Analytical solitons for the space-time conformable differential equations using two efficient techniques

Exact solutions to nonlinear differential equations play an undeniable role in various branches of science. These solutions are often used as reliable tools in describing the various quantitative and qualitative features of nonlinear phenomena observed in many fields of mathematical physics and nonlinear sciences. In this paper, the generalized exponential rational function method and the extended sinh-Gordon equation expansion method are applied to obtain approximate analytical solutions to the space-time conformable coupled Cahn–Allen equation, the space-time conformable coupled Burgers equation, and the space-time conformable Fokas equation. Novel approximate exact solutions are obtained. The conformable derivative is considered to obtain the approximate analytical solutions under constraint conditions. Numerical simulations obtained by the proposed methods indicate that the approaches are very effective. Both techniques employed in this paper have the potential to be used in solving other models in mathematics and physics.


Introduction
Despite the recent extensive advances in the theory of differential equations, it can generally be said that it is still a complex task to determine an analytical solution for many ordinary and partial differential equations [1][2][3][4][5][6][7][8][9]. One of the events that led to the introduction of a wide range of new methods was the emergence and use of computers. So today it is almost impossible to use most of the existing techniques in solving differential equations, numerically or analytically, without the use of suitable computer software [10][11][12][13][14][15][16][17][18][19].
Khalil in [53] proposed an interesting definition of a derivative, namely the conformable derivative that generalizes the classical concept of derivative. This definition is wellbehaved and obeys the Leibniz rule and the chain rule. Nonlinear conformable differential and integral equations have been the focus of many studies due to their applications in various applications in physics, biology, engineering, signal processing, control theory, finance, etc. [54][55][56][57][58]. More precisely, the extended Zakharov-Kuzetsov equation with conformable derivative using the generalized exponential rational function method was solved in [59]. In [60], a generalized type of conformable local fractal derivative (GCFD) was employed to investigate some nonlinear evolution equations. They also set up a general technique to find exact solutions for their under studied PDEs. In [61] the first integral method was employed to construct the solutions to the conformable Burgers equation, modified Burgers equation, and Burgers-Korteweg-de Vries equation. In [62], several wave solutions for Burgers' type equations in the sense of conformable fractional derivative have been obtained via the residual power series method. Moreover, in [63] the auxiliary equation method has been employed to solve (2 + 1)-dimensional time-fractional Zoomeron equation and the time-fractional third order modified KdV equation. Abundant solitary wave solutions to an extended nonlinear Schrödinger's equation with conformable derivative using an efficient integration method called the generalized exponential rational function method have been reported in [64]. Very recently, the conformable derivative and adequate fractional complex transform have been implemented to discuss the conformable higher-dimensional Ito equation [65].
In this paper, we apply both the generalized exponential rational function method and the extended sinh-Gordon equation expansion method for solving space-time conformable partial differential equations. Approximate analytical solutions for the coupled Cahn-Allen equation, coupled Burgers equation, and Fokas equation are obtained. Several exact solutions for them are successfully established. The solutions obtained by the methods indicate that they are easy to implement and effective. This article has been arranged as follows. In Sect. 2, we propose some mathematical definitions and prerequisites required later in the article. The section also illustrates general principles of the conformable derivative along with basic steps of techniques. In Sect. 3, three equations including the space-time conformable coupled Cahn-Allen equation, the space-time coupled Burgers equation, and the space-time conformable Fokas equation are examined, and the exact solution for them is determined using two techniques. This section also contains several numerical simulations of acquired solutions. Finally, the article ends with some conclusions.

Preliminaries and definitions
In this section, we review some of the necessary prerequisites that will be employed in the article.
It should be noted that taking α = 1 in this derivative yields the standard definition for derivative. Therefore, this method can be considered a natural generalization for the conventional derivative.
This new definition satisfies the following properties. Let α ∈ (0, 1], f , g be α-differentiable at a point t, then holds. As stated in [64], many of the existing definitions for derivative do not meet some of these mentioned properties. Enjoying these features is one of the valuable and distinctive points for the conformable derivative.
1. Let us take the following problem with the conformable derivative: 2. Using the transformations ψ = ψ(ξ ) and ξ = σ x α (α)l t α (α) , we reduce the nonlinear partial differential equation to the following ordinary differential equation: where the values of σ and l will be found later. 3. Now, consider that Eq. (3) has the solution of the form where (ξ ) = p 1 e q 1 ξ + p 2 e q 2 ξ p 3 e q 3 ξ + p 4 e q 4 ξ .
The values of constants

The extended sinh-Gordon equation expansion method
The extended sinh-Gordon equation expansion method (EShGEEM) is a robust method that may easily derive dark, bright, combined dark-bright, singular, combined singular soliton, and other trigonometric function solutions to nonlinear PDEs of an integer or noninteger order [83]. This technique has had many successful applications in solving various problems. For example, the authors of [84] used EShGEEM to study the conformable version of Biswas-Milovic equation with the Kerr law and parabolic law nonlinearity. Another application of EShGEEM can be found in [85], where they considered a nonlinear partial differential equation describing the wave propagation in nonlinear low-pass electrical transmission lines.
Following the works of [84,85], we outline the main steps of EShGEEM as follows.
1. Let us take the following problem with the conformable derivative: Using the transformations = (ξ ) and ξ = σ x α (α)l t α (α) , it is possible reduce the NPDE to the following ordinary differential equation: where the values of σ and l will be found later, and the prime notation means the derivative of with respect to ξ . 2. Consider Eq. (7) has the solution of the form where A 0 , A j , B j (j = 1, 2, . . . , M) are constants to be determined later and θ is a function of ξ that satisfies the following ordinary differential equation: By considering the homogenous balance principle in (7), the value of M can be determined. Equation (9) possesses the following solutions: and where i = √ -1. 3. Substituting Eq. (8) along with Eqs. (10) and (11) into Eq. (7) and collecting all terms, we obtain a polynomial in terms of θ l sinh i (θ ) cosh j (θ ) for l = 0, 1, i, j = 0, 1, 2, . . . . Setting each coefficient of such a polynomial equal to zero, a system of nonlinear equations in terms of σ , l, Solving the above algebraic equations using any symbolic computation software, the values of σ , l and A 0 , A j , B j (1 ≤ j ≤ M) are determined. 5. Based on Eqs. (10) and (11), one can obtain the soliton solutions of Eq. (6) as follows:

Applications of techniques and the main results
In this section, to illustrate the applicability of the generalized exponential rational function method and the extended sinh-Gordon equation expansion method to solve nonlinear conformable partial differential equations, three examples are considered.

The space-time conformable coupled Cahn-Allen equation
Consider the space-time conformable Cahn-Allen equation [86] Using the transformation where c and ν are two nonzero constants. Utilizing the wave transformation (15) converts Eq. (14) into the following NODE: Using the balance principle on the terms U 3 and U in Eq. (16), we have M + 2 = 3M, so M = 1.

Application of GERFM for (14)
Using Eq. (5) together with M = 1, we have Proceeding as outlined in the second section, we acquire the following sets of solutions to Eq. (14).
Putting values in Eqs. (17) and (18) yields the following solution: Consequently, we get the solution of Eq. (14) as Figure 1 depicts the dynamic behavior of solution u 1 (x, t) presented in (19).
Putting values in Eqs. (17) and (18) yields the following solution: Consequently, we get the solution of Eq. (14) as .
We obtain Putting values in Eqs. (17) and (21) yields the following solution: Consequently, we get the solution of Eq. (14) as Figure 3 depicts the dynamic behavior of solution u 3 (x, t) presented in (22).
We obtain Putting values in Eqs. (17) and (23) yields the following solution: Consequently, we get the solution of Eq. (14) as We obtain Putting values in Eqs. (17) and (25) yields the following solution: Consequently, we get the solution of Eq. (14) as Figure 4 depicts the dynamic behavior of solution u 5 (x, t) presented in (26). We obtain Putting values in Eqs. (17) and (27) yields the following solution: Consequently, we get the solution of Eq. (14) as

Application of EShGEEM for (14)
According to what was discussed above, we obtain M = 1. Taking M = 1 into account in Eqs. (8), (12), and (13), we respectively obtain and Set 1: Using these values, the following solution for (16) is obtained: Consequently, we get the solution of Eq. (14) as ) .

(32)
Set 2: Using these values, the following solution for (16) is obtained: Consequently, we get the solution of Eq. (14) as ) .
Using these values, the following solution for (16) is obtained: Consequently, we get the solution of Eq. (14) as ) .

The space-time coupled Burgers equation
Consider the space-time conformable coupled Burgers equations [87] D α t u -D 2α Using the transformation where c is a nonzero constant. Utilizing the wave transformation (38) converts Eq. (37) into the following NODE: Using the balance principle on the terms UU and U in Eq. (39), we have M +2 = M +M +1, so M = 1.

Application of GERFM for (37)
Using Eq. (5) together with M = 1, we have Proceeding as outlined in the second section, we acquire the following sets of solutions to Eq. (37).

Application of EShGEEM for (37)
The initial assumption of the solution structure of (39) is taken to be: and Applying the extended EShGEEM with the help of Eqs. Set 1: Using these values, the following solution for (16) is obtained: and Consequently, we get the solution of Eq. (14) as where ξ = 1 (α) (x α + 2A 0 (pq-1) p-1 t α ). Figure 10 depicts the dynamic behavior of solution u 5 (x, t), v 5 (x, t) presented in (56).

The space-time conformable Fokas equation
Consider the space-time conformable Fokas equation [88] 4 Let us introduce the wave transformation as where c, k 1 , k 2 , l 1 , l 2 are nonzero constants. Utilizing Eq. (66) converts Eq. (65) into the following NODE: If we apply the balance principle on the terms UU and U in Eq. (67), we have 2M = M +2, so M = 2.

Application of GERFM for (65)
Using Eq. (5) together with M = 2, we have Proceeding as outlined in the second section, we acquire the following sets of solutions to Eq. (65).
(90) that have not been previously explored in previous references. Since the techniques are direct, powerful, and efficient, they can be efficiently used to find the exact solutions of different nonlinear differential equations in several branches of nonlinear sciences.