The Extended Trial Equation Method for Some Time Fractional Differential Equations

Nonlinear fractional partial differential equations have been solved with the help of the extended trial equation method. Based on the fractional derivative in the sense of modified Riemann-Liouville derivative and traveling wave transformation, the fractional partial differential equation can be turned into the nonlinear nonfractional ordinary differential equation. For illustrating the reliability of this approach, we apply it to the generalized third order fractional KdV equation and the fractional K(n, n) equation according to the complete discrimination system for polynomial method. As a result, some new exact solutions to these nonlinear problems are successfully constructed such as elliptic integral function solutions, Jacobi elliptic function solutions, and soliton solutions.


Introduction
Many physical phenomena in mathematical biology, electromagnetic theory, fluid mechanics, signal processing, engineering, and other fields of science can be modeled using the fractional derivatives.For example, the use of fractional derivative is analyzed in the fields of controller tuning, redundant robots, legged robots, digital circuit synthesis, and heat diffusion [1].Also, we refer the reader to [1,2] for details concerning fractional derivative or calculus and their applications in science and engineering.In some sense fractional differential equations could represent various reallife problems [3][4][5].However, an effective and general method for solving them cannot be found even in the most useful works.Also, remarkable progress has been made in the construction of the approximate solutions for fractional nonlinear partial differential equations.There are various mathematical methods which can constitute the exact solutions for some time fractional differential equations [6,7].Single kink soliton solutions, multiple-soliton solutions, compacton-like solutions, singular solitons, and other solutions have been found by use of these approaches.Apart from all these, some new exact solutions are obtained by using the trial equation methods.Some of them are elliptic integral , , and Π functions, and Jacobi elliptic function solutions.
In Section 2, primarily we give some definitions and properties of the fractional calculus theory and also produce the extended trial equation method for fractional nonlinear evolution equations with higher order nonlinearity.The power of this steerable method showed that this method can be applied to two nonlinear fractional problems.In Section 3, as applications, we solve the fractional nonlinear partial differential equations such as the generalized third order fractional KdV equation [8]      +   +     = 0, −∞ <  < ∞,  > 0, (1) where 0 <  ≤ 1 and  ≥ 2 model the subdiffusion and the subdispersion term, and the fractional (, ) equation [9]      + (  )  + (  )  = 0, −∞ <  < ∞,  > 0, 0 <  ≤ 1, (2) where  > 1. Equation (1) has a wide range of applications in plasma physics, fluid physics, and capillary-gravity waves.
Although there are a lot of studies for the classical generalized third order KdV and the (, ) equation, this is the first study with respect to the fractional versions of these problems.The aim of this paper is to obtain exact solutions of the generalized third order fractional KdV and the fractional (, ) equation by extendend trial equation method and to determine the accuracy of extended trial equation method in solving these kinds of problems.The exact solutions obtained by the interesting wave transformation  =  − (  /Γ(1 + )) are used to determine some new phenomena of the fractional nonlinear evolution equations.Based on the derived solutions for (1) and ( 2), the phenomena of some solutions are shown graphically (see Figures 1, 2, and 3).

Preliminaries and the Extended Trial Equation Method
In this part of the paper, it would be helpful to give some definitions and properties of the fractional calculus theory.
For an introduction to the classical fractional calculus we indicate the reader to [3][4][5].Here, we shortly review the modified Riemann-Liouville derivative from the recent fractional calculus proposed by Jumarie [10][11][12].Let  : [0, 1] → R be a continuous function and  ∈ (0, 1).The Jumarie modified fractional derivative of order  and  may be defined by expression of [13] as follows: In addition to this expression, we may give a summary of the fractional modified Riemann-Liouville derivative properties which are used further in this paper.Some of the useful formulas are given as Some new trial equation methods were defined in the literature [14][15][16][17][18][19][20][21][22][23][24][25][26].In this paper, the extended trial equation method will be given.In order to apply this method to fractional nonlinear partial differential equations, we consider the following steps.
Step 4. Reduce (9) to the elementary integral form Using a complete discrimination system for polynomial to classify the roots of Φ(Υ), we solve (13) with the help of Mathematica and classify the exact solutions to (7).In addition, we can write the exact traveling wave solutions to (5), respectively.

Applications to the Extended Trial Equation Method
In this section, we apply the method given in Section 2 to the generalized third order fractional KdV and the fractional (, ) equation, respectively.

Application to Generalized Fractional KdV Equation.
In the case of  = 1, (1) reduces to the classical nonlinear generalized third order KdV equation.Many researchers have tried to get the exact solutions of this equation by using a variety of methods.But, we solve the generalized fractional KdV equation for 0 <  ≤ 1. Compactons, solitary patterns, and periodic and solitary traveling plane waves solutions of this equation are found.
Let us consider the travelling wave solutions of (1), and we perform the transformation (, ) = (),  =  − (  /Γ(1 + )), where  is constant to be determined later.Then, integrating this equation with respect to  and setting the integration constant to zero, we get By use of the following transformation: (14) turns into the equation From ( 8)-( 10), we can write the following highest order nonlinear terms in order to determine the balance procedure: Balancing the highest order nonlinear terms of VV  and V 3 in ( 16), we have hence This resolution procedure is applied, and we obtain results as follows.
Remark 1.We provide a proof by verifying the exact solution (31) by direct substitution.If we take  = 1 and  = 2, then becomes where ( Substituting the above relations into the partial differential equation, we obtain the following equation: Otherwise, we can write where Substituting this relation into the above equation yields 0 = 0. Thus, it is seen that the obtained solution ( 31) is an exact solution of the fractional KdV equation.
Remark 2. The solutions (30)-(33) obtained by using the extended trial equation method for (1) have been checked by Mathematica.To our knowledge, rational function solution, 1-soliton solution, the singular soliton solution, and the Jacobi elliptic function that we find in this paper are not shown in the previous literature.These results are new traveling wave solutions of (1).
Case 2. If we take  = 0,  = 2, and  = 4, then where  4 ̸ = 0,  0 ̸ = 0. Respectively, solving the algebraic equation system (12) yields as follows: Substituting these results into (13), we get ) . (41) Integrating (41), we obtain the solutions to (1) as follows: where Also  1 ,  2 ,  3 , and  4 are the roots of the polynomial equation Substituting the solutions (42) into ( 8) and ( 15), we have ) ) For simplicity, if we take  0 = 0, then we can write the solutions (45) as follows: where Here,  2 is the amplitude of the soliton, while  2 is the velocity and  3 and  are the inverse width of the solitons.
Remark 3. To our knowledge, the solutions (46) that we find in this paper have not been found in the literature, and these are new exact solutions of (1).

Application to the Fractional 𝐾(𝑛,𝑛) Equation.
In the case of  = 1, (2) reduces to the classical nonlinear generalized (, ) equation.Many authors have studied to find the exact solutions of this equation by using a variety of methods [27,28].In this paper, we consider the generalized fractional (, ) equation for the case of 0 <  ≤ 1. Compactons, solitary patterns, and periodic and solitary traveling plane waves solutions of this equation are obtained.
Let us consider the travelling wave solutions of ( 2), and we perform the transformation (, ) = (),  =  − (  /Γ(1 + )), where  is constant to be determined later.Then, integrating this equation with respect to  and setting the integration constant to zero, we get By use of the following transformation: (47) turns into the equation From ( 8)- (10), we can write the following highest order nonlinear terms in order to determine the balance procedure: Balancing the highest order nonlinear terms of VV  and V 2 in (49), we have hence This resolution procedure is applied and we obtain results as follows.
Remark 4. The solutions (59)-(60) obtained by using the extended trial equation method for (2) have been checked by Mathematica.To our knowledge, the hyperbolic function solution and single kink solution, that we find in this paper, are not shown in the previous literature.These results are new traveling wave solutions of (2).

Conclusions
In this study, the extended trial equation method is applied successfully for solving the nonlinear fractional differential equations.We used it to obtain some soliton and elliptic function or elliptic integral function solutions, which are one-wave solutions, to the generalized third order fractional KdV equation and the fractional (, ) equation.The performance of this method is reliable and effective, and also this method gives more general solutions.We think that the proposed method can also be applied to other fractional nonlinear differential equations with nonlinear evolution.In