K m 1 ( x , t ) 3 K m 2 ( x , t ) 4 K m 3 ( x , t ) m + 1 K 0 ( x , t ) .
For any m , n N , n > m, we have
W m ( x , t ) W n ( x , t ) = K m + n ( x , t ) = ( W m ( x , t ) W m 1 ( x , t ) ) + ( W m 1 ( x , t ) W m 2 ( x , t ) ) + ( W m 2 ( x , t ) W m 3 ( x , t ) ) + + ( W n + 1 ( x , t ) W n ( x , t ) ) W m ( x , t ) W m 1 ( x , t ) + W m 1 ( x , t ) W m 2 ( x , t ) + W m 2 ( x , t ) W m 3 ( x , t ) + + W n + 1 ( x , t ) W n ( x , t ) m K 0 ( x , t ) + m 1 K 0 ( x , t ) + m 2 K 0 ( x , t ) + m 3 K 0 ( x , t ) + + m + 1 K 0 ( x , t ) = K 0 ( x , t ) 1 m n 1 n + 1 .
Then
lim m , n K m + n ( x , t ) = 0 ,
since 0 < < 1. Thus, we produces the required result. □ □
Theorem 12 Error analysis. Suppose i = 0 j K i ( x , t ) < and K ( x , t ) be its truncated solution. Let > 0 such that K i + 1 ( x , t ) K i ( x , t ) , ( 0 , 1 ) , for i, then
K ( x , t ) i = 0 j K i ( x , t ) j + 1 1 K 0 ( x , t ) .
Proof Since i = 0 j K i ( x , t ) < , then
K ( x , t ) i = 0 j K i ( x , t ) = i = j + 1 K i ( x , t ) i = j + 1 K i ( x , t ) i = j + 1 i K 0 ( x , t ) K 0 ( x , t ) j + 1 ( 1 + + 2 + ) j + 1 1 K 0 ( x , t ) .We obtain the required result. □
The analytical and the numerical solutions of the fractional GKSE are illustrated in the following section.

5. Applications

Example 1 Consider Eqs. (50) to (52), where α = 2 , β = 1 , γ = 0 , a n d F ( x , t ) = 0 .
C D t ( ξ ) W ( x , t ) + W W x + 2 2 W x 2 + 3 W x 3 = 0
C F C D t ( ξ ) W ( x , t ) + W W x + 2 2 W x 2 + 3 W x 3 = 0
A B C D t ( ξ ) W ( x , t ) + W W x + 2 2 W x 2 + 3 W x 3 = 0 .
with the initial condition
W ( x , 0 ) = 418 11 270 361 418 tanh ( 418 38 x ) + 330 361 418 tanh 3 ( 418 38 x ) .
Applying the J-transform of the Caputo, the Caputo-Fabrizio, and the Atangana-Baleanu fractional derivative on Eqs. (77), (78), and (79) we get
( s u ) ξ J [ W ( x , t ) ] s ξ 1 u ξ 2 W ( x , 0 ) + S [ W W x + 2 2 W x 2 + 3 W x 3 ] = 0 ,
u M ( ξ ) s + ξ ( u s ) [ s u J [ W ( x , t ) ] ( s , u ) u W ( x , 0 ) ] + J [ W W x + 2 2 W x 2 + 3 W x 3 ] = 0 ,
u M ( ξ ) s ξ + ξ ( u ξ s ξ ) [ ( s u ) ξ J [ W ( x , t ) ] ( s , u ) u W ( x , 0 ) ] + J [ W W x + 2 2 W x 2 + 3 W x 3 ] = 0 ,
respectively.
Simplifying Eqs. (81) to (83), gives
J [ W ( x , t ) ] u 2 s W ( x , 0 ) + Ψ ( ) J [ W W x + 2 2 W x 2 + 3 W x 3 ] = 0 ,
where
Ψ ( C ) = u ξ s ξ , Ψ ( C F C ) = s + ξ ( u s ) M ( ξ ) s ,and
Ψ ( A B C ) = s ξ + ξ ( u ξ s ξ ) M ( ξ ) s ξ .
We defined the nonlinear operator as
N * [ Θ ( x , t ; q ) ] = J [ Θ ( x , t ; q ) ] u 2 s Θ ( x , 0 ) + Ψ ( ) J [ Θ Θ x + 2 2 Θ x 2 + 3 Θ x 3 ] .
Now the 0th-order deformation equation is
( 1 q ) J [ Θ ( x , t ; q ) W 0 ( x , t ) ] = q H ^ ( x , t ) N * [ Θ ( x , t ) ] ,
Multiplying Eq. (86) with 1 Γ ( m + 1 ) after m-times differentiation with respect to q, and setting q = 0, we get the higher order ( Mth) deformation equation as
J [ W m ( x , t ) χ ^ m W m 1 ( x , t ) ] = H ^ ( x , t ) R ^ m ( W m 1 , x , t ) ,
where
R ^ m ( W m 1 , x , t ) = J ( W m 1 ( x , t ) ) ( 1 χ ^ m ) u 2 s ( 418 11 270 361 418 tanh ( 418 38 x ) + 330 361 418 tanh 3 ( 418 38 x ) ) + Ψ ( ) J ( i = 0 m 1 W i W m 1 i x + 2 2 W m 1 x 2 + 3 W m 1 x 3 ) .
Inverting Eq. (87), we get
W m ( x , t ) = χ ^ m W m 1 ( x , t ) + J 1 ( H ^ ( x , t ) R ^ m ( W m 1 , x , t ) ) .
Upon solving Eq. (89) with H ^ ( x , t ) = M ( ξ ) = 1, we get the following approximations
W 0 ( ) ( x , t ) = 418 11 270 361 418 tanh ( 418 38 x ) + 330 361 418 tanh 3 ( 418 38 x ) W 1 C ( x , t ) = 1 260642 Γ ( 1 + ξ ) 45 t ξ sech ( 11 38 x ) 7 ( 19 ( 17424 + 247 418 ) cosh ( 11 38 x ) + ( 211508 361 418 ) cosh ( 3 11 38 x ) 9196 cosh ( 5 11 38 x ) + 722 418 cosh ( 5 11 38 x ) 133584 418 sinh ( 11 38 x ) + 25652 418 sinh ( 3 11 38 x ) 484 418 sinh ( 5 11 38 x ) ) cosh ( 5 11 38 x ) 722 418 cosh ( 5 11 38 x ) + 133584 418 sinh ( 11 38 x ) 25652 418 sinh ( 3 11 38 x ) + 484 418 sinh ( 5 11 38 x ) ) + 1 Γ ( 1 + 2 ξ ) t ξ × sech 4 ( 11 38 x ) ( 110352 ( 865766 + 3743 418 ) cosh ( 11 38 x ) + 73568 × ( 1393601 + 1748 418 ) cosh ( 3 11 38 x ) 18989004320 cosh ( 5 11 38 x ) + 132790240 418 cosh ( 5 11 38 x ) + 606733688 cosh ( 7 11 38 x ) 19044916 418 cosh ( 7 11 38 x ) 2225432 cosh ( 9 11 38 x ) + 174724 418 cosh ( 9 11 38 x ) + 1307983864 sinh ( 11 38 x ) 16746447147 418 sinh ( 11 38 x ) + 1706354584
sinh ( 3 11 18 x ) + 5841551043 418 sinh ( 3 11 38 x ) + 232382920 sinh ( 5 11 38 x ) 589091385 418 sinh ( 5 11 38 x ) 162668044 sinh ( 7 11 38 x ) + 14014962 418 sinh ( 7 11 38 x ) + 3319756 sinh ( 9 11 38 x ) 239463 418 sinh ( 9 11 38 x ) ) ) W 1 C F C ( x , t ) = 1 260642 45 ( 1 ξ + t ξ ) sech 7 ( 11 38 x ) ( 19 ( 17424 + 247 418 ) cosh ( 11 38 x ) + ( 211508 361 418 ) cosh ( 3 11 38 x ) 9196 cosh ( 5 11 38 x ) + 722 418 cosh ( 5 11 38 x ) 133584 418 sinh ( 11 38 x ) + 25652 418 sinh ( 3 11 38 x ) 484 418 sinh ( 5 11 38 x ) ) W 2 C F C ( x , t ) = 1 1505468192 45 sech 11 ( 11 38 x ) ( 19 ( 361 ( 119064 + 2527 418 ) ( 1 + ( 1 + t ) ξ ) + ( 20070493608 + 86045129 418 + 5 ( 8036793864 + 34600501 418 ) ( 1 + t ) ξ + 11616 ( 865766 + 3743 418 ) ( 2 4 t + t 2 ) ξ 2 ) ) cosh ( 11 38 x ) + 19 ( 361 ( 22264 + 1197 418 ) ( 1 + ( 1 + t ) ξ ) + ( 21592129592 + 27505141 418 + 5 ( 8635244376 + 10915633 418 ) ( 1 + t ) ξ + 7744 ( 1393601 + 1748 418 ) ( 2 4 t + t 2 ) ξ 2 ) ) cosh ( 3 11 38 x ) 165987800 cosh ( 5 11 38 x ) + 651605 418 cosh ( 5 11 38 x ) 76122005080 cosh ( 5 11 38 x )
+ 531812565 418 cosh ( 5 11 38 x ) + 165987800 ξ cosh ( 5 11 38 x ) + 651605 418 cosh ( 5 11 38 x ) 76122005080 cosh ( 5 11 38 x ) + 531812565 418 h cosh ( 5 11 38 x ) + 165987800 ξ cosh ( 5 11 38 x ) 651605 418 ξ cosh ( 5 11 38 x ) + 152078022360 ξ cosh ( 5 11 38 x ) 1062973525 418 ξ cosh ( 5 11 38 x ) 165987800 t ξ cosh ( 5 11 38 x ) + 651605 418 t ξ cosh ( 5 11 38 x ) 152078022360 t ξ cosh ( 5 11 38 x ) + 1062973525 418 t ξ cosh ( 5 11 38 x ) 75956017280 ξ 2 cosh ( 5 11 38 x ) + 531160960 418 ξ 2 cosh ( 5 11 38 x ) + 151912034560 t ξ 2 cosh ( 5 11 38 x ) 1062321920 418 t ξ 2 cosh ( 5 11 38 x ) 37978008640 t 2 ξ 2 cosh ( 5 11 38 x ) + 265580480 418 t 2 ξ 2 cosh ( 5 11 38 x ) 63075364 cosh ( 7 11 38 x ) 912247 418 cosh ( 7 11 38 x ) + 2363859388 cosh ( 7 11 38 x ) 77091911 418 cosh ( 7 11 38 x ) + 63075364 ξ cosh ( 7 11 38 x ) + 912247 418 ξ cosh ( 7 11 38 x ) 4790794140 ξ cosh ( 7 11 38 x ) + 153271575 418 ξ cosh ( 7 11 38 x ) 63075364 t ξ cosh ( 7 11 38 x ) 912247 418 t ξ cosh ( 7 11 38 x ) + 4790794140 t ξ cosh ( 7 11 38 x )
153271575 418 t ξ cosh ( 7 11 38 x ) + 2426934752 ξ 2 cosh ( 7 11 38 x ) 76179664 418 ξ 2 cosh ( 7 11 38 x ) 4853869504 t ξ 2 cosh ( 7 11 38 x ) + 152359328 418 t ξ 2 cosh ( 7 11 38 x ) + 1213467376 t 2 ξ 2 cosh ( 7 11 38 x ) 38089832 418 t 2 ξ 2 cosh ( 7 11 38 x ) + 3319756 cosh ( 9 11 38 x ) 260642 418 cosh ( 9 11 38 x ) 5581972 cosh ( 9 11 38 x ) + 438254 418 cosh ( 9 11 38 x ) 3319756 ξ cosh ( 9 11 38 x ) + 260642 418 ξ cosh ( 9 11 38 x ) + 14483700 ξ cosh ( 9 11 38 x ) 1137150 418 ξ cosh ( 9 11 38 x ) + 3319756 t ξ cosh ( 9 11 38 x ) 260642 418 t ξ cosh ( 9 11 38 x ) 14483700 t ξ cosh ( 9 11 38 x ) + 1137150 418 t ξ cosh ( 9 11 38 x ) 8901728 ξ 2 cosh ( 9 11 38 x ) + 698896 418 ξ 2 cosh ( 9 11 38 x ) + 17803456 t ξ 2 cosh ( 9 11 38 x ) 1397792 418 t ξ 2 cosh ( 9 11 38 x ) 4450864 t 2 ξ 2 cosh ( 9 11 38 x ) + 349448 418 t 2 ξ 2 cosh ( 9 11 38 x ) + 68841256 418 sinh ( 11 38 x ) + 5231935456 sinh ( 11 38 x ) 66916947332 418 sinh ( 11 38 x )
68841256 418 ξ sinh ( 11 38 x ) 10463870912 ξ sinh ( 11 38 x ) + 133902735920 418 ξ sinh ( 11 38 x ) + 68841256 418 t ξ sinh ( 11 38 x ) + 10463870912 t ξ sinh ( 11 38 x ) 133902735920 418 t ξ sinh ( 11 38 x ) + 5231935456 ξ 2 sinh ( 11 38 x ) 66985788588 418 ξ 2 sinh ( 11 38 x ) 10463870912 t ξ 2 sinh ( 11 38 x ) + 133971577176 418 t ξ 2 sinh ( 11 38 x ) + 2615967728 t 2 ξ 2 sinh ( 11 38 x ) 33492894294 418 h t 2 ξ 2 sinh ( 11 38 x ) + 89808136 418 sinh ( 3 11 38 x ) + 6825418336 sinh ( 3 11 38 x ) + 23456012308 418 sinh ( 3 11 38 x ) 89808136 418 ξ sinh ( 3 11 38 x ) 13650836672 ξ sinh ( 3 11 38 x ) 46822216480 418 ξ sinh ( 3 11 38 x ) + 89808136 418 t ξ sinh ( 3 11 38 x ) + 13650836672 t ξ sinh ( 3 11 38 x ) + 46822216480 418 t ξ sinh ( 3 11 38 x ) + 6825418336 ξ 2 sinh ( 3 11 38 x ) + 23366204172 418 ξ 2 sinh ( 3 11 38 x ) 13650836672 t ξ 2 sinh ( 3 11 38 x ) 46732408344 418 t ξ 2 sinh ( 3 11 38 x ) + 3412709168 t 2 ξ 2 sinh ( 3 11 38 x ) + 11683102086 418 t 2 ξ 2 sinh ( 3 11 38 x ) + 12230680 418 sinh ( 5 11 38 x ) +
929531680 sinh ( 5 11 38 x ) 2344134860 418 sinh ( 5 11 38 x ) 12230680 418 ξ sinh ( 5 11 38 x ) 1859063360 ξ sinh ( 5 11 38 x ) + 4700500400 418 ξ sinh ( 5 11 38 x ) + 12230680 418 t ξ sinh ( 5 11 38 x ) + 1859063360 t ξ sinh ( 5 11 38 x ) 4700500400 418 t ξ sinh ( 5 11 38 x ) + 929531680 ξ 2 sinh ( 5 11 38 x ) 2356365540 418 ξ 2 sinh ( 5 11 38 x ) 1859063360 t ξ 2 sinh ( 5 11 38 x ) + 4712731080 418 t ξ 2 sinh ( 5 11 38 x ) + 464765840 t 2 ξ 2 sinh ( 5 11 38 x ) 1178182770 418 t 2 ξ 2 sinh ( 5 11 38 x ) 8561476 418 sinh ( 7 11 38 x ) 650672176 sinh ( 7 11 38 x ) + 47498372 418 sinh ( 7 11 38 x ) + 8561476 418 ξ sinh ( 7 11 38 x ) + 1301344352 ξ sinh ( 7 11 38 x ) 103558220 418 ξ sinh ( 7 11 38 x ) 8561476 418 t ξ sinh ( 7 11 38 x ) 1301344352 t ξ sinh ( 7 11 38 x ) + 103558220 418 t ξ sinh ( 7 11 38 x ) 650672176 ξ 2 sinh ( 7 11 38 x ) + 56059848 418 ξ 2 sinh ( 7 11 38 x ) + 1301344352 t ξ 2 sinh ( 7 11 38 x ) 112119696 418 t ξ 2 sinh ( 7 11 38 x ) 325336088 t 2 ξ 2 sinh ( 7 11 38 x ) +
28029924 418 t 2 ξ 2 sinh ( 7 11 38 x ) + 174724 418 sinh ( 9 11 38 x ) + 13279024 sinh ( 9 11 38 x ) 783128 418 sinh ( 9 11 38 x ) 174724 418 ξ sinh ( 9 11 38 x ) 26558048 ξ sinh ( 9 11 38 x ) + 1740980 418 ξ sinh ( 9 11 38 x ) + 174724 418 t ξ sinh ( 9 11 38 x ) + 26558048 t ξ sinh ( 9 11 38 x ) 1740980 418 t ξ sinh ( 9 11 38 x ) + 13279024 ξ 2 sinh ( 9 11 38 x ) 957852 418 ξ 2 sinh ( 9 11 38 x ) 26558048 t ξ 2 sinh ( 9 11 38 x ) + 1915704 418 t ξ 2 sinh ( 9 11 38 x ) + 6639512 t 2 ξ 2 sinh ( 9 11 38 x ) 478926 418 t 2 ξ 2 sinh ( 9 11 38 x ) ) W 1 A B C ( x , t ) = 1 260642 45 ( 1 + ξ t ξ ξ Γ ( 1 + ξ ) ) sech 7 ( 11 38 x ) ( 19 ( 17424 + 247 418 ) cosh ( 11 38 x ) + ( 211508 361 418 ) cosh ( 3 11 38 x ) 9196 cosh ( 5 11 38 x ) + 722 418 cosh ( 5 11 38 x ) 133584 418 sinh ( 11 38 x ) + 25652 418 sinh ( 3 11 38 x ) 484 418 sinh ( 5 11 38 x ) ) W 2 A B C ( x , t ) = 1 1505468192 Γ ( 1 + ξ ) Γ ( 1 + 2 ξ ) 45 sech 11 ( 11 38 x )
( t ξ ξ Γ ( 1 + 2 ξ ) ( 19 ( 361 ( 119064 + 2527 418 ) 5 ( 8036793864 + 34600501 418 ) + 46464 ( 865766 + 3743 418 ) ξ ) cosh ( 11 38 x ) 19 ( 361 ( 22264 + 1197 418 ) 5 ( 8635244376 + 10915633 418 ) + 30976 ( 1393601 + 1748 418 ) ξ ) cosh ( 3 11 38 x ) 165987800 cosh ( 5 11 38 x ) + 651605 418 cosh ( 5 11 38 x ) 152078022360 cosh ( 5 11 38 x ) + 1062973525 418 cosh ( 5 11 38 x ) + 151912034560 ξ cosh ( 5 11 38 x ) 1062321920 418 ξ cosh ( 5 11 38 x ) 63075364 cosh ( 7 11 38 x ) 912247 418 cosh ( 7 11 38 x ) + 4790794140 cosh ( 7 11 38 x ) 153271575 418 cosh ( 7 11 38 x ) 4853869504 ξ cosh ( 7 11 38 x ) + 152359328 418 ξ cosh ( 7 11 38 x ) + 3319756 cosh ( 9 11 38 x ) 260642 418 cosh ( 9 11 38 x ) 14483700 cosh ( 9 11 38 x ) + 1137150 418 cosh ( 9 11 38 x ) + 17803456 ξ cosh ( 9 11 38 x ) 1397792 418 ξ cosh ( 9 11 38 x ) + 68841256 418 sinh ( 11 38 x ) + 10463870912 sinh ( 11 38 x ) 133902735920 418 sinh ( 11 38 x ) 10463870912 ξ sinh ( 11 38 x ) + 133971577176 418 ξ sinh ( 11 38 x ) +
89808136 418 sinh ( 3 11 38 x ) + 13650836672 sinh ( 3 11 38 x ) + 46822216480 418 sinh ( 3 11 38 x ) 13650836672 ξ sinh ( 3 11 38 x ) 46732408344 418 ξ sinh ( 3 11 38 x ) + 12230680 418 sinh ( 5 11 38 x ) + 1859063360 sinh ( 5 11 38 x ) 4700500400 418 sinh ( 5 11 38 x ) 1859063360 ξ sinh ( 5 11 38 x ) + 4712731080 418 ξ sinh ( 5 11 38 x ) 8561476 418 sinh ( 7 11 38 x ) 1301344352 sinh ( 7 11 38 x ) + 103558220 418 sinh ( 7 11 38 x ) + 1301344352 ξ sinh ( 7 11 38 x ) 112119696 418 ξ sinh ( 7 11 38 x ) + 174724 418 sinh ( 9 11 38 x ) + 26558048 sinh ( 9 11 38 x ) 1740980 418 sinh ( 9 11 38 x ) 26558048 ξ sinh ( 9 11 38 x ) + 1915704 418 h ξ sinh ( 9 11 38 x ) ) + Γ ( 1 + ξ ) ( 4 t 2 ξ ξ 2 ( 110352 ( 865766 + 3743 418 ) cosh ( 11 38 x ) + 73568 ( 1393601 + 1748 418 ) cosh ( 3 11 38 x ) 18989004320 cosh ( 5 11 38 x ) + 132790240 418 cosh ( 5 11 38 x ) + 606733688 cosh ( 7 11 38 x ) 19044916 418 cosh ( 7 11 38 x ) 2225432 cosh ( 9 11 38 x ) +
174724 418 cosh ( 9 11 38 x ) + 1307983864 sinh ( 11 38 x ) 16746447147 418 sinh ( 11 38 x ) + 1706354584 sinh ( 3 11 38 x ) + 5841551043 418 sinh ( 3 11 38 x ) + 232382920 sinh ( 5 11 38 x ) 589091385 418 sinh ( 5 11 38 x ) 162668044 sinh ( 7 11 38 x ) + 14014962 418 sinh ( 7 11 38 x ) + 3319756 sinh ( 9 11 38 x ) 239463 418 sinh ( 9 11 38 x ) ) ( 1 + ξ ) Γ ( 1 + 2 ξ ) ( 19 ( 361 ( 119064 + 2527 418 ) + ( 20070493608 86045129 418 + 23232 ( 865766 + 3743 418 ) ξ ) ) cosh ( 11 38 x ) 19 ( 361 ( 22264 + 1197 418 ) + ( 21592129592 27505141 418 + 15488 ( 1393601 + 1748 418 ) ξ ) ) cosh ( 3 11 38 x ) 165987800 cosh ( 5 11 38 x ) + 651605 418 cosh ( 5 11 38 x ) 76122005080 cosh ( 5 11 38 x ) + 531812565 418 cosh ( 5 11 38 x ) + 75956017280 ξ cosh ( 5 11 38 x ) 531160960 418 ξ cosh ( 5 11 38 x ) 63075364 cosh ( 7 11 38 x ) 912247 418 cosh ( 7 11 38 x ) +
2363859388 cosh ( 7 11 38 x ) 77091911 418 cosh ( 7 11 38 x ) 2426934752 ξ cosh ( 7 11 38 x ) + 76179664 418 ξ cosh ( 7 11 38 x ) + 3319756 cosh ( 9 11 38 x ) 260642 418 cosh ( 9 11 38 x ) 5581972 cosh ( 9 11 38 x ) + 438254 418 cosh ( 9 11 38 x ) + 8901728 ξ cosh ( 9 11 38 x ) 698896 418 ξ cosh ( 9 11 38 x ) + 68841256 418 sinh ( 11 38 x ) + 5231935456 sinh ( 11 38 x ) 66916947332 418 sinh ( 11 38 x ) 5231935456 ξ sinh ( 11 38 x ) + 66985788588 418 ξ sinh ( 11 38 x ) + 89808136 418 sinh ( 3 11 38 x ) + 6825418336 sinh ( 3 11 38 x ) + 23456012308 418 sinh ( 3 11 38 x ) 6825418336 ξ sinh ( 3 11 38 x ) 23366204172 418 ξ sinh ( 3 11 38 x ) + 12230680 418 sinh ( 5 11 38 x ) + 929531680 sinh ( 5 11 38 x ) 2344134860 418 sinh ( 5 11 38 x ) 929531680 ξ sinh ( 5 11 38 x ) + 2356365540 418 ξ sinh ( 5 11 38 x ) 8561476 418 sinh ( 7 11 38 x ) 650672176 sinh ( 7 11 38 x ) + 47498372 418 sinh ( 7 11 38 x ) +
650672176 ξ sinh ( 7 11 38 x ) 56059848 418 ξ sinh ( 7 11 38 x ) + 174724 418 sinh ( 9 11 38 x ) + 13279024 sinh ( 9 11 38 x ) 783128 418 sinh ( 9 11 38 x ) 13279024 ξ sinh ( 9 11 38 x ) + 957852 418 ξ sinh ( 9 11 38 x ) ) ) )and so on.
Using the same procedure we obtain the remaining terms. Thus, the approximate solution of Eqs. (77), (78), and (79) using the proposed HAJTM are given by
{ W ( C ) ( x , t ) = W 0 ( C ) ( x , t ) + W 1 ( C ) ( x , t ) + W 2 ( C ) ( x , t ) + W ( C F C ) ( x , t ) = W 0 ( C F C ) ( x , t ) + W 1 ( C F C ) ( x , t ) + W 2 ( C F C ) ( x , t ) + W ( A B C ) ( x , t ) = W 0 ( A B C ) ( x , t ) + W 1 ( A B C ) ( x , t ) + W 2 ( A B C ) ( x , t ) +
As M , ξ = 1 , = 1 , the exact solution of Eqs. (77) to (79) is given by
W ( x , t ) = 1 ζ + 60 19 ζ ( 38 β ζ 2 + α ) tanh ( Θ ) + 120 β ζ 3 tanh 3 ( Θ ) ,
where Θ = ζ x + t and ζ = 0.5 22 19.
Example 2 Consider Eqs. (50) to (52), where α = β = 1 , γ = 4 , a n d F ( x , t ) = 0 .
C D t ( ξ ) W ( x , t ) + W W x + 2 W x 2 + 3 W x 3 + 4 4 W x 4 = 0
C F C D t ( ξ ) W ( x , t ) + W W x + 2 W x 2 + 3 W x 3 + 4 4 W x 4 = 0
A B C D t ( ξ ) W ( x , t ) + W W x + 2 W x 2 + 3 W x 3 + 4 4 W x 4 = 0 .
with the initial condition
W ( x , 0 ) = 11 + 15 tanh ( x 2 ) 15 tanh 2 ( x 2 ) 15 tanh 3 ( x 2 ) .
Computing the J-transform of the Caputo, the Caputo-Fabrizio, and the Atangana-Baleanu fractional derivative on Eqs. (92), (93), and (94) we get
( s u ) ξ J [ W ( x , t ) ] s ξ 1 u ξ 2 W ( x , 0 ) + S [ W W x + 2 W x 2 + 3 W x 3 + 4 4 W x 4 ] = 0 ,
u M ( ξ ) s + ξ ( u s ) [ s u J [ W ( x , t ) ] ( s , u ) u W ( x , 0 ) ] + J [ W W x + 2 W x 2 + 3 W x 3 + 4 4 W x 4 ] = 0 ,
u M ( ξ ) s ξ + ξ ( u ξ s ξ ) [ ( s u ) ξ J [ W ( x , t ) ] ( s , u ) u W ( x , 0 ) ] + J [ W W x + 2 W x 2 + 3 W x 3 + 4 4 W x 4 ] = 0 ,
respectively.
Table 1. Comparison of the numerical solutions of Eqs. (77) to (79) using the HAJTM and the LVIM [41] at x = 5.5 , ξ = 1 , 0.85 , 0.95 and different time intervals.
HAJTM LVIM [41] & HAJTM HAJTM HAJTM
Exact solutions ABC CFC C
t ξ = 1 ξ = 0.85 ξ = 0.95 ξ = 0.85 ξ = 0.95 ξ = 0.85 ξ = 0.95
0.1 1.36131 1.31202 1.31402 1.31273 1.31455 1.31377 1.31532
0.2 1.39338 1.31089 1.31065 1.31122 1.31111 1.30952 1.31063
0.3 1.41971 1.31065 1.30856 1.31035 1.30874 1.30736 1.30755
0.4 1.44131 1.31101 1.30754 1.31004 1.30734 1.30668 1.30583
0.5 1.45903 1.31178 1.30742 1.31022 1.30683 1.30711 1.30527
0.6 1.47356 1.31284 1.30806 1.31083 1.30711 1.30838 1.30569
0.7 1.48547 1.31407 1.30933 1.3118 1.30808 1.31029 1.30696
0.8 1.49523 1.31539 1.31112 1.31307 1.30964 1.31266 1.30893
0.9 1.50323 1.3167 1.31332 1.31455 1.3117 1.31533 1.31147
1 1.50978 1.31794 1.31582 1.31619 1.31418 1.31817 1.31445
Table 2. Comparison of the numerical simulations of Eqs. (92) to (94) using the HAJTM and the LVIM [41], at x = 6.5 , ξ = 1 , 0.85 , 0.95 and different time intervals.
HAJTM LVIM [41] & HAJTM HAJTM HAJTM
Exact solutions ABC CFC C
t ξ = 1 ξ = 0.85 ξ = 0.95 ξ = 0.85 ξ = 0.95 ξ = 0.85 ξ = 0.95
0.1 -3.9996 -4.74491 -4.08462 -4.47938 -4.06525 -4.02729 -4.00794
0.2 -3.9994 -5.61008 -4.29293 -4.99779 -4.22522 -4.1884 -4.07518
0.3 -3.99911 -6.7831 -4.67612 -5.72863 -4.52608 -4.56248 -4.26213
0.4 -3.99867 -8.28573 -5.27878 -6.70216 -5.01007 -5.20899 -4.62403
0.5 -3.99802 -10.1381 -6.14383 -7.94862 -5.71942 -6.17822 -5.21314
0.6 -3.99705 -12.3593 -7.31289 -9.49826 -6.69634 -7.51478 -6.07971
0.7 -3.99562 -14.9676 -8.82653 -11.3813 -7.98308 -9.25925 -7.27238
0.8 -3.99349 -17.9806 -10.7244 -13.6281 -9.62186 -11.4492 -8.8386
0.9 -3.99034 -21.4152 -13.0456 -16.2687 -11.6549 -14.1197 -10.8248
1 -3.98567 -25.2879 -15.8282 -19.3335 -14.1244 -17.3039 -13.2763
After simplifying Eqs. (96) to (98), we get
J [ W ( x , t ) ] u 2 s W ( x , 0 ) + Ψ ( ) J [ W W x + 2 W x 2 + 3 W x 3 + 4 4 W x 4 ] = 0 ,
where
Ψ ( C ) = u ξ s ξ , Ψ ( C F C ) = s + ξ ( u s ) M ( ξ ) s ,and
Ψ ( A B C ) = s ξ + ξ ( u ξ s ξ ) M ( ξ ) s ξ .
We defined the nonlinear operator as
N * [ Θ ( x , t ; q ) ] = J [ Θ ( x , t ; q ) ] u 2 s Θ ( x , 0 ) + Ψ ( ) J ( Θ Θ x + 2 Θ x 2 + 3 Θ x 3 + 4 4 Θ x 4 ) .
The 0th-order deformation equation is
( 1 q ) J ( Θ ( x , t ; q ) W 0 ( x , t ) ) = q H ^ ( x , t ) N * ( Θ ( x , t ) ) .
Fig. 5. Comparison of the analytical and numerical solutions of the FGKS Eq. (93) using the Caputo-Fabrizio fractional derivative for different x , t and ξ s.
Multiplying Eq. (101) with 1 Γ ( m + 1 ) after m-times differentiation with respect to q, and setting q = 0, we get the higher order ( Mth) deformation equation as
J [ W m ( x , t ) χ ^ m W m 1 ( x , t ) ] = H ^ ( x , t ) R ^ m ( W m 1 , x , t ) ,
where
R ^ m ( W m 1 , x , t ) = J [ W m 1 ( x , t ) ] ( 1 χ ^ m ) u 2 s ( 11 + 15 tanh ( x 2 ) 15 tanh 2 ( x 2 ) 15 tanh 3 ( x 2 ) ) + Ψ ( ) J ( i = 0 m 1 W i W m 1 i x + 2 W m 1 x 2 + 3 W m 1 x 3 + 4 4 W m 1 x 4 ) .
Inverting Eq. (102), we get
W m ( x , t ) = χ ^ m W m 1 ( x , t ) + J 1 ( H ^ ( x , t ) R ^ m ( W m 1 , x , t ) ) .
Upon solving Eq. (104) with H ^ ( x , t ) = M ( ξ ) = 1, we get the following approximations
W 0 ( ) ( x , t ) = 11 + 15 tanh ( x 2 ) 15 tanh 2 ( x 2 ) 15 tanh 3 ( x 2 ) W 1 C ( x , t ) = 240 exp ( x ) ( 1 35 exp ( x ) + 231 exp ( 2 x ) 239 exp ( 3 x ) + 34 exp ( 4 x ) ) t ξ ( 1 + exp ( x ) ) 7 Γ ( 1 + ξ ) W 2 C ( x , t ) = 240 exp ( x ) t ξ ( 1 + exp ( x ) ) 11 ( 1 Γ ( 1 + ξ ) ( 1 + exp ( x ) ) 4 ( 1 35 exp ( x ) + 231 exp ( 2 x ) 239 exp ( 3 x ) + 34 exp ( 4 x ) ) ( 1 + ) + 2 t ξ Γ ( 1 + 2 ξ ) ( 1 1297 exp ( x ) + 65263 exp ( 2 x ) 632059 exp ( 3 x ) + 1852225 exp ( 4 x ) 1866349 exp ( 5 x ) + 637723 exp ( 6 x ) 63127 exp ( 7 x ) + 1156 exp ( 8 x ) ) ) W 1 C F C ( x , t ) = 240 exp ( x ) ( 1 ξ + t ξ ) ( 1 + exp ( x ) ) 7 ( 1 35 exp ( x ) + 231 exp ( 2 x ) 239 exp ( 3 x ) + 34 exp ( 4 x ) ) W 2 C F C ( x , t ) = 1 ( 1 + exp ( x ) ) 11 240 exp ( x ) ( 1 ξ + t ξ + ( 3 + 5 ( 1 + t ) ξ + ( 2 4 t + t 2 ) ξ 2 ) + 34 exp ( 8 x ) ( 1 + ( 1 + t ) ξ + ( 69 + 137 ( 1 + t ) ξ + 34 ( 2 4 t + t 2 ) ξ 2 ) ) exp ( x ) ( 31 ( 1 + ( 1 + t ) ξ ) + ( 2625 + 5219 ( 1 + t ) ξ + 1297 ( 2 4 t + t 2 ) ξ 2 ) ) exp ( 7 x ) ( 103 ( 1 + ( 1 + t ) ξ ) + ( 126357 + 252611 ( 1 + t ) ξ + 63127 ( 2 4 t + t 2 ) ξ 2 ) ) + exp ( 2 x ) ( 97 ( 1 + ( 1 + t ) ξ ) + ( 130623 + 261149 ( 1 + t ) ξ + 65263 ( 2 4 t + t 2 ) ξ 2 ) ) + 25 exp ( 4 x ) ( 13 ( 1 + ( 1 + t ) ξ ) + ( 148191 + 296369 ( 1 + t ) ξ + 74089 ( 2 4 t + t 2 ) ξ 2 ) )
+ exp ( 3 x ) ( 479 ( 1 + ( 1 + t ) ξ ) ( 1263639 + 2527757 ( 1 + t ) ξ + 632059 ( 2 4 t + t 2 ) ξ 2 ) ) + exp ( 6 x ) ( 521 ( 1 + ( 1 + t ) ξ ) + ( 1274925 + 2550371 ( 1 + t ) ξ + 637723 ( 2 4 t + t 2 ) ξ 2 ) ) exp ( 5 x ) ( 409 ( 1 + ( 1 + t ) ξ ) + ( 3733107 + 7465805 ( 1 + t ) ξ + 1866349 ( 2 4 t + t 2 ) ξ 2 ) ) ) W 1 A B C ( x , t ) = 1 ( 1 + exp ( x ) ) 7 240 exp ( x ) ( 1 + exp ( x ) ( 35 + exp ( x ) ( 231 + exp ( x ) ( 239 + 34 exp ( x ) ) ) ) ) ( 1 ξ + t ξ ξ Γ ( 1 + ξ ) ) W 2 A B C ( x , t ) = 1 ( 1 + exp ( x ) ) 11 240 exp ( x ) ( ( 1 + exp ( x ) ) 4 ( 1 + exp ( x ) ( 35 + exp ( x ) ( 231 + exp ( x ) ( 239 + 34 exp ( x ) ) ) ) ) ( 1 ξ + t ξ ξ Γ ( 1 + ξ ) ) + ( 3 5 ξ + 2 ξ 2 + exp ( 5 x ) ( 3733107 + 7465805 ξ 3732698 ξ 2 ) + exp ( 3 x ) ( 1263639 + 2527757 ξ 1264118 ξ 2 ) + exp ( 7 x ) ( 126357 + 252611 ξ 126254 ξ 2 ) + exp ( x ) ( 2625 + 5219 ξ 2594 ξ 2 ) + 34 exp ( 8 x ) ( 69 137 ξ + 68 ξ 2 ) + exp ( 2 x ) ( 130623 261149 ξ + 130526 ξ 2 ) + 25 exp ( 4 x ) ( 148191 296369 ξ + 148178 ξ 2 ) + exp ( 6 x ) ( 1274925 2550371 ξ + 1275446 ξ 2 ) 1 Γ ( 1 + ξ ) t ξ ξ ( 5 + exp ( 5 x ) ( 7465805 7465396 ξ ) + exp ( 3 x ) ( 2527757 2528236 ξ ) + exp ( 7 x ) ( 252611 252508 ξ ) + exp ( x ) ( 5219 5188 ξ ) + 4 ξ + 34 exp ( 8 x ) ( 137 + 136 ξ ) + exp ( 2 x ) ( 261149 + 261052 ξ ) + 25 exp ( 4 x ) ( 296369 + 296356 ξ ) +
exp ( 6 x ) ( 2550371 + 2550892 ξ ) ) + 1 Γ ( 1 + 2 ξ ) 2 ( 1 1297 exp ( x ) + 65263 exp ( 2 x ) 632059 exp ( 3 x ) + 1852225 exp ( 4 x ) 1866349 exp ( 5 x ) + 637723 exp ( 6 x ) 63127 exp ( 7 x ) + 1156 exp ( 8 x ) ) t 2 ξ ξ 2 ) ) and so on.
Using the same procedure we obtain the remaining terms. Thus, the approximate solution of (93), (93), and (94) using the proposed HAJTM are given by
{ W ( C ) ( x , t ) = W 0 ( C ) ( x , t ) + W 1 ( C ) ( x , t ) + W 2 ( C ) ( x , t ) + W ( C F C ) ( x , t ) = W 0 ( C F C ) ( x , t ) + W 1 ( C F C ) ( x , t ) + W 2 ( C F C ) ( x , t ) + W ( A B C ) ( x , t ) = W 0 ( A B C ) ( x , t ) + W 1 ( A B C ) ( x , t ) + W 2 ( A B C ) ( x , t ) +
As M , ξ = 1 , = 1 , the exact solution of Eqs. (92) to (94) is given by
W ( x , t ) = 11 + 15 tanh ( x 2 + t ) 15 tanh 2 ( x 2 + t ) 15 tanh 3 ( x 2 + t ) .

5.1. Results and discussion

We discuss the analytical and numerical simulations of the fractional generalized Kuramoto-Sivashinky equation via the Atangana-Baleanu (AB), the Caputo-Fabrizio (CF), and the Caputo (C) fractional derivatives respectively. Figs. 1 to 6, show the result of the fractional GKSE obtained via none-singular kernel exhibits a new features or characteristic compared to the well-known Caputo fractional derivative. Moreover, the results of the Atangana-Baleanu fractional derivative is in excellent agreement with the Caputo fractional derivative. In Fig. 1(a) to (h), we plot the graphical solution behavior of Eq. (77) for different values of ξ at different time interval in the Caputo sense. When ξ = 1 , 0.45 and = 1, the graphical solutions of Eq. (77) are presented in Fig. 1(a) and (b) respectively. The solutions of the fractional FGKSE for ξ = 0.5 , = 1 and ξ = 0.65 , = 1.5 are given in Fig. 1(c) and (d) respectively. In Fig. 1(e) and (f), the graphical solutions of the FGKS Eq. (77) is depicted for ξ = 0.75 , 0.95 , = 1.5. Besides, the exact solutions of Eqs. (77) to (79) were compared in Fig. 1(g). In Fig. 1(h), the approximate solutions of Eqs. (77) to (79) are compared when ξ = 0.25 and = 1 at different time intervals. The 3D error analysis of the FGKS E 20 ( W ( x , t ) ) = | W e x t . ( x , t ) W a p p r . ( x , t ) | is presented in Fig. 1(i). The 2D error estimate of the FGKS Eq. (77) is provided in Fig. 1(j). Moreover, we observe that the increase in the values of ξ result in the increase in the turbulence behavior and vice-versa. The numerical simulations behavior of Eqs. (77) to (79) at x = 5.5 and different values of ξ were computed in Table 1, and the obtained results are in excellent agreement with all the three fractional derivatives and the LVIM [41].
Fig. 6. Comparison of the analytical and numerical solutions of the FGKS Eq. (94) using the Atangana-Baleanu fractional derivative for different x , t and ξ s.
Using the Caputo-Fabrizio fractional derivative, in Fig. 2(a) to (h) we provided the graphical solutions of Eq. (78). For different values of ξ and different time intervals, in Fig. 2(a) and (b) the graphical solutions of Eq. (78) are depicted when ξ = 1 , 0.45 and = 1. The graphical solutions of Eq. (78) are presented in Fig. 2(b) and Fig. 1(c) for ξ = 0.5 , = 1 and ξ = 0.65 , = 1.5 respectively. The surface solutions behavior of the FGKS Eq. (78) when ξ = 0.75 , 0.95 , = 1.5 are provided in Fig. 2(e) and (f). The 3D error analysis of the FGKS E 20 ( W ( x , t ) ) = | W e x t . ( x , t ) W a p p r . ( x , t ) | is presented in Fig. 2(g). The 2D error estimate of the FGKSE is plotted in Fig. 2(h). It is worth noting that the increase in the value of ξ is directly proportional to increase in the chaotic behavior and vice-versa.
In Fig. 3(a) to (h), we depict the graphical solution behavior of Eq. (79) for different values of ξ at different time intervals using Atangana-Baleanu fractional derivative. When ξ = 1 , 0.45 and = 1, the graphical solutions of Eq. (79) are provided in Fig. 3(a) and (b) respectively. The numerical solutions of the FGKSE for ξ = 0.5 , = 1 and ξ = 0.65 , = 1.5 are presented in Fig. 3(c) and (d). In Fig. 3(e) and (f), the analytical solution behavior of the FGKS Eq. (79) is depicted for ξ = 0.75 , 0.95 , = 1.5. It is noticeable that the increase in the values of ξ result in the increase in the chaotic behavior and vice-versa. The 3D error analysis of the FGKS Eq. (79) E 20 ( W ( x , t ) ) = | W e x t . ( x , t ) W a p p r . ( x , t ) | is presented in Fig. 3(g). The 2D error estimate of the FGKSE is plotted in Fig. 3(h).
In Fig. 4(a) to (h), the graphical solutions of Eq. (92) are computed using the Caputo fractional derivative. When ξ = 1 , 0.5 and = 1, the graphical solutions behavior of Eq. (92) are provided in Fig. 4(a) and (b) respectively. In Fig. 4(c) and (d), we Plotted the graphical solutions of the FGKSE for ξ = 0.45 , 0.65 , = 1. In Fig. 4(e) and (f), the solutions of the FGKS Eq. (92) are provided for ξ = 0.15 , 2.5 , = 1.5. Besides, we observe that the increase in the values of ξ result in the increase in the turbulence behavior and vice-versa. Fig. 4(g) shows the result obtain by comparing the exact solutions of Eqs. (92) to (94). Also, the approximate solutions of Eqs. (92) to (94) were obtained in Fig. 4(h) with ξ = 0.95 and = 1 at different time. The 3D error analysis of the FGKS Eq. (92) E 20 ( W ( x , t ) ) = | W e x t . ( x , t ) W a p p r . ( x , t ) | is presented in Fig. 4(i). The 2D error estimate of the FGKS Eq. (92) is presented in Fig. 4(j). The numerical simulations of Eqs. (92) to (94) at x = 6.5, ξ = 0.25, and different time intervals were computed in Table 2, and the obtained results are in excellent agreement with all the three fractional derivatives and the LVIM [41].
Using the fractional derivative with non-singular kernel the CF, we depicted the graphical solutions behavior of Eq. (93) in Fig. 5(a) to (h). In Fig. 5(a) and (b), the graphical solutions behavior of Eq. (93) for different values of ξ = 1 , 0.45 and = 1 are provided. When ξ = 0.5 , = 1 and ξ = 0.65 , = 1.5, we plot the graphical solutions of Eq. (93) in Fig. 5(c) and (d) respectively. In Fig. 5(e) and (f), the surface solutions behavior of the FGKS Eq. (93) are given when ξ = 0.75 , 0.95 , = 1.5. It is worth noting that the increase in the value of ξ is directly proportional to increase in the chaotic behavior and vice-versa. The 3D error estimate of the FGKS Eq. (93) E 20 ( W ( x , t ) ) = | W e x t . ( x , t ) W a p p r . ( x , t ) | is presented in Fig. 5(g). The 2D error of Eq. (93) is plotted in Fig. 5(h).
In Fig. 6(a) to (h), we compute the graphical solution behavior of Eq. (94) for different values of ξ at different time intervals using the Atangana-Baleanu fractional derivative. The graphical solutions behavior of Eq. (94) when ξ = 1 , 0.45 and = 1 are provided in Fig. 6(a) and (b) respectively. In Fig. 6(c) and (d), the numerical solutions of the GKSE for ξ = 0.5 , = 1 and ξ = 0.65 , = 1.5 are presented. In Fig. 6(e) and (f), the analytical solution behavior of the FGKS Eq. (94) for ξ = 0.75 , 0.95 , = 1.5 are depicted. The 3D error analysis of the FGKS E 20 ( W ( x , t ) ) = | W e x t . ( x , t ) W a p p r . ( x , t ) | is presented in Fig. 6(g). The 2D error estimate of Eq. (94) is plotted in Fig. 6(h).

6. Conclusions

Using fractional derivatives with non-singular kernel-the CF and the AB operators, we proposed a new semi-analytical method called the homotopy analysis J-transform method for solving fractional GKSE. The GKSE is considered as a large class of generalized Burger’s equation mimics the well-known Navier-Stokes equations of fluid motion-an important aspect of oceanography which covers a wide range of topics such as waves, geophysical fluid dynamics, the ocean currents, and the geology of the sea floor. Our proposed analytical method proved to be highly efficient and the non-zero convergence-control parameter was used to accelerate and adjust the convergence of the series solutions. We proved the existence and uniqueness of the fractional GKSE via fixed-point theorem. Furthermore, the error estimate and convergence of the method are established. Moreover, using Mathematica 12, the numerical and analytical solutions behaviors of the fractional GKSE at different values of ξ and are presented. It is imperative to note that the analytical and numerical solutions obtained via the non-singular kernel fractional operators (CF and AB) are in mutual agreement with the well-known singular kernel operator (Caputo). Finally, in the future, we plan to apply the proposed HAJTM to explore the analytical and numerical solutions of other fractional models arising in ocean engineering and its related areas.

7. Funding

This work was partially supported by the National Natural Science Foundation of China (12071261, 11871068, and 11831010), the National Key R&D Program (2018YFA0703900).

Declaration of Competing Interest

We have no conflicts of interest to disclose.
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