A New Analysis of Fractional-Order Equal-Width Equations via Novel Techniques

: In this paper, the new iterative transform method and the homotopy perturbation transform method was used to solve fractional-order Equal-Width equations with the help of Caputo-Fabrizio. This method combines the Laplace transform with the new iterative transform method and the homotopy perturbation method. The approximate results are calculated in the series form with easily computable components. The fractional Equal-Width equations play an essential role in describe hydromagnetic waves in cold plasma. Our object is to study the nonlinear behaviour of the plasma system and highlight the critical points. The techniques are very reliable, effective, and efﬁcient, which can solve a wide range of problems arising in engineering and sciences.


Introduction
Many researchers have studied fractional evaluation equations in the last decade because of their significant applicability in various fields of modern technology and science. It has been demonstrated that time-fractional equations define certain physical processes and that their application solves different problems. In this regard, it is critical to develop more implementations of innovative for fractional calculus [1-6]. Ford and Simpson found the fractional Caputo derivative [7] to be the best technique for finding time-fractional problems since it consistently contains the initial specifications that are missing in different individual models [8]. According to Spanier and Oldham, integrals and fractional derivatives can be utilized to demonstrate much more useful synthetic models than traditional approaches [9]. Furthermore, later on, fractional theory commitments and implementation, such as fractal mathematics, can be discussed in the literature. Readers interested are referred to [10][11][12][13][14][15][16][17][18].
Many researchers have focused on partial differential equations in recent years due to their wide range of technology and research implementations. These fractional equations are appropriate for identifying different important inventions in fluid dynamics, magnetic fields, nuclear physics, acoustics, electrodynamics, particle physics, optical structures, viscoelasticity, and other fields [19][20][21]. The fractional-order nonlinear Equal-Width equations where p is a positive integer; α and β are the positive constant, which require the boundary conditions ω → 0 as φ → ±∞; and is a parameter presenting order of fractional derivative. The derivative is understood in Caputo-Fabrizio form. Function ω(φ, ) is probability density function, φ is the spatial coordinate, and is the temporal coordinate. This expression carries a parameter that describes fractional-order derivative. For = 1, fractional-order equations convert into classical equations. In this paper, we shall incorporate periodic boundary conditions for a region a ≤ φ ≤ b. The form of the initial wave will be taken so that at large distances from the wave, |ω| is very small and follows the free-space boundary conditions ω = 0.
Numerous researchers have used various techniques to solve nonlinear fractional differential equations. Many investigators have used various methods to solve a variety of problems in previously implemented different analytical and numerical methods, such as the Adomian decomposition technique, finite difference method, generalized differential transform technique, finite element technique, perturbation methods, fractional differential transform technique, homotopy analysis strategy, iterative technique, etc., see [26][27][28][29][30][31][34][35][36] for more details. The homotopy analysis method (HAM) is a brilliant mathematical strategy proposed and applied by Liao [32,33,37]. Some scientists have shown the promise of using the HAM to study different mathematical modeling [38]. Furthermore, a good fundamental method identified as the homotopy analysis transform technique is as an important example of the homotopy analysis method that is used by combining the Laplace transformation technique. This inventive convergence of HAM and Laplace transformation is used to examine a wide variety of different problems [39,40]. When compared to traditional methods, these changes promote and strengthen the problem-solving methodology.
Several authors have suggested techniques for find the solution of fractional partial differential equations applying the fractional order Caputo and Fabrizio operators. The fractional-order wave equations was analyzed analytically by Xu in [41], who reduced the governing equation to two fractional ordinary differential equations. Dehghan et al. in [42], for example, used the HAM to solve linear partial differential equations; fractional derivatives are expressed by Liouville-Caputo sense in this work. Using the CF fractional derivative, Goufo et al. [43] developed a mathematical analysis of a model of rock fracture in the environment and achieved computational and analytical techniques. In [44], Jafari et al. utilized the HAM to solve a multi-order fractional differential equation investigated by Diethelm and Ford [45]. The chinese mathemation JH He introduce the homotopy perturbation method in 1998 [46]. This technique is efficient and accurate and eliminates an unconditione matrix, complicate integrals, and infinite series form. This method does not need of the problem a specific parameters. The homotopy perturbation transformation method (HPTM) combines the Elzaki transformation and the Homotopy perturbation method. Many researcher have been implemented HPTM to solving differential equations, such as Navier-Stokes problems [47], heat-like problems [48], gas dynamic model [49], Fisher's and hyperbolic equation [50].
Daftardar-Gejji and Jafari [51,52] developed a new iterative approach for solving nonlinear equations in 2006. Jafari et al. [53] first applied Laplace transform in the iterative technique. They proposed a new straightforward method called the iterative Laplace transform method to look for the numerical solution of the fractional partial differential equation (FPDE) system. the iterative Laplace transform method to solve linear and nonlinear partial differential equations such as time-fractional Fokker Planck equation [54], Zakharov Kuznetsov equation [55] and Fornberg Whitham equation [56], etc.
In this paper, we use the Iterative and homotopy perturbation transform methods to solved fractional Equal-Width equations with the help of Caputo-Fabrizio. The fractional calculus fundamental definitions are defined in Section 2, write the general methodologies in Sections 3 and 4, many test models to show the effectiveness of suggested techniques are given in Sections 5 and 6, and finally, the conclusion is given in Section 7.

Preliminaries Concepts
This section provides some fundamental concepts of fractional calculus.
Definition 1 ( [57]). The Liouville-Caputo fractional derivative of order is given as: is the integer n-th order derivative of ω(φ, θ) and n ∈ N. For 0 < ≤ 1, we defined the Laplace transform for the Liouville-Caputo fractional derivative of order as: where M(α) is a normalization form such that M(0) = M(1) = 1. The exponential law is used as the nonsingular kernel in this fractional derivative. For 0 < ≤ 1, we defined the Laplace transform for the Caputo-Fabrizio fractional derivative of order as:

Homotopy Perturbation Transform Method
Consider the general form fractional-order partial differential equation of the form where ∂ the fractional-order Caputo operator of , M and N are linear and nonlinear functions, and the source term is h( , ).
Next, we apply the Laplace transform to (5), and we get Further, simplification through Laplace differentiation leads to Now, taking inverse Laplace transform converts (7) into (8) or Now, the perturbation procedure in terms of power series with parameter p is presented as where perturbation term is p and p ∈ [0, 1]. The nonlinear terms can be defined as where H m are He's polynomials of ω 0 , ω 1 , ω 2 , . . . , ω m , and can be determined as [46] H m (ω 0 , ω 1 , . . . , where m ∈ N ∪ {0}. Putting (11) and (12) into (9), we have By comparing the coefficient of p on both sides of (13), we get . . . where

The New Iterative Transform Method Basic Procedure
Consider a particular type of a FPDE of the form with the initial conditions where M and N are linear and nonlinear functions, respectively. By applying the Laplace transformation to (16), we get Applying the Laplace differentiation is given as (19) using inverse Laplace transformation to (19) into Using the iterative technique, we obtain Further, the operator M is linear; therefore, and the operator N is nonlinear, so we have By substituting (21), (22) and (23) into (20), we obtain The new iterative transform method is defined as and , m ≥ 1. (27) Lastly, (16) and (17) provide the m-term solution in series form, defined as 5. Implementation of the HPTM Example 1. Consider the fractional-order nonlinear Equal-Width equation; if α = 1, β = 1 and p = 1 is given as with the initial condition Applying the Laplace transform to (29) with initial condition (30), we have Now, using the inverse Laplace transform to (32), we have Now, by implementing HPM, we get The nonlinear terms can be defined with the help of He's polynomials He's polynomials are defined as With the coefficients comparing p-like, we get p 0 : ω 0 (φ, ) = 3sech 2 φ − 15 2 , . . ..

Provided the series form solution is
Then, we have The exact solution of this problem as follows: In Figure 1, the exact and the HPTM solutions of Example 1 at = 1 are shown by subgraphs. From the given figure, it can be seen that both the HPTM and exact results are in close contact with each other. Furthermore, in Figure 2, the HPTM results of Example 1 are investigated at different fractional-order at = 1, 0.8, 0.6 and 0.4 of the 3D graph. It is analyzed that fractional-order problem solutions converge to an integer-order effect as fractional-order analysis to integer-order.
with the initial condition Applying Laplace transform to (39), we get Using the initial condition (40) into (41), we get By applying inverse Laplace transform to (42), we get Now, we implement HPM, and we get The nonlinear terms can be defined with the help of He's polynomials He's polynomial are defined as By comparing p-like coefficients, we get . . ..

The series form solution is
Then, we have (46), we obtain the solution of this problem as The exact solution of this problem is In Figure 3, the exact and the HPTM solutions of Example 2 at = 1 are shown by subgraphs. From the given figure, it can be seen that both the HPTM and exact results are in close contact with each other. Furthermore, in Figure 4, the HPTM results of Example 2 are investigated at different fractional-order at = 1, 0.8, 0.6 and 0.4 of 3D graph. According to the analysis, that fractional-order problem solutions converge to an integer-order effect as fractional-order analysis to integer-order.

Example 3. Consider the fractional nonlinear fractional-order modified equal width equation is given as follows. Consider the fractional nonlinear Equal-Width equation of the form
with the initial condition ω(φ, 0) = cosh 2/5 5φ 6 .
Then, we have Putting = 1 into (63), we obtain the solution of this problem as The exact solution of this problem is In Figure 7, it is shown that the exact and the NITM solutions graph with respect to φ and of Example 4 at = 1. From the given figures, it can be seen that both the NITM and exact results are in close contact with each other. Furthermore, in Figure 8, the NITM results of Example 4 are investigated at different fractional-order at = 1, 0.8, 0.6 and 0.4 of 2D graph with respect to φ and .
Then, we have Putting = 1 into (70), we obtain the solution of this problem as The exact solution of this problem is In Figure 9, it is shown that the exact and the NITM solutions graph with respect to φ and of Example 5 at = 1. From the given figures, it can be seen that both the NITM and exact results are in close contact with each other. Furthermore, in Figure 10, the NITM results of Example 5 are investigated at different fractional-order at = 1, 0.8, 0.6, and 0.4 of 2D graph with respect to φ and .

Conclusions
In this paper, we have presented a homotopy perturbation transform method and iterative transform method for solving fractional-order Equal-Width equations. The derivative is considered in the Caputo-Fabrizio sense. The figures analysis of the fractional-order results achieved has verified the convergence towards the results of the integer order.

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