A Variety of Novel Exact Solutions for Different Models With the Conformable Derivative in Shallow Water

For different nonlinear time-conformable derivative models, a versatile built-in gadget, namely the generalized exp(−φ(ξ))-expansion (GEE) method, is devoted to retrieving different categories of new explicit solutions. These models include the time-fractional approximate long-wave equations, the time-fractional variant-Boussinesq equations, and the time-fractional Wu-Zhang system of equations. The GEE technique is investigated with the help of fractional complex transform and conformable derivative. As a result, we found four types of exact solutions involving hyperbolic function, periodic function, rational functional, and exponential function solutions. The physical significance of the explored solutions depends on the choice of arbitrary parameter values. Finally, we conclude that the GEE method is more effective in establishing the explicit new exact solutions than the exp(−φ(ξ))-expansion method.


INTRODUCTION
Analytical solutions of the non-linear partial differential equation (NPDEs) are significantly more important for describing the physical meaning for any real-world problems. Due to the rapid expansion of computer technologies and computer-based symbolic tools, researchers have concentrated increasingly on the analytical and numerical solutions for the NPDEs, including integer and fractional orders. During recent decades, several analytical and semi-analytical methods, such as the improved fractional sub-equation [1], the exp function method [2,3], the G ′ /G-expansion [4-7], the tan ( (ξ ) /2)-expansion [8], the modified Kudryashov [9, 10], the new extended direct algebraic [11], the extended exp(−ϕ(ξ ))-expansion [12], the RB sub-ODE [13], the sine-Gordon expansion [14][15][16], the unified [17,18], and the generalized unified [19,20] methods, have been investigated and also employed for acquiring the new exact solutions of the well-known NPDEs that arise in applied sciences. Presenting new exact solution of PDEs provides a better understanding of the phenomena, which are governed by three special form of time-fractional WKB equations.
The time-fractional Whitham-Broer-Kaup (WBK) equation has the following structure [21] D α t u + uu x + v x + βu xx = 0 D α t v + (uv) x − βv xx + γ u xxx = 0 , t ≥ 0, 0 < α ≤ 1. (1) Eq. (1) describes the dispersive long wave in shallow water [22] where u = u(x, t) is the velocity field in the horizontal direction, v = v(x, t) is he height which deviates from the liquid balance position, and β and γ are real parameters [23]. D α t (.) is conformable derivative of order α. In the past, many researchers studied the WBK equation via different analytical approaches according to their field, particularly within mathematical physics and ocean engineering. For instance, Guo et al. [24] employed the improved sub-equation method to extract analytical solutions for space-and time-fractional WBK equations. El-Borai et al. [25] applied the exp-function method under the sense of the modified Riemann-Liouville derivative for solving the timefractional coupled WBK equations.
For the simplicity of the solutions, we did not consider solving the time-fractional WKB equations by the generalized exp (−ϕ(ξ ))-expansion method. The main aim of this work is to construct the new exact traveling wave solutions of the threespecial form of time-fractional WKB equations, such as the timefractional approximate long-wave equations, the time-fractional variant Boussinesq equations, and the time-fractional Wu-Zhang system of equations using the generalized exp (−ϕ(ξ ))expansion method with a conformable derivative sense. The generalized exp (−ϕ(ξ ))-expansion method is an effectual and easily applicable technique that is used to investigate the new exact solution for different integer-and fractional-order PDEs. Very recently, Lu et al. [36] used the generalized exp (−ϕ(ξ ))expansion method and construct the exact solutions of spacetime-fractional generalized fifth-order KdV equation with Jumarie's modified Riemann-Liouville derivatives.
The rest of the paper is arranged as follows. In section Conformable derivative and the generalized exp (−ϕ(ξ ))expansion method, some basic definitions of conformable derivative and the main steps of the generalized exp (−ϕ(ξ ))expansion method are given. In section Application of the generalized exp (−ϕ(ξ ))-expansion method, we look for the exact solutions of Eq. (2) to Eq. (4) via the generalized exp (−ϕ(ξ ))expansion method. Finally, a brief conclusion is provided in the last section.

THE CONFORMABLE DERIVATIVE AND THE GENERALIZED exp(−ϕ(ξ ))-EXPANSION METHOD
Khalil et al. [37] started to give us the first definition of the conformable derivative (CD) with a limit operator as follows.
Theorem 2. Let f :(0, ∝) → R be a function such that f is differentiable and α-conformable differentiable. Also, let g be a differentiable function defined in the range of f . Then where prime denotes the classical derivatives with respect to t. Now, we impose the generalized exp (−ϕ(ξ ))-expansion method for solving some fractional differential equations. In this respect, we described the essential steps of the generalized exp (−ϕ(ξ ))-expansion method [36] as follows.
Step-1: Suppose that a general form of the non-linear FDEs, say in two independent variables x and t, is given by where D α t u and D α t v are conformable derivatives of u and v, respectively, u = u(x, t) and v = v(x, t) are an unknown functions, and P 1 and P 2 are a polynomial in their arguments.
Step-2: To construct the exact solution of Eq. (5), we introduce the variable transformation, combine the real variables x and t by a compound variable ξ where, c is a constant which is determined later. The traveling wave transformation of Eq. (6) converts Eq. (5) into an ordinary differential equation (ODE) for u = U (ξ ) and v = V (ξ ): where Q 1 and Q 2 are a polynomial of U, V, and its derivatives with respect to ξ.
Step 3: Suppose that the traveling wave solution of system Eq. (7) can be presented as follows where the arbitrary constants a i (i = 1, 2 . . . , m) and b i (i = 1, 2 . . . , n) are determined latter, but a m = 0 and b n = 0 and also m and n are a positive integer, which can be determined by using homogeneous balance principle on Eq. (7), and ϕ = ϕ(ξ ) satisfies the following new ansatz equation where p, q, and r are constant. The general solutions of the equation are the following. Case-I: When p = 1 and = r 2 − 4q, one obtains 8.510.5 and Case-II: When r = 0, one obtains Case-III: When q = 0 and r = 0, one obtains For all cases, E is the integrating constant.
Step 4: Inserting Eq. (9) in Eq. (8) and compiling the terms in the resulting equation yields a set of algebraic non-linear equations. Finally, by solving this set we reach the exact solutions of the non-linear fractional PDEs.

APPLICATION OF THE GENERALIZED exp(−ϕ(ξ ))-EXPANSION METHOD
In this part, we will execute the generalized exp(−ϕ(ξ ))expansion method to solve three well-known non-linear fractional partial differential equations in shallow water, namely, the time-fractional approximate long wave (ALW) equations, the time-fractional variant-Boussinesq equations, and the timefractional Wu-Zhang system of equations. All the above mentioned equations are the special-form WBK equations that describe the physical phenomena arising in fluid mechanics.

The Time-Fractional Variant-Boussinesq Equations
Let us consider the time-fractional variant-Boussinesq equations Now, applying under the traveling wave transformation of Eq. (6), Eq. (45) reduces to a non-linear ODE as This integrates with respect to ξ of Eq. (46) and considers the integration constant to be zero. Eq. (46) then yields From the balancing condition in Eq. (47), we have m = 1 and n = 2. Now, the formal solution of (47) in the existence of (8) will be where a 1 = 0 and b 2 = 0. By inserting Eq. (48) into Eq. (47) along with Eq. (9) and using the same techniques investigated in the previous section we get Set-1: and c = ± −4pq + r 2 , −4pq + r 2 > 0. Therefore, by substituting the values of Set-1 into Eq. (48), along with the Eq. (10) to Eq. (18), we generate the following traveling wave solutions for the time-fractional variant-Boussinesq equations.

CONCLUSION
This research successfully applied the generalized exp(−ϕ(ξ ))expansion method combined with the complex fractional transformation and conformable derivative to exactly solve a special class of time-fractional WBK equations in shallow water, such as the time-fractional ALW equations, the timefractional variant-Boussinesq equations, and the time fractional Wu-Zhang system of equations. Afterwards, a sequence of new analytical wave solutions for these models were established. Finally, some 3D and 2D plots were added for some of the gained solutions for every model to illustrate the effect of the parameter α on the behaviors of these solutions. In conclusion, we found that the method mentioned here-with the aid of symbolic computations-is aspiring and efficient, and it is a superior mathematical construction with which to deal with the NPDEs.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding author/s.

AUTHOR CONTRIBUTIONS
All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.