An efficient technique for higher order fractional differential equation

In this study, we establish exact solutions of fractional Kawahara equation by using the idea of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp \,\left( { - \varphi \left( \eta \right)} \right)$$\end{document}exp-φη-expansion method. The results of different studies show that the method is very effective and can be used as an alternative for finding exact solutions of nonlinear evolution equations (NLEEs) in mathematical physics. The solitary wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Graphical representations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very effective, expedient for fractional PDEs, and could be extended to other physical problems.

In the present paper, we applied the exp (−ϕ(η))-expansion method to construct the appropriate solutions of fractional Kawahara equation and demonstrate the straightforwardness of the method. The fractional derivatives are used in modified Riemann-Liouville sense. The subject matter of this method is that the traveling wave solutions of nonlinear fractional differential equation can be expressed by a polynomial in exp (−ϕ(η)).
The article is organized as follows: In "Caputo's fractional derivative" section, the exp (−ϕ(η))-expansion method is discussed. In "Description of exp (−ϕ(η)) expansion method" section, we exert the method to the nonlinear evolution equation pointed out above, in "Solution procedure" section, interpretation and graphical representation of results, and in "Graphical representation of the solutions" section conclusion and references are given.

Caputo's fractional derivative
In modelling physical phenomena, using differential equation of fractional order some drawbacks of Riemann-Liouville derivatives were observed In this section we set up the notations and recall some significant possessions.
Definition 3 Let f ∊ C α and α ≥ −1, then the (left-sided) Riemann-Liouville integral of order µ, µ > 0 is given by Definition 4 The (left sided) Caputo partial fractional derivative of f with respect to t, f ∈ C m −1 , m ∈ N ∪ {0}, is defined as: Note that

Description of exp (−ϕ(η)) expansion method
Now we explain the exp (−ϕ(η))-expansion method for finding traveling wave solutions of nonlinear evolution equations. Let us consider the general nonlinear FPDE of the type where D α t u, D α x u, D α xx u are the modified Riemann-Liouville derivatives of u with respect to t, x, xx respectively.
Step1. Combining the real variables x and t by a compound variable η we assume using the traveling wave variable Eqs. (10) and (8) is reduced to the following ODE for where Q is a function of u(η) and its derivatives, prime denotes derivative with respect to η Step2. Suppose the solution of Eq. (11) can be expressed by a polynomial in exp (−ϕ(η)) as follows where a n , a n−1 , . . . and V are constants to determined later such that a n ≠ 0 and φ(η) satisfies equation Eq. (8) Step3. By using the homogenous principal, we can evaluate the value of positive integer n between the highest order linear terms and nonlinear terms of the highest order in u(η) = a n (exp(−ϕ(η))) n + a n−1 (exp(−ϕ(η))) n−1 + · · · , Eq. (11). Our solutions now depend on the parameters involved in Eq. (1). So Eq. (1) provides the solutions from (13) to (16) Case 1 λ 2 − 4μ > 0 and μ ≠ 0, where c 1 is a constant of integration.
Step5. Eventually solving the algebraic system of equations obtained in step 4 by the use of Maple, we obtain the values of the constants a n , . . . , and μ. Substituting a n ,… and the general solution of Eq. (8) into solution Eq. (11), we obtain some valuable traveling wave solutions of Eq. (8).

Solution procedure
Consider the generalized form of fractional order nonlinear Kawahara equation.
where the prime denotes the derivative with respect to η. Now integrating equation Eq. (19), we have, Balancing the u ′′′′ and u 2 by using homogenous principal, we have Then the trial solution of equation Eq. (19) can be expressed as follows, where a 4 � = 0, a 0 , a 1 , a 2 and a 3 are constants to determined, while λ, μ are arbitrary constants.

Solution 4
Similarly, we can find the other exact solution of remaining solutions, while one solution is analyzed.

Graphical representation of the solutions
The graphical illustrations of the solutions are given below in the figures with the aid of Maple.

Physical interpretation
The proposed method provides more general and abundant new solitary wave solutions with some free parameters. The traveling wave solutions have its extensive significance to interpret the inner structures of the natural phenomena. We have explained the different types of solitary wave solutions by setting the physical parameters as special values. In this paragraph, we will explain the physical elucidation of the solutions for the Kawahara equation for a 0 = 11.1, µ = −0.0002, x = 15, α = 0.50, u 1 shows the singular solitary wave solution as shown in Figs. 1, 2, 3). Figure 4 shows the shape of the singular kink wave solution of u 2 for a 0 = 5.1, µ = 0.002, x = 2, α = 0.75. Again singular Kink solution obtained in Fig. 5 of u 2 for a 0 = 5.1, µ = 0.002, x = 2, α = 0.50 (Figs. 6,7,8,9,10,11). Finally simple kink solution got from u 5 for the choice of a 0 = −2, µ = 14, x = 18, α = 0.75. which is shown in Fig. 12. In one asymptotic state to another asymptotic state, kink solitons are upsurge or descent. Such solitons are called topological solitons. The other exact solutions could be obtained from the remaining solution sets. (29)

Numerical discussion
We have obtained the exact solutions (29), (30) and (31) in the above study and to know the correctness we have matched those solutions with the exact solution (Bongsoo 2009).We note that the absolute errors given in the tables from the solutions we have obtained are very precise and accurate. where

Conclusion
With the help of a suitable transformation and the exp (−ϕ(η))-expansion method, we obtained different types of exact solutions for fractional Kawahara equation. The obtained results show that the proposed technique is effective and capable for solving nonlinear fractional partial differential equations. In this research, some exact solitary wave solutions, mostly solitons and kinks solutions are obtained through the hyperbolic, trigonometric, exponential and rational functions. It is observed that the proposed method fully validate the competence and reliability of computational work as evident from Tables 1, 2 and 3 and may be utilized for other physical problems.