Research Article | Open Access
Exact Solutions of the Time Fractional BBM-Burger Equation by Novel -Expansion Method
The fractional derivatives are used in the sense modified Riemann-Liouville to obtain exact solutions for BBM-Burger equation of fractional order. This equation can be converted into an ordinary differential equation by using a persistent fractional complex transform and, as a result, hyperbolic function solutions, trigonometric function solutions, and rational solutions are attained. The performance of the method is reliable, useful, and gives newer general exact solutions with more free parameters than the existing methods. Numerical results coupled with the graphical representation completely reveal the trustworthiness of the method.
Differential equations of noninteger order are generalizations of conventional differential equations of integer order . Exploration and applications of integrals and derivatives of random order are efficiently dealt with the field of mathematical analysis, called as fractional calculus, which has engrossed in considerable interest in many disciplines, nowadays. The behavior of many physical systems can be perfectly defined by the fractional theory. In recent years, we cannot rebuff the importance of fractional differential equations because of their numerous applications in the areas of physics and engineering. For example, the nonlinear fluctuation of earthquakes can be modeled with the help of fractional derivatives and the fluid-dynamic traffic model with fractional derivatives can eradicate the problems arising from the huge traffic flow [2, 3]. We apply a generalized fractional complex transform [4–7] to convert the given fractional order differential equation to ordinary differential equation. On account of development of the computer and its exact description of countless real-life problems, fractional calculus has touched the height of fame and success nowadays, although it was invented three centuries ago by Newton and Leibniz. Many important phenomena in electromagnetic, acoustics, viscoelasticity, electrochemistry, and material science are better described by differential equations of noninteger order [8–11]. A physical interpretation of the fractional calculus was given in [12–14]. With the development of symbolic computation software, likeMaple, many researchers developed and established many numerical and analytical methods to search for exact solutions of nonlinear evolution equations (NLEEs), for example, Cole-Hopf transformation , Tanh-function method [16–19], inverse scattering transform method , variational iteration method [21, 22], Exp-function method [23–26], homogeneous balance method [27, 28], and F-expansion method [29, 30], which are used for searching the exact solutions.
Lately, a straight and crisp method, called -expansion method, was introduced by Wang et al.  and confirmed that it is a powerful method for seeking analytic solutions of nonlinear evolution equations (NLEEs). For additional references see the articles [32–37]. In order to establish the efficiency and diligence of -expansion method and to extend the range of applicability, further research has been carried out by several researchers. For illustration, Zhang et al.  made an overview of -expansion method for the evolution equations with variable coefficients. Zhang et al.  also presented an improved -expansion method to look for more broad traveling wave solutions. Zayed  offered a new technique of -expansion method where gratifies the Jacobi elliptic equation, , where , , and are random constants, and obtained new exact solutions. Zayed  for a second time offered a different approach of this method in which satisfies the Riccati equation , where and are casual constants.
The -expansion method and the transformed rational function method used by Ma and Lee  have a common idea. That is, we initially put the given nonlinear evolution equation (NLEE) into the equivalent ordinary differential equation (ODE), and then ODE can be transformed into a system of arithmetical polynomials with the influential constants. By the solutions of the ordinary differential equation, we can obtain the exact traveling solutions and rational solution of the nonlinear evolution equations.
In this article, we will apply novel -expansion method introduced by Alam et al.  to solve the time fractional BBM-Burger equation, whereas, the modified Riemann-Liouville derivative given by Jumarie  is used and abundant new families of exact solutions are found. The Jumarie’s modified Riemann-Liouville derivative of order is defined by the following expression: Some important properties of Jumarie’s derivative are
2. Description of the Method
Consider the fractional partial differential equation in the form, where is Jumarie’s modified Riemann-Liouville derivatives of is an unknown function, and is a polynomial in and its various partial derivatives including fractional derivatives in which the highest order derivatives and nonlinear terms are involved.
The main steps of the method are as follows.
Step 1. Li and He  projected a fractional complex transformation to convert fractional partial differential equation into ordinary differential equation (ODE), so all analytical methods devoted to the advanced calculus can be easily applied to the fractional calculus. The following complex transformation: where are arbitrary constants with , permits us to convert (5) into an ordinary differential equation of integer order in the form where the primes stand for the ordinary derivatives with respect to .
Step 2. Integrate (7) term by term one or more times, if possible; capitulate constant(s) of integration which can be calculated later on.
Step 3. Suppose that the solution of (7) can be expressed as
Herein or may be zero, but both of them cannot be zero simultaneously. and are constants to be determined later and satisfies the second order nonlinear ordinary differential equation
where primes denote the derivative with respect ; , , and are genuine constants.
The Cole-Hopf transformation reduces (10) into the Riccati equation Equation (11) has individual twenty-five solutions (see Zhu,  for details).
Step 4. The positive integer can be calculated by balancing the highest order linear term with the highest order nonlinear term in (7).
Step 5. Inserting (8) together with (9) and (10) into (7), we obtain polynomials in and , . Collecting each coefficient of the resulted polynomials to zero acquiesces an overdetermined set of algebraic equations for , , , and .
Step 6. After solving the system of algebraic equations obtained in Step 5, we can obtain the values of the constants. The solutions of (10) together with the obtained values of the constants yield abundant exact traveling wave solutions of the nonlinear evolution equation (5).
3. Application of the Method to the Time Fractional BBM-Burger Equation
Consider the following BBM-Burger equation in time fractional operator form as: S. Kumar and D. Kumar  used new fractional homotopy analysis transform method to time fractional BBM-Burger equation and found the series solution of this equation. Song and Zhang  and Fakhari et al.  find different type of solutions of the fractional BBM-Burger equation by using homotopy analysis method.
Considering the homogeneous balance between and in (13), we obtain . Therefore, the trial solution (8) becomes Using (14) into (13), left-hand side is converted into polynomials in and . Equating the coefficients of same power of the resulted polynomials to zero, we attain a set of algebraic equations (which are omitted for the sake of simplicity) for , , , , , , , , and . Solving the overdetermined set of algebraic equations by using the symbolic computation software, such as Maple 13, we obtain the following solution sets:
The Set 1. We consider that where , , , , , and are arbitrary constants.
The Set 2. We consider that where , , , , , and are arbitrary constants.
The Set 3. We consider that where , , , , and are arbitrary constants.
When and (or ), and its graph is shown in Figure 1. Consider where and are real constants. Consider the following: When and (or ), where and are real constants such that . Consider the following: When and , and its graph is shown in Figure 2. Consider where is an arbitrary constant.
When and , the solution of (12) is where is an arbitrary constant.
When and (or ), the other families of exact solutions of (12) are omitted for convenience.
We can write down the other families of exact solutions of (12) which are omitted for practicality.
When and (or ), Others families of exact solutions are omitted for the sake of simplicity.
Other exact solutions of (12) are omitted here for convenience.
If we replace by and by and put in (10), then the novel -expansion method coincides with Akbar et al.’s generalized and improved -expansion method . On the other hand if we put in (8) and in (10) then the method is identical to the improved -expansion method presented by Zhang et al. . Again if we set , and the negative exponents of are zero in (8), then the method turn out into the basic -expansion method introduced by Wang et al. . At the end, if we put in (10) and are functions of and instead of constants then the method is transformed into the generalized -expansion method developed by Zhang et al. . Thus the methods presented in [31, 36, 38, 39] are only special cases of the novel -expansion method. So our method is more general having more free parameters than the existing methods.
A novel -expansion method is applied to fractional partial differential equation successfully. As applications, abundant new exact solutions for time fractional BBM-Burger equation have been successfully obtained. The nonlinear fractional complex transformation for is very important, which ensures that a certain fractional partial differential equation can be converted into another ordinary differential equation of integer order. The obtained solutions are more general with more free parameters. Thus novel -expansion method would be a powerful mathematical tool for solving nonlinear evolution equations. To the best of our knowledge, the solutions obtained in this article are not reported in literature previously.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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