New Exact Solutions of the Conformable Space-Time Sharma–Tasso–Olver Equation Using Two Reliable Methods

The major purpose of this article is to seek for exact traveling wave solutions of the nonlinear space-time Sharma–Tasso–Olver equation in the sense of conformable derivatives. The novel (G ′ G )-expansion method and the generalized Kudryashov method, which are analytical, powerful, and reliable methods, are used to solve the equation via a fractional complex transformation. The exact solutions of the equation, obtained using the novel (G ′ G )-expansion method, can be classified in terms of hyperbolic, trigonometric, and rational function solutions. Applying the generalized Kudryashov method to the equation, we obtain explicit exact solutions expressed as fractional solutions of the exponential functions. The exact solutions obtained using the two methods represent some physical behaviors such as a singularly periodic traveling wave solution and a singular multiple-soliton solution. Some selected solutions of the equation are graphically portrayed including 3-D, 2-D, and contour plots. As a result, some innovative exact solutions of the equation are produced via the methods, and they are not the same as the ones obtained using other techniques utilized previously.


Introduction
Many nonlinear physical phenomena, such as those found in solid-state physics, plasma physics, optical fibers, shallow water waves, fluid dynamics, and biology, have been modeled by nonlinear partial differential equations (NPDEs) of integer-or fractional-order. Therefore, finding explicit exact solutions and approximate analytical solutions of NPDEs is one of the most significant and active areas of investigation in pure and applied mathematics. Several powerful methods based on using symbolic software packages such as Mathematica or Maple 17 have been proposed and developed to analytically solve NPDEs. Some examples of approaches for obtaining approximate solutions in an analytical form to NPDEs are the Adomian decomposition method (ADM) [1,2], the variational iteration method (VIM) [3,4], the differential transform method (DTM) [5], and the homotopy perturbation method (HPM) [6,7]. Some algorithms for obtaining explicit exact solutions of NPDEs are, for instance, the ( G G )-expansion method [8,9], the ( G G , 1 G )-expansion method [10], the fractional Riccati expansion method [11,12], the improved extended tanh-coth method [13], the Kudryashov method [14,15], and the sub-equation method [16,17]. All of the above methods are based on the homogeneous balance principle. as follows. In Section 2, the definition and some properties of the conformable derivative are given. The description of the two methods is included in this section as well. In Section 3, we illustrate the application of the used methods to the nonlinear conformable space-time Sharma-Tasso-Olver equation. In Section 4, we give some plots and their physical explanations of some chosen exact solutions of the equation. Some conclusions and discussions for the results obtained using the methods are given in Section 5.

Mathematical Preliminaries
In this section, we will provide fundamental concepts including the definition and vital properties of the conformable derivative and the algorithms of the novel ( G G )-expansion method and the generalized Kudryashov method. They are required for constructing explicit exact solutions of the conformable space-time Sharma-Tasso-Olver equation using the methods.

Conformable Derivative and Its Properties
The definition of the conformable derivative and its important properties are provided as follows.
Definition 1. Let f : [0, ∞) → R be a function. Then, the conformable derivative of f of order α is defined by [44,45] The function f is α-conformable differentiable at a point t > 0 if the limit in Equation (1) exists. (1) was initially called the conformable fractional derivative and had been utilized in numerous applications of fractional differential equations (FDEs) [46][47][48]. Until 2018, Tarasov [49] showed that the conformable fractional derivative in Equation (1) does not provide innovative ideas in the spaces of differentiable functions and is not a fractional-order derivative. In particular, some of the following properties demonstrate that the conformable fractional derivative can be written in terms of an ordinary derivative. Throughout this work, we thus call the derivative in Equation (1) the conformable derivative.

Remark 2.
Conformable derivatives of some interesting functions are as follows [44].
Theorem 2. [45,46] Suppose f : (0, ∞) → R is a function such that f is differentiable and α-conformable differentiable. Further, suppose that g is a differentiable function defined in the range of f . Then, where the prime notation ( ) denotes the ordinary derivative.

Description of the Methods
Consider the following nonlinear evolution partial differential equation for two independent variables x and t, where D α t u, D β x u are the conformable derivatives of a dependent variable u with respect to variables t and x, respectively, with 0 < α, β < 1. The function F is a polynomial of the unknown function u = u(x, t), and its diverse partial derivatives in which the highest order derivatives and nonlinear terms are involved. There is a common step between the novel ( G G )-expansion method and the generalized Kudryashov method which is a conversion from the partial differential equation in (3) to an ordinary differential equation (ODE) via using a traveling wave variable ξ [50][51][52]. We suppose that where V is a non-zero arbitrary constant. Converting (3) via transformation (4) and then integrating the resulting equation with respect to ξ (if possible), Equation (3) is reduced to an ODE in the variable U = U(ξ) as where P is a polynomial function of U(ξ) and its various derivatives. The prime notation ( ) represents the ordinary derivative with respect to ξ. Next, we provide the remaining steps of each of the methods.
2.2.1. Description of the Novel ( G G )-Expansion Method Step 1: Suppose that a solution of Equation (5) can be expressed in powers of ψ(ξ) as follows, where The unknown constants a −N or a N may be zero, but both of them cannot be zero simultaneously. The constants a j (j = 0, ±1, ±2, . . . , ±N) and d are computed at a following step and the function G = G(ξ) satisfies the following nonlinear second-order ODE, where the prime notation ( ) is the ordinary derivative with respect to ξ, and where λ, µ, and v are real parameters.
Step 2: The value of the positive integer N in solution (6) can be computed by balancing the highest-order derivative term with the nonlinear terms of the highest order occurring in Equation (5). The degree formulas of some terms are given as where N is the degree of U(ξ), i.e., Deg[U(ξ)] = N.
Step 4: Assuming that the unknown constants of the algebraic equations in Step 3 can be possibly obtained, we then substitute the values of these constants together with the solution φ(ξ) of Equation (9) into Equation (6) to get exact traveling wave solutions of (3) when ξ is defined in (4).

Description of the Generalized Kudryashov Method
Step 1: Assume that the solution of Equation (5) can be expressed in a rational form as where a i (i = 0, 1, 2 . . . , N), b j (j = 0, 1, 2 . . . , M) are constants to be determined at a later step such that a N = 0, b M = 0 and the function Q = Q(ξ) is a solution of It is obviously found that the solution of Equation (12) is where C is a constant of integration.
Step 2: We find the positive integers N and M in Equation (11) by employing the homogeneous balance method, i.e, equating between the highest order derivative and the highest power nonlinear term in Equation (5). The formulas in Equation (10) can also be used in this step.
Step 3: Substituting Equation (11) into Equation (5) along with Equation (12), we obtain a polynomial R(Q) of Q. Next, equating all of the coefficients of R(Q) to zero, a system of algebraic equations is obtained. Solving this system with the aid of symbolic software packages such as Maple, we can get the values of a i (i = 0, 1, 2 . . . , N), b j (j = 0, 1, 2 . . . , M). When we substitute these values and the function Q(ξ) in Equation (13) into Equation (11), then we attain the exact solutions of the reduced Equation (5). In consequence, exact solutions of (3) are finally obtained using ξ defined in (4).

Application of the Methods
The Sharma-Tasso-Olver equation of integer order is expressed as where ρ is an arbitrary real parameter. In this paper, we, however, consider the conformable space-time Sharma-Tasso-Olver equation described as where D κ s u represents the conformable derivative of the function u with respect to an independent variable s of order κ. This function u(x, t) is unknown and depends on the variables x and t, and ρ is an arbitrary real parameter. We will apply the novel G G -expansion method and the generalized Kudryashov method to produce exact traveling wave solutions of (15). However, the common step of both methods is to convert (15) into an ODE via the traveling wave transformation in the variables x and t as follows, Utilizing Theorem 2, we obtain D α where the prime notation ( ) denotes the derivative with respect to ξ. Integrating Equation (17) with respect to ξ along with algebraically manipulating some terms and then letting the constant of integration to be zero, we get the following ODE, 3.1. Obtaining Exact Solutions of Equation (15) Using the Novel ( G G )-Expansion Method Utilizing the formulas (10) to balance the highest order derivative U with the nonlinear term of the highest order U 3 in Equation (18), we obtain N = 1. By Equation (6) in Section 2.2.1, the solution of Equation (18) has the following form, where ψ(ξ) = d + φ(ξ) and φ(ξ) = G G is a solution of the generalized Riccati equation in Equation (9). Substituting Equation (19) into Equation (18), the left hand side of Equation (18) is converted into polynomials of (ψ(ξ)) j = (d + φ(ξ)) j , where j = 0, ±1, ±2, ±3.
Equating the coefficients of the same power of the resulting polynomials to zero, we have the following set of nonlinear algebraic equations, Using the symbolic computation software Maple 17 to solve the above system (20), we obtain three independent cases of the unknown constants a 0 , a 1 , a −1 , d, and V. For the sake of convenience, we set Consequently, the exact solutions of Equation (15), depending upon the following three cases of the unknowns in Equation (20) and the families of the solution φ(ξ) as shown in Appendix of [53], are expressed as follows.

Family 1: When
, the solutions of Equation (15) written in terms of the hyperbolic functions are as follows, where A and B are two non-zero real constants that satisfy B 2 − A 2 > 0, Family 2: When ∆ < 0 and λ(v − 1) = 0 (or µ(v − 1) = 0), the exact solutions of Equation (15) expressed in terms of the trigonometric functions are as follows, where A and B are two non-zero real constants satisfying Family 3: When µ = 0 and λ(v − 1) = 0, the exact solutions of Equation (15) written as the hyperbolic functions are as follows, where c 1 is an arbitrary constant. Family 4: When µ = λ = 0 and v − 1 = 0, the rational function solution of Equation (15) is as follows, where c 2 is an arbitrary constant. Case 2: The second set of the unknown constants is where µ, λ, v, and ρ are arbitrary constants. Substituting Equation (51) into Equation (19), and then using Equations (16) and (21), we obtain the following explicit exact solutions in which ξ = Family 1: When ∆ > 0 and λ(v − 1) = 0 (or µ(v − 1) = 0), the exact solutions of Equation (15) expressed as the hyperbolic functions are where A and B are non-zero real constants satisfying B 2 − A 2 > 0, Family 2: When ∆ < 0 and λ(v − 1) = 0 (or µ(v − 1) = 0), the exact solutions of Equation (15) written in terms of the trigonometric functions are shown as follows, where A and B are two non-zero real constants such that A 2 − B 2 > 0, Family 3: When µ = 0 and λ(v − 1) = 0, Equation (15) has the hyperbolic function solutions as follows, where c 1 is an arbitrary constant. Family 4: When µ = λ = 0 and v − 1 = 0, Equation (15) has the rational function solution as follows, where c 2 is an arbitrary constant. Case 3: The last set of the unknown constants is expressed as where µ, λ, v, d, and ρ are arbitrary constants. Substituting Equation (79) into Equation (19), and then using Equations (16) and (21), we obtain the following exact solutions in which ξ = x β β − ρ∆t α α .
Family 1: When ∆ > 0 and λ(v − 1) = 0 (or µ(v − 1) = 0), the exact solutions of Equation (15) written as the hyperbolic function solutions are as follows, where A and B are two non-zero real constants satisfying the condition B 2 − A 2 > 0, where A and B are two non-zero real constants satisfying the condition A 2 − B 2 > 0, Family 3: When µ = 0 and λ(v − 1) = 0, Equation (15) has the hyperbolic function solutions as follows, where c 1 is an arbitrary constant. Family 4: When µ = λ = 0 and v − 1 = 0, the exact solution of Equation (15) expressed as the rational function is where c 2 is an arbitrary constant.

Obtaining Exact Solutions of Equation (15) Using the Generalized Kudryashov Method
Substituting solution form (11) into (18) and then applying the homogeneous balance principle to the resulting equation, we have If we choose M = 1, then N = 2. Using Equation (11) in Section 2.2.2, the exact solution of Equation (18) takes the form where Q = Q(ξ) satisfies Equation (12). The parameters a 0 , a 1 , a 2 , b 0 , and b 1 are determined at the next step. Substituting Equation (108) into Equation (18), and utilizing Equation (12) and then setting all coefficients of the functions Q k to zero, we get Solving the above algebraic system with the aid of the Maple package program, we get the following results. Case 1: where b 1 is an arbitrary constant. From Equations (13), (108), and (110), we obtain the simplified exact solution of Equation (15) as follows, where ξ = x β β − 4ρt α α . Case 2: where b 0 , b 1 are arbitrary constants. From Equations (13), (108), and (112), the simplified exact solution of Equation (15) can be obtained as where ξ = x β β − ρt α α . Case 3: where b 0 , b 1 are arbitrary constants. From Equations (13), (108), and (114), the simplified exact solution of Equation (15) can be expressed as where ξ = x β β − ρt α 4α .

Graphical Representations of Some Exact Solutions and Their Physical Explanations
In this section, we will give some graphical representations of the above-determined exact solutions of the conformable space-time Sharma-Tasso-Olver Equation (15), which were obtained using the novel ( G G )-expansion method and the generalized Kudryashov method in Section 3. Here, we take ρ = 2, and use the following sets of the fractional orders, {β = 1, α = 1}, {β = 0.8, α = 0.2}, and {β = 0.5, α = 0.5}, for the equation. Some selected explicit exact solutions will be plotted as two-and three-dimensional graphs in which the used domain is 0 ≤ x ≤ 100 and 0 ≤ t ≤ 10. Their corresponding contours are drawn as well. Furthermore, their physical explanations of the solutions are included. Figures 1-3 demonstrate the graphical representations of some chosen exact solutions of the problem obtained using the novel ( G G )-expansion method. They are described below. In Figure 1, we show different plots of the exact solution u 1 9 (x, t) in Equation (28) using the following parameter values, µ = 0.5, λ = 1, v = 0.5, d = 1, A = 0.5, and B = 1. In particular, Figure 1a-c shows the 3-D plot, the 2-D plot with t fixed at t = 1, and the contour plot of solution (28), respectively, when the set of the fractional orders {β = 1, α = 1} is used. Using the same parameter values as shown above except using {β = 0.8, α = 0.2}, the 3-D graph, the 2-D graph while t is held fixed at t = 1 and the contour graph of solution (28) are plotted in Figure 1d-f, respectively. Proceeding in a manner analogous to the above plots except using the fractional order set {β = 0.5, α = 0.5}, we obtain the 3-D plot, the 2-D plot with t = 1, and the contour plot of solution (28) in Figure 1g-i, respectively. By characterizing the shapes of the plots in Figure 1, solution (28) behaves as a singular kink-type solution. The selected exact solution u 3 38 (x, t) in Equation (105) is plotted in Figure 3 using the three sets of the fractional orders and the following parameter values, λ = 0.5, v = 1.6, d = c 1 = 1. In particular, Figure 3a-c shows the 3-D plot, the 2-D plot when t = 1, and the contour plot of solution (105), respectively, when we use {β = 1, α = 1}. Employing the same parameter values as shown above except using {β = 0.8, α = 0.2}, the 3-D graph, the 2-D graph as t is held fixed at t = 1, and the contour graph of solution (105) are plotted in Figure 3d-f, respectively. Proceeding in a manner analogous to the above plots, except using the fractional order set {β = 0.5, α = 0.5}, we obtain the 3-D plot, the 2-D plot with t = 1, and the contour graph of solution (105) in Figure 3g-i, respectively. From these plots, solution (105) is characterized as a kink-type solution.  Figure 4d-f, respectively. Proceeding in like manner to the above plots, except using the fractional-order set {β = 0.5, α = 0.5}, we have the 3-D plot, the 2-D plot with t = 2, and the contour graph of solution (113) in Figure 4g-i, respectively. By characterizing the shapes of the plots in Figure 4, solution (113) shows the behavior of a kink-type solution. Using the second method, the exact solution u(x, t) in Equation (119) is graphically shown in Figure 5 with the three different sets of the fractional orders and the constant C = 1. In particular, Figure 5a-c displays the 3-D plot, the 2-D plot with t = 2, and the contour plot of solution (119), respectively, when {β = 1, α = 1} is used. Employing the same parameter values above, except using {β = 0.8, α = 0.2}, the 3-D graph, the 2-D graph with t = 2, and the contour graph of solution (119) are plotted in Figure 5d-f, respectively. Proceeding in a manner analogous to the previous plots, except using the fractional-order set {β = 0.5, α = 0.5}, we obtain the 3-D plot, the 2-D plot with t = 2, and the contour graph of solution (119) in Figure 5g-i, respectively. By observing the shapes of the plots in Figure 5, solution (119) is classified into a singular multiple-soliton solution.

Conclusions
In this article, we have constructed explicit exact solutions of the (1+1)-dimensional conformable space-time Sharma-Tasso-Olver equation expressed in Equation (15) using the novel G G -expansion method and the generalized Kudryashov method with the help of the fractional complex transform and the symbolic computation package Maple 17. The obtained results have revealed that the methods are straightforward, reliable, and powerful. In particular, the novel G G -expansion method gives hyperbolic, trigonometric, and rational function solutions for the equation; however, the generalized Kudryashov method provides fractional solutions of the exponential functions, which are possibly converted into hyperbolic function solutions. Some of these obtained exact solutions have been graphically characterized into a variety of various physical structures such as the single-kink wave solution, the singular periodic wave solution, and the singular multiple-soliton solution. The applications of these exact solutions have been discovered in several physical phenomena such as plasma waves and optical fibers. All calculations in this investigation have been made and verified using the Maple package program. Previously, many authors had tackled the fractional Sharma-Tasso-Olver equation in different approaches. For example, finding exact solutions of the fractional Sharma-Tasso-Olver equation in the sense of the modified Riemann-Liouville derivative using the (G /G, 1/G)-expansion method [56], the tanh ansatz method [57], the exp(−Φ(ξ)) method [58], the sub-equation method [59], and the improved extended tanh-coth method [13]. In addition, constructing exact solutions of the nonlinear conformable time Sharma-Tasso-Olver equation via conformable derivatives was done using the simplest equation method [60] and the direct algebraic method [61]. The Sharma-Tasso-Olver equation as shown in the above articles involves with both of only the conformable time derivative and the conformable space-time derivatives. However, in this paper, we construct exact solutions of Equation (15) in the sense of conformable derivatives with respect to x and t. To the best of our knowledge, in [62], the authors analytically solved (15) using the Exp-function method and only two exact solutions, which are expressed in terms of the exponential function solutions and characterized as kink-type solutions, were constructed. Comparing our results obtained using the novel G G -expansion method with the results in [62], not only the number of our exact solutions is considerably more than the number of their results, but our solutions are also classified into more different types. This is because the novel G G -expansion method is generalized from many similar methods such as the G G -expansion method, improved G G -expansion method, and the generalized and improved G G -expansion method. In consequence, some of our exact solutions of (15) via the two methods are novel and reported here for the first time. According to the mentioned advantages of the methods for obtaining exact traveling wave solutions, they could be applied efficiently for a wide range of nonlinear conformable partial differential equations or other fractional-order PDEs, which appear in several branches of the applied sciences and engineering.
Author Contributions: All authors contributed equally to the manuscript. All authors have read and agreed to the published version of the manuscript.