Analytical Solution for the Time Fractional BBM-Burger Equation by Using Modified Residual Power Series Method

School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China Shaanxi Key Laboratory of Network Data Analysis and Intelligent Processing, Xi’an University of Posts and Telecommunications, Xi’an, Shaanxi 710121, China School of Mathematics and Computer Science, Shaanxi University of Technology, Hanzhong 723000, China Institute of IOT and IT-based Industrialization, Xi’an University of Posts and Telecommunications, Xi’an 710061, China


Introduction
Today, fractional differential equations are more and more important in many fields, such as mathematics and dynamic systems [1,2]. The persons who firstly proposed fractional differential equations were Leibniz and L'Hopital in 1695. Lakshmikantham and Vatsala [3] discussed the basic theory for the initial value problem involving Riemann-Liouville differential operators by fractional differential equations. Diethelm and Ford [4] proposed the analytical questions of existence and uniqueness of solutions by fractional differential equations. And many other academics studied different theories in fractional differential equations. However, most of the problems do not possess analytical solution, and thus, a lot of numerical methods have been developed to solve these fractional differential equations.
Many different methods are introduced to develop an approximate analytical solution for the fractional differential equations and systems, such as the variational iteration method [5], homotopy analysis transform [6], homotopy asymptotic method [7], G′/G expansion method [8], polynomial least squares method [9], and finite difference method [10]. Recently, an analytical method based on power series expansion without linearization, discretization, or perturbation has been introduced and successfully applied to many kinds of fractional differential equations arising in strongly nonlinear and dynamic problems. The method was named residual power series method (RPSM) [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27], which was used to find the analytical solution for several classes of time fractional differential equations. The residual power series method has been widely used in different fields. In [11], some important theorems that are related to the classical power series have been generalized to the fractional power series by El-Ajou et al. These theorems are constructed by using Caputo fractional derivatives. They also presented and discussed the explicit and approximate solutions of the nonlinear fractional KdV-Burgers equation with timespace-fractional derivatives in [12]. Moaddy et al. [13] proposed that the residual power series method can be applied to differential algebraic equation systems. Jaradat et al. [14] solved the time fractional Drinfeld-Sokolov-Wilson system by residual power series method. In [15][16][17][18][19][20][21], residual power series method, as a powerful method, was used to solve the other time fractional differential equations. Residual power series method was also used for the time fractional Gardner [23] and Kawahara equations in [22], the time fractional Phi-4 equation in [24], the fractional population diffusion model [25], the generalized Burger-Huxley equation [26], and the time fractional two-component evolutionary system of order 2 [27].
In this paper, an analytical solution of the time fractional Benjamin-Bona-Mahony-Burger equation (called BBM-Burger equation) is proposed by residual power series method. The BBM-Burger equation describes the mathematical model of propagation of small-amplitude long waves in nonlinear dispersive media. It is well known that the BBM equation is a refinement of the KdV equation. The BBM-Burger equation and the KdV equation are relevant to the wave breaking models [28]. The KdV equation came from water waves, and the KdV equation as a model was used for long waves in many other physical systems. But in some physical system of long waves, the KdV equation was not applicable. So, the BBM-Burger was proposed; it described unidirectional propagation of long waves in a certain nonlinear dispersive system [28][29][30]. The integer order of the BBM-Burger equation can be written as where α and β are positive constants and x ∈ x L , x R is a domain partition. In order to discuss the dynamic physical system, the time fractional BBM-Burger equation was proposed. The BBM-Burger equation can be written in time fractional operator form as [31] where α is a parameter, which is the order of the time fractional derivative and is located in the range of (0,1]. The initial condition is If α = 1, the exact solution [32] is The rest of the paper is as follows. In Section 2, some basic definitions about the Caputo and modified residual power series method are introduced. In Section 3, we use residual power series method to solve the time fractional BBM-Burger equation specifically. Numerical results and discussions are presented by graphics and charts in Section 4. At last, the conclusion was drawn in Section 5.

Modified Residual Power Series Method
In this section, the definition of the Caputo fractional is introduced systematically. And this section also presents the most details of the modified residual power series method. Fractional residual power series method is used to solve many kinds of differential equations, and this method is effective in calculating these equations.
Definition 1 (see [33]). Let f t : 0, +∞ → R be a function and n be the upper positive integer of α α > 0 . The Caputo fractional derivative is defined by Theorem 1 (see [33]). The Caputo fractional derivative of the power function satisfies Below, we introduce some definitions and theorems related to the fractional power series used in this paper. These important theorems that are related to the fractional power series were presented by El-Ajou et al. [11,12]. These theorems are constructed by using Caputo fractional derivatives.
Definition 2 (see [11,12]). A power series expansion of the form for 0 ≤ n − 1 < α ≤ n and t ≥ t 0 , is called fractional power series about t = t 0 , where t is a variable and c m are constants called the coefficients of the series.
Theorem 2 (see [11]). Suppose that f has a fractional power series representation at t = t 0 of the form If D mα f t ∈ t 0 , t 0 + R , m = 0,1,2, … , then coefficients c m of (8) are given by the formula Complexity where D mα = D α ⋅ D α ⋯ D α m − times and R is the radius of convergence.
Definition 3 (see [12]). A power series of the form where t is a variable and f m are functions of x called the coefficients of the series.
Theorem 3 (see [11,12]). Suppose that u x, t has a multiple power series representation at t = t 0 of the form , then coefficients f m x of (11) are given as where D mα t = ∂ mα /∂t mα = ∂ α /∂t α ⋅ ∂ α /∂t α ⋯ ∂ α /∂t α m − times and R = min C∈I R C , in which R C is the radius of convergence of the fractional power series ∑ ∞ m=0 f m c t − t 0 mα . Now, the RPSM can be proposed by In order to obtain the approximate value of (13), the form of the ith series of u x, t is proposed. Then the truncated series u i x, t is defined by We define the ith residual function as follows: In order to get f n x , n ∈ N * , we look for the solution of where N * = 1,2,3, … .

Solution of the Time Fractional BBM-Burger Equation by Residual Power Series Method
The purpose of this paper is to use modified residual power series method to solve the time fractional BBM-Burger equation. The initial condition of the time fractional BBM-Burger equation is (3), and the exact solution of the time fractional BBM-Burger equation is (4). In this section, we use residual power series method to solve the time fractional BBM-Burger equation specifically.
Res i x, t is the ith residual function of (2), which is defined as Step 1. For i = 1, the residual function of the time fractional BBM-Burger equation can be written as where u 1 x, t can be written by (13) as Then we get For t = 0, we have In addition, since 3 Complexity then according to Res 1 x, 0 = 0, we have Step 2. For i = 2, the residual function of the time fractional BBM-Burger equation can be written as with the condition Therefore, we can attain Then we solve D α t Res 2 x, 0 = 0; thus, Step 3. For i = 3, the residual function of the time fractional BBM-Burger equation can be written by with the condition Then we can get Step 4. For i = 4, the residual function of the time fractional BBM-Burger equation can be written as with the condition Using the same method, through the equation of D 3α t Res 4 x, 0 = 0, we can get f 4 x as follows: where f 0 x is given in the initial condition at (3) and

Results and Discussion
In this section, the approximate analytical solution of the time fractional BBM-Burger equation by using residual power series method is calculated. We can compare the exact solution of the BBM-Burger equation with the analytical approximate solution by graphics and charts. In Figure 1, the approximate solutions and the exact solutions are presented by drawing three-dimensional graphics. Figure 1(a) presents the approximate solution, where α = 0 9, and Figure 1(b) presents the exact solution at α = 1.
In Figure 1, when α approaches 1, the approximate solution is close to the exact solution. So, we can conclude that when α approaches 1, the three-dimensional graphic is accurate and when α approaches 0, the three-dimensional graphic is inaccurate. In such phenomena, one can say that when alpha approaches 0, the solution bifurcates or admits chaotic behavior.
In Figures 2 and 3, we find that the larger the value of α is, the smoother is the plane. As parameter α increases, the graphics get closer and closer to the exact solution of the graphic.
For any α ∈ 0, 1 , the exact value of Res x, t is 0. The difference between the 4th approximate solutions and the exact solutions can be shown by the value of Res x, t . In Figure 4, the two-dimensional graphics show the influence of different α and x on the value of Res x, t . We can find the impact of different α and x on Res 4 x, t at t = 0 01 and t = 0 001. In Figure 4, the different colors present different curves. In Figure 4(a), the parameter t is 0.01; in Figure 4 In Figure 5, the two-dimensional graphics show the impact of different x and α on the value of Res 4 x, t . If the value of x is fixed, the relationship of t and Res 4 x, t is presented in the each panel of Figure 5. Different colors present different α.
In Figure 5, we can see that if the values of α and t are fixed, the value of Res 4 x, t decreases with an increase in constant x. If the value of t and x are fixed, the value of Res 4 x, t decreases with an increase in α. If the value of α and x are fixed, the value of Res 4 x, t decreases with an increase in t ∈ 0,0 2 . As shown in Figures 5(a)-5(d), the value of Res 4 x, 0 is a constant and Res 4 x, 0 ≠ 0. However, if t → 0, the value of Res 4 x, t → 0, which is because the power series approximate solution of u 4 x, t is a generalized Taylor expansion at t 0 = 0. If t → t 0 , the precision of the power series approximate solution of u 4 x, t is higher.
Then we compare the exact solution with the approximate solution for the time fractional BBM-Burger equation. The absolute error is In Table 1, we present the exact solutions, the 4th-term approximate solutions by residual power series method, and the absolute errors. We can find the relationship between x and t and the solutions of u x, t exact and u x, t RPSM .
In Table 1, we can conclude that the greater the absolute value of x is, the smaller the u x, t exact and u 4 x, t RPSM values are. At the same condition of x, we can find that the smaller the value of t is, the smaller the absolute errors are. There are two conditions at the same value of t-one is that the absolute errors are smallest when the value of x is zero and the other is that the smaller the values of absolute x are, the larger are the values of absolute errors. In general, we can find that the absolute errors with different x and t between the 4th-term analytical approximate solutions and the exact solutions are within the acceptable range. The range of magnitude of absolute errors is from 10 −3 to 10 −7 .
As shown in Table 2, we compare the 4th-term approximate solutions by residual power series method (RPSM) with the 5th-term approximate solutions by fractional homotopy analysis transform method (FHATM) in [31].
By comparing 4th-term approximate solutions by residual power series method (RPSM) with the 5th-term approximate solutions by fractional homotopy analysis transform 8 Complexity method (FHATM), we can find the absolute errors are smaller than the results in [31]. The absolute error by using RPSM is one order of magnitude smaller than that by using FHATM. So, residual power series method is efficient and accurate for solving the time fractional BBM-Burger equation. In addition, in Table 2, we can conclude that when the   value of parameter t is smaller, the absolute error is smaller and when the value of parameter x is larger, the absolute error is smaller; in contrast, when the value of parameter α approaches 1, the absolute error is smaller.
We compare the value of the 4th residual function for different α at t = 0 01 in Table 3.
In Table 3, we get the solutions of the 4th residual function Res 4 x, t . If the value of t and α are fixed, the value of Res 4 x, t decreases with an increase in x. If the value of t and x are fixed, the value of Res 4 x, t decreases with an increase in α. So, when the value of Res 4 x, t is close to 0, the approximate solutions are close to exact solutions and the approximate solutions are accurate. In Table 3, we can conclude that when the value of α is approaching 1, the solutions are more accurate.
In conclusion, the residual power series method is a powerful method to solve the analytical approximate solution of the time fractional BBM-Burger equation in x > 10 and t ∈ 0,0 2 .

Conclusion
In this paper, we discuss the analytical solution of the time fractional BBM-Burger equation by using residual power series method (RPSM). The time fractional BBM-Burger equation is calculated by Maple in Windows 7 (64 bit). The analytical solution is presented by graphics and datum. Results show that the analytical solutions by residual power series method are close to the exact solution. In general, RPSM is an effective and convenient method in finding analytical solution for the time fractional BBM-Burger equation and other long waves in certain nonlinear dispersive systems.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that they have no conflicts of interest.