An Extension of the Double ( G 9 / G, 1 / G ) -Expansion Method for Conformable Fractional Differential Equations

The phenomena, molecular path in a liquid or a gas, ﬂuctuating price stoke, ﬁssion and fusion, quantum ﬁeld theory, relativistic wave motion, etc., are modeled through the nonlinear time fractional clannish random Walker’s parabolic (CRWP) equation, nonlinear time fractional SharmaTassoOlver (STO) equation, and the nonlinear space-time fractional KleinGordon equation. The fractional derivative is described in the sense of conformable derivative. From there, the ( G ′ / G, 1/ G ) -expansion method is found to be ensuing, eﬀective, and capable to provide functional solutions to nonlinear models concerning physical and engineering problems. In this study, an extension of the ( G ′ / G, 1/ G ) -expansion method has been introduced. This enhancement establishes broad-ranging and adequate fresh solutions. In addition, some existing solutions attainable in the literature also conﬁrm the validity of the suggested extension. We believe that the extension might be added to the literature as a reliable and eﬃcient technique to examine a wide variety of nonlinear fractional systems with parameters including solitary and periodic wave solutions to nonlinear FDEs.

In this study, we introduce an extension of the (G ′ /G, 1/G)-expansion method for analyzing nonlinear FDEs in mathematical physics, engineering, and applied mathematics. To demonstrate the reliability and advantages of the suggested extension, the time-fractional CRWP equation, the time fractional STO equation, and the spacetime fractional KleinGordon equation are examined and further broad-ranging and new families of exact wave solutions are established. is new extension can be applied to further nonlinear FDEs which can be done in forthcoming work.

Conformable Fractional Derivative and Its Important Properties
In the last years, Khalil et al. [4] introduced a simple, interesting, and compatible with typical definition of derivative named conformable fractional derivative, which can rectify the deficiencies of the other definitions. One can also find several useful studies related to this new definition in [5][6][7][8]. In [39], the geometrical and physical interpretations of this definition are investigated and the potential applications in science and engineering are pointer out. e conformable fractional derivative of a function g of order β is defined as where g: [0, ∞) ⟶ R, t > 0 and β ∈ (0, 1). Some important properties of above definition are given below:

Methodology
In this section, we will suggest an extension of the (G ′ /G, 1/G)-expansion method to ascertain the analytic solutions to nonlinear FDEs. We begin with considering the second-order linear ODE: We choose us, from equation (2) and (3), it is found From the different general solutions of the linear ODE (2), we attain the following: Case 1: when λ < 0, the hyperbolic function solution is and thus we obtain where σ � A 2 1 − A 2 2 and A 1 , A 2 are arbitrary constants. Case 2: when λ > 0, the trigonometric function solution is and corresponding relation is where σ � A 2 1 + A 2 2 and A 1 and A 2 are arbitrary constants. Case 3: when λ � 0, the rational function solution is and thus it is found where A 1 and A 2 are arbitrary constants. Let us consider a general nonlinear FDE: where D α t u, D α x u and D α y u are the conformable fractional derivatives of the wave function u with respect to spatial variables x and y and the temporal variable t, and P is a polynomial of u � u(x, y, t, . . .) and its various partial derivatives.
e main steps of the new extension of the (G ′ /G, 1/G)-expansion method to seek exact solutions of nonlinear FDEs are as follows: Step 1: we estimate the new form of the fractional wave variable: where k and l are nonzero constants and v is the wave velocity to be determined later. e complex wave transformation (12) translates equation (11) into an ODE as follows: 2 Complexity Step 2: suppose that the general solution of nonlinear ODE (13) can be expressed by a polynomial in (1/ϕ) and ψ as where G � G(ξ) satisfies the auxiliary linear ODE (2) and a i and b i are arbitrary constants to be determined later, and balancing the highest order derivative with the nonlinear terms in equation (13), we can find the positive integer N.
Step 3: inserting solution (14) into equation (13) and utilizing (4) and (6) (here case 1 is selected as an example), the left-hand side of equation (13) can be converted into a polynomial in (1/ϕ) and ψ, where the degree of ψ is not more than one. Equating all coefficients of this polynomial to zero yield a set of algebraic equations for a i , b i , k, l, v, A 1 , A 2 , λ (λ < 0), and μ.
Step 4: the solution of the algebraic equations found in the step 3 can be found with the aid of Maple software package. Making use of the values of a i , b i , k, l, v, A 1 , A 2 , λ, and μ into (14), we might determine exact solutions expressed by hyperbolic functions of equation (13).

Determination of Solutions
In this paragraph, we will search out solutions to three conformable FDEs as appertain of the new extension of the (G ′ /G, 1/G)-expansion method.

Time Fractional CRWP Equation.
First, we consider the time fractional CRWP equation [22,40]: e ensuing wave transformation is where k is the wave number and c is the velocity; reducing the equation (15) into the subsequent ODE, we get Integrating (17) once, we find where ξ 0 is a constant of integration. It is clear that the homogeneous balance between U 2 and U ′ present in equation (18) gives N � 1. erefore, the shape of the exact solution of equation (18) is ere are three cases to be considered: Case 1: when λ < 0, substituting solution (19) into equation (18) and utilizing (4) and (6), equation (18) can be transmuted to a polynomial in (1/ϕ) and ψ.
Equalizing its coefficients to zero yields a set of algebraic equations in a 0 , a 1 , b 1 , k, c, σ, λ, μ, and ξ 0 : Resolving these algebraic equations by Maple software package, we attain the following values: where k and c are free constants.
Computing the action similar to case 1 and after resolving the algebraic equations, we ascertain the following values: where k and c are arbitrary constants.
Substituting the values scheduled in (29) into solution (19) along with (8) and (16), the succeeding trigonometric solution to the CRWP equation (15) is established: Since A 1 and A 2 are free constants, we may set (31), we derive the under mentioned periodic wave solutions to the CRWP equation:
Utilizing the values of the parameters arranged in (71), we obtain Now setting A 1 � 0, A 2 ≠ 0, and μ � 0 (or A 2 � 0 and μ � 0), the solution (85) becomes where ξ � x + βλ(t α /α). Embedding the parametric values compiled in (72), we find out Suppose that A 1 � 0, A 2 ≠ 0, and μ � 0 (or A 2 � 0, A 1 ≠ 0, and μ � 0), then the solution (88) developed into where ξ � x + 4βλ(t α /α). From the obtained broad-ranging solutions, it is observed that setting definite values of the associated parameters, we manage to determine some particular solutions which coincide with those accessible in the literature and some fresh solutions are established. It is seen that, by putting β � d, the obtained solutions (53), (54), (77), and (78) completely agree with the solutions (4.8), (4.7), (4.11), 8 Complexity and (4.10), respectively, found in [43]. In addition to these solutions, we found many more solutions that were not found in any other studies.

Space-Time Fractional KleinGordon Equation.
In this subsection, we extract the closed form solutions to the spacetime fractional KleinGordon equation. Let us consider the KleinGordon equation with space-time fractional order [44,45]: where β and c are nonzero constants. We apply the following transformation for reducing equation (91) to an ODE: where k and c are nonzero constants. us, the space-time fractional KleinGordon equation turns out as follows: Balancing U ″ and U 3 in equation (93), we found N � 1. On account of this, the structure of the solution of equation (93) is identical to the shape of the solution of equation (19) and therefore has not been repeated. ere are three cases should be discussed as described in Section 3: Case 1: when λ < 0, plugging in (19) into equation (93) and utilizing (4) and (6), equation (93) will be converted into a polynomial in (1/ϕ) and ψ. Equalizing the coefficients of this polynomial to zero yields a set of algebraic equations for a 0 , a 1 , b 1 , c, k, σ, λ, μ, β, and c. Resolving these equations with the assistance of computer algebra, like Maple software package, we found the following values for the constants: a 0 � 0, where k, β, c, and μ are free parameters.
For the values arranged in (100), we obtain the general solution: Similarly, for the values organized in (101), we accomplish where ξ � k(x α /α)∓ ����������� ((λk 2 − β)/λ) (t α /α). And, for the values laid out in (102), we ascertain e obtained results can be compared with the exact solutions accessible in the literature. In [45], the exact solutions of the space-time fractional KleinGordon equation are established by using the (G ′ /G, 1/G)-expansion method. It is seen that the solutions established in this study are different than the solutions found in [45].

Physical Explanations
In this section, we put forth the physical explanation and the 2D and 3D graphical representation of the solutions ob-      Figure 3 shows the singular periodic solution of (104). e remaining graphs are left for minimalism.
Solution (98) is the soliton solution. Figure 4 shows the shape of the exact soliton solution of (98) of the space-time fractional KleinGordon equation.

Conclusion
In this article, we have introduced an extension of the (G ′ /G, 1/G)-expansion method to look into nonlinear fractional differential equations in the sense of conformable derivative. Taking the advantage of this extension, the time fractional CRWP equation, the general time fractional STO equation, and the space-time fractional Klein-Gordon equation have been investigated. Scores of broad-ranging exact solutions have successfully been found as a linear combination of hyperbolic, trigonometric, and rational function associated with free parameters. For definite values of these parameters, some known periodic, kink, and solitary wave solutions accessible in the literature are derived from the general solutions and some fresh solutions are originated.
is study shows that the proposed extension is quite efficient, useful, direct, and easily computable with the aid of Maple software package and practically well suited to be used in finding analytical exact solutions to many other nonlinear FDEs, and this is our scheme in the future.

Data Availability
No data were used to support the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.