Abstract
Exact travelling wave solutions to the space and time fractional Benjamin-Bona-Mahony (BBM) equation defined in the sense of Jumarie’s modified Riemann-Liouville derivative via the expansion and the modified simple equation methods are presented in this paper. A fractional complex transformation was applied to turn the fractional BBM equation into an equivalent integer order ordinary differential equation. New complex type travelling wave solutions to the space and time fractional BBM equation were obtained with Liu’s theorem. The modified simple equation method is not effective for constructing solutions to the fractional BBM equation.
1. Introduction
Nonlinear partial differential equations arise in a large number of physics, mathematics, and engineering problems. In the Soliton theory, the study of exact solutions to these nonlinear equations plays a very germane role, as they provide much information about the physical models they describe. In recent times, it has been found that many physical, chemical, and biological processes are governed by nonlinear partial differential equations of noninteger or fractional order [1–4].
Various powerful methods have been employed to construct exact travelling wave solutions to nonlinear partial differential equations. These methods include the inverse scattering transform [5], the Backlund transform [6, 7], the Darboux transform [8], the Hirota bilinear method [9], the tanh-function method [10, 11], the sine-cosine method [12], the exp-function method [13], the generalized Riccati equation [14], the homogenous balance method [15], the first integral method [16, 17], the expansion method [18, 19], and the modified simple equation method [20–22].
In this paper, we apply the expansion method and the modified simple equation method to construct travelling wave solutions to the space and time fractional Benjamin-Bona-Mahony (BBM) equation in the sense of Jumarie’s modified Riemann-Liouville derivative via a fractional complex transformation and we further get new complex type solutions to the equation by applying Liu’s theorem [23]. The Benjamin-Bona-Mahony equation is of the following form: This equation was introduced in [24] as an improvement of the Korteweg-de Vries equation (KdV equation) for modelling long waves of small amplitude in 1 + 1 dimensions. It is used in modelling surface waves of long wavelength in liquids, acoustic gravity waves in compressible fluids, and acoustic waves in anharmonic crystals.
Jumarie’s modified Riemann-Liouville derivative of order with respect to is defined as [25] Some useful properties of the modified Riemann-Liouville derivative are listed below [25, 26]:
2. Description of the Expansion Method
Here, we provide a brief explanation of the expansion method for finding travelling wave solutions of nonlinear fractional partial differential equations. Suppose that the nonlinear evolution equation is in the following form: where is a polynomial of and its derivatives (integer and fractional) with respect to and . and are parameters that describe the order of the space and time derivatives, respectively.
Theorem 1. Fractional Complex Transformation. To transform (4) into a nonlinear ordinary differential equation (ODE) of integer order by applying a fractional complex transformation proposed by Li and He [27],
where is an arbitrary constant and (4) reduces to a nonlinear integer order ODE of the following form:
The method assumes that the solution to (6) can be expressed as a polynomial in :
where are constants to be determined and is a solution of the linear ordinary differential equation of the following form:
where and are arbitrary constants. The general solution of (8) gives [28] the following: Equation (6) is integrated as long as all the terms contain derivatives, where integration constants are considered to be zero. To determine , one considers the homogenous balance between the highest order derivative and the highest order nonlinear term(s).
Substitute (7) with the determined value of into (6), and collect all terms with the same order of together. If the coefficients of vanish separately, one has a set of algebraic equations in , , and that is solved with the aid of Mathematica.
Finally, substituting , and the general solution to (8) into (7) yields the travelling wave solution of (4).
Theorem 2 (Liu’s theorem [23]). If a nonlinear evolution equation has a kink-type solution in the form of where is a polynomial of degree , then it has a certain kink-bell-type solution in the following form: where is the imaginary number unit.
3. Description of the Modified Simple Equation Method
Here, we highlight the basic ideas of the modified simple equation method. Suppose that the fractional nonlinear evolution equation is in the following form: where is a polynomial of and its fractional partial derivatives with respect to and . Suppose a fractional complex transformation (see (5)) which allows us to reduce (12) to an ordinary differential equation (ODE) of the following form: We suppose that the solution to (13) can be expressed as a polynomial of in the following form: where is a positive integer obtained by balancing the highest order derivatives and the highest order nonlinear terms in (13). are arbitrary constants to be determined such that and is an unknown function to be determined and is not a solution of any predefined differential equation.
We substitute (14) and its derivatives into (13) taking into account the function . As a result of this substitution we obtain a polynomial of with the derivatives of . Equating all the coefficients of to zero yields a system of equations which can be solved to obtain and . Finally, substituting the values of , and its derivative into (14) gives the exact solution of (12).
4. Application
4.1. Solutions for Space and Time Fractional BBM Equation via Expansion Method
In this section, we apply the expansion method to construct travelling wave solution of the space and time fractional Benjamin-Bona-Mahony equation. Consider the space and time fractional BBM equation (1) which can be written in subscript notation as We make the transformation , . Equation (15) becomes Integrating (16) once with respect to and setting the integration constant to zero yields To get , we consider the homogenous balance between the highest order derivative and the highest order nonlinear term : Then (7) becomes where , , and are constants to be determined later. From (19), Substituting (19) and its derivatives into (17) and collecting all terms with the same power of together yields a simultaneous set of nonlinear algebraic equations as follows: Solving this algebraic system of equations with the aid of Mathematica yields the following solution: Substituting the solution to the algebraic equation and the general solution to (8) into (19), we obtain three types of travelling wave solutions of the space and time fractional Benjamin-Bona-Mahony equation.
Case 1. When , we obtain the following hyperbolic function solutions:
where If we set and in (23), we obtain
If we set and in (23), we obtain
Applying Theorem 1 to (24) yields two further solutions:
where .
Applying Theorem 1 to (25) yields two further solutions:
where .
Case 2. When , we obtain the following trigonometric function solutions: where . If we set and in (28), we obtain If we set and in (28), we obtain
Case 3. When , we obtain a rational function solution: where . If we set and in (31), we obtain If we set and in (31), we obtain
4.2. Solutions for Space and Time Fractional BBM Equation via MSE Method
Now, we apply the modified simple equation method to construct travelling wave solutions of the space and time fractional Benjamin-Bona-Mahony equation. The procedure for transforming the nonlinear fractional PDE into an ODE in the MSE method is the same as that of the expansion method. Hence, from (17), we have We get by considering the homogenous balance between the highest order derivative and the highest order nonlinear term. Then (14) becomes where , , and are constants to be determined such that and is an unidentified function to be determined. It is easy to find that Substituting these values into (17) and equating the coefficients of , , , , and , respectively, to zero, we obtain Solving (37) and (41), respectively, yields From (38), (39), and (40), we have
Case 1. When , we get . This yields an absurd solution and hence this case is discarded.
Case 2. When and , we have where is a constant of integration. Integrating (44) with respect to , we obtain where is a constant of integration. Substituting the value of and from (44) and (45), respectively, into (35) yields when , we obtain the exact solution of (15) as when , we obtain the exact solution of (15) as Since and are arbitrary constants, therefore, if we set and , we have the solutions (47) and (48) as Bulent and Erdal [29] used direct algebraic method to get new complex travelling wave solutions to the BBM equation (15) by employing a transformation given by , . The results reported by [29] are A generalized expansion method which uses the Klein-Gordon equation as the auxiliary equation was applied by Yanhong and Baodan [30] to a form of BBM equation given by The travelling wave solutions to the BBM equation obtained by [30] using a transformation given by , are where , , , , , and are Jacobi elliptic functions and gives the modulus of the Jacobi elliptic functions. These functions generate hyperbolic functions when and trigonometric functions when as follows:
Remark 3. When the arbitrary constants and are taken to be zero separately in (23), (28), and (31), we get (24), (25), (29), (30), and (32) which are the travelling wave solutions to the space and time fractional BBM equation obtained via the expansion method. Equations (49) are the travelling wave solution to the space and time fractional BBM equation obtained through the modified simple equation method. We see that the modified simple equation is not very effective for constructing travelling wave solutions to the space and time fractional BBM equation because the integer obtained from considering the homogenous balance between the highest nonlinear terms and the highest derivative is greater than 1.
Remark 4. Applying Liu’s theorem to (24) and (25), we obtain new complex travelling wave solutions in the form of (26) and (27).
Remark 5. All the travelling wave solutions to the space and time fractional Benjamin-Bona-Mahony equation obtained were checked by putting them back into (1) with the aid of Mathematica.
Remark 6. For , , and , the complex travelling wave solutions , , , and in (50)–(54) by [29] reduce to some of the solutions we obtained via the expansion method; that is, (see (50)), (see (51)), (see (52)), and (see (53)) are the same as (see (25)), (see (24)), (see (30)), and (see (29)), respectively. Hence, the complex direct algebra method does not produce new complex solutions as suggested in [29]. Solution (see (54)) reported by [29] does not satisfy the space and time fractional BBM equation for .
Remark 7. We also examine the solutions to the BBM equation from [30] for . For and , (see (56)) reduces to (see (32)). For , (see (58)) and (see (59)) reduce to (see (24)) and (see (25)), respectively. When the modulus of the Jacobi elliptic functions is , then both (see (63)) and (see (64)) reduce to (see (25)) for . We observe that (see (29)) and (see (30)) are, respectively, the same as (see (62)) and (see (63)) when and . It should be noted that (see (57)), (see (60)), and (see (61)) are not valid solutions to the space and time fractional BBM equation for . Also (see (62)) and (see (64)) for and , respectively, do not satisfy the space and time fractional BBM equation for .
The shapes of (24), (25), (29), and (30) for selected values of, , and at different and values are presented in Figures 1–4. The solution (see (24)) is a singular kink shaped travelling wave solution of (15). Figure 1 shows the shape of (24) with , , and for the interval , at different and values. From Figure 1 and , we observe that, for , the propagation of the wave will be in the positive -direction. If we take , then the propagation of the wave will be in the negative -direction. The solution (see (25)) is a bell-shaped Soliton solution of (15). Figure 2 shows the shape of (25) with , , and for the interval at different and values. From Figure 2 and , we also observe that, for , the propagation of the wave will be in the positive -direction. If we take , then the propagation of the wave will be in the negative -direction. Figures 3 and 4 show, respectively, the shape of (see (29)) and (see (30)) with and for the interval , at different and values. (see (29)) and (see (30)) are exact periodic travelling wave solutions of the space and time fractional Benjamin-Bona-Mahony equation.
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5. Conclusion
Hyperbolic, trigonometric, and rational function travelling wave solutions to the space and time fractional Benjamin-Bona-Mahony equation defined in the sense of Jumarie’s modified Riemann-Liouville derivative have been obtained using the expansion method and modified simple equation method. Liu’s theorem was also applied to obtain new complex travelling wave solutions to the space and time fractional BBM equation. The results were compared to those obtained by Bulent and Erdal [29] using the direct algebraic method for complex travelling wave solution and those by Yanhong and Baodan [30] using the generalized expansion method with the Klein-Gordon equation as an auxiliary equation. The expansion method is equivalent to the direct algebraic method and the generalized expansion method with some of their reported solutions found not to satisfy the BBM equation. Also, the modified simple equation method is not effective for constructing travelling wave solutions to the space and time fractional BBM equation because the index obtained from considering the homogenous balance between the highest nonlinear term and the highest derivative is greater than 1. The improvement of the modified simple equation scheme for evolution equations with is still an open problem.
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.