Novel Analysis of Fractional-Order Fifth-Order Korteweg–de Vries Equations

In this paper, the ρ -homotopy perturbation transformation method was applied to analysis of fifth-order nonlinear fractional Korteweg–de Vries (KdV) equations. This technique is the mixture form of the ρ -Laplace transformation with the homotopy perturbation method. The purpose of this study is to demonstrate the validity and efficiency of this method. Furthermore, it is demonstrated that the fractional and integer-order solutions close in on the exact result. The suggested technique was effectively utilized and was accurate and simple to use for a number of related engineering and science models.


Introduction
A number of researchers have recently become interested in fractional calculus, which was rst developed during Newton's period. Within the fractional calculus structure, many interesting and signi cant steps have been discovered within the last thirty decades. A fractional derivative was invented as a result of the complexity of a heterogeneous phenomenon.
e fractional derivative operators, by incorporating di usion methods, are capable of capturing the attitudes of multidimensional media [1][2][3][4]. e use of di erential equations of any scale has proved useful in showing a number of problems more quickly and accurately. Increasingly, scholars turned to generalized calculus to convey their viewpoints while analyzing complex phenomena in the context of mathematical methods using software [5][6][7][8][9][10].
as quantam mechanics, fluid dynamics, optics, and plasma physics. Fifth-order KdV form equations were utilized to analyze many nonlinear phenomena in particle physics [23][24][25]. It plays a vital role in the distribution of waves [26]. In their analysis, the KdV form equation has dispersive terms of the third and fifth-order relevant to the magnetoacoustic wave problem in cold plasma free collision plasma and dispersive terms appear near-critical angle propagation [27]. Plasma is a dynamic, quasineutral, and electrically conductive fluid. It consists of neutral particles, electrons, and ions. It consists of magnetic and electric areas due to the electrically conducting behavior of plasma. e mixture of particles and areas supports plasma waves of various forms. A magnetic lock is a less longitudinal ion dispersion. e magnetoacoustic wave behaves as an ionacoustic wave in the low magnetic field range, while in the low-temperature capacity, it acts as an Alfven wave [28,29]. e general model for the analysis of magnetic propertiesacoustic waves in plasma and shallow water waves with surface tension is equated with the fifth order of KdV. Recent study reveals that the solutions to this equation for travelling waves do not vanish at infinity [30,31]. Consider the well-known three types of the fifth-order KdV equations as follows [32,33]: with initial condition V(ζ, 0) � 1/ζ, with initial condition V(ζ, 0) � e ζ , and with initial condition V(ζ, 0) � 105/169sech 4 (ζ − ϕ/2 �� 13 √ ). (1) and (2) are called fifth-order KdV equations and (3) is called the Kawahara equation. Analytic techniques for these mathematical model are particularly difficult to come across due to their severe nonlinearity. Several researchers have employed various analytical and computational strategies to the solution of linear and nonlinear KdV equations throughout the last decade, such as the multisymplectic method [34], variational iteration method [33], He's homotopy perturbation method [35], and Exp-function method [36].
Recently, Fahd and Abdeljawad [37] developed the Laplace transform of the generalized fractional Caputo derivatives. We established a novel methodology with ρ-Laplace transform for solving fractional differential equations with a generalized fractional Caputo derivative. e homotopy perturbation method is merged with the Laplace transform method to create a highly effective method for handling nonlinear terms which is known as the homotopy perturbation transformation technique. is technique can provide the result in quick convergent series. Ghorbani pioneered the use of He's polynomials in nonlinear terms [38][39][40]. Later on, many scholars utilized the homotopy perturbation transformation method for linear and nonlinear differential equations such as heat-like equations [41], Navier-Stokes equations [42], hyperbolic equation and Fisher's equation [43], and gas dynamic equation [44].

Definition.
e ρ-Laplace transform of a CF g: [0, +∞] ⟶ R is defined as [37] e fractional generalized Caputo derivative of ρ-Laplace transformation of a CF g is given by [37]

The Rod Map of the Proposed Method
Consider the general partial differential equation given as (10) Applying ρ-Laplace transformation of (10), we get Now, applying the inverse ρ-Laplace transform, we get where Now, the perturbation procedure in terms of power series with parameter p is presented as where p is the perturbation parameter and p ∈ [0, 1]. e nonlinear term can be defined as where H n are He's polynomials in terms of . . , V n and can be calculated as where D κ p � z κ /zp κ . Substituting (15) and (16) in (12) e coefficients comparison on both sides of p, we have e Vκ(ζ, τ) component can be determined easily which quickly leads us to the convergent series. We can get p ⟶ 1:

Numerical Implementations
Example 1. Consider the fifth-order nonlinear KdV equation with the IC Journal of Mathematics Applying the ρ-Laplace transform on (20), we get Next, using the inverse of ρ-Laplace transform of (22), Now, we apply HPM where H n (x) represents the nonlinear function of He's polynomial. For the first few components, we present He's polynomials Comparing the P-like coefficients, we have e analytical solution of V(ζ, τ) is defined as en, put β � 1 in (27): e exact result is V(ζ, τ) � 1/ζ − τ. In Figure 1, the three-dimensional figures of ρ-HPTM and exact results in graphs (a) and (b) respectively at β � 1 and the close contact of the exact and ρ-HPTM solutions are investigated. In Figure 2, represent that various fractional 4 Journal of Mathematics order of ρ-HPTM results at β � 1, 0.8, 0.6, 0.4. e nonclassical results are investigated to be converge to an integerorder result of the given problem.

Example 2. Consider the fifth-order nonlinear fraction KdV equation
with the IC Applying the ρ-Laplace transform on (29), we get Next, using the inverse of ρ-Laplace transform of (31), Now, we apply HPM ∞ n�0 p n V n (ζ, τ) � e ζ + p L −1

Journal of Mathematics
where H n (x) represents the nonlinear term of He's polynomial. For the first few components, we present He's polynomials Comparing the P-like coefficients, we have erefore, the analytic solution of V(ζ, τ) is defined as en, β � 1 for (36), and we get e exact solution is V(ζ, τ) � e ζ− τ .
In Figure 3, the three-dimensional figures of ρ-HPTM and exact results in graphs (a) and (b) respectively at β � 1 and the close contact of the exact and ρ-HPTM solutions are investigated. In Figure 4, represent that various fractional order of ρ-HPTM results at β � 1, 0.8, 0.6, 0.4. e nonclassical results are investigated to be converge to an integerorder result of the given problem.

Example 3. Consider nonlinear fractional-order Kawahara equation
with the IC Applying the ρ-Laplace transform on (38), we get Next, using the inverse of ρ-Laplace transform of (40), Now, we apply HPM ∞ n�0 p n V n (ζ, τ) � 105 169 where H n (V) represent the nonlinear terms of He's polynomial. For the first few components, we present He's polynomials Comparing the P-like coefficients, we get 6 Journal of Mathematics  In Figure 5, the three-dimensional figures of ρ-HPTM and exact results in graphs (a) and (b) respectively at β � 1 and the close contact of the exact and ρ-HPTM solutions are investigated. In Figure 6, represent that various fractional order of ρ-HPTM results at β � 1, 0.8, 0.6, 0.4. e nonclassical results are investigated to be converge to an integer-order result of the given problem.

Conclusions
is paper determined the fractional-order Kawahara and fifth-order KdV equations, applying the ρ-homotopy perturbation transform method. e present method is used to describe the results for specific examples. e ρ-HPTM result is highly congruent with the precise solution of the suggested problems. Additionally, the proposed method estimated the results of the cases using fractional-order derivatives. e graphical examination of the resulting fractional-order results proved their convergence to integer-order outcomes. Additionally, the ρ-HPTM technique is straightforward, simple, and computationally efficient; the suggested method can be adapted to solve additional fractional-order partial differential equations.

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

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.