A new computational investigation to the new exact solutions of ( 3 + 1 )- dimensional WKdV equations via two novel procedures arising in shallow water magnetohydrodynamics

: Various new exact solutions to ( ) + 3 1 - dimen sional Wazwaz – KdV equations are obtained in this work via two techniques: the modi ﬁ ed Kudryashov procedure and modi ﬁ ed simple equation method. The 3D plots, con tour plots, and 2D plots of some obtained solutions are provided to describe the dynamic characteristics of the obtained solutions. Our employed techniques are very helpful in constructing new exact solutions to several nonlinear models encountered in ocean scienti ﬁ c phe nomena arising in strati ﬁ ed ﬂ ows, shallow water, plasma physics, and internal waves.

For the fractional version of NPDEs and other types of differential equations, a newly proposed definition of generalized fractional derivative, named Abu-Shady-Kaabar fractional derivative, [18], can be utilized further in studying these equations due to the simplicity and efficiency of obtained analytical solutions using this new definition.
This article is organized as follows: Basic preliminaries about our adapted algorithms are reviewed in Section 2. The utilized procedures, particularly the modified KdV and MSE procedures, are discussed in Section 3. The illustrations of some obtained solutions are represented graphically in Section 4. A conclusion is drawn in Section 5.

Adopted algorithms
The needed tools are presented here to help in a NPDE's reduction to an ODE. We suppose that NPDE is expressed as follows: where P is a polynomial of u and its partial derivatives. We will take a transformation as follows: = + + − ξ kx ry sz wt. (2) Here, k r s , , , and w are constants. By substituting Eq. (2) to Eq. (1), we obtain an ODE as follows, which will be integrated with respect to ξ possible times [19].
In Sections 2.1 and 2.2, we describe the modified KdV and the modified simple eqaution (MSE) procedures, respectively.

The procedure of modified Kudryashov
The exact solutions of Eq. (3) are assumed as follows follows: and Eq. (5) satisfies: Nonlinear algebraic equations' system is obtained for ( ) = … α n m a k r s 0, , , , , , n , and w by substituting Eq. (4) into Eq. (3) associated with Eq. (6) and then setting the collection of all the coefficients of ( ) ϕ ξ n to be 0. By solving the obtained system via MAPLE, a variety of exact solutions [20][21][22] is found.

The method of MSE
We present the MSE method's main steps along with its fundamental ideas [27]. Through the transformation Eq.

The modified version of (+ 1)dimensional KdV equations
The modified (3 + 1)-dimensional KdV equations' exact solutions are presented in this section. These equations are expressed as follows [23,24]: The above equations are essential in mathematical physics topics. The first equation is given by Hereman [25], while the second and third equations are given by Wazwaz [26].

Application of the modified Kudryashov procedure
We will employ the modified KdV procedure to the adopted equations.

First equation's exact solutions
Let the wave variable: = + + − ξ kx ry sz wt be applied to Eq. (8). Then, by integrating the obtained ODE, we obtain: Eq. (12) is substituted via Eq. (6)'s help into Eq. (11). Then, by collecting all terms with the same power of ( ) ϕ ξ , we obtain: The exact solutions are obtained by solving the aforementioned system as follows:

The second equation's exact solutions
Let the wave variable: = + + − ξ kx ry sz wt be applied to Eq. (9). Then, by integrating the obtained ODE, we obtain: Here, the balancing number is 1. So, the ODE's solution is same as Eq. (12). Eq. (12) is substituted via Eq. (6)'s help into Eq. (16). Then, by collecting all terms with the same power of ( ) ϕ ξ , we obtain:  If we solve the aforementioned system, we obtain following values of the constant:

The third equation's exact solutions
Let the wave variable: = + + − ξ kx ry sz wt be applied to Eq. (10), and we obtain: Here, the balancing number is 1. So, the ODE's solution is same as Eq. (12). Eq. (12) is substituted via Eq. (6)'s help into Eq. (19). Then, by collecting all terms with the same power of ( ) ϕ ξ , we obtain:  The values of constants are obtained by solving the aforementioned system as follows:

Application of the modified simple equation procedure
We will employ the MSE procedure to the adopted equations.

The first equation's exact solutions
From the employed technique, Eq. (11)'s exact solution is assumed as follows: Eq. (21) is substituted into Eq. (11), and all terms with the same power of ( ) ϕ ξ are collected. Then, we obtain: By solving Eqs. (25) and (22), we obtain the following values of the constants:

The second equation's exact solutions
From the employed technique, Eq. (16)'s exact solution is assumed as follows: Eq. (29) is substituted into Eq. (16), and all terms with the same power of ( ) ϕ ξ are collected. Then, we obtain: Eq. (37) is substituted into Eq. (19), and all terms with the same power of ( ) ϕ ξ are collected. Then, we obtain:

Graphical representation of the obtained solutions
The figures of some obtained solutions are given in this section, which are obtained by the discussed methods. We give graphical illustrations by 3D plots, contour plots, and 2D plots. First, we have given graphs for solution (8) in Figure 1.

Conclusion
Exploring the exact solutions of NPDEs is essential in studying various modeling scenarios. In our work, we have obtained the ( ) + 3 1 -dimensional Wazwaz-KdV (WKdV) equations' new exact solutions via the modified Kudryashov procedure and the MSE method. The graphical representations involving 3D plots, contour plots and 2D plots have been provided for some obtained solutions to understand the nonlinear model's behavior. These investigated techniques can provide exact solutions to many nonlinear models in physics and engineering. Future research works can be based on our results by extending this work to study the fractional version of ( ) + 3 1 -dimensional WKdV equations via some new fractional definitions such as Abu-Shady-Kaabar fractional derivative [18].