Fractional view analysis of Kersten-Krasil’shchik coupled KdV-mKdV systems with non-singular kernel derivatives

: The approximate solution of the Kersten-Krasil’shchik coupled Korteweg-de Vries-modified Korteweg-de Vries system is obtained in this study by employing a natural decomposition method in association with the newly established Atangana-Baleanu derivative and Caputo-Fabrizio derivative of fractional order. The Korteweg-de Vries equation is considered a classical super-extension in this system. This nonlinear model scheme is commonly used to describe waves in tra ffi c flow, electromagnetism, electrodynamics, elastic media, multi-component plasmas, shallow water waves and other phenomena. The acquired results are compared to exact solutions to demonstrate the suggested method’s e ff ectiveness and reliability. Graphs and tables are used to display the numerical results. The results show that the natural decomposition technique is a very user-friendly and reliable method for dealing with fractional order nonlinear problems.


Introduction
Fractional calculus is as old as ordinary calculus, i. e. at three centuries; however, it is not widely used in research and engineering. In a letter to Leibniz dated September 30, 1695, L'Hopital introduced the concept of fractional-order derivatives. In Lacroix's writings, P. S. Laplace defined a fractional derivative of arbitrary order in 1812. Several researchers have developed excellent literature on fractional differentiation and integration operators for the purposes of extending scientific and technical areas, including Caputo [1], Oldham and Spanier [2], Carpinteri and Mainardi [3], Samko et al. [4], Ahmed et al. [5], Podlubny [6], Atangana and Alabaraoye [7], Kumar et al. [8], Yin et al. [9] and Arife et al. [10]. The beauty of fractional derivatives is that they are not local point properties. The genetic and nonlocal dispersed effects are taken into account in fractional calculus. This property makes it more accurate than the integer-order derivative description.
Due to their proven applications in science and engineering, fractional differential equations have grown in prominence and popularity. These equations, for example, are increasingly being utilized to simulate problems in signal processing, biology, fluid mechanics, acoustics, diffusion, electromagnetism and a wide range of other physical phenomena. The nonlocal quality of fractional differential equations is the most essential advantage of utilizing them in these and other applications. The integer order differential operator is well-known to be a local operator, whereas the fractional order differential operator is not. This indicates that a system's next state is determined not just by its current state, but also by all of its previous states. The theory of fractional differential equations better and more systematically describes natural occurrences [11][12][13][14][15][16][17][18][19].
Fractional coupled systems are commonly used to investigate the complex behavior of plasma that contains several components such as atoms, free electrons and ions. Many scholars have tried to assess this behavior. Recently, Paul Kersten and Joseph Krasil'shchik investigated the Korteweg-de Vries (KdV) equation and modified KdV (mKdV) equation, proposing absolute complexity between coupled KdV-mKdV nonlinear systems for the investigation of nonlinear system behavior [20][21][22][23]. Many scholars have proposed numerous variants of this Kersten-Krasil'shchik coupled KdV-mKdV nonlinear system [24][25][26][27][28]. Among these variants, the nonlinear fractional Kersten-Krasil'shchik linked KdV-mKdV system provides a mathematical model for understanding the behavior of multi-component plasma for waves travelling along the positive zeta axis: where ϑ is a spatial coordinate and χ is a time coordinate. The fractional operator's order is represented by the factor α. The Caputo form is used to study this operator. When α = 1, the fractional coupled system becomes a classical system, as follows: (1.2) If G = 0, the Kersten-Krasil'shchik linked KdV-mKdV system is converted to the well-known KdV system as follows: When F = 0, the Kersten-Krasil'shchik coupled KdV-mKdV system becomes the well-known mKdV system as follows: As a result, the Kersten-Krasil'shchik linked KdV-mKdV system can be considered of as a combination of the KdV system and the mKdV system, which are represented by (1.2) to (1.4). We also investigate the following fractional nonlinear two component homogeneous time fractional coupled third order KdV system in this work as follows: where χ is the temporal coordinate, ϑ is the spatial coordinate and α is the fractional operator's order factor. The Caputo form is used to study this operator. When α = 1, the fractional coupled system becomes a classical system, as follows: (1.6) To solve the differential equations, we employ the natural decomposition method (NDM), which combines the natural transform (NT) and Adomian decomposition method and offers the approximate solution in series form. The proposed method has been implemented with the aid of two different fractional derivatives to solve two nonlinear systems. Many researchers have employed the NDM to obtain approximate analytical solutions, and it has provided reliable and closely convergent results. The calculations were done in Maple. The convergence of the proposed technique was also achieved by extending the concept discussed in [29,30].
The organization of the present paper is as follows: Section 2 gives some basic definitions and the properties of the natural transform that is used in our present work. Section 3 handles the methodology of the proposed technique. In Section 4, we presented the convergence analysis of the suggested technique. Section 5 gives the implementation of the suggested approach to approximate the solution of the above systems. Finally, we discuss the obtained results and conclusion.

Preleminaries
In this section, we recall some basic definitions and results from fractional calculus.
(2.1) Definition 2.2. The derivative of j(ρ) with fractional-order in the Caputo sense is given as [31]
Definition 2.4. The derivative of j(ρ) with fractional-order in the Atangana-Baleanu-Caputo operator (ABC) manner is given as [31] D α ρ j(ρ) = Definition 2.5. For the function F(χ), the natural transformation is given as
Definition 2.7. The NT of D α χ F(χ) in the Caputo sense is stated as [31] Definition 2.9. The NT of D α χ F(χ) in the ABC manner is given as [31] N[D α

Methodology
This section is concerned with the general procedure for numerical treatment of the below equation.
where the initial condition has L and N linear and nonlinear terms, and where h(ϑ, χ) represents the source term.

Case I (NT DM CF )
By means of the NT and CF fractional derivative, Eq (3.1) can be stated as By taking the natural inverse transform, we get Assume that for F(ϑ, χ), the series form solution is determined as 6) and the N(F(ϑ, χ)) decomposition is given as with A i representing the Adomian polynomials, which is illustrated as By putting Eqs (3.6) and (3.7) into Eq (3.5), we have (3.9) In this way, the solution of Eq (3.1) is obtained by putting Eq (3.9) into Eq (3.6) to solve for the NT DM CF , (3.10)

Case II (NT DM ABC )
By means of the NT and ABC fractional derivative, Eq (3.1) can be stated as .
By taking the natural inverse transform, we get In terms of the Adomain decomposition, we have From Eq (3.8), we have the following: (3. 15) In this way, the solution of Eq (3.1) is obtained to solve for the NT DM ABC : (3.16)

Convergence analysis
This section is concerned with the NT DM ABC and NT DM CF convergence and uniqueness.
and F * := F * (µ, χ) are two variable functions values, and γ 1 , γ 2 are Lipschitz constants. The operators L and N are given in Eq (3.1). Thus, the solution for the NT DM CF is unique for Proof. Now, from the theorem above, let K = (C[J], ||.||) be the Banach space that is ∀ continuous on the J = [0, T ] interval. Let I : K → K be the nonlinear mapping with So, I is a contraction as 0 < (γ 1 + γ 2 )(1 − α + α χ α Γ(α+1) ) < 1. Thus, by means of the Banach fixed point theorem, the solution of Eq (3.1) is unique. □ Theorem 4.3. Let L and N be Lipschitz functions as given in the above theorems; then, the solution for the NT DM CF is convergent for Eq (3.1).
Proof. Let us consider K to be the Banach space as stated above and let F m = m r=0 F r (µ, χ). To prove that F m is a Cauchy sequence in H, let As 0 < γ < 1, we have that 1 − γ m−n < 1. Thus, Since ||F 1 || < ∞, ||F m − F n || → 0 when n → ∞. In this way, F m is a Cauchy sequence in K and is convergent. □ Theorem 4.4. Let L and N be Lipschitz functions as given in the above theorems; then, the solution for the NT DM ABC is convergent for Eq (3.1).

Applications
Example 5.1. Let us consider the fractional Kersten-Krasil'shchik coupled KdV-mKdV nonlinear system as where the initial conditions are as follows: By taking the NT, we get Thus, we have that By simplification, we get By taking the inverse NT, we have that Solution for the NDM CF : Assume that for the unknown functions F(ϑ, χ) and G(ϑ, χ), the series form solutions are given as The nonlinear components in terms of the Adomian polynomials are given as With the help of these terms, Eq (5.6) can be stated as By comparing both sides of Eq (5.8), we have that (5.10) In this way, given (l ≥ 3), the remaining terms for F l and G l are easy to get. Thus, the series form solutions are given as (5.11)

Solution for the NDM ABC :
Assume that for the unknown functions F(ϑ, χ) and G(ϑ, χ), the series form solutions respectively are given as The nonlinear components in terms of the Adomian polynomials are given as With the help of these terms, Eq (5.6) can be stated as By comparing both sides of Eq (5.13), we have that , 14)

(5.15)
In this way, given (l ≥ 3), the remaining terms for F l and G l are easy to get. Thus, the series form solutions are given as where the initial conditions are as follows: By taking the NT, we get (5.20) Thus, we have By simplification, we get By taking the inverse NT, we have that

(5.23)
Solution for the NT DM CF : Assume that for the unknown functions F(ϑ, χ) and G(ϑ, χ), the series form solution are given as The nonlinear components in terms of the Adomian polynomials are given as −FF ϑ − GG ϑ = ∞ m=0 A m and FG ϑ = ∞ m=0 B m . With the help of these terms, Eq (5.23) can be stated as (5.25) By comparing both sides of Eq (5.25), we have that (5.27) In this way, given (l ≥ 3), the remaining terms for F l and G l are easy to get. Thus, the series form solutions are given as (5.28) Solution for the NT DM ABC : Assume that for the unknown functions F(ϑ, χ) and G(ϑ, χ), the series form solutions are given as F l (ϑ, χ).

(5.29)
The nonlinear components in terms of the Adomian polynomials are given as −FF ϑ − GG ϑ = ∞ m=0 A m and FG ϑ = ∞ m=0 B m . Given these terms, Eq (5.23) can be stated as By comparing both sides of Eq (5.30), we have that , , (5.32) In this way, given (l ≥ 3), the remaining terms for F l and G l are easy to get. Thus, the series form solutions are given as (5.33) When α = 1, we get the exact solution as (5.34)

Results and discussion
In Figure 1, the actual and suggested methods solutions of F(ϑ, χ) are calculated at α = 1. Figure 2 gives the graphical layouts of F(ϑ, χ) when α = 0.8 and 0.6. In Figure 3, the 2D and 3D behaviors of F(ϑ, χ) for different fractional orders are respectively given. In Figure 4, the actual and suggested methods solutions of G(ϑ, χ) are calculated at α = 1. Figure 5 gives the graphical layouts of G(ϑ, χ) when α = 0.8 and 0.6. In Figure 6, the 2D and 3D behaviors of G(ϑ, χ) for different fractional orders are respectively given. The graphical layout of Example 5.

Conclusions
The main goal of this work as to develop a fractional-order Kersten-Krasil' shchik coupled KdV-mKdV nonlinear system approximate analytical solution. Using the NDM, we were able to achieve this goal. The numerical solutions can be achieved in two steps. The targeted problems were first simplified using the NT; then, the decomposition approach was employed to get the solutions. The method's fundamental benefit is that it gives the user an analytical approximation, and in many cases an exact solution, in a quickly converging sequence with elegantly computed terms. The suggested method has been applied to obtain the solution of the given two problems. The method's small computational size in comparison to the computational size required by other numerical methods, as well as its rapid convergence, demonstrate that it is reliable and a significant improvement on existing methods in terms of solving the generalized Kersten-Krasil' shchik coupled KdV-mKdV equation. The results we obtained have been illustrated with the help of plots and tables which confirm the validity of the proposed method. Furthermore, the proposed method is simple, straightforward, and requires minimal computational time; it may also be extended to solve other fractional-order partial differential equations.