Extracting the Ultimate New Soliton Solutions of Some Nonlinear Time Fractional PDEs via the Conformable Fractional Derivative

: Nonlinear fractional-order differential equations have an important role in various branches of applied science and fractional engineering. This research paper shows the practical application of three such fractional mathematical models, which are the time-fractional Klein–Gordon equation (KGE), the time-fractional Sharma–Tasso–Olever equation (STOE), and the time-fractional Clannish Random Walker’s Parabolic equation (CRWPE). These models were investigated by using an expansion method for extracting new soliton solutions. Two types of results were found: one was trigonometric and the other one was an exponential form. For a profound explanation of the physical phenomena of the studied fractional models, some results were graphed in 2D, 3D, and contour plots by imposing the distinctive results for some parameters under the oblige conditions. From the numerical investigation, it was noticed that the obtained results referred smooth kink-shaped soliton, ant-kink-shaped soliton, bright kink-shaped soliton, singular periodic solution, and multiple singular periodic solutions. The results also showed that the amplitude of the wave augmented with the pulsation in time, which derived the order of time fractional coefficient, remarkably enhanced the wave propagation, and influenced the nonlinearity impacts.


Introduction
The mysteries of various complex phenomena occurring in nature can be explained by providing fractional mathematical models with the help of fractional-order derivatives and by providing exact soliton solutions of these models.Studying such mathematical models has a firm place in various fields of engineering, modern medical science, and physical science.As an evolution of integer-order perception, fractional calculus is valuable for modeling real-world occurrences with complex dynamics [1,2].Many studies have shown that fractional-order differential equations can successfully express complex forcible characteristics [3,4].Moreover, by applying fractional derivatives, we can explain the reliability of the memory effects, called a fundamental approach, to various real-world occurrences [5].Over the past few years, numerous interpretations of fractional derivatives have been proposed due to the rewards provided by giving models of real-world phenomena.It has been shown that fractional differential equations, especially fractional-order partial differential equations (PDEs), have made an indispensable contribution in many scientific and engineering sectors.The use of such fractional PDEs in fields like theoretical physics, plasma physics, computational biology, applied physics, biochemistry, signal processing, systems identification, electronic communication, including blogs and Facebook, electromagnetism, electrochemistry, nanotechnology, nonlinear optics, fluid mechanics, control theory, finance, fractional dynamics, etc., is continuously increasing [6,7].Since fractional operators have a single kernel in the classical frame, many nonlinear complex phenomena cannot be exactly characterized by traditional illustrations.Therefore, new fractional derivatives with nonsingular kernels were developed to investigate nonlocal dynamics, such as Atangana-Baleanu-Caputo [8,9] and fractal-fractional operator kernels [10].
Moreover, the applications of the spectral approach in the procedure for obtaining solutions to the space-time non-integer order reaction-diffusion situation can be noticed in [11,12].We note that fractional-order derivatives [13,14] are difficult to deal with analytically, primarily those that describe real-world operations, and investigators sometimes must depend on the numerical approach to solve these equations.Many investigators have used various operators of fractional derivatives like Caputo-Fabrizio [15,16], Riemann-Liouville [17,18], conformable fractional derivatives [19,20], etc., in fractional-order PDEs of any system.One of the most familiar fractional-order PDE is the nonlinear KGE [21], which has widespread implementations in condensed-type matter physics, nonlinear optics, quantum mechanics, etc., and is also suitable for modeling real-world occurrences.The second one is the nonlinear STOE [22], which is essential in applied science and engineering.The nonlinear STOE explains the factual naturalistic model of fusion and fission, which Kupershmidt introduces as the idea of dark equations in model equations [23].The third one is the nonlinear CRWPE [24], which is vital in understanding the quantitative and qualitative features of various natural phenomena.For example, the nonlinear oscillation of an earthquake can be modeled by the nonlinear CRWPE.Our cited model equations can gain complete shock-wave or topological-wave solutions.The fast progress of systematic methods for getting the solution to fractional PDEs has been of considerable help in addressing physical-world problems of a complicated nature.Consequently, researchers have focused on investigating fractional-order calculus and detecting exact and methodical techniques for finding the perfect solutions for fractional PDEs.Recently, many researchers have investigated these types of equations by applying different methods, such as the Generalized Kudryashov method [25,26], the residual-power-series method [27,28], the exp-function method [29,30], the long-wave method [31], the variational iteration method [32,33], the extended direct algebraic method [34,35], the sine-Gordon expansion approach [36], the Jacobi elliptic function method [37], the Sarder sub-equation method [38], the G ′ G , 1 G -expansion method [39][40][41], and many other techniques.Now, there is a more well-organized method called the [42,43] to solve nonlinear fractional-order PDEs.This research article's prime goal is to find the exact solitary-wave solutions of our proposed three model equations through this simple mathematical expansion method.
The three proposed nonlinear time-fractional PDEs are given below: -The nonlinear KGE is [44,45] where k 1 and k 2 are any constants and β represents the time derivative of fractional order.
-The nonlinear CRWPE is [48,49] The configuration of this article is split into six parts as follows: In Section 2, we give a short explanation of the conformable fractional derivative with some characteristics.The general study of our proposed expansion method is described in Section 3. The analytical implementation of our proposed technique to extract the exact solitary wave solutions of the KGE, the STOE, and the CRWPE is in Section 4. In Section 5, the outcomes and graphical explanations are given.Ultimately, the conclusion is conferred in Section 6.

Definition of Conformable Fractional Derivative and Its Characteristics
The generally accepted notion of the fractional-order integral differential operator is defined as follows [50][51][52][53][54]: where β is the fractional order and f (β) implies the real part of β, and a is the lower limit of the operation, which is constant.On the contrary, the upper limit x varies with x > a.
Recently, some researchers have developed definitions of some essential fractional derivatives, such as the conformable fractional derivative, the Riemann-Liouville derivative, the modified Riemann-Liouville derivative, the Caputo derivative, the generalized Riemann-Liouville-Caputo derivative, the Caputo-Fabrizio derivative, Atangana-Baleanu derivative, etc.Here, the characteristics and definitions of a simple fractional-order derivative described by Khalil et al. [55] are given, called the conformable fractional derivative.
In our article, we used the G ′ G ′ +G+A -expansion method for solving our proposed three nonlinear model equations in the sense of the conformable fractional derivative [56,57].
If F : R + → R is a continuous function, then the definition of the conformable fractional derivative of order β is written as where t is positive and δ ∈ (0, 1).Characteristics: Consider the two β-differentible functions F and G; then, (i) (ii) where r 1 , r 2 are real constants. (iv) (vii) If F and G are β-differentiable function of t in the domain (0, ∞) and G(t) ̸ = 0 then,

Discussion of the Expansion Method
The main steps of the G ′ +G+A -expansion method are given in detail in this portion.Consider the following nonlinear PDEs as where polynomial ψ is a function of v = v(x, t) and its derivatives.Now, the mechanisms of this method are given periodically.Mechanism I: Using the wave conversion technique, we first wrote all the variables in the proposed PDEs as a linear combination of a unique variable as follows: Here, the coefficient of space derivative m is called the wave number, and the coefficient of time derivative µ is called the wave velocity and has constant values.Applying the wave conversion technique from Equation ( 14) to ( 13), a simple ordinary differential equation (ODE) can be written as Here, Mechanism II: Now, we must consider an equation like the following form, which will be the solution of Equation ( 15) where a r are the coefficients of the polynomial , r = 0, 1, 2, . . .M and consider the function G(ρ) that satisfies the following ODE: Now, the value of balance number M will be determined by applying the homogeneous balance technique between the supreme-order derivative and the supreme-degree convective terms in Equation (15).By assigning the value of the balance number M in Equation ( 16) and simplifying the resultant, we performed the derivative with respect to ρ as often as was needed, then set these derivatives in Equation (15).
Mechanism III: From Equation ( 15), the coefficients of the successive power of G ′ +G+A are taken to be zero, which provides some equations in terms of H, R, A, µ, m, and a r (r = 0, 1, 2, . . .M).After solving these equations via the program Mathematica, the values of a r (r = 0, 1, 2, . . .M) will be found.Now, we find the exponential and trigonometric function solutions for the positive and negative values of the discriminant of the auxiliary equation in Equation (17).Then, we put the values of the solution G in the term G ′ G ′ +G+A and finally, for the values of G ′ G ′ +G+A , a r , and ρ, we have the ultimate solution of Equation ( 13).

Investigation of the KGE
By setting the following transformation in Equation (1), we obtain the following second equation: By applying the homogenous balance rule between the terms v 3 and d2 v dρ 2 in Equation ( 19), the balance number M = 1.Equation ( 16) moves as Inserting Equation ( 20) along with Equation ( 17) into Equation ( 19) and then placing the coefficients of (r = 0, 1, 2, . . .M) to zero into the resultant, a set of equations in terms of a 0 , a 1 , m, µ, H, R, k 1 and k 2 will be obtained as follows: By solving the above system via the software Mathematica 11, we obtain the following set of values: Now, the soliton solutions to the KGE are given for two occurrences as follows: Using the wave transformation from Equation ( 18), we can write the soliton solution of Equation (1) as Using the wave transformation from Equation ( 18), the other soliton solution of Equation ( 1) is written as

Investigation of the STOE
By setting the following transformation in Equation ( 2), we obtain the following second equation: Integrating the above equation, we obtain where N is the integration constant.By applying the balance rule between the terms v 3 and dρ 2 in Equation ( 27), the balance number, M = 1.Equation ( 16) moves as Inserting Equation ( 28) along with Equation ( 17) into (27) and then placing the coefficients of (r = 0, 1, 2, . . .M) to zero into the resultant, a set of equations in terms of a 0 , a 1 , µ, H, R, N, and d will be obtained, and after solving the obtaining set of equations via the software Mathematica, two sets of results are attained as follows.
Set 1: Set 2: For set 1, the soliton solutions to the STOE are given for two occurrences as follows: Using the wave transformation from Equation (25), we can write the soliton solution of Equation (2) as where Again, using the wave transformation from Equation ( 25), the soliton solution of Equation ( 2) is written as where µ = −3dHR − dR + 6da 0 R + 3dR 2 + 3da 2 0 − 3da 0 H + dH 2 .For set 2, the soliton solutions to the STOE are given for two occurrences as follows, Occurrence 1.
Using the wave transformation from Equation ( 25), the other soliton solution of Equation ( 2) is written as Again, using the wave transformation from Equation ( 25), the other soliton solution of Equation ( 2) is written as

Investigation of the CRWPE
By setting the following transformation in Equation (3), we obtain the following second equation: Integrating the above equation, we obtain where ρ 0 is the integration constant.By applying the balance rule between terms v 2 and dv dρ in Equation ( 39), the balance number, M = 1.Equation ( 16) moves as Inserting Equation ( 40) along with Equation ( 17) into Equation ( 39) and then placing the coefficients of , (r = 0, 1, 2, . . .M) to zero into the resultant, a set of equations in terms of a 0 , a 1 , η, H, R, p, and ρ 0 will be obtained, and after solving the obtaining set of equations via the software Mathematica, the following set of particular values are obtained as follows: For the above set, the soliton solutions to the nonlinear CRWP equation are given for two occurrences as follows: Occurrence 1.
Using the wave transformation from Equation ( 37), the soliton solution of Equation ( 3) is written as where Again, using the wave transformation from Equation (37), the other soliton solution of Equation ( 3) is written as where η = −p + 2p 2 R + 2pa 0 − p 2 H.

Interpretation of the Numerical Outputs along with Their Graphical Portraiture
In this graphical section, we want to figure out some soliton solutions of the timefractional nonlinear PDEs which are investigated in this article.This article aims to solve the KGE, the STOE, and the CRWPE in the sense of the conformable fractional derivative by applying the expansion method.The results obtained in this research paper are consistent, ordinary, and more accurate than the results of all previous papers.The accuracy and stability were looked over by setting the attained results back into the main equations, and we found exactness.The proposed expansion method gives two types of solitary-wave solutions: one is the exponential-function solution, and the other is the trigonometric-function solution.The above Equations ( 22), ( 30), (34), and ( 42) represent the exponentialfunction solutions, and Equations ( 24), ( 32), (36), and (44) represent trigonometric-function solutions.For simplicity of the article, the graphical interpretations of Equations ( 34), (36), and ( 44) are omitted.The data in the graph displays the characteristics of the traveling wave solutions by inserting various parametric values.
Figures 1-5 exhibits the exponential and trigonometric solitary-wave solutions of the mentioned three model equations in the shape of a 3D plot (for β = 0.5), contour plot (for β = 0.5), and 2D plot (for β = 0.25, 0.5, 0.75).Figure 1 implies the smooth kink-shaped soliton solution with the parameters as 01, and T = 0.6 in the case of the 3D plot (for −100 ≤ x ≤ 100 and 0 ≤ t ≤ 100), contour plot (for −100 ≤ x ≤ 100 and 0 ≤ t ≤ 100), and 2D plot (for x = 0 and 0 ≤ t ≤ 10).These figure shows that when increasing the order β, the slope of the graph is increasing.
Figure 3 implies the anti-kink soliton solution with the parameters as 75, and T = 3 in the case of the 3D plot (for −100 ≤ x ≤ 100 and 0 ≤ t ≤ 100), contour plot (for −100 ≤ x ≤ 100 and 0 ≤ t ≤ 100), and 2D plot (for x = 0 and 0 ≤ t ≤ 3).These figure shows that when increasing the order β, the graph is shifted right.
Figure 4 implies the singular periodic solution with the parameters as 2475 and T = −9 in the case of the 3D plot (for −3 ≤ x ≤ 3 and 0 ≤ t ≤ 3), contour plot (for −3 ≤ x ≤ 3 and 0 ≤ t ≤ 3), and 2D plot (for x = 0 and 0 ≤ t ≤ 50).These figure shows that when increasing the order β, the graph closes together.
From the overall outcomes, it is understood that the fractional-order derivative forms the nature of the results, narrates the continuous behavior of the waveform and h a remarkable influence on the nonlinear propagation related to the soliton solutions.

Comparison of the Results
To investigate the novelty of this work, we compared the findings we obtained with the previously obtained results of the time-fractional KGE [44], the time-fractional STOE [44], and the time-fractional CRWPE [48] in Tables 1-3, respectively.(a)

Comparison of the Results
To investigate the novelty of this work, we compared the findings we obtained with the previously obtained results of the time-fractional KGE [44], the time-fractional STOE [44], and the time-fractional CRWPE [48] in Tables 1-3, respectively.

Table 1. Comparison (I).
Comparison of the obtained results with those of Taghizadeh et al. [44] Results of Taghizadeh et al. [44] using the simplest equation method on time-fractional KGE Results obtained using the -expansion method (ii)  =  − 4 < 0 gives trigonometric function solution as . From the overall outcomes, it is understood that the fractional-order derivative reforms the nature of the results, narrates the continuous behavior of the waveform and has a remarkable influence on the nonlinear propagation related to the soliton solutions.

Comparison of the Results
To investigate the novelty of this work, we compared the findings we obtained with the previously obtained results of the time-fractional KGE [44], the time-fractional STOE [44], and the time-fractional CRWPE [48] in Tables 1-3, respectively.

Table 1. Comparison (I).
Comparison of the obtained results with those of Taghizadeh et al. [44] Results of Taghizadeh et al. [44] using the simplest equation method on time-fractional KGE Results obtained using the , a > 0 and b < 0, then the result (i) gives exponential function solution as For a 0 = a 0 , a 1 = 2a 0 (H−R−1) .

Table 2. Comparison (II).
Comparison of the obtained results with those of Taghizadeh et al. [44] Results of Taghizadeh et al. [44] using the simplest equation method on time-fractional STOE Results obtained using the