Application of two different algorithms to the approximate long water wave equation with conformable fractional derivative

Abstract The current paper devoted on two different methods to find the exact solutions with various forms including hyperbolic, trigonometric, rational and exponential functions of fractional differential equations systems with conformable farctional derivative. We have employed the modified simple equation and exp( method here for the approximate long water wave equation. We have adopted here the fractional complex transform accompanied by properties of conformable fractional calculus for reduction of fractional partial differential equation systems to ordinary differential equation systems.


Introduction
Fractional differential equations, containing the fractional differentiation are generalizations of classical differential equations of integer order. Recently, these equations have a great deal of attention and applied in many research fields, such as nonlinear control theory, electromagnetic theory, fluid mechanics, signal processing, electrochemistry and mathematical biology (Kilbas, Srivastava, & Trujillo, 2006;Miller & Ross, 1993). Moreover, various applications of the fractional calculus can be found in plasma physics turbulence, stochastic dynamical system, image processing, fluid dynamics and astrophysics. Fractional partial differential equations are becoming increasingly popular due to their practical applications in various fields of science and engineering. There are two major approaches in the theoretical formulation of initial value problems for fractional differential equations (Dai, Wang, & Liu, 2016;Imran, Khan, Ahmad, Shah, & Nazar, 2017;Saqib, Ali, Khan, Sheikh, & Jan, in press;Shah & Khan, 2016). One of them is based on the interpretation of the initial condition of fractional systems as a distributed initial condition (Agarwal, O'Regan, Hristova, & Cicek, 2017).
In this paper, by use of the properties of fractional calculus, we propose two different methods to seek exact solutions of fractional partial differential equations with conformable derivative. Based on a traveling wave transformation, certain fractional partial differential equation systems can be turned into another fractional ordinary differential equation systems with respect to one new variable.

Brief of conformable fractional derivative
Recently, the authors Khalil et al. introduced a new simple and intriguing definition of the fractional derivative called conformable fractional derivative (Khalil, Al Horani, Yousef, & Sababheh, 2014). This derivative is well-behaved and obeys the Leibniz rule and chain rule. Let us review the conformable fractional derivative (Cenesiz, Baleanu, Kurt, & Tasbozan, 2017;Chung 2015).
Definition 1. Suppose f : 0; 1 ½ Þ ! R be a function. Then, the conformable fractional derivative of f of order a; 0<a 1; is defined as for all t > 0: Some useful properties can be listed as follows: 1. Linearity: T a ðaf þ bgÞ ¼ aðT a f Þ þ bðT a gÞ, for all a; b 2 R 2. Leibniz rule: T a ðfgÞ ¼ fT a ðgÞ þ gT a ðf Þ 3. Let f be a differentiable and aÀconformable differentiable function and g be a differentiable function defined in the range of f. Then T a ðf gÞðtÞ ¼ t 1Àa g 0 ðtÞf 0 ðgðtÞÞ: Moreover, the following rules are hold. T a ðt p Þ ¼ pt pÀa ; for all p 2 R T a ðkÞ ¼ 0, for all constant functions f ðtÞ ¼ k Additively, if f is differentiable, then T a ðf ÞðtÞ ¼ t 1Àa df dt ðtÞ:

The fractional complex transformation and the modified simple equation method
Consider the fractional differential equation with conformable derivative: To find the exact solution of Equation (1), the following fractional complex transformation can be introduced: where k and c are constants to be determined. Under the transformation Equation (2), we can rewrite Equation (1) in the following nonlinear ordinary differential equation QðU; U 0 ; U 00 ; U 000 ; . . .Þ ¼ 0: (3) Then we integrate Equation (3) as many times as possible with respect to n and set the integration constant as zero.
Firstly, according to the modified simple equation method (Kaplan, Bekir, Akbulut, & Aksoy, 2015), the exact solution of Equation (3) can be represented by a polynomial in w 0 n ð Þ w n ð Þ as follows where a n ; n ¼ 0; 1; 2; . . .; m ð Þ are unknown constants such that a m 6 ¼ 0; and w is an unknown function of n to be calculated. Here the positive integer m, called as the balancing number is determined by thinking the homogeneous balance principle between the highest order nonlinear term with the highest order derivative term which appears in Equation (3).
We obtain a polynomial of w Àj n ð Þ with the derivatives of w n ð Þ.by substituting Equation (4) into Equation (3). Then, by equating the coefficients of w Àj n ð Þ to zero, where j ! 0; we get a system which can be solved to find a n n ¼ 0; 1; 2; . . .; m ð Þ ; c and u n ð Þ: Finally we substitute the values of a n ; k, c and w n ð Þ into Equation (4) to find the exact solution of Equation (1).

The fractional complex transformation and the exp ðÀUðnÞÞ method
Secondly, we introduce the expðÀUðnÞÞ method for finding different types of exact solutions to nonlinear fractional differential equations with conformable fractional derivative (Kaplan & Bekir, 2017). To reduce the considering Equation (1) to a nonlinear ordinary differential equation, we follow the same procedure.
According to the expðÀUðnÞÞ method, we seek the exact solution of Equation (3) in the following form: where a n a m 6 ¼ 0 ð Þ are constants to be determined later, and UðnÞ satisfies the following auxiliary ordinary differential equation: One can know that the auxiliary equation Equation (6) has different solutions as follows: Case 1 (Hyperbolic function solutions): When k 2 À 4l > 0 and l 6 ¼ 0; Case 2 (Trigonometric function solutions): When k 2 À 4l < 0 and l 6 ¼ 0, Case 3 (Hyperbolic function solutions): When k 2 À 4l > 0; m ¼ 0 and k 6 ¼ 0, Case 4 (Rational function solutions): When k 2 À 4l ¼ 0; l 6 ¼ 0 and k 6 ¼ 0, Case 5: When k 2 À 4l ¼ 0; l ¼ 0 and k ¼ 0, Here C is the integration constant. Also we balance the highest order linear term with the highest order nonlinear term in Equation (5) to find the balancing number N.
Substituting Equation (5) into Equation (3) and collecting all terms with the same order of expðÀUðnÞÞ n ðn ¼ 0; 1; 2; . . .Þ together, we get a polynomial in expðÀUðnÞÞ. Equating each coefficient of this polynomial to zero yields a set of algebraic equations for a n ; k; k; l and c. Solving the equation system, we can construct a variety of exact solutions for Equation (1).

Modified simple equation method
The fractional approximate long water wave (ALW) equation is known as Yan has found three types of travelling wave solutions of this equation via the fractional sub-equation method (Yan, 2015). Also Guner et al. used ðG 0 =GÞÀ expansion method to establish the exact solutions of Equation (12). By using the transformation: uðx; tÞ ¼ UðnÞ; vðx; tÞ ¼ VðnÞ; Equation (12) reduces a nonlinear ordinary differential equation system, which reads ÀcU 0 À kUU 0 À kV 0 þ ak 2 U 00 ¼ 0; ÀcV 0 À k UV ð Þ 0 À ak 2 V 00 ¼ 0: Here prime denotes the derivative with respect to n: Then, we can integrate this system once with respect to n ÀcU À k 2 U 2 À kV þ ak 2 U 0 ¼ 0; ÀcV À kUV À ak 2 V 0 ¼ 0: If we balance the highest-order derivative term U 0000 and the non-linear term ðU 0 Þ 2 of Equation (14), we obtain the balancing number as m ¼ 1: So, we assume that our solution is in the following form: By substituting Equation (15) into Equation (14) and collecting all the terms with the same power of e ÀUðnÞ ; ðn ¼ À4; À3; . . .; 0Þ. Then by equating each coefficient of the above system to zero yields a set of the following algebraic equations as follows: From the first equation, we find: w 0 : À ka 2 0 2 À ca 0 À kb 0 ¼ 0; w À1 : Àkb 1 À ca 1 À ka 0 a 1 ð Þ w 0 þ ak 2 a 1 w 00 ¼ 0; w À2 : À 1 2 ka 2 1 À ak 2 a 1 À kb 2 w 0 À Á 2 ¼ 0; and from the second equation, we get Then it is easy to obtain from the first equations of the systems above: from the last equations of the systems. Then, we only consider the result: a 1 ¼ 2ak; b 2 ¼ À4a 2 k 3 satisfied. Since a 1 6 ¼ 0; b 2 6 ¼ 0 the first and second cases are omitted.
Case 1: a 0 ¼ 0; b 0 ¼ 0: By substituting these solutions into the remaining equations we get: b 1 ¼ À4ac: Therefore, we find the exact solutions of the approximate long water wave equation with conformable fractional derivative as where n ¼ k x a 1 a 1 À c t a 2 a 2 (Figures 1 and 2). Case 2: a 0 ¼ À 2c k ; b 0 ¼ 0; By substituting these solutions into the remaining equations we get: b 1 ¼ 4ac: Then we find w n ð Þ ¼ C 1 þ C 2 exp cn ak 2 : Therefore, we find the exact solutions of the approximate long water wave equation with conformable fractional derivative as where n ¼ k x a 1 a 1 À c t a 2 a 2 (Figures 3 and 4). Case 3: When a 0 ¼ À c k ; b 0 ¼ c 2 2k 2 ; w n ð Þ yields an absurd solution. Hence, the case is discarded.
Note that, our solutions are different from the given ones in (Guner, Atik, & Aytugan Kayyrzhanovich, 2017). Also, we can say that the solutions obtained in this paper via two different methods are different from each others. If we compare them, the solution process in the modified simple equation method is difficult from the expðÀUðnÞÞ method and it gives more fresh solutions. On the other hand, expðÀUðnÞÞ method gives the exact solutions with different forms.

Conclusions
In this work, by use of the fractional calculus for conformable fractional derivative, and the modified simple equation and a new expðÀUðnÞÞ method is proposed to seek exact solutions of the fractional differential equation systems. The modified simple equation method has proved its validity in that more fresh solutions for fractional partial differential equation systems can be obtained. Since, the auxiliary equation of this method is not a solution of any predefined functions. Moreover, by using the expðÀUðnÞÞ method more general exact solutions with different forms including solitary wave solutions, periodic wave solutions and rational solutions. As an application the approximate long water wave equation has been considered and abundant exact solutions are verified.

Disclosure statement
No potential conflict of interest was reported by the authors.