Searching closed form analytic solutions to some nonlinear fractional wave equations

Abstract Numerous tangible incidents in physics, chemistry, applied mathematics and engineering are described successfully by means of models making use of the theory of derivatives of fractional order and research in this area has grown significantly. In this article, we establish exact solutions to some nonlinear fractional differential equations. The recently established rational ( )-expansion method with the help of fractional complex transform is used to examine abundant further general and new closed form wave solutions to the nonlinear space-time fractional mBBM equation, the space-time fractional Burger’s equation and the space-time fractional ZKBBM in the sense of the Jumarie modified Riemann-Liouville derivative. The fractional complex transform reduces the nonlinear fractional differential equations into nonlinear ordinary differential equations and then the theories of ordinary differential equations are implemented effectively. It is observed that the performance of this method is reliable, useful and gives new and broad-ranging closed form solutions than the existing methods.

Recently, Khalil et al. (2014) proposed a definition of derivative named conformable fractional derivative defined as: Our assessment is highly satisfactory to obtain new and more general exact traveling wave solutions to the equations mentioned above by the suggested method.

Methodology
In this section, the rational ðG 0 =GÞ-expansion method is discussed to study nonlinear partial differential equations of fractional order.
Step 1: We utilize the nonlinear fractional composite transformation n ¼ nðt, x 1 , x 2 , :::, x n Þ, u i ¼ u i ðt, x 1 , x 2 , :::, Making use of Kilbas et al. (2006), Eq. (1) turns to the following ordinary differential equation of integer order with respect to the variable n : QðU 1 , :::, U k , U 1 0 , :::, U k 0 , U 11 00 , :::, U kk 00 , ::: where the primes in U i 0 s denote the order of derivatives of U with respect to n: Step 2: For convenience, integrate Eq. (3) one or more times as possibility and integral constant can be set to zero as soliton solutions are sought.
Step 3: In conformity with the rational ðG 0 =GÞ-expansion method the solution of (Miller & Ross, 1993) can be revealed in terms of ðG 0 =GÞ as follows (Islam, Akbar, & Azad, 2015): wherea n and b n are non-zero real constants to be determined later and G ¼ GðnÞ satisfies the second order linear ODE: G 00 ðnÞ þ kG 0 ðnÞ þ lGðnÞ ¼ 0, wherek and l are real constants. Eq. (5) can be rewritten as: The general solutions of (6) are referred to below: where A and B are constants of integration.
Step 4: To determine the positive integer n, we substitute (4) along with (5) into (3) and balance between the highest order derivatives and the highest order nonlinear terms appearing in (3). Furthermore, if the degree of uðnÞ is defined as deg½uðnÞ ¼ n, the degree of the other terms are as follows: Step 5: Substituting (4) together with (5) into (3), we will obtain a polynomial equation with indeterminate ðG 0 =GÞ: Setting each coefficient of ðG 0 =GÞ to zero gives a system of algebraic equations. This system of equations can be unraveled fora i , b i , k, and l by means of the symbolic computation software Maple.
Step 6: We use the values of a i , b i , k, and l together with (7) into (4) to obtain the closed form traveling wave solutions of the nonlinear fractional partial differential equation (1).

Formulation of the solutions
The rational ðG 0 =GÞ-expansion approach is exploited in this section to search for general closed-form traveling wave solutions to the nonlinear space-time fractional mBBM equation, the space-time fractional Burger's equation and the space-time fractional ZKBBM equation.

The space-time fractional mBBM equation
Let us consider the space-time fractional mBBM equation where v is a nonzero positive constant. This equation was first derived to describe an approximation for surface long waves in nonlinear dispersive media. It can also characterize the hydro-magnetic waves in cold plasma, acoustic waves in anharmonic crystals and acoustic gravity waves in compressible fluids. Now we make use of the transformation of the fractional compound Along with the chain rule (He et al., 2012), Eq. (8) is converted into the following ODE, where r is fractal index to be determined. Equation (10) is integrable and hence integrating, yields ðm þ lÞuÀ l 3 vu 3 þ l 3 r 2 u 00 ¼ 0, where the integral constant is supposed to be zero. The homogeneous balance between nonlinear term u 3 and the linear term u 00 appearing in (11) gives n ¼ 1: Thus, the solution Eq. (4) has the form Substituting (12) into (11), the left-hand side becomes a polynomial in ðG 0 =GÞ: We obtain an overdetermined set of algebraic equations by zeroing each coefficient of this polynomial (for simplicity, we will omit to display them) for a 0 , b 0 , a 1 , b 1 , m, l: Solving this set of equations by using the symbolic computation software Maple, provide the following results: where b 0 , b 1 , l, k, r and l are free parameters.
where b 1 , l, k, r and l are free parameters.
Set 3: a 0 ¼ 6 where b 0 , l, k, r and l are free parameters.
Inserting the values provided in (13)-(15) into solution (12), three general solutions can be found, but for brevity, we have used only the solution Set 1. Thus, the subsequent solution is found: where n ¼ l Cðaþ1Þ x a Àð2 þ 4l 2 lr 2 Àl 2 k 2 r 2 Þt a È É : The use of the general solutions presented in (7) into the solutions (16) results in the closed-form wave solutions to the space-time fractional mBBM equation via rational functions, hyperbolic function, and trigonometric. For simplicity, we record here only the obtained results taking into account the solution Set 1.
When k 2 À 4 l > 0, letting the arbitrary constants A 6 ¼ 0 and B ¼ 0, the hyperbolic function solution is found as where n ¼ l Cðaþ1Þ x a Àð2 þ 4l 2 lr 2 Àl 2 k 2 r 2 Þt a È É : (17) and simplifying, we attain u 1 1, 2 ðnÞ ¼ 6tanh When k 2 À 4 l < 0, treating the arbitrary constants as A 6 ¼ 0 and B ¼ 0, the trigonometric function solution is constructed as: where n ¼ l Cðaþ1Þ x a Àð2 þ 4l 2 lr 2 Àl 2 k 2 r 2 Þt a È É : When k 2 À 4 l ¼ 0, the rational solution is derived as: where n ¼ l Cðaþ1Þ x a Àð2 þ 4l 2 lr 2 Àl 2 k 2 r 2 Þt a È É : For terseness, we have estimated solutions for the values of Set 1 presented in (13). Thus, solution (12) provides the broad-spectrum solution (16) of the mBBM equation. For the different values of the free parameters, the solution (16) yields hyperbolic function solution (18), trigonometric solution (20) and rational function solution (22). Many other solutions can be sought for other choices of the parameters, but the remaining solutions are not written for minimalism and conciseness. It is worth noting that the hyperbolic solution represents the soliton of the kink shape soliton that descends from left to rising right and is constant towards infinity, and the periodic wave is represented by the triangular function solution. The rational function solution depicts a periodic type, breather type and rogue (unusually large, unexpected and suddenly appearing surface waves that can be extremely dangerous, even to large ships) type wave for different values of the free parameter.

The space-time fractional burger's equation
We now determine the traveling wave solutions of the nonlinear space-time fractional Burgers equation through the introduced method. Consider the spacetime fractional Burger's equation in the underneath D a t u þ auD b x u þ bD 2b x u ¼ 0, t>0, x>0, 0<a, b<1, (23) wherea and b are nonzero arbitrary constants. This equation has been found to apply in diverse fields, like, gas dynamics, heat conduction, elasticity, continuous stochastic processes, etc.
The fractional complex transformation where k and l are constants, with the help of the chain rule (He, Elagan, & Li, 2012) permits us to reduce the Eq. (23) into the following ODE ðlu 0 þ akuu 0 Þr þ bk 2 r 2 u 00 ¼ 0, where r is fractal index to be determined. After integrating, Eq. (25) becomes lu þ a 2 ku 2 þ bk 2 ru 0 ¼ 0, where the integral constant is supposed to be zero. The balance between u 2 and u 0 in Eq. (26) yields n ¼ 1: Then, the solution Eq. (4) is turned into the form Now, Eq. (26) with the help of Eq. (27) becomes a polynomial equation in ðG 0 =GÞ: Equating the coefficients of like powers of ðG 0 =GÞ to zero derives a system of equations (for straightforwardness, the equations are omitted to display) for a 0 , b 0 , a 1 , b 1 , k, l: This system of equations can be solved by the symbolic computation software Maple, and attain the subsequent solutions where a 1 ,b 1 ,k,k,r and l are arbitrary constants. Making use of the results given in Eq. (28), the solution Eq. (27) becomes Substituting the general solutions provided in (7) into (29), yields the subsequent traveling wave solutions: When k 2 À 4 l > 0, assigning the arbitrary constants as A 6 ¼ 0 and B ¼ 0, the hyperbolic function solution is derived as: (31) When k 2 À 4 l < 0, selecting the integral constants as A 6 ¼ 0 and B ¼ 0, the trigonometric function solution is obtained as: For the values a ¼ ffiffi ffi (33) When k 2 À 4 l ¼ 0, we obtain rational solution as: Solution (29) offers hyperbolic, trigonometric, and rational function solutions based on the parameters k and l, shown in u 1, 2 ðx, tÞ, u 3, 4 ðx, tÞ and u 5, 6 ðx, tÞ, respectively. The solutions described in this study are wide-ranging and typical than existing solutions, investigated in earlier research. If we sort distinct values of the comprising parameters, further closedform solutions to the space-time fractional Burger's equation can be extracted, but for simplicity and conciseness the residual solutions have not been marked out. The solutions u 1, 2 ðx, tÞ, u 3, 4 ðx, tÞ characterize topological wave, smooth kink, ideal kink waves and u 5, 6 ðx, tÞ typify the breather type soliton for making use the various values of the free parameters.

The space-time fractional ZKBBM equation
In this subsection, we study the following nonlinear space-time fractional ZKBBM equation: where a and b are nonzero arbitrary constants. This equation arises as a description of gravity water waves in the long-wave regime. Applying the fractional compound transformation together with the chain rule (He et al., 2012) reduces Eq. (36) into the following ODE ðmu 0 þ lu 0 À2aluu 0 ÞrÀbml 2 r 3 u 000 ¼ 0, where r is fractal index to be determined. Equation (38) is integrable and hence integrating, we get l þ m ð ÞuÀalu 2 Àbml 2 r 2 u 00 ¼ 0, where integral constant is supposed to be zero. Using the homogeneous balance theory it is found n ¼ 2, the solution Eq. (4) can be written as 2 , a 2 , b 2 6 ¼ 0: (40) Equation (39) with the aid of (40) reduces to a polynomial equation ðG 0 =GÞ: Associating the coefficients of the same power of ðG 0 =GÞ to zero, we get a set of algebraic equations (for simplicity, the equa- When k 2 À 4 l < 0, for the particular values a ¼ b ¼ l ¼ 1, k ¼ 2, l ¼ 1=2, r ¼ 1 of the parameters, we obtain the following trigonometric function solution: When we construct the rational solution as: We have assessed the solutions for the values in set 1 for conciseness, provided in (41). The solution (43) thus provides wide-spectrum solutions namely, u 1 1 ðx, tÞ, u 2 1 ðx, tÞ and u 3 1 ðx, tÞ: In addition, for set 2, it might be achieved three types of exact traveling wave solutions, namely the hyperbolic, trigonometric and rational function solutions which are not documented here to avoid repetition. The characteristic of solution u 1 1 ðx, tÞ and u 2 1 ðx, tÞ represent kink shape wave and periodic wave, respectively, but gives rogue wave with several soliton for different values of parameters. The solution u 3 1 ðx, tÞ depicts irregular spike type soliton for different values of related constants.

Conclusions
In this article, the exact traveling wave solutions to NPDEs of fractional order, namely the space-time fractional mBBM equation, the space-time fractional Burger's equation and the space-time fractional ZKBBM equation are established. To do this, we employ the recently established rational ðG 0 =GÞ-expansion method which gives new and further general traveling wave solutions. Three types of closed form analytical solutions including the generalized hyperbolic function solutions, the generalized trigonometric function solutions and rational solutions for each of the above NPDEs are obtained successfully. These solutions might be further useful and effective for understanding the mechanisms of the intricate nonlinear physical phenomena occur in science and engineering. So far we know, the results gained in this article have not been reported in the literature.