Analysis of the Fractional-OrderKaup–Kupershmidt Equation via Novel Transforms

Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi Arabia Department of Basic Sciences, Preparatory Year Deanship, King Faisal University, Al-Ahsa 31982, Saudi Arabia Department of Computer Science and Mathematics, Mouhamed Cherif Messadia University, Souk Ahras, Algeria Central Department of Mathematics, Tribhuvan University Kritipur, Kathmandu, Nepal Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt


Introduction
Fractional calculus is regarded as an important branch of science, particularly for phenomena that cannot be defined by basic nonlinear ordinary differential equations or partial differential equations with integer-order operators. e use of memory is one of the main advantages of fractional-order derivatives over standard derivatives. In recent years, there have been numerous applications of fractional-order ordinary and partial differential equations in many fields of physics and engineering. ere have been several key works discovered, particularly in genetic mechanics and in the viscoelasticity concept, where fractional-order derivatives are utilized for a good explanation of the properties of materials. is is the main benefit of fractional derivatives compared with traditional integer-order models in which such effects are neglected. e computational modeling and analysis of structures and procedures, based on the explanation of their properties in concepts of fractional derivatives, obviously result in differential equations of fractional order and the requirement of finding solutions such as mathematical equations [1][2][3][4][5][6][7][8][9][10].
e fractional-order Kaup-Kupershmidt equation is used to investigate the analysis of capillary gravity waves' attitude and nonlinear dispersive waves. e extensive fifthorder nonlinear development equation is written as D ρ τ μ(ζ, τ) + αμμ ζζζ + βpμ ζ μ ζζ + cμ 2 μ ζ + μ ζζζζζ � 0, (1) with the initial condition where α, β, and c are real constants and 0 < ρ ≤ 1 is the parameter symbolizing the order of the fractional-order derivative. By considering different values for α, β, and c, the overload nonlinear fifth-order development model can be scaled down to the fifth-order fractional-order Kaup-Kupershmidt equation.
is study is concerned with the analysis of the modified fractionalorder Kaup-Kupershmidt (KK) equation. In recent decades, excellent scientific work has been devoted to the analysis of the classical KK equation. e modern KK equation can be integrated at p � 5/2 [13] and is considered to have bilinear representation [14]. Soliton and solitary wave results can be obtained for general nonlinear development problems by importing four diverse techniques autonomously. Nonlaopon et al. [15] used the inverse scattering approach to establish soliton results to analyze nonlinear equations with physical implications. Two integrable differential-difference equations exhibit soliton solutions of the Kaup-Kupershmidt equation type [16]. Musette introduced the fifth-order KK equation, and Verhoeven was one of the combined instances of the Henon-Heiles method; see [17] for more details. Prakasha et al. [18] used the q-homotopy analysis transform method which is implemented to obtain the result for the fractionalorder KK equation.
Daftardar-Gejji and Jafari [19] introduced a new iterative methodology for investigating nonlinear equations in 2006. Jafari [20] was the first to use the Laplace transform in an iterative technique. In [21], Jafari et al. suggested a modified straightforward methodology, named iterative Laplace transformation technique, to look for the numerical effects of the fractional partial differential equation system. Iterative Laplace transformation technique is used to solve linear and nonlinear partial differential equations such as time-fractional Zakharov-Kuznetsov equation [22], fractional-order Fokker-Planck equation [23], and Fornberg-Whitham equation [24].
is article modified the iterative method with the Elzaki transform; the novel approach is named the iterative transformation technique. e new iterative transformation technique is implemented to evaluate the fractional order of the system of the KK equation. e outcome of several illustrative cases is described to demonstrate the effectiveness of the proposed technique. e present method is used to obtain the results of fractional-order and integral-order models.
e new method reduces computing costs while increasing rate convergence. e proposed method is also helpful in dealing with other fractional-order linear and nonlinear partial differential equations.
Definition 6 (see [25][26][27]). e inverse Elzaki transform is given as 2 Journal of Mathematics e inverse Elzaki transform of some of the functions is given by

The General Discussion of the Proposed Method
Consider the particular type of the fractional partial differential equation: where n ∈ N, M and N are linear and nonlinear functions, and h is a source function. e initial condition is Applying the Elzaki transform of (14), we obtain as e differentiation property is defined as using the inverse Elzaki transform of equation (17), we have rough the iterative technique, we have M is a linear operator: and N is the nonlinear function; we get Substituting (19)- (21) in (18), we obtain the following solution: Applying the iterative method, we get Finally, equations (14) and (15) provide the series form solution which is defined as 3.1. Error Analysis of the Projected Technique. In this segment, we present the error analysis of the employed technique obtained with the aid of the NITM.
which proves the theorem.

Numerical Results
Example 1. Consider the following fractional Kaup-Kupershmidt equation which is given as with the initial condition Using the Elzaki transform to (24), we obtain Applying the inverse Elzaki transform of (29), we have Now, by applying the proposed semianalytical technique, we get , e series form result is (33) For ρ � 1, the exact results of (27) are given by Analytical approximate solutions with some free parameters are provided by the proposed technique. e analytical findings are extremely useful in deciphering the internal components of acts of nature. Depending on the physical factors, the explicit solutions represented several forms of approximate solutions. Figure 1 Figure 3 shows the error plot of three-and twodimensional graphs.
with the initial condition Using the Elzaki transform to (35), we get Applying the inverse Elzaki transform of (38), we have Now, by applying the proposed semianalytical technique, we get

Journal of Mathematics
e series form result is For ρ � 1, the exact results of (35) are given by Analytical approximate solutions with some free parameters are provided by the proposed technique. e analytical findings are extremely useful in deciphering the internal components of acts of nature. Depending on the physical factors, the explicit solutions represented several forms of approximate solutions. Figure 4 compares the result obtained by the help of the proposed technique to the exact and analytical result for the fractional-order KK equation. Figure 5 shows different fractional orders of ρ with respect to ζ and τ comparison which show that they have close contact with each other.

Conclusion
In this article, the iterative transformation technique is utilized to achieve analytical solutions of the fractional-order Kaup-Kupershmidt equations, which are broadly utilized as problems for spatial effects in applied sciences. e method gave a series type of solutions that converge very quickly in the mathematical model. It is predicted that the results obtained in this paper will be effective for more evaluation of the complicated nonlinear physical models. e analyses of this method are very clear and straightforward. As a result, we conclude that this method can be used to solve a variety of nonlinear fractional-order partial differential equation schemes.

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.