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Article

Fractional View Analysis of Kuramoto–Sivashinsky Equations with Non-Singular Kernel Operators

1
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2
School of Huminities and Social Sciences (H&SS), Bahria University, Karachi 75300, Pakistan
3
Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan
4
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
*
Author to whom correspondence should be addressed.
Symmetry 2022, 14(7), 1463; https://doi.org/10.3390/sym14071463
Submission received: 23 June 2022 / Revised: 11 July 2022 / Accepted: 15 July 2022 / Published: 18 July 2022

Abstract

:
In this article, we investigate the nonlinear model describing the various physical and chemical phenomena named the Kuramoto–Sivashinsky equation. We implemented the natural decomposition method, a novel technique, mixed with the Caputo–Fabrizio (CF) and Atangana–Baleanu deriavatives in Caputo manner (ABC) fractional derivatives for obtaining the approximate analytical solution of the fractional Kuramoto–Sivashinsky equation (FKS). The proposed method gives a series form solution which converges quickly towards the exact solution. To show the accuracy of the proposed method, we examine three different cases. We presented proposed method results by means of graphs and tables to ensure proposed method validity. Further, the behavior of the achieved results for the fractional order is also presented. The results we obtain by implementing the proposed method shows that our technique is extremely efficient and simple to investigate the behaviour of nonlinear models found in science and technology.

1. Introduction

Although fractional calculus (FC) has been in existence since classical calculus, it has recently attracted a lot of attention as a result of how it relates to the fundamental principles. Fractional calculus was first introduced by Leibniz and L’Hospital, but the idea has since been adopted by many scholars with numerous applications in several fields. Subsequently, it has been frequently employed to look at a range of phenomena. However, many researchers have emphasised the drawbacks of using this operator, specifically the derivative of a non-zero constant and the physical significance of the initial condition. Caputo then presented a new and unique fractional operator that covered all of the previous restrictions. The Caputo operator is used to study the majority of the models studied and examined under the FC framework. Some basic works of fractional calculus on different aspects are given by Podlubny [1], Momani and Shawagfeh [2], Kiryakova [3], Jafari and Seifi [4,5], Miller and Ross [6], Oldham and Spanier [7], Diethelm et al. [8], Kilbas and Trujillo [9], and Kemple and Beyer [10].
Physical and engineering processes have been modeled using fractional calculus, and fractional differential equations have been determined to give the best description of these models. Fractional differential equations (FDEs) have grown in popularity and significance as a result of their proven applications in a wide range of largely disparate fields of science and engineering. Fractional differential equations are applied to model a wide range of physical problems, including signal processing [11], electrodynamics [12], fluid and continuum mechanics [13], chaos theory [14], biological population models [15], finance [16], optics [17] and financial models [18]. Here, in particular, [19] presents a homotopy perturbation technique for nonlinear transport equations, papers [20,21,22,23,24,25,26] give the application of ADM to different transport models, also including fractional and nonlinear cases, works [27,28,29,30,31,32] provide reviews or/and developments of various numerical approaches to transport/advection-diffusion problems, while [33] proposes perturbational approach to construct analytical approximations based on the double-parameter transformation perturbation expansion method. Finally, the review paper [34] contains an exhaustive review of various modern fractional calculus applications. Symmetry is the cornerstone of nature, yet the vast majority of natural phenomena lack symmetry. Breaking unexpected symmetry is an effective approach for concealing symmetry. Two types of symmetry exist: finite and infinitesimal. Finite symmetries can have discrete or continuous symmetries. Space is a constant transformation, whereas natural symmetries such as symmetry and time reversal are discrete. Patterns have interested mathematicians for millennia. In the nineteenth century, systematic classifications of planar and spatial patterns emerged. Solving fractional nonlinear differential equations with precision has proven to be rather challenging [29,35,36].
In nature, most of the complicated phenomenons are nonlinear. The world’s most important processes are represented by nonlinear equations. Nonlinear FDEs solutions are of tremendous interest in both mathematics and real-world applications. In applied mathematics and physics, it is still a major problem to get the the nonlinear differential equations exact solution importance of finding the nonlinear differential equations exact solution that needs new approaches to obtain new exact or approximate solutions [37,38,39]. In the literature, several analytical techniques have been introduced for solving these equations, such as the fractional Adomian decomposition method (FADM) [40], reduced differential transform method (RDTM) [41], iterative Laplace transform method (ILTM) [42], the fractional variational iteration method (FVIM) [43], the fractional natural decomposition method (FNDM) [44], Elzaki transform decomposition method (ETDM) [45,46], Yang transform decomposition method (YTDM) [47], homotopy transform method and the Laplace transform (HLTM) [48], and the homotopy perturbation transform method (HPTM) [49], among many others.
Plasma instabilities, chemical reaction-diffusion, flame front propagation viscous flow problems, and magnetised plasmas are all modeled by the Kuramoto–Sivashinsky equation [50,51]. Our focus in this article is to investigate the FKS equation [52,53]
D τ ρ ξ ( χ , τ ) + ξ ξ χ + x ξ χ χ + y ξ χ χ χ + z ξ χ χ χ χ = 0 , 0 < ρ 1 , χ [ u , v ] , τ > 0
having the initial source
ξ ( χ , 0 ) = g ( χ ) ,
and boundary conditions ξ ( u , τ ) = Ψ 1 ( χ ) ,   ξ ( v , τ ) = Ψ 2 ( χ ) ,   ξ χ ( u , τ ) = ξ χ ( v , τ ) = ξ χ χ ( u , τ ) = ξ χ χ ( v , τ ) where g ( χ ) , Ψ 1 ( χ ) and Ψ 2 ( χ ) are known functions, and x, y and z are constants. The nonlinear advection term ξ ξ χ , as well as the dissipation terms, show the energy transfer mechanism.
To investigate the classical order KS equation, many authors used techniques such as finite-difference discretization [54], Chebyshev spectral collocation methods [55], homotopy analysis method [56], He’s variational iteration method [57], the Lattice–Boltzmann method [58], cubic B-spline finite difference-collocation method [59] and other schemes [60,61]. To address the difficulties in this issue, these approaches require a complex procedure and a large amount of computing; however, it is relatively easy to use the proposed method to obtain and examine the solution for nonlinear problems. As a result, we used the natural transform decomposition method (NTDM) to determine the approximate solution of the Kuramoto–Sivashinsky problem with arbitrary order using two distinct fractional derivatives. The natural transform decomposition method (NTDM) combines the well-known natural transform and the Adomian decomposition method. This new method is regarded to be the finest tool for solving specific classes of coupled nonlinear partial differential equations in a straightforward and fast manner. This method produces a solution in the form of a quick convergence series, which can be exact or approximate. Many physical phenomena which are modeled by fractional PDEs are solved by using NTDM [62,63,64].
The present work is structured as follows: In Section 2, we provide some useful definitions for fractional derivatives, which we will employ in our current work. The basic concept of the natural decomposition approach in connection with two separate fractional derivatives is described in Section 3 for the FKS equation. Section 5 gives the execution of the proposed method for solving various FKS equation problems. A short summary of the entire work is provided at the end.

2. Basic Preliminaries

The natural transform, the natural transform of fractional derivatives, and the fractional calculus that will be used throughout the study are introduced here along with certain definitions and basic properties.
Definition 1.
A real function j ( κ ) , κ > 0 , will be in the space C v , v R if there exists a real number q > v with j ( κ ) = κ q g ( κ ) , where g C [ 0 , ) , and will be in the space C v m if j ( m ) C v , m N .
Definition 2.
The fractional Riemann–Liouville integral for a function j C v , v 1 is stated as [65]
I ρ j ( κ ) = 1 Γ ( ρ ) 0 κ ( κ ψ ) ρ 1 j ( ψ ) d ψ , ρ > 0 , κ > 0 . a n d I 0 j ( κ ) = j ( κ )
Definition 3.
The Caputo derivative of j ( κ ) having fractional order is stated as [65]
C D κ ρ j ( κ ) = I m ρ D m j ( κ ) = 1 Γ ( m ρ ) 0 κ ( κ ψ ) m ρ 1 D ( j ( ψ ) ) d ψ
for m 1 < ρ m , m N , κ > 0 ,   j C v m ,   v 1 .
Definition 4.
The Caputo–Fabrizio derivative of j ( κ ) having fractional order is stated as [65]
C F D κ ρ j ( κ ) = F ( ρ ) 1 ρ 0 κ exp ρ ( κ ψ ) 1 ρ D ( j ( ψ ) ) d ψ
where 0 < ρ < 1 and F ( ρ ) is the normalization function having F ( 0 ) = F ( 1 ) = 1 .
Definition 5.
The Atangana–Baleanu–Caputo derivative of j ( κ ) having fractional order is stated as [65]
A B C D κ ρ j ( κ ) = B ( ρ ) 1 ρ 0 κ E ρ ρ ( κ ψ ) ρ 1 ρ D ( j ( ψ ) ) d ψ
where 0 < ρ < 1 , where B ( ρ ) represents the normalization function and E ρ ( z ) = m = 0 z m Γ ( ρ m + 1 ) represents the Mittag–Leffler function.
Definition 6.
On employing the natural transform to ξ ( τ ) , we have
N ( ξ ( τ ) ) = V ( , γ ) = e τ ξ ( γ τ ) d τ , , γ ( , ) .
The natural transform of ξ ( τ ) for τ ( 0 , ) is given as
N ( ξ ( τ ) H ( τ ) ) = N + = V + ( , γ ) = e τ ξ ( γ τ ) d τ , , γ ( 0 , ) .
where H ( τ ) is the Heaviside function.
Definition 7.
On employing the inversenNatural transform the function V ( , γ ) is defined as
N 1 [ V ( , γ ) ] = ξ ( τ ) , τ 0 .
Lemma 1.
The linearity property of the natural transform for ξ 1 ( τ ) is ξ 1 ( , γ ) and ξ 2 ( τ ) is ξ 2 ( , γ ) . Then
N [ c 1 ξ 1 ( τ ) + c 2 ξ 2 ( τ ) ] = c 1 N [ ξ 1 ( τ ) ] + c 2 N [ ξ 2 ( τ ) ] = c 1 ξ 1 ( , γ ) + c 2 ξ 2 ( , γ ) ,
where c 1 and c 2 are constants.
Lemma 2.
If the inverse natural transforms of V 1 ( , γ ) and V 2 ( , γ ) are ξ 1 ( τ ) and ξ 2 ( τ ) , respectively, then
N 1 [ c 1 V 1 ( , γ ) + c 2 V 2 ( , γ ) ] = c 1 N 1 [ V 1 ( , γ ) ] + c 2 N 1 [ V 2 ( , γ ) ] = c 1 ξ 1 ( τ ) + c 2 ξ 2 ( τ ) ,
where c 1 and c 2 are constants.
Definition 8.
The natural transform of D τ ρ ξ ( τ ) in the Caputo sense is defined as [65]
N [ C D τ ρ ξ ( τ ) ] = γ ρ N [ ξ ( τ ) ] 1 ξ ( 0 ) , ρ 1 .
Definition 9.
The natural transform of D τ ρ ξ ( τ ) in the sense of Caputo–Fabrizio is defined as [65]
N [ C F D τ ρ ξ ( τ ) ] = 1 1 ρ + ρ ( γ ) N [ ξ ( τ ) ] 1 ξ ( 0 ) , ρ 1 .
Definition 10.
The natural transform of D τ ρ ξ ( τ ) in the sense of Atangana–Baleanu–Caputo is defined as [65]
N [ A B C D τ ρ ξ ( τ ) ] = B ( ρ ) 1 ρ + ρ ( γ ) ρ N [ ξ ( τ ) ] 1 ξ ( 0 ) , ρ 1 .
Definition 11.
The inverse natural transform N 1 is stated as
N 1 [ V ( , γ ) ] = ξ ( τ ) = lim T 1 2 π ı σ ı T σ + ı T e τ γ V ( , γ ) d .

3. General Procedure

Here, we discuss the general procedure of the proposed method to solve the below equation.
D τ ρ ξ ( χ , τ ) = L ( ξ ( χ , τ ) ) + N ( ξ ( χ , τ ) ) + h ( χ , τ ) ,
subjected to the initial condition
ξ ( χ , 0 ) = ϕ ( ψ ) ,
with L , N representing the linear and nonlinear terms and h ( χ , τ ) as the source function.

3.1. Case I ( N T D M C F )

On taking the natural transform and using the fractional Caputo–Fabrizio derivative, Equation (15) is determined as
1 p ( ρ , γ , ) N [ ξ ( χ , τ ) ] ϕ ( χ ) = N L ( ξ ( χ , τ ) ) + N ( ξ ( χ , τ ) ) + h ( χ , τ ) ,
with
p ( ρ , γ , ) = 1 ρ + ρ ( γ ) .
On taking the inverse natural transform, Equation (17) is stated as
ξ ( χ , τ ) = N 1 ϕ ( χ ) + p ( ρ , γ , ) N [ h ( χ , τ ) ] + N 1 p ( ρ , γ , ) N L ( ξ ( χ , τ ) ) + N ( ξ ( χ , τ ) ) ,
N ( ξ ( χ , τ ) ) can be decomposed as
N ( ξ ( χ , τ ) ) = i = 0 A i .
The approximate solution for ξ C F ( χ , τ ) in series form is defined as
ξ C F ( χ , τ ) = i = 0 ξ i C F ( χ , τ ) .
Substituting Equations (20) and (21) into (19), we get
i = 0 ξ i ( χ , τ ) = N 1 ϕ ( χ ) + p ( ρ , γ , ) N [ h ( χ , τ ) ] + N 1 p ( ρ , γ , ) N i = 0 ( L ( ξ i ( χ , τ ) ) + A i ) .
From (22), we have
ξ 0 C F ( χ , τ ) = N 1 ϕ ( χ ) + p ( ρ , γ , ) N [ h ( χ , τ ) ] , ξ 1 C F ( χ , τ ) = N 1 p ( ρ , γ , ) N L ( ξ 0 ( χ , τ ) ) + A 0 , ξ l + 1 C F ( χ , τ ) = N 1 p ( ρ , γ , ) N L ( ξ l ( χ , τ ) ) + A l , l = 1 , 2 , 3 ,
Hence, we get the solution to (15) in the N T D M C F manner by substituting (23) into (21):
ξ C F ( χ , τ ) = ξ 0 C F ( χ , τ ) + ξ 1 C F ( χ , τ ) + ξ 2 C F ( χ , τ ) +

3.2. Case II ( N T D M A B C )

On taking the natural transform and using the fractional Atangana–Baleanu–Caputo derivative, Equation (15) is determined as
1 q ( ρ , γ , ) N [ ξ ( χ , τ ) ] ϕ ( χ ) = N L ( ξ ( χ , τ ) ) + N ( ξ ( χ , τ ) ) + h ( χ , τ ) ,
with
q ( ρ , γ , ) = 1 ρ + ρ ( γ ) ρ B ( ρ ) .
On taking the inverse natural transform, Equation (25) is stated as
ξ ( χ , τ ) = N 1 ϕ ( χ ) + q ( ρ , γ , ) N [ h ( χ , τ ) ] + N 1 q ( ρ , γ , ) N L ( ξ ( χ , τ ) ) + N ( ξ ( χ , τ ) ) .
N ( ξ ( χ , τ ) ) can be decomposed as
N ( ξ ( χ , τ ) ) = i = 0 A i .
The approximate solution for ξ A B C ( χ , τ ) in series form is defined as
ξ A B C ( χ , τ ) = i = 0 ξ i A B C ( χ , τ ) .
Substituting Equations (28) and (29) into (27), we get
i = 0 ξ i ( χ , τ ) = N 1 ϕ ( χ ) + q ( ρ , γ , ) N [ h ( χ , τ ) ] + N 1 q ( ρ , γ , ) N i = 0 ( L ( ξ i ( χ , τ ) ) + A i ) .
From (22), we have
ξ 0 A B C ( χ , τ ) = N 1 ϕ ( χ ) + q ( ρ , γ , ) N [ h ( χ , τ ) ] , ξ 1 A B C ( χ , τ ) = N 1 q ( ρ , γ , ) N L ( ξ 0 ( χ , τ ) ) + A 0 , ξ l + 1 A B C ( χ , τ ) = N 1 q ( ρ , γ , ) N L ( ξ l ( χ , τ ) ) + A l , l = 1 , 2 , 3 ,
Hence, we get the solution to (15) in an N T D M A B C manner by substituting (31) into (29):
ξ A B C ( χ , τ ) = ξ 0 A B C ( χ , τ ) + ξ 1 A B C ( χ , τ ) + ξ 2 A B C ( χ , τ ) +

4. Applications

In this part, we obtain the approximate solution of fractional Kuramoto–Sivashinsky equations.
Example 1.
Consider the FKS Equation (1) forx= 1 ,y= 0,z= 1,
D τ ρ ξ ( χ , τ ) + ξ ξ χ ξ χ χ + ξ χ χ χ χ = 0 , 0 < ρ 1 ,
subjected to the initial condition
ξ ( χ , 0 ) = δ + 15 tanh 3 [ ( χ γ ) ] 45 tanh [ ( χ γ ) ] 19 19 .
where δ , and γ are real constants. On taking the inverse natural transform, Equation (33) is stated as
N [ D τ ρ ξ ( χ , τ ) ] = N ξ ξ χ + N ξ χ χ N ξ χ χ χ χ .
By the transformation property, we have
1 ρ N [ ξ ( χ , τ ) ] 2 ρ ξ ( χ , 0 ) = N ξ ξ χ + ξ χ χ ξ χ χ χ χ .
We obtain after simplification
N [ ξ ( χ , τ ) ] = 2 δ + 15 tanh 3 [ ( χ γ ) ] 45 tanh [ ( χ γ ) ] 19 19 + ρ ( ρ ( ρ ) ) 2 ξ ξ χ + ξ χ χ ξ χ χ χ χ .
On taking the inverse natural transform, Equation (37) is stated as
ξ ( χ , τ ) = δ + 15 tanh 3 [ ( χ γ ) ] 45 tanh [ ( χ γ ) ] 19 19 + N 1 ρ ( ρ ( ρ ) ) 2 N ξ ξ χ + ξ χ χ ξ χ χ χ χ .
Now we implement NDM CF .
The approximate solution for ξ ( χ , τ ) in series form is defined as
ξ ( χ , τ ) = l = 0 ξ l ( χ , τ ) .
Additionally, by Adomian polynomials, the nonlinear terms ξ ξ χ = l = 0 A l , are calculated as
A 0 = ξ 0 ( ξ 0 ) χ , A 1 = ξ 1 ( ξ 0 ) χ + ξ 0 ( ξ 1 ) χ .
Thus, Equation (38) is stated as
l = 0 ξ l + 1 ( χ , τ ) = δ + 15 tanh 3 [ ( χ γ ) ] 45 tanh [ ( χ γ ) ] 19 19 + N 1 ρ ( ρ ( ρ ) ) 2 N l = 0 A l + l = 0 ξ l χ χ l = 0 ξ l χ χ χ χ .
On comparison of Equation (40) on both sides, we get
ξ 0 ( χ , τ ) = δ + 15 tanh 3 [ ( χ γ ) ] 45 tanh [ ( χ γ ) ] 19 19 ,
ξ 1 ( χ , τ ) = 45 sech 7 [ ( χ γ ) ] 27436 19 ( 1083 δ cosh [ ( χ γ ) ] 361 δ cosh [ 3 ( χ γ ) ] + 4 ( 30 19 722 31768 3 ) + 15 19 722 + 11552 3 cosh [ 2 ( χ γ ) ] sinh [ ( χ γ ) ] ) ρ ( τ 1 ) + 1 ,
ξ 2 ( χ , τ ) = 45 sech 11 [ ( χ γ ) ] 39617584 ( 21660 27 + 38 19 ( 1 + 32 2 ) δ cosh [ ( χ γ ) ] + 1444 ( 45 + 152 19 ( 1 + 31 2 ) ) δ cosh [ 3 ( χ γ ) ] + 1444 ( 75 + 76 19 ( 1 + 74 2 ) ) δ cosh [ 5 ( χ γ ) ] 1444 ( 15 + 38 19 ( 1 + 16 2 ) ) δ cosh [ 7 ( χ γ ) ] 5 ( 5776 ( 69 + 2 ( 19 19 + 4 ( 2031 + 19 19 ( 49 + 11014 2 ) ) ) ) ) + 19 ( 10620 6859 δ 2 ) sinh [ ( χ γ ) ] + 3 ( 2888 ( 220 + 19 ( 19 19 + 8 65 + 19 109 + 12442 2 ) ) ) + 3 19 ( 1500 + 6859 δ 2 ) sinh [ ( χ γ ) ] 5 ( 2888 ( 12 + 19 19 + 8 111 + 19 19 ( 31 + 734 2 ) ) ) 19 ( 900 + 6859 δ 2 ) sinh [ 5 ( χ γ ) ] + ( 5776 ( 1 + 16 2 ) ) 15 + 19 19 ( 1 + 16 2 ) + 19 ( 900 + 6859 δ 2 ) sinh [ 7 ( χ γ ) ] ( 1 ρ ) 2 + 2 ρ ( 1 ρ ) τ + ρ 2 τ 2 2 .
In this way, the other terms ξ l for ( l 3 ) are easy to obtain.
ξ ( χ , τ ) = l = 0 ξ l ( χ , τ ) = ξ 0 ( χ , τ ) + ξ 1 ( χ , τ ) + ξ 2 ( χ , τ ) + , ξ ( χ , τ ) = δ + 15 tanh 3 [ ( χ γ ) ] 45 tanh [ ( χ γ ) ] 19 19 45 sech 7 [ ( χ γ ) ] 27436 19 ( 1083 δ cosh [ ( χ γ ) ] 361 δ cosh [ 3 ( χ γ ) ] + 4 ( 30 19 722 31768 3 ) + 15 19 722 + 11552 3 cosh [ 2 ( χ γ ) ] sinh [ ( χ γ ) ] ) ρ ( τ 1 ) + 1 + 45 sech 11 [ ( χ γ ) ] 39617584 ( 21660 27 + 38 19 ( 1 + 32 2 ) δ cosh [ ( χ γ ) ] + 1444 45 + 152 19 ( 1 + 31 2 ) δ cosh [ 3 ( χ γ ) ] + 1444 ( 75 + 76 19 ( 1 + 74 2 ) ) δ cosh [ 5 ( χ γ ) ] 1444 ( 15 + 38 19 ( 1 + 16 2 ) ) δ cosh [ 7 ( χ γ ) ] 5 ( 5776 ( 69 + 2 ( 19 19 + 4 2031 + 19 19 ( 49 + 11014 2 ) ) ) ) + 19 ( 10620 6859 δ 2 ) sinh [ ( χ γ ) ] + 3 ( 2888 ( 220 + 19 19 19 + 8 65 + 19 ( 109 + 12442 2 ) ) ) + 3 19 ( 1500 + 6859 δ 2 ) sinh [ ( χ γ ) ] 5 2888 12 + 19 19 + 8 111 + 19 19 ( 31 + 734 2 ) 19 ( 900 + 6859 δ 2 ) sinh [ 5 ( χ γ ) ] + ( 5776 ( 1 + 16 2 ) ) 15 + 19 19 ( 1 + 16 2 ) + 19 ( 900 + 6859 δ 2 ) sinh [ 7 ( χ γ ) ] ( 1 ρ ) 2 + 2 ρ ( 1 ρ ) τ + ρ 2 τ 2 2 + .
Now we implement NDM ABC .
The approximate solution for ξ ( χ , τ ) in series form is defined as
ξ ( χ , τ ) = l = 0 ξ l ( χ , τ ) .
Additionally, by Adomian polynomials, the nonlinear terms ξ ξ χ = l = 0 A l are calculated as
A 0 = ξ 0 ( ξ 0 ) χ , A 1 = ξ 1 ( ξ 0 ) χ + ξ 0 ( ξ 1 ) χ .
Thus, Equation (38) is stated as
l = 0 ξ l + 1 ( χ , τ ) = δ + 15 tanh 3 [ ( χ γ ) ] 45 tanh [ ( χ γ ) ] 19 19 + N 1 γ ρ ( ρ + ρ ( γ ρ ρ ) ) 2 ρ N l = 0 A l + l = 0 ξ l χ χ l = 0 ξ l χ χ χ χ .
On comparison of Equation (45) on both sides, we obtain
ξ 0 ( χ , τ ) = δ + 15 tanh 3 [ ( χ γ ) ] 45 tanh [ ( χ γ ) ] 19 19 ,
ξ 1 ( χ , τ ) = 45 sech 7 [ ( χ γ ) ] 27436 19 ( 1083 δ cosh [ ( χ γ ) ] 361 δ cosh [ 3 ( χ γ ) ] + 4 ( 30 19 722 31768 3 ) + 15 19 722 + 11552 3 cosh [ 2 ( χ γ ) ] sinh [ ( χ γ ) ] ) 1 ρ + ρ τ ρ ρ ( ρ + 1 ) ,
ξ 2 ( χ , τ ) = 45 sech 11 [ ( χ γ ) ] 39617584 ( 21660 27 + 38 19 ( 1 + 32 2 ) δ cosh [ ( χ γ ) ] + 1444 ( 45 + 152 19 ( 1 + 31 2 ) ) δ cosh [ 3 ( χ γ ) ] + 1444 ( 75 + 76 19 ( 1 + 74 2 ) ) δ cosh [ 5 ( χ γ ) ] 1444 ( 15 + 38 19 ( 1 + 16 2 ) ) δ cosh [ 7 ( χ γ ) ] 5 ( 5776 ( 69 + 2 ( 19 19 + 4 ( 2031 + 19 19 ( 49 + 11014 2 ) ) ) ) ) + 19 ( 10620 6859 δ 2 ) sinh [ ( χ γ ) ] + 3 ( 2888 ( 220 + 19 ( 19 19 + 8 ( 65 + 19 ( 109 + 12442 2 ) ) ) ) ) + 3 19 ( 1500 + 6859 δ 2 ) sinh [ ( χ γ ) ] 5 ( 2888 ( 12 + ( 19 19 + 8 ( 111 + 19 19 ( 31 + 734 2 ) ) ) ) ) 19 ( 900 + 6859 δ 2 ) sinh [ 5 ( χ γ ) ] + ( 5776 ( 1 + 16 2 ) ) ( 15 + 19 19 ( 1 + 16 2 ) ) + 19 ( 900 + 6859 δ 2 ) sinh [ 7 ( χ γ ) ] ρ 2 τ 2 ρ ρ ( 2 ρ + 1 ) + 2 ρ ( 1 ρ ) τ ρ ρ ( ρ + 1 ) + ( 1 ρ ) 2 .
In this way, the other terms ξ l for ( l 3 ) are easy to obtain.
ξ ( χ , τ ) = l = 0 ξ l ( χ , τ ) = ξ 0 ( χ , τ ) + ξ 1 ( χ , τ ) + ξ 2 ( χ , τ ) + , ξ ( χ , τ ) = δ + 15 tanh 3 [ ( χ γ ) ] 45 tanh [ ( χ γ ) ] 19 19 45 sech 7 [ ( χ γ ) ] 27436 19 ( 1083 δ cosh [ ( χ γ ) ] 361 δ cosh [ 3 ( χ γ ) ] + 4 ( 30 19 722 31768 3 ) + 15 19 722 + 11552 3 cosh [ 2 ( χ γ ) ] sinh [ ( χ γ ) ] ) 1 ρ + ρ τ ρ ρ ( ρ + 1 ) + 45 sech 11 [ ( χ γ ) ] 39617584 ( 21660 27 + 38 19 ( 1 + 32 2 ) δ cosh [ ( χ γ ) ] + 1444 ( 45 + 152 19 ( 1 + 31 2 ) ) δ cosh [ 3 ( χ γ ) ] + 1444 ( 75 + 76 19 ( 1 + 74 2 ) ) δ cosh [ 5 ( χ γ ) ] 1444 ( 15 + 38 19 ( 1 + 16 2 ) ) δ cosh [ 7 ( χ γ ) ] 5 ( 5776 ( 69 + 2 ( 19 19 + 4 2031 + 19 19 ( 49 + 11014 2 ) ) ) ) + 19 ( 10620 6859 δ 2 ) sinh [ ( χ γ ) ] + 3 ( 2888 ( 220 + 19 19 19 + 8 65 + 19 ( 109 + 12442 2 ) ) ) + 3 19 ( 1500 + 6859 δ 2 ) sinh [ ( χ γ ) ] 5 2888 12 + 19 19 + 8 111 + 19 19 ( 31 + 734 2 ) 19 ( 900 + 6859 δ 2 ) sinh [ 5 ( χ γ ) ] + ( 5776 ( 1 + 16 2 ) ) 15 + 19 19 ( 1 + 16 2 ) + 19 ( 900 + 6859 δ 2 ) sinh [ 7 ( χ γ ) ] ρ 2 τ 2 ρ ρ ( 2 ρ + 1 ) + 2 ρ ( 1 ρ ) τ ρ ρ ( ρ + 1 ) + ( 1 ρ ) 2 + .
By taking ρ = 1 , we obtain the exact solution as
ξ ( χ , τ ) = δ + 15 tanh 3 [ ( χ δ τ γ ) ] 45 tanh [ ( χ δ τ γ ) ] 19 19 ,
Example 2.
Consider the FKS Equation (1) forx=1,y=0,z=1,
D τ ρ ξ ( χ , τ ) + ξ ξ χ + ξ χ χ + ξ χ χ χ χ = 0 , 0 < ρ 1 ,
subjected to the initial condition
ξ ( χ , 0 ) = δ + 15 9 11 9 ( 9 tanh [ ( χ γ ) ] + 11 tanh 3 [ ( χ γ ) ] ) .
On taking the inverse natural transform, Equation (50) is stated as
N [ D τ ρ ξ ( χ , τ ) ] = N ξ ξ χ N ξ χ χ N ξ χ χ χ χ .
By the transformation property, we have
1 ρ N [ ξ ( χ , τ ) ] 2 ρ ξ ( χ , 0 ) = N ξ ξ χ ξ χ χ ξ χ χ χ χ .
We obtain after simplification
N [ ξ ( χ , τ ) ] = 2 δ + 15 9 11 9 ( 9 tanh [ ( χ γ ) ] + 11 tanh 3 [ ( χ γ ) ] ) + ρ ( ρ ( ρ ) ) 2 [ ξ ξ χ ξ χ χ ξ χ χ χ χ ] .
On taking the inverse natural transform, Equation (54) is stated as
ξ ( χ , τ ) = δ + 15 9 11 9 ( 9 tanh [ ( χ γ ) ] + 11 tanh 3 [ ( χ γ ) ] ) + N 1 ρ ( ρ ( ρ ) ) 2 N ξ ξ χ ξ χ χ ξ χ χ χ χ .
Now, we apply NDM CF .
The approximate solution for ξ ( χ , τ ) in series form is defined as
ξ ( χ , τ ) = l = 0 ξ l ( χ , τ ) .
Additionally, by Adomian polynomials, the nonlinear terms ξ ξ χ = l = 0 A l are calculated as
A 0 = ξ 0 ( ξ 0 ) χ , A 1 = ξ 1 ( ξ 0 ) χ + ξ 0 ( ξ 1 ) χ .
Thus, Equation (55) is stated as
l = 0 ξ l + 1 ( χ , τ ) = δ + 15 9 11 9 ( 9 tanh [ ( χ γ ) ] + 11 tanh 3 [ ( χ γ ) ] ) + N 1 ρ ( ρ ( ρ ) ) 2 N l = 0 A l l = 0 ξ l χ χ l = 0 ξ l χ χ χ χ .
On comparison of Equation (57) on both sides, we obtain
ξ 0 ( χ , τ ) = δ + 15 9 11 19 ( 9 tanh [ ( χ γ ) ] + 11 tanh 3 [ ( χ γ ) ] ) , ξ 1 ( χ , τ ) = 45 11 sech 7 h [ ( χ γ ) ] 27436 19 ( 1444 δ cosh 3 [ ( χ γ ) ] ( 7 + cosh [ 2 ( χ γ ) ] ) + 6 ( 955 209 722 ( 3 + 368 2 ) ) sinh [ ( χ γ ) ] + 2 735 209 + 2888 ( 2 + 53 2 ) sinh [ 3 ( χ γ ) ] + 4 15 209 361 ( + 4 3 ) sinh [ 5 ( χ γ ) ] ) ρ ( τ 1 ) + 1
ξ 2 ( χ , τ ) = 45 2 sech 11 [ ( χ γ ) ] 39617584 ( 4332 39105 + 38 209 ( 41 + 1576 2 ) δ cosh [ ( χ γ ) ] 33212 2145 + 38 209 ( 1 + 64 2 ) δ cosh [ 3 ( χ γ ) ] 137180 ( 429 + 304 2 ) ) × δ cosh [ 5 ( χ γ ) ] + 1444 8580 + 19 209 ( 19 + 436 2 ) δ cosh [ 7 ( χ γ ) ] 1444 ( 165 +
19 209 ( 1 + 4 2 ) ) δ cosh [ 9 ( χ γ ) ] + ( 2888 ( 456060 + ( 3743 209 + 8 ( 22074195 + 19 209 ( 5173 + 1200214 2 ) ) ) ) + 209 ( 356390100 281219 δ 2 ) ) × sinh [ ( χ γ ) ] + ( 2888 ( 267960 + 4883 209 + 8 8086155 + 19 209 × ( 3313 + 417866 2 ) ) 209 ( 147341700 + 486989 δ 2 ) ) sinh [ 3 ( χ γ ) ] + 5 ( 2888 34584 + 133 209 + 8 191631 + 19 209 ( 349 + 8834 2 ) + 209 ( 4468860 48013 δ 2 ) ) sinh [ 5 ( χ γ ) ] 4 ( 361 ( 30030 + ( 931 209 + 8 ( 53625 + 19 209 ( 229 + 1898 2 ) ) ) ) + 209 ( 247500 + 6859 δ 2 ) ) sinh [ 7 ( χ γ ) ] + ( 1444 ( 1 + 4 2 ) ( 330 + 19 209 ( 1 + 4 2 ) ) + 209 ( 9900 + 6859 δ 2 ) ) sinh [ 9 ( χ γ ) ] ) ( 1 ρ ) 2 + 2 ρ ( 1 ρ ) τ + ρ 2 τ 2 2 .
In this way, the other terms ξ l for ( l 3 ) are easy to obtain.
ξ ( χ , τ ) = l = 0 ξ l ( χ , τ ) = ξ 0 ( χ , τ ) + ξ 1 ( χ , τ ) + ξ 2 ( χ , τ ) + , ξ ( χ , τ ) = δ + 15 9 11 19 ( 9 tanh [ ( χ γ ) ] + 11 tanh 3 [ ( χ γ ) ] ) 45 11 sech 7 h [ ( χ γ ) ] 27436 19 ( 1444 δ cosh 3 [ ( χ γ ) ] ( 7 + cosh [ 2 ( χ γ ) ] ) + 6 ( 955 209 722 ( 3 + 368 2 ) ) sinh [ ( χ γ ) ] + 2 735 209 + 2888 ( 2 + 53 2 ) sinh [ 3 ( χ γ ) ] + 4 15 209 361 ( + 4 3 ) sinh [ 5 ( χ γ ) ] ) ( ρ ( τ 1 ) + 1 ) 45 2 sech 11 [ ( χ γ ) ] 39617584 ( 4332 39105 + 38 209 ( 41 + 1576 2 ) δ cosh [ ( χ γ ) ] 33212 2145 + 38 209 ( 1 + 64 2 ) δ cosh [ 3 ( χ γ ) ] 137180 ( 429 + 304 2 ) ) × δ cosh [ 5 ( χ γ ) ] + 1444 8580 + 19 209 ( 19 + 436 2 ) δ cosh [ 7 ( χ γ ) ] 1444 165 + 19 209 ( 1 + 4 2 ) δ cosh [ 9 ( χ γ ) ] + ( 2888 456060 + 3743 209 + 8 22074195 + 19 209 ( 5173 + 1200214 2 ) + 209 ( 356390100 281219 δ 2 ) ) × sinh [ ( χ γ ) ] + ( 2888 ( 267960 + ( 4883 209 + 8 ( 8086155 + 19 209 × ( 3313 + 417866 2 ) ) ) ) 209 ( 147341700 + 486989 δ 2 ) ) sinh [ 3 ( χ γ ) ] + 5 ( 2888 ( 34584 + 133 209 + 8 191631 + 19 209 ( 349 + 8834 2 ) ) + 209 ( 4468860 48013 δ 2 ) ) sinh [ 5 ( χ
γ ) ] 4 ( 361 30030 + 931 209 + 8 53625 + 19 209 ( 229 + 1898 2 ) + 209 ( 247500 + 6859 δ 2 ) ) sinh [ 7 ( χ γ ) ] + 1444 ( 1 + 4 2 ) 330 + 19 209 ( 1 + 4 2 ) + 209 ( 9900 + 6859 δ 2 ) sinh [ 9 ( χ γ ) ] ) ( 1 ρ ) 2 + 2 ρ ( 1 ρ ) τ + ρ 2 τ 2 2 + .
Now, we apply NDM ABC
The approximate solution for ξ ( χ , τ ) in series form is defined as
ξ ( χ , τ ) = l = 0 ξ l ( χ , τ ) .
Additionally, by Adomian polynomials, the nonlinear terms ξ ξ χ = l = 0 A l are calculated as
A 0 = ξ 0 ( ξ 0 ) χ , A 1 = ξ 1 ( ξ 0 ) χ + ξ 0 ( ξ 1 ) χ .
Thus, Equation (55) is stated as
l = 0 ξ l ( χ , τ ) = δ + 15 9 11 9 ( 9 tanh [ ( χ γ ) ] + 11 tanh 3 [ ( χ γ ) ] ) + N 1 γ ρ ( ρ + ρ ( γ ρ ρ ) ) 2 ρ N l = 0 A l l = 0 ξ l χ χ l = 0 ξ l χ χ χ χ .
On comparison of Equation (61) on both sides, we obtain
ξ 0 ( χ , τ ) = δ + 15 9 11 19 ( 9 tanh [ ( χ γ ) ] + 11 tanh 3 [ ( χ γ ) ] ) , ξ 1 ( χ , τ ) = 45 11 sech 7 h [ ( χ γ ) ] 27436 19 ( 1444 δ cosh 3 [ ( χ γ ) ] ( 7 + cosh [ 2 ( χ γ ) ] ) + 6 ( 955 209 722 ( 3 + 368 2 ) ) sinh [ ( χ γ ) ] + 2 735 209 + 2888 ( 2 + 53 2 ) sinh [ 3 ( χ γ ) ] + 4 15 209 361 ( + 4 3 ) sinh [ 5 ( χ γ ) ] ) 1 ρ + ρ τ ρ ρ ( ρ + 1 ) ,
ξ 2 ( χ , τ ) = 45 2 sech 11 [ ( χ γ ) ] 39617584 ( 4332 39105 + 38 209 ( 41 + 1576 2 ) δ cosh [ ( χ γ ) ] 33212 2145 + 38 209 ( 1 + 64 2 ) δ cosh [ 3 ( χ γ ) ] 137180 ( 429 + 304 2 ) ) × δ cosh [ 5 ( χ γ ) ] + 1444 8580 + 19 209 ( 19 + 436 2 ) δ cosh [ 7 ( χ γ ) ] 1444 165 + 19 209 ( 1 + 4 2 ) δ cosh [ 9 ( χ γ ) ] + ( 2888 ( 456060 + ( 3743 209 + 8 ( 22074195 + 19 209 ( 5173 + 1200214 2 )
) ) ) + 209 ( 356390100 281219 δ 2 ) ) × sinh [ ( χ γ ) ] + ( 2888 ( 267960 + ( 4883 209 + 8 8086155 + 19 209 × ( 3313 + 417866 2 ) ) ) 209 ( 147341700 + 486989 δ 2 ) ) sinh [ 3 ( χ γ ) ] + 5 ( 2888 34584 + 133 209 + 8 191631 + 19 209 ( 349 + 8834 2 ) + 209 ( 4468860 48013 δ 2 ) ) sinh [ 5 ( χ γ ) ] 4 ( 361 ( 30030 + ( 931 209 + 8 ( 53625 + 19 209 ( 229 + 1898 2 ) ) ) ) + 209 ( 247500 + 6859 δ 2 ) ) sinh [ 7 ( χ γ ) ] + ( 1444 ( 1 + 4 2 ) 330 + 19 209 ( 1 + 4 2 ) + 209 ( 9900 + 6859 δ 2 ) ) sinh [ 9 ( χ γ ) ] ρ 2 τ 2 ρ ρ ( 2 ρ + 1 ) + 2 ρ ( 1 ρ ) τ ρ ρ ( ρ + 1 ) + ( 1 ρ ) 2 .
In this way, the other terms ξ l for ( l 3 ) are easy to obtain.
ξ ( χ , τ ) = l = 0 ξ l ( χ , τ ) = ξ 0 ( χ , τ ) + ξ 1 ( χ , τ ) + ξ 2 ( χ , τ ) + , ξ ( χ , τ ) = δ + 15 9 11 19 ( 9 tanh [ ( χ γ ) ] + 11 tanh 3 [ ( χ γ ) ] ) 45 11 sech 7 h [ ( χ γ ) ] 27436 19 ( 1444 δ cosh 3 [ ( χ γ ) ] ( 7 + cosh [ 2 ( χ γ ) ] ) + 6 ( 955 209 722 ( 3 + 368 2 ) ) sinh [ ( χ γ ) ] + 2 ( 735 209 + 2888 ( 2 + 53 2 ) ) sinh [ 3 ( χ γ ) ] + 4 15 209 361 ( + 4 3 ) sinh [ 5 ( χ γ ) ] ) 1 ρ + ρ τ ρ ρ ( ρ + 1 ) 45 2 sech 11 [ ( χ γ ) ] 39617584 ( 4332 39105 + 38 209 ( 41 + 1576 2 ) δ cosh [ ( χ γ ) ] 33212 ( 2145 + 38 209 ( 1 + 64 2 ) ) δ cosh [ 3 ( χ γ ) ] 137180 ( 429 + 304 2 ) ) × δ cosh [ 5 ( χ γ ) ] + 1444 ( 8580 + 19 209 ( 19 + 436 2 ) ) δ cosh [ 7 ( χ γ ) ] 1444 165 + 19 209 ( 1 + 4 2 ) δ cosh [ 9 ( χ γ ) ] + ( 2888 456060 + 3743 209 + 8 22074195 + 19 209 ( 5173 + 1200214 2 ) + 209 ( 356390100 281219 δ 2 ) ) × sinh [ ( χ γ ) ] + ( 2888 ( 267960 + ( 4883 209 + 8 ( 8086155 + 19 209 × ( 3313 + 417866 2 ) ) ) ) 209 ( 147341700 + 486989 δ 2 ) ) sinh [ 3 ( χ γ ) ] + 5 ( 2888 ( 34584 + ( 133 209 + 8 191631 + 19 209 ( 349 + 8834 2 ) ) ) + 209 ( 4468860 48013 δ 2 ) ) sinh [ 5 ( χ γ ) ] 4 ( 361 30030 + 931 209 + 8 53625 + 19 209 ( 229 + 1898 2 ) + 209 ( 247500 + 6859 δ 2 ) ) sinh [ 7 ( χ γ ) ] + 1444 ( 1 + 4 2 ) 330 + 19 209 ( 1 + 4 2 ) + 209 ( 9900 + 6859 δ 2 )
sinh [ 9 ( χ γ ) ] ρ 2 τ 2 ρ ρ ( 2 ρ + 1 ) + 2 ρ ( 1 ρ ) τ ρ ρ ( ρ + 1 ) + ( 1 ρ ) 2 + .
By taking ρ = 1 , we obtain the exact solution as
ξ ( χ , τ ) = δ + 15 9 11 19 ( 9 tanh [ ( χ δ τ γ ) ] + 11 tanh 3 [ ( χ δ τ γ ) ] ) .
Example 3.
Consider the FKS Equation (1) atx= 1,y= 4,z= 1,
D τ ρ ξ ( χ , τ ) + ξ ξ χ + ξ χ χ + 4 ξ χ χ χ + ξ χ χ χ χ = 0 , 0 < ρ 1 ,
subjected to the initial condition
ξ ( χ , 0 ) = δ + 9 15 ( tanh [ ( χ γ ) ] + tanh 2 [ ( χ γ ) ] tanh 3 [ ( χ γ ) ] ) .
On taking the inverse natural transform, Equation (65) is stated as
N [ D τ ρ ξ ( χ , τ ) ] = N ξ ξ χ N ξ χ χ N 4 ξ χ χ χ N ξ χ χ χ χ .
By the transformation property, we have
1 ρ N [ ξ ( χ , τ ) ] 2 ρ ξ ( χ , 0 ) = N ξ ξ χ ξ χ χ 4 ξ χ χ χ ξ χ χ χ χ .
We obtain after simplification
N [ ξ ( χ , τ ) ] = 2 δ + 9 15 ( tanh [ ( χ γ ) ] + tanh 2 [ ( χ γ ) ] tanh 3 [ ( χ γ ) ] ) + ρ ( ρ ( ρ ) ) 2 ξ ξ χ ξ χ χ 4 ξ χ χ χ ξ χ χ χ χ .
On taking the inverse natural transform, Equation (69) is stated as
ξ ( χ , τ ) = δ + 9 15 ( tanh [ ( χ γ ) ] + tanh 2 [ ( χ γ ) ] tanh 3 [ ( χ γ ) ] ) + N 1 ρ ( ρ ( ρ ) ) 2 N ξ ξ χ ξ χ χ 4 ξ χ χ χ ξ χ χ χ χ .
Now, we apply NDM CF .
The approximate solution for ξ ( χ , τ ) in series form is defined as
ξ ( χ , τ ) = l = 0 ξ l ( χ , τ ) .
Additionally, by Adomian polynomials, the nonlinear terms ξ ξ χ = l = 0 A l , are calculated as
A 0 = ξ 0 ( ξ 0 ) χ , A 1 = ξ 1 ( ξ 0 ) χ + ξ 0 ( ξ 1 ) χ .
Thus, Equation (70) is stated as
l = 0 ξ l + 1 ( χ , τ ) = δ + 9 15 ( tanh [ ( χ γ ) ] + tanh 2 [ ( χ γ ) ] tanh 3 [ ( χ γ ) ] ) + N 1 ρ ( ρ ( ρ ) ) 2 N l = 0 A l l = 0 ξ l χ χ 4 l = 0 ξ l χ χ χ l = 0 ξ l χ χ χ χ .
On comparison of Equation (72) on both sides, we obtain
ξ 0 ( χ , τ ) = δ + 9 15 ( tanh [ ( χ γ ) ] + tanh 2 [ ( χ γ ) ] tanh 3 [ ( χ γ ) ] ) , ξ 1 ( χ , τ ) = 15 sech 6 [ ( χ γ ) ] 8 ( 4 ( 36 + 2 32 2 + 344 3 + δ ) cosh [ 2 ( χ γ ) ] + ( 6 10 + 112 2 136 3 + δ ) cosh [ 4 ( χ γ ) ] + 3 ( 74 + 6 80 2 456 3 + δ + 2 ( 14 2 112 2 + 120 3 + δ ) sinh [ 2 ( χ γ ) ] + ( 6 2 + 48 2 40 3 + δ ) sinh [ 4 ( χ γ ) ] ) ) × ( 1 + tanh [ ( χ γ ) ] ) ρ ( τ 1 ) + 1
ξ 2 ( χ , τ ) = 15 2 sech 10 [ ( χ γ ) ] 64 ( 2 ( 5 ( 4 ( 966 + ( 77693 + 8 ( 25495 + ( 2205 + 31238 ( 4 + ) ) ) ) ) + 4 ( 19 + 4 ( 38 245 ) ) δ ) 5 δ 2 + 60 ( 693 + 5 δ ) ) 2 ( 127332 + 4 ( 282 + ( 242299 + 8 ( 70727 + ( 693 + 88234 ( 4 + ) ) ) ) ) + 276 δ + 4 ( 43 + 4 ( 86 + 77 ) ) δ + 17 δ 2 ) cosh [ 2 ( χ γ ) ] + 8 ( 6624 + 4 ( 471 + 4 ( 3313 + ( 5503 + 2 ( 1071 + 3652 ( 4 + ) ) ) ) ) 198 δ + 16 ( 1 + ( 8 + 119 ) ) δ δ 2 ) cosh [ 4 ( χ γ ) ] 2 ( 1404 + 4 ( 246 + ( 2917 + 8 ( 329 + ( 531 + 502 ( 4 + ) ) ) ) ) 228 δ + 4 ( 11 + 4 ( 22 + 59 ) ) δ δ 2 ) cosh [ 6 ( χ γ ) ] + ( 6 + 2 ( 1 + 4 ( 2 + ) ) + δ ) 2 cosh [ 8 ( χ γ ) ] + 2 ( 4 ( 3558 + ( 157321 + 8 ( 87713 + ( 50967 + 45304 + 504046 2 ) ) ) ) 4 ( 23 + 868 ( 2 + ) ) δ + 23 δ 2 + 36 ( 3223 + 59 δ ) ) sinh [ 2 ( χ γ ) ] 2 ( 4 ( 1 + 2 ) ( 6507 + ( 11136 + ( 34637 + 2 ( 77025 + 81222 + 97708 2 ) ) ) ) + 4 ( 1 + 2 ) ( 159 + 67 ( 5 + 2 ) ) δ 17 δ 2 ) × sinh [ 4 ( χ γ ) ] + 6 ( 516 + 4 ( 66 + ( 1223 + 8 ( 99 + ( 369 + 328 + 338 2 ) ) ) ) 92 δ + 4 ( 1 + 36 ( 2 + ) ) δ + δ 2 ) sinh [ 6 ( χ γ ) ] ( 6 + 2 ( 1 + 4 ( 2 + ) ) + δ ) 2 × sinh [ 8 ( χ γ ) ] 43200 ( 5 208 3 + 1344 6 ) tanh [ ( χ γ ) ] ) ( 1 ρ ) 2 + 2 ρ ( 1 ρ ) τ + ρ 2 τ 2 2 .
In this way, the other terms ξ l for ( l 3 ) are easy to obtain.
ξ ( χ , τ ) = l = 0 ξ l ( χ , τ ) = ξ 0 ( χ , τ ) + ξ 1 ( χ , τ ) + ξ 2 ( χ , τ ) + , ξ ( χ , τ ) = δ + 9 15 ( tanh [ ( χ γ ) ] + tanh 2 [ ( χ γ ) ] tanh 3 [ ( χ γ ) ] ) 15 sech 6 [ ( χ γ ) ] 8 ( 4 ( 36 + 2 32 2 + 344 3 + δ ) cosh [ 2 ( χ γ ) ] + ( 6 10 + 112 2 136 3 + δ ) cosh [ 4 ( χ γ ) ] + 3 ( 74 + 6 80 2 456 3 + δ + 2 ( 14 2 112 2 + 120 3 + δ ) sinh [ 2 ( χ γ ) ] + ( 6 2 + 48 2 40 3 + δ ) sinh [ 4 ( χ γ ) ] ) ) × ( 1 + tanh [ ( χ γ ) ] ) ρ ( τ 1 ) + 1 + 15 2 sech 10 [ ( χ γ ) ] 64 ( 2 ( 5 ( 4 ( 966 + ( 77693 + 8 ( 25495 + ( 2205 + 31238 ( 4 + ) ) ) ) ) + 4 ( 19 + 4 ( 38 245 ) ) δ ) 5 δ 2 + 60 ( 693 + 5 δ ) ) 2 ( 127332 + 4 ( 282 + ( 242299 + 8 ( 70727 + ( 693 + 88234 ( 4 + ) ) ) ) ) + 276 δ + 4 ( 43 + 4 ( 86 + 77 ) ) δ + 17 δ 2 ) cosh [ 2 ( χ γ ) ] + 8 ( 6624 + 4 ( 471 + 4 ( 3313 + ( 5503 + 2 ( 1071 + 3652 ( 4 + ) ) ) ) ) 198 δ + 16 ( 1 + ( 8 + 119 ) ) δ δ 2 ) cosh [ 4 ( χ γ ) ] 2 ( 1404 + 4 ( 246 +
( 2917 + 8 ( 329 + ( 531 + 502 ( 4 + ) ) ) ) ) 228 δ + 4 ( 11 + 4 ( 22 + 59 ) ) δ δ 2 ) cosh [ 6 ( χ γ ) ] + ( 6 + 2 ( 1 + 4 ( 2 + ) ) + δ ) 2 cosh [ 8 ( χ γ ) ] + 2 ( 4 ( 3558 + ( 157321 + 8 ( 87713 + ( 50967 + 45304 + 504046 2 ) ) ) ) 4 ( 23 + 868 ( 2 + ) ) δ + 23 δ 2 + 36 ( 3223 + 59 δ ) ) sinh [ 2 ( χ γ ) ] 2 ( 4 ( 1 + 2 ) ( 6507 + ( 11136 + ( 34637 + 2 ( 77025 + 81222 + 97708 2 ) ) ) ) + 4 ( 1 + 2 ) ( 159 + 67 ( 5 + 2 ) ) δ 17 δ 2 ) × sinh [ 4 ( χ γ ) ] + 6 ( 516 + 4 ( 66 + ( 1223 + 8 ( 99 + ( 369 + 328 + 338 2 ) ) ) ) 92 δ + 4 ( 1 + 36 ( 2 + ) ) δ + δ 2 ) sinh [ 6 ( χ γ ) ] ( 6 + 2 ( 1 + 4 ( 2 + ) ) + δ ) 2 × sinh [ 8 ( χ γ ) ] 43200 ( 5 208 3 + 1344 6 ) tanh [ ( χ γ ) ] ) ( 1 ρ ) 2 + 2 ρ ( 1 ρ ) τ + ρ 2 τ 2 2 + .
Now, we apply NDM ABC .
The approximate solution for ξ ( χ , τ ) in series form is defined as
ξ ( χ , τ ) = l = 0 ξ l ( χ , τ ) .
Additionally, by Adomian polynomials, the nonlinear terms ξ ξ χ = l = 0 A l are calculated as
A 0 = ξ 0 ( ξ 0 ) χ , A 1 = ξ 1 ( ξ 0 ) χ + ξ 0 ( ξ 1 ) χ .
l = 0 ξ l ( χ , τ ) = δ + 9 15 ( tanh [ ( χ γ ) ] + tanh 2 [ ( χ γ ) ] tanh 3 [ ( χ γ ) ] ) + N 1 γ ρ ( ρ + ρ ( γ ρ ρ ) ) 2 ρ N l = 0 A l l = 0 ξ l χ χ 4 l = 0 ξ l χ χ χ l = 0 ξ l χ χ χ χ .
On comparison of Equation (76) on both sides, we obtain
ξ 0 ( χ , τ ) = δ + 9 15 ( tanh [ ( χ γ ) ] + tanh 2 [ ( χ γ ) ] tanh 3 [ ( χ γ ) ] ) , ξ 1 ( χ , τ ) = 15 sech 6 [ ( χ γ ) ] 8 ( 4 ( 36 + 2 32 2 + 344 3 + δ ) cosh [ 2 ( χ γ ) ] + ( 6 10 + 112 2 136 3 + δ ) cosh [ 4 ( χ γ ) ] + 3 ( 74 + 6 80 2 456 3 + δ + 2 ( 14 2 112 2 + 120 3 + δ ) sinh [ 2 ( χ γ ) ] + ( 6 2 + 48 2 40 3 + δ ) sinh [ 4 ( χ γ ) ] ) ) × ( 1 + tanh [ ( χ γ ) ] ) 1 ρ + ρ τ ρ ρ ( ρ + 1 ) ,
ξ 2 ( χ , τ ) = 15 2 sech 10 [ ( χ γ ) ] 64 ( 2 ( 5 ( 4 ( 966 + ( 77693 + 8 ( 25495 + ( 2205 + 31238 ( 4 + ) ) ) ) ) + 4 ( 19 + 4 ( 38 245 ) ) δ ) 5 δ 2 + 60 ( 693 + 5 δ ) ) 2 ( 127332 + 4 ( 282 + ( 242299 + 8 ( 70727 + ( 693 + 88234 ( 4 + ) ) ) ) ) + 276 δ + 4 ( 43 + 4 ( 86 + 77 ) ) δ + 17 δ 2 ) cosh [ 2 ( χ γ ) ] + 8 ( 6624 + 4 ( 471 + 4 ( 3313 + ( 5503 + 2 ( 1071 + 3652 ( 4 + ) ) ) ) ) 198 δ + 16 ( 1 + ( 8 + 119 ) ) δ δ 2 ) cosh [ 4 ( χ γ ) ] 2 ( 1404 + 4 ( 246 + ( 2917 + 8 ( 329 + ( 531 + 502 ( 4 + ) ) ) ) ) 228 δ + 4 ( 11 + 4 ( 22 + 59 ) ) δ δ 2 ) cosh [ 6 ( χ γ ) ] + ( 6 + 2 ( 1 + 4 ( 2 + ) ) + δ ) 2 cosh [ 8 ( χ γ ) ] + 2 ( 4 ( 3558 + ( 157321 + 8 ( 87713 + ( 50967 + 45304 + 504046 2 ) ) ) ) 4 ( 23 + 868 ( 2 + ) ) δ + 23 δ 2 + 36 ( 3223 + 59 δ ) ) sinh [ 2 ( χ γ ) ] 2 ( 4 ( 1 + 2 ) ( 6507 + ( 11136 + ( 34637 + 2 ( 77025 + 81222 + 97708 2 ) ) ) ) + 4 ( 1 + 2 )
( 159 + 67 ( 5 + 2 ) ) δ 17 δ 2 ) × sinh [ 4 ( χ γ ) ] + 6 ( 516 + 4 ( 66 + ( 1223 + 8 ( 99 + ( 369 + 328 + 338 2 ) ) ) ) 92 δ + 4 ( 1 + 36 ( 2 + ) ) δ + δ 2 ) sinh [ 6 ( χ γ ) ] ( 6 + 2 ( 1 + 4 ( 2 + ) ) + δ ) 2 × sinh [ 8 ( χ γ ) ] 43200 ( 5 208 3 + 1344 6 ) tanh [ ( χ γ ) ] ) ρ 2 τ 2 ρ ρ ( 2 ρ + 1 ) + 2 ρ ( 1 ρ ) τ ρ ρ ( ρ + 1 ) + ( 1 ρ ) 2 .
In this way, the other terms ξ l for ( l 3 ) are easy to obtain.
ξ ( χ , τ ) = l = 0 ξ l ( χ , τ ) = ξ 0 ( χ , τ ) + ξ 1 ( χ , τ ) + ξ 2 ( χ , τ ) + , ξ ( χ , τ ) = δ + 9 15 ( tanh [ ( χ γ ) ] + tanh 2 [ ( χ γ ) ] tanh 3 [ ( χ γ ) ] ) 15 sech 6 [ ( χ γ ) ] 8 ( 4 ( 36 + 2 32 2 + 344 3 + δ ) cosh [ 2 ( χ γ ) ] + ( 6 10 + 112 2 136 3 + δ ) cosh [ 4 ( χ γ ) ] + 3 ( 74 + 6 80 2 456 3 + δ + 2 ( 14 2 112 2 + 120 3 + δ ) sinh [ 2 ( χ γ ) ] + ( 6 2 + 48 2 40 3 + δ ) sinh [ 4 ( χ γ ) ] ) ) × ( 1 + tanh [ ( χ γ ) ] ) 1 ρ + ρ τ ρ ρ ( ρ + 1 ) + 15 2 sech 10 [ ( χ γ ) ] 64 ( 2 ( 5 ( 4 ( 966 + ( 77693 + 8 ( 25495 + ( 2205 + 31238 ( 4 + ) ) ) ) ) + 4 ( 19 + 4 ( 38 245 ) ) δ ) 5 δ 2 + 60 ( 693 + 5 δ ) ) 2 ( 127332 + 4 ( 282 + ( 242299 + 8 ( 70727 + ( 693 + 88234 ( 4 + ) ) ) ) ) + 276 δ + 4 ( 43 + 4 ( 86 + 77 ) ) δ + 17 δ 2 ) cosh [ 2 ( χ γ ) ] + 8 ( 6624 + 4 ( 471 + 4 ( 3313 + ( 5503 + 2 ( 1071 + 3652 ( 4 + ) ) ) ) ) 198 δ + 16 ( 1 + ( 8 + 119 ) ) δ δ 2 ) cosh [ 4 ( χ γ ) ] 2 ( 1404 + 4 ( 246 + ( 2917 + 8 ( 329 + ( 531 + 502 ( 4 + ) ) ) ) ) 228 δ + 4 ( 11 + 4 ( 22 + 59 ) ) δ δ 2 ) cosh [ 6 ( χ γ ) ] + ( 6 + 2 ( 1 + 4 ( 2 + ) ) + δ ) 2 cosh [ 8 ( χ γ ) ] + 2 ( 4 ( 3558 + ( 157321 + 8 ( 87713 + ( 50967 + 45304 + 504046 2 ) ) ) ) 4 ( 23 + 868 ( 2 + ) ) δ + 23 δ 2 + 36 ( 3223 + 59 δ ) ) sinh [ 2 ( χ γ ) ] 2 ( 4 ( 1 + 2 ) ( 6507 + ( 11136 + ( 34637 + 2 ( 77025 + 81222 + 97708 2 ) ) ) ) + 4 ( 1 + 2 ) ( 159 + 67 ( 5 + 2 ) ) δ 17 δ 2 ) × sinh [ 4 ( χ γ ) ] + 6 ( 516 + 4 ( 66 + ( 1223 + 8 ( 99 + ( 369 + 328 + 338 2 ) ) ) ) 92 δ + 4 ( 1 + 36 ( 2 + ) ) δ + δ 2 ) sinh [ 6 ( χ γ ) ] ( 6 + 2 ( 1 + 4 ( 2 + ) ) + δ ) 2 × sinh [ 8 ( χ γ ) ] 43200 ( 5 208 3 + 1344 6 ) tanh [ ( χ γ ) ] ) [ ρ 2 τ 2 ρ ρ ( 2 ρ + 1 ) + 2 ρ ( 1 ρ ) τ ρ ρ ( ρ + 1 ) + ( 1 ρ ) 2 ] + .
By taking ρ = 1 , we obtain the exact solution as
ξ ( χ , τ ) = δ + 9 15 ( tanh [ ( χ δ τ γ ) ] + tanh 2 [ ( χ δ τ γ ) ] tanh 3 [ ( χ δ τ γ ) ] ) .

Result and Discussion

In this part of the paper, we discuss the numerical study of nonlinear Fractional Kuramoto–Sivashinsky equations by using the natural transform decomposition method. The tabular and graphical views for the given problems are obtained through Maple. In Figure 1, we present the exact and analytical behavior of the FKS equation by means of our proposed scheme, while Figure 2 shows the nature of analytical results at various fractional orders. Figure 3 illustrates the exact and proposed method scheme behavior, whereas Figure 4 shows the fractional layout of the proposed scheme. It is clear from Figure 4 that the solution becomes closer towards the exact solution as the value of ρ approaches integer-order. In the same manner, Figure 5 show the graphical behaviour of the exact solution and our method’s solution, while Figure 6 gives the layout of the analytical solution in terms of different fractional orders. Similarly, in Table 1, Table 2 and Table 3, we presented the absolute error analysis of the FKS equation obtained with the help of the proposed method at different values of χ and τ . The results in the tables reveal that our method is highly promising and accurate.

5. Conclusions

In this article, we have successfully employed the natural decomposition method in connection with the two different fractional derivatives to obtain the analytical solution of the fractional Kuramoto–Sivashinsky equation. To illustrate the validity of the suggested technique, we examined the FKS equation in three different cases. The numerical simulations confirm that our method results are in good agreement with the exact solution. The present scheme is very simple, effective, and appropriate for obtaining numerical solutions ofthe FKS equation. The suggested scheme’s main benefit is the series form solution, which quickly converges to the exact solution. As a result, we can conclude that the proposed approach is highly systematic and powerful for analysing fractional-order mathematical models in a systematic and better manner.

Author Contributions

Data curation, A.S.A. and A.K.; formal analysis, R.S. and M.I.; funding acquisition, W.W.; methodology, R.S. and A.K; project administration, A.S.A.; resources, M.I. and W.W.; supervision, W.W.; writing—original draft, R.S. All authors have read and agreed to the published version of the manuscript.

Funding

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R183), Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The numerical data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this article.

References

  1. Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications; Academic Press: New York, NY, USA, 1999. [Google Scholar]
  2. Momani, S.; Shawagfeh, N.T. Decomposition method for solving fractional Riccati differential equations. Appl. Math. Comput. 2006, 182, 1083–1092. [Google Scholar] [CrossRef]
  3. Kiryakova, S.V. Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J. Comput. Appl. Math. 2000, 118, 441–452. [Google Scholar] [CrossRef] [Green Version]
  4. Jafari, H.; Seifi, S. Homotopy Analysis Method for solving linear and nonlinear fractional diffusion-wave equation. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 2006–2012. [Google Scholar] [CrossRef]
  5. Jafari, H.; Seifi, S. Solving a system of nonlinear fractional partial differential equations using homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 2009, 14, 1962–1969. [Google Scholar] [CrossRef]
  6. Millerand, K.S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; Wiley: New York, NY, USA, 1993. [Google Scholar]
  7. Oldham, K.B.; Spanier, J. The Fractional Calculus; Academic Press: New York, NY, USA, 1974. [Google Scholar]
  8. Diethelm, K.; Ford, N.J.; Freed, A.D. A predictor-corrector approach for the numerical solution of fractional differential equation. Nonlinear Dyn. 2002, 29, 3–22. [Google Scholar] [CrossRef]
  9. Kilbas, A.A.; Trujillo, J.J. Differential equations of fractional order: Methods, results problems. Appl. Anal. 2001, 78, 153–192. [Google Scholar] [CrossRef]
  10. Shah, R.; Khan, H.; Kumam, P.; Arif, M. An analytical technique to solve the system of nonlinear fractional partial differential equations. Mathematics 2019, 7, 505. [Google Scholar] [CrossRef] [Green Version]
  11. Cruz-Duarte, J.M.; Rosales-Garcia, J.; Correa-Cely, C.R.; Garcia-Perez, A.; Avina-Cervantes, J.G. A closed form expression for the Gaussian-based Caputo-Fabrizio fractional derivative for signal processing applications. Commun. Nonlinear Sci. Numer. Simul. 2018, 61, 138–148. [Google Scholar] [CrossRef]
  12. Mukhtar, S.; Noor, S. The Numerical Investigation of a Fractional-Order Multi-Dimensional Model of Navier-Stokes Equation via Novel Techniques. Symmetry 2022, 14, 1102. [Google Scholar] [CrossRef]
  13. Shah, N.A.; Alyousef, H.A.; El-Tantawy, S.A.; Chung, J.D. Analytical Investigation of Fractional-Order Korteweg-De-Vries-Type Equations under Atangana-Baleanu-Caputo Operator: Modeling Nonlinear Waves in a Plasma and Fluid. Symmetry 2022, 14, 739. [Google Scholar] [CrossRef]
  14. Baleanu, D.; Wu, G.C.; Zeng, S.D. Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos Solitons Fractals 2017, 102, 99–105. [Google Scholar] [CrossRef]
  15. Singh, B.K. A novel approach for numeric study of 2D biological population model. Cogent Math. 2016, 3, 1261527. [Google Scholar] [CrossRef]
  16. Scalas, E.; Gorenflo, R.; Mainardi, F. Fractional calculus and continuous-time finance. Phys. A 2000, 284, 376–384. [Google Scholar] [CrossRef] [Green Version]
  17. Esen, A.; Sulaiman, T.A.; Bulut, H.; Baskonus, H.M. Optical solitons and other solutions to the conformable space-time fractional Fokas-Lenells equation. Optik 2018, 167, 150–156. [Google Scholar] [CrossRef]
  18. Sweilam, N.H.; Hasan, M.M.A.; Baleanu, D. New studies for general fractional financial models of awareness and trial advertising decisions. Chaos Solitons Fractals 2017, 104, 772–784. [Google Scholar] [CrossRef]
  19. Ahmad, S.; Ullah, A.; Akgul, A.; De la Sen, M. A Novel Homotopy Perturbation Method with Applications to Nonlinear Fractional Order KdV and Burger Equation with Exponential-Decay Kernel. J. Funct. Spaces 2021, 2021, 8770488. [Google Scholar] [CrossRef]
  20. Basto, M.; Semiao, V.; Calheiros, F.L. Numerical study of modified Adomian’s method applied to Burgers equation. J. Comput. Appl. Math. 2007, 206, 927–949. [Google Scholar] [CrossRef] [Green Version]
  21. Adomian, G. Solutions of Nonlinear PDE. Appl. Math. Lett. 1998, 11, 121–123. [Google Scholar] [CrossRef] [Green Version]
  22. Yee, E. Application of the Decomposition Method to the Solution of the Reaction-Convection-Diffusion Equation. Appl. Math. Comput. 1993, 56, 1–27. [Google Scholar] [CrossRef]
  23. Inc, M.; Cherruault, Y. A new approach to solve a diffusion-convection problem. Kybernetes 2002, 31, 536–549. [Google Scholar] [CrossRef]
  24. Adomian, G. Analytical solution of Navier-Stokes flow of a viscous compressible fluid. Found. Phys. Lett. 1995, 8, 389–400. [Google Scholar] [CrossRef]
  25. Krasnoschok, M.; Pata, V.; Siryk, S.V.; Vasylyeva, N. A subdiffusive Navier-Stokes-Voigt system. Phys. D Nonlinear Phenom. 2020, 409, 132503. [Google Scholar] [CrossRef]
  26. Wang, Y.; Zhao, Z.; Li, C.; Chen, Y.Q. Adomian’s method applied to Navier-Stokes equation with a fractional order. In Proceedings of the ASME 2009 IDETC/CIE, San Diego, CA, USA, 30 August–2 September 2009; pp. 1047–1054. [Google Scholar] [CrossRef]
  27. Siryk, S.V.; Salnikov, N.N. Numerical solution of Burger’s equation by Petrov-Galerkin method with adaptive weighting functions. J. Autom. Inf. Sci. 2012, 44, 50–67. [Google Scholar] [CrossRef]
  28. Roos, H.-G.; Stynes, M.; Tobiska, L. Robust Numerical Methods for Singularly Perturbed Differential Equations; Springer: Berlin/Heidelberg, Germany, 2008; 604p. [Google Scholar]
  29. Nonlaopon, K.; Naeem, M.; Zidan, A.M.; Alsanad, A.; Gumaei, A. Numerical investigation of the time-fractional Whitham–Broer–Kaup equation involving without singular kernel operators. Complexity 2021, 2021, 7979365. [Google Scholar] [CrossRef]
  30. Siryk, S.V.; Salnikov, N.N. Construction of Weight Functions of the Petrov-Galerkin Method for Convection-Diffusion-Reaction Equations in the Three-Dimensional Case. Cybern. Syst. Anal. 2014, 50, 805–814. [Google Scholar]
  31. Siryk, S.V. A note on the application of the Guermond-Pasquetti mass lumping correction technique for convection-diffusion problems. J. Comput. Phys. 2019, 376, 1273–1291. [Google Scholar] [CrossRef] [Green Version]
  32. John, V.; Knobloch, P.; Novo, J. Finite elements for scalar convection-dominated equations and incompressible flow problems: A never ending story? Comput. Vis. Sci. 2018, 19, 47–63. [Google Scholar] [CrossRef] [Green Version]
  33. Xu, Y. Similarity solution and heat transfer characteristics for a class of nonlinear convection-diffusion equation with initial value conditions. Math. Probl. Eng. 2019, 2019, 3467276. [Google Scholar] [CrossRef]
  34. Sun, H.; Zhang, Y.; Baleanu, D.; Chen, W.; Chen, Y. A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simulat 2018, 64, 213–231. [Google Scholar] [CrossRef]
  35. Elsayed, E.M.; Shah, R.; Nonlaopon, K. The analysis of the fractional-order Navier-Stokes equations by a novel approach. J. Funct. Spaces 2022, 2022, 8979447. [Google Scholar] [CrossRef]
  36. Sunthrayuth, P.; Ullah, R.; Khan, A.; Shah, R.; Kafle, J.; Mahariq, I.; Jarad, F. Numerical analysis of the fractional-order nonlinear system of Volterra integro-differential equations. J. Funct. Spaces 2021, 2021, 1537958. [Google Scholar] [CrossRef]
  37. Seadawy, A.R.; Rizvi, S.T.R.; Ahmad, S.; Younis, M.; Baleanu, D. Lump, lump-one stripe, multiwave and breather solutions for the Hunter-Saxton equation. Open Phys. 2021, 19, 1–10. [Google Scholar] [CrossRef]
  38. Gonzalez-Gaxiola, O.; Leon-Ramirez, A.; Chacon-Acosta, G. Application of the Kudryashov Method for Finding Exact Solutions of the Schamel-Kawahara Equation. Russ. J. Nonlinear Dyn. 2022, 18, 203–215. [Google Scholar] [CrossRef]
  39. Akinyemi, L. Two improved techniques for the perturbed nonlinear Biswas-Milovic equation and its optical solitons. Optik 2021, 243, 167477. [Google Scholar] [CrossRef]
  40. Aljahdaly, N.H.; Akgul, A.; Mahariq, I.; Kafle, J. A comparative analysis of the fractional-order coupled Korteweg-De Vries equations with the Mittag-Leffler law. J. Math. 2022, 2022, 8876149. [Google Scholar] [CrossRef]
  41. Keskin, Y.; Oturanc, G. Reduced differential transform method for partial differential equations. Int. J. Nonlinear Sci. Numer. Simul. 2009, 10, 741–750. [Google Scholar] [CrossRef]
  42. Khan, H.; Khan, A.; Al-Qurashi, M.; Shah, R.; Baleanu, D. Modified modelling for heat like equations within Caputo operator. Energies 2020, 13, 2002. [Google Scholar] [CrossRef]
  43. Wu, G.C.; Lee, E.W.M. Fractional variational iteration method and its application. Phys. A Lett. 2010, 374, 2506–2509. [Google Scholar] [CrossRef]
  44. Rawashdeh, M.S. The fractional natural decomposition method: Theories and applications. Math. Methods Appl. Sci. 2017, 40, 2362–2376. [Google Scholar] [CrossRef]
  45. Srivastava, H.M.; Shah, R.; Khan, H.; Arif, M. Some analytical and numerical investigation of a family of fractional-order Helmholtz equations in two space dimensions. Math. Methods Appl. Sci. 2020, 43, 199–212. [Google Scholar] [CrossRef]
  46. Nonlaopon, K.; Alsharif, A.M.; Zidan, A.M.; Khan, A.; Hamed, Y.S.; Shah, R. Numerical investigation of fractional-order Swift-Hohenberg equations via a Novel transform. Symmetry 2021, 13, 1263. [Google Scholar] [CrossRef]
  47. Alaoui, M.K.; Fayyaz, R.; Khan, A.; Shah, R.; Abdo, M.S. Analytical Investigation of Noyes-Field Model for Time-Fractional Belousov-Zhabotinsky Reaction. Complexity 2021, 2021, 3248376. [Google Scholar] [CrossRef]
  48. Taneco-Hernández, M.A.; Morales-Delgado, V.F.; Gómez-Aguilar, J.F. Fractional Kuramoto-Sivashinsky equation with power law and stretched Mittag-Leffler kernel. Phys. Stat. Mech. Its Appl. 2019, 527, 121085. [Google Scholar] [CrossRef]
  49. Qin, Y.; Khan, A.; Ali, I.; Al Qurashi, M.; Khan, H.; Shah, R.; Baleanu, D. An efficient analytical approach for the solution of certain fractional-order dynamical systems. Energies 2020, 13, 2725. [Google Scholar] [CrossRef]
  50. Kuramoto, Y.; Tsuzuki, T. Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys. 1976, 55, 356–369. [Google Scholar] [CrossRef] [Green Version]
  51. Sivashinsky, G.L. Instabilities, pattern-formation, and turbulence in flames. Ann. Rev. Fluid Mech. 1983, 15, 179–199. [Google Scholar] [CrossRef] [Green Version]
  52. Xu, Y.; Shu, C.-W. Local discontinuous Galerkin methods for the Kuramoto-Sivashinsky equations and the Ito-type coupled KdV equations. Comput. Methods Appl. Mech. Eng. 2006, 195, 3430–3447. [Google Scholar] [CrossRef]
  53. Shah, R.; Khan, H.; Baleanu, D. Fractional Whitham–Broer–Kaup equations within modified analytical approaches. Axioms 2019, 8, 125. [Google Scholar] [CrossRef] [Green Version]
  54. Akrivis, G.D. Finite difference discretization of the Kuramoto-Sivashinsky equation. Numer. Math. 1992, 63, 1–11. [Google Scholar] [CrossRef]
  55. Khater, A.H.; Temsah, R.S. Numerical solutions of the generalized Kuramoto-Sivashinsky equation by Chebyshev spectral collocation methods. Comput. Math. Appl. 2008, 56, 1465–1472. [Google Scholar] [CrossRef] [Green Version]
  56. Kurulay, M.; Secer, A.; Akinlar, A. A new approximate analytical solution of Kuramoto-Sivashinsky equation using Homotopy analysis method. Appl. Math. Inf. Sci. 2013, 7, 267–271. [Google Scholar] [CrossRef]
  57. Porshokouhi, M.G.; Ghanbari, B. Application of He’s variational iteration method for solution of the family of Kuramoto-Sivashinsky equations. J. King Saud Univ. Sci. 2011, 23, 407–411. [Google Scholar] [CrossRef] [Green Version]
  58. Ye, L.; Yan, G.; Li, T. Numerical method based on the Lattice Boltzmann model for the Kuramoto-Sivashinsky equation. J. Sci. Comput. 2011, 49, 195–210. [Google Scholar] [CrossRef]
  59. Lakestania, M.; Dehghan, M. Numerical solutions of the generalized Kuramoto-Sivashinsky equation using B-spline functions. Appl. Math. Model. 2012, 36, 605–617. [Google Scholar] [CrossRef]
  60. Singh, B.K.; Arora, G.; Kumar, P. A note on solving the fourth-order Kuramoto-Sivashinsky equation by the compact finite difference scheme. Ain Shams Eng. J. 2016, 9, 1581–1589. [Google Scholar] [CrossRef]
  61. Sahoo, S.; Ray, S.S. New approach to find exact solutions of time-fractional Kuramoto-Sivashinsky equation. Physica A 2015, 434, 240–245. [Google Scholar] [CrossRef]
  62. Kbiri Alaoui, M.; Nonlaopon, K.; Zidan, A.M.; Khan, A.; Shah, R. Analytical investigation of fractional-order cahn-hilliard and gardner equations using two novel techniques. Mathematics 2022, 10, 1643. [Google Scholar] [CrossRef]
  63. Botmart, T.; Agarwal, R.P.; Naeem, M.; Khan, A.; Shah, R. On the solution of fractional modified Boussinesq and approximate long wave equations with non-singular kernel operators. AIMS Math. 2022, 7, 12483–12513. [Google Scholar] [CrossRef]
  64. Shah, N.A.; Hamed, Y.S.; Abualnaja, K.M.; Chung, J.D.; Shah, R.; Khan, A. A comparative analysis of fractional-order kaup-kupershmidt equation within different operators. Symmetry 2022, 14, 986. [Google Scholar] [CrossRef]
  65. Zhou, M.X.; Kanth, A.S.V.; Aruna, K.; Raghavendar, K.; Rezazadeh, H.; Inc, M.; Aly, A.A. Numerical Solutions of Time Fractional Zakharov-Kuznetsov Equation via Natural Transform Decomposition Method with Nonsingular Kernel Derivatives. J. Funct. Spaces 2021, 2021, 9884027. [Google Scholar] [CrossRef]
Figure 1. The exact and approximate solution at ρ = 1 of Example 1.
Figure 1. The exact and approximate solution at ρ = 1 of Example 1.
Symmetry 14 01463 g001
Figure 2. The approximate solution at various fractional orders of ρ and τ = 0.5 ; for Example 1.
Figure 2. The approximate solution at various fractional orders of ρ and τ = 0.5 ; for Example 1.
Symmetry 14 01463 g002
Figure 3. The exact and approximate solution at ρ = 1 of Example 2.
Figure 3. The exact and approximate solution at ρ = 1 of Example 2.
Symmetry 14 01463 g003
Figure 4. The approximate solution at various orders of ρ and τ = 0.5 for Example 2.
Figure 4. The approximate solution at various orders of ρ and τ = 0.5 for Example 2.
Symmetry 14 01463 g004
Figure 5. The exact and approximate solution at ρ = 1 of Example 3.
Figure 5. The exact and approximate solution at ρ = 1 of Example 3.
Symmetry 14 01463 g005
Figure 6. The approximate solution at various orders of ρ and τ = 0.5 ; for Example 3.
Figure 6. The approximate solution at various orders of ρ and τ = 0.5 ; for Example 3.
Symmetry 14 01463 g006
Table 1. The comparison on the basis of error at various fractional orders of ρ for problem 1.
Table 1. The comparison on the basis of error at various fractional orders of ρ for problem 1.
τ χ ρ = 0.4 ρ = 0.6 ρ = 0.8 ρ = 1 ( NTDM CF ) ρ = 1 ( NTDM ABC )
0.23.8763631750 × 10 5 2.5603465160 × 10 5 1.2452127610 × 10 5 5.2357898000 × 10 9 5.2357898000 × 10 9
0.43.5383874710 × 10 5 2.3370987630 × 10 5 1.1366159900 × 10 5 5.1930879000 × 10 9 5.1930879000 × 10 9
0.10.63.2298917490 × 10 5 2.1333562800 × 10 5 1.0375564680 × 10 5 4.1981581000 × 10 9 4.1981581000 × 10 9
0.82.9481307140 × 10 5 1.9472375180 × 10 5 9.4701581360 × 10 6 4.2586011000 × 10 9 4.2586011000 × 10 9
12.6909407330 × 10 5 1.7773682470 × 10 5 8.6440866950 × 10 6 3.7555413000 × 10 9 3.7555413000 × 10 9
0.23.9051956730 × 10 5 2.6213609880 × 10 5 1.2791603100 × 10 5 2.1671579400 × 10 8 2.1671579400 × 10 8
0.43.5647273900 × 10 5 2.3928150910 × 10 5 1.1676250400 × 10 5 1.9986175500 × 10 8 1.9986175500 × 10 8
0.20.63.2539153040 × 10 5 2.1841946000 × 10 5 1.0658420020 × 10 5 1.7896316100 × 10 8 1.7896316100 × 10 8
0.82.9700833590 × 10 5 1.9936660840 × 10 5 9.7285869800 × 10 6 1.6517202000 × 10 8 1.6517202000 × 10 8
12.7109715270 × 10 5 1.8197396140 × 10 5 8.8799030700 × 10 6 1.5011082500 × 10 8 1.5011082500 × 10 8
0.23.9074452580 × 10 5 2.6557920700 × 10 5 1.3018392060 × 10 5 4.9107369000 × 10 8 4.9107369000 × 10 8
0.43.5667759500 × 10 5 2.4242396230 × 10 5 1.1883219200 × 10 5 4.5079263000 × 10 8 4.5079263000 × 10 8
0.30.63.2557659070 × 10 5 2.2128595690 × 10 5 1.0847147790 × 10 5 4.0994474000 × 10 8 4.0994474000 × 10 8
0.82.9717650370 × 10 5 2.0198233190 × 10 5 9.9007783700 × 10 6 3.7675803000 × 10 8 3.7675803000 × 10 8
12.7125122030 × 10 5 1.8436205320 × 10 5 9.0371291300 × 10 6 3.4266623600 × 10 8 3.4266623600 × 10 8
0.23.8968119170 × 10 5 2.6758919850 × 10 5 1.3174768270 × 10 5 8.7943159000 × 10 8 8.7943159000 × 10 8
0.43.5570453110 × 10 5 2.4425629700 × 10 5 1.2025719880 × 10 5 8.0772351000 × 10 8 8.0772351000 × 10 8
0.40.63.2469220340 × 10 5 2.2296233550 × 10 5 1.0977604780 × 10 5 7.3192632000 × 10 8 7.3192632000 × 10 8
0.82.9636656620 × 10 5 2.0350980460 × 10 5 1.0019587770 × 10 5 6.7334404000 × 10 8 6.7334404000 × 10 8
12.7051309960 × 10 5 1.8575742050 × 10 5 9.1456888300 × 10 6 6.1222165000 × 10 8 6.1222165000 × 10 8
0.23.8780476890 × 10 5 2.6863820070 × 10 5 1.3278652660 × 10 5 1.3857894800 × 10 7 1.3857894800 × 10 7
0.43.5399297510 × 10 5 2.4521513270 × 10 5 1.2120676190 × 10 5 1.2686543800 × 10 7 1.2686543800 × 10 7
0.50.63.2312565220 × 10 5 2.2383331580 × 10 5 1.1063856400 × 10 5 1.1569079000 × 10 7 1.1569079000 × 10 7
0.82.9494198080 × 10 5 2.0431014380 × 10 5 1.0098849110 × 10 5 1.0559300400 × 10 7 1.0559300400 × 10 7
12.6920845170 × 10 5 1.8648358860 × 10 5 9.2176004600 × 10 6 9.6577706000 × 10 8 9.6577706000 × 10 8
Table 2. The comparison on the basis of error at various fractional orders of ρ for problem 2.
Table 2. The comparison on the basis of error at various fractional orders of ρ for problem 2.
τ χ ρ = 0.4 ρ = 0.6 ρ = 0.8 ρ = 1 ( NTDM CF ) ρ = 1 ( NTDM ABC )
0.21.1612667280 × 10 6 8.5363755450 × 10 7 4.6919140870 × 10 7 6.8704703900 × 10 8 6.8704703900 × 10 8
0.41.0028798100 × 10 6 7.3867676970 × 10 7 4.0850054000 × 10 7 6.4548098400 × 10 8 6.4548098400 × 10 8
0.10.68.5796886930 × 10 7 6.3106175850 × 10 7 3.4749449090 × 10 7 5.2095715600 × 10 8 5.2095715600 × 10 8
0.87.3807877630 × 10 7 5.4320275680 × 10 7 2.9966494870 × 10 7 4.5965813960 × 10 8 4.5965813960 × 10 8
16.3324213410 × 10 7 4.6587557540 × 10 7 2.5671651650 × 10 7 3.8830543490 × 10 8 3.8830543490 × 10 8
0.21.1160913160 × 10 6 8.6892447100 × 10 7 5.1845663050 × 10 7 1.1940940770 × 10 7 1.1940940770 × 10 7
0.49.5808260400 × 10 7 7.4580679400 × 10 7 4.4481235990 × 10 7 1.0209619810 × 10 7 1.0209619810 × 10 7
0.20.68.2234346360 × 10 7 6.4003333330 × 10 7 3.8152844530 × 10 7 8.7191431200 × 10 8 8.7191431200 × 10 8
0.87.0730673850 × 10 7 5.5073221600 × 10 7 3.2871888450 × 10 7 7.5931629400 × 10 8 7.5931629400 × 10 8
16.0790881850 × 10 7 4.7343696780 × 10 7 2.8276390620 × 10 7 6.5661087000 × 10 8 6.5661087000 × 10 8
0.21.0067944450 × 10 6 8.0987732900 × 10 7 5.0016428380 × 10 7 1.1911411150 × 10 7 1.1911411150 × 10 7
0.48.6901621500 × 10 7 6.9989668720 × 10 7 4.3390394210 × 10 7 1.0664429710 × 10 7 1.0664429710 × 10 7
0.30.67.4103967680 × 10 7 5.9579371980 × 10 7 3.6734954620 × 10 7 8.6287146800 × 10 8 8.6287146800 × 10 8
0.86.3622255760 × 10 7 5.1148008530 × 10 7 3.1528398080 × 10 7 7.3897444000 × 10 8 7.3897444000 × 10 8
15.4643673440 × 10 7 4.3930339730 × 10 7 2.7080310190 × 10 7 6.3491630400 × 10 8 6.3491630400 × 10 8
0.28.3045099900 × 10 7 6.7775158900 × 10 7 4.1126291900 × 10 7 5.7818811000 × 10 8 5.7818811000 × 10 8
0.47.1575681000 × 10 7 5.8461304340 × 10 7 3.5574295280 × 10 7 5.2192393400 × 10 8 5.2192393400 × 10 8
0.40.66.1227594040 × 10 7 4.9964493820 × 10 7 3.0308304070 × 10 7 4.2382858300 × 10 8 4.2382858300 × 10 8
0.85.2830801080 × 10 7 4.3157643950 × 10 7 2.6276198470 × 10 7 3.8863255800 × 10 8 3.8863255800 × 10 8
14.4867499390 × 10 7 3.6559842940 × 10 7 2.2061448310 × 10 7 2.8322174000 × 10 8 2.8322174000 × 10 8
0.25.5952845800 × 10 7 4.4690695900 × 10 7 2.2473572590 × 10 7 9.4476480800 × 10 8 9.4476480800 × 10 8
0.44.8042354800 × 10 7 3.8370013900 × 10 7 1.9289146280 × 10 7 8.1259504900 × 10 8 8.1259504900 × 10 8
0.50.64.1587226470 × 10 7 3.3280270810 × 10 7 1.6892932770 × 10 7 6.6521422000 × 10 8 6.6521422000 × 10 8
0.83.5512618890 × 10 7 2.8378305440 × 10 7 1.4304267840 × 10 7 5.9170930300 × 10 8 5.9170930300 × 10 8
13.0596603570 × 10 7 2.4469397700 × 10 7 1.2382106280 × 10 7 4.9847282600 × 10 8 4.9847282600 × 10 8
Table 3. The comparison on the basis of error at various fractional orders of ρ for problem 3.
Table 3. The comparison on the basis of error at various fractional orders of ρ for problem 3.
τ χ ρ = 0.4 ρ = 0.6 ρ = 0.8 ρ = 1 ( NTDM CF ) ρ = 1 ( NTDM ABC )
0.27.60000000 × 10 7 7.10000000 × 10 8 3.40000000 × 10 9 1.10000000 × 10 8 1.10000000 × 10 8
0.45.13000000 × 10 7 5.10000000 × 10 8 6.10000000 × 10 9 1.20000000 × 10 8 1.20000000 × 10 8
0.10.63.35000000 × 10 7 4.10000000 × 10 8 3.00000000 × 10 9 0.00000000 × 10 00 0.0000000 × 10 00
0.82.31000000 × 10 7 3.80000000 × 10 8 8.20000000 × 10 9 1.00000000 × 10 9 1.00000000 × 10 9
11.52000000 × 10 7 1.80000000 × 10 8 1.00000000 × 10 9 1.10000000 × 10 9 1.10000000 × 10 9
0.25.49000000 × 10 7 5.10000000 × 10 8 1.00000000 × 10 8 1.40000000 × 10 8 1.40000000 × 10 8
0.43.63000000 × 10 7 2.80000000 × 10 8 1.20000000 × 10 8 1.30000000 × 10 8 1.30000000 × 10 8
0.20.62.38000000 × 10 7 3.60000000 × 10 8 4.20000000 × 10 9 1.50000000 × 10 8 1.50000000 × 10 8
0.81.70000000 × 10 7 3.10000000 × 10 8 7.50000000 × 10 9 1.10000000 × 10 8 1.10000000 × 10 8
19.80000000 × 10 8 1.00000000 × 10 8 2.00000000 × 10 9 0.00000000 × 10 00 0.00000000 × 10 00
0.29.90000000 × 10 8 1.10000000 × 10 8 9.10000000 × 10 9 1.30000000 × 10 8 1.30000000 × 10 8
0.47.60000000 × 10 8 7.00000000 × 10 9 3.40000000 × 10 9 0.00000000 × 10 00 0.00000000 × 10 00
0.30.64.60000000 × 10 8 9.00000000 × 10 9 5.20000000 × 10 9 1.30000000 × 10 8 1.30000000 × 10 8
0.83.60000000 × 10 8 1.30000000 × 10 8 1.90000000 × 10 9 0.00000000 × 10 00 0.00000000 × 10 00
11.20000000 × 10 8 2.00000000 × 10 9 2.00000000 × 10 9 1.10000000 × 10 8 1.10000000 × 10 8
0.27.06000000 × 10 7 1.15000000 × 10 7 1.80000000 × 10 8 1.20000000 × 10 8 1.20000000 × 10 8
0.44.64000000 × 10 7 7.30000000 × 10 8 1.20000000 × 10 8 1.50000000 × 10 9 1.50000000 × 10 9
0.40.63.17000000 × 10 7 4.80000000 × 10 8 5.00000000 × 10 9 1.30000000 × 10 8 1.30000000 × 10 8
0.82.00000000 × 10 7 1.50000000 × 10 8 2.60000000 × 10 8 0.00000000 × 10 00 0.00000000 × 10 00
11.55000000 × 10 7 2.70000000 × 10 8 1.80000000 × 10 8 1.10000000 × 10 8 1.10000000 × 10 8
0.22.19300000 × 10 6 3.09000000 × 10 7 4.70000000 × 10 8 1.20000000 × 10 8 1.20000000 × 10 8
0.41.45500000 × 10 6 2.09000000 × 10 7 3.10000000 × 10 8 8.10000000 × 10 9 8.10000000 × 10 9
0.50.69.81000000 × 10 7 1.36000000 × 10 7 1.40000000 × 10 8 1.30000000 × 10 8 1.30000000 × 10 8
0.86.66000000 × 10 7 8.40000000 × 10 8 2.60000000 × 10 8 0.00000000 × 10 00 0.00000000 × 10 00
14.53000000 × 10 7 6.60000000 × 10 8 7.00000000 × 10 9 1.40000000 × 10 8 1.40000000 × 10 8
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Saad Alshehry, A.; Imran, M.; Khan, A.; Shah, R.; Weera, W. Fractional View Analysis of Kuramoto–Sivashinsky Equations with Non-Singular Kernel Operators. Symmetry 2022, 14, 1463. https://doi.org/10.3390/sym14071463

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Saad Alshehry A, Imran M, Khan A, Shah R, Weera W. Fractional View Analysis of Kuramoto–Sivashinsky Equations with Non-Singular Kernel Operators. Symmetry. 2022; 14(7):1463. https://doi.org/10.3390/sym14071463

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Saad Alshehry, Azzh, Muhammad Imran, Adnan Khan, Rasool Shah, and Wajaree Weera. 2022. "Fractional View Analysis of Kuramoto–Sivashinsky Equations with Non-Singular Kernel Operators" Symmetry 14, no. 7: 1463. https://doi.org/10.3390/sym14071463

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