Abundant Wave Accurate Analytical Solutions of the Fractional Nonlinear Hirota–Satsuma–Shallow Water Wave Equation

: This research paper targets the fractional Hirota’s analytical solutions–Satsuma ( HS ) equations. The conformable fractional derivative is employed to convert the fractional system into a system with an integer–order. The extended simplest equation (ESE) and modiﬁed Kudryashov (MKud) methods are used to construct novel solutions of the considered model. The solutions’ accuracy is investigated by handling the computational solutions with the Adomian decomposition method. The solutions are explained in some different sketches to demonstrate more novel properties of the considered model.

In this article, we study a well-known model in INLPDEs presented by Hirota et al. [27][28][29]. This model is known by HS shallow water wave equation which is given by where E = E(Ξ, τ), E = E (Ξ, τ), 0 < ≤ 1. System (1) describes the dynamical behavior of the solitary wave in the shallow water. Applying the next wave transformation E = S(Γ), E = S(Γ), Γ = Ξ + λ t where λ is an arbitrary constant, then substituting the second equation in the system into the first, convert the above-fractional system into the following equation with an-integer order Handling Equation (2) , gets the value of balance equal two. Consequently, the general solutions of Equation (2) are given by where a −2 , a −1 , a 0 , a 1 , a 2 are arbitrary constants to be calculated later.
The rest sections are ordered as follows, we test, by means of two suggested analytical techniques [30][31][32], the analytical solutions to the nonlinear HS fractional equation. We search for the accuracy of the solutions we obtain in conjunction with the semi-analytical AD schema [33,34] in Section 2 part. In Section 3 we clarify the innovation of our approach and its physical interpretation. In theSection 4 portion, the outcome of a paper is summed up.

Computational Solutions vs. Accuracy
Here, we employ two recent analytical schemes (ESE and MKud methods) to formulate some novel computational wave solutions of the considered model. Additionally, the evaluated solutions are used to calculate the initial and boundary conditions. These conditions allow applying the AD method to test the accuracy of the obtained solutions and used schemes. This investigation takes the following steps:

Analytical Solutions
Applying the ESE and MKud methods' framework gets the values of the abovementioned parameters as following:

1.
Through the ESE method's steps gets the next values:

Set IV
Consequently, the exact solutions of the fractional nonlinear HS equation are constructed in the following For h 2 = 0, h 1 h 3 > 0, we find For For For For 4h 1 h 3 > h 2 2 , we find

Set II
Consequently, the exact solutions of the fractional nonlinear HS equation are constructed in the following

Solutions' Accuracy
Checking the accuracy of the obtained exact solutions of the HSI equation along with ESE and MKud methods with respect to Equations (12) and (32) for , gets the following semi-analytical solutions; , (39) Consequently, the semi-analytical solutions are given by Calculating the exact, semi-analytical solutions based on Equations (42) and (43) gets the following value in Tables 1 and 2:

Results' Explanation
This paper has constructed some novel solutions of the fractional HSI equation by implementing ESE and MKud methods. These solutions have been represented through some different forms (Figures 1-4) in three-dimension, density and spherical plot threedimensional to illustrate more novel properties of the considered model. Comparing our results with that obtained in [35] which has applied the Hirota bilinear method and symbolic computation on the integer-order of the same model, explains our results' novelty where all our solutions are entirely different from their obtained solutions. Additionally, employing the AD method explains our solutions' accuracy, where the analytical and semi-analytical solutions are almost matching. This matching has been cleared along with Tables 1 and 2 and Figures 5 and 6. Still, it also shows the superiority of the MKud method's solution over the ESE method, as shown in Figure 7.

Conclusions
This article has successfully implemented two recent analytical schemes (ESE and MKud techniques), and many novel solutions have been obtained for the considered model. The conformable fractional derivative has been employed to convert the fractional system to a system with an integer. The exact solutions have been demonstrated through 3D, density, spherical plot 3D sketches. Moreover, the accuracy of the obtained solutions has been illustrated by calculating the absolute value of error between the exact and semianalytical methods accepted by the AD method. The novelty of the obtained results in this article has been explained by comparing our results with the previously published research paper.

Data Availability Statement:
The data that support the findings of this study are available from the corresponding author upon reasonable request.