Research Article  Open Access
Analytical Solution for the Time Fractional BBMBurger Equation by Using Modified Residual Power Series Method
Abstract
In this study, a generalized Taylor series formula together with residual error function, which is named the residual power series method (RPSM), is used for finding the series solution of the time fractional BenjaminBonaMahonyBurger (BBMBurger) equation. The BBMBurger equation is useful in describing approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems. The numerical solution of the BBMBurger equation is calculated by Maple. The numerical results show that the RPSM is reliable and powerful in solving the numerical solutions of the BBMBurger equation compared with the exact solutions as well as the solutions obtained by homotopy analysis transform method through different graphical representations and tables.
1. 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 Lâ€™Hopital in 1695. Lakshmikantham and Vatsala [3] discussed the basic theory for the initial value problem involving RiemannLiouville 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], 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) [11â€“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 ElAjou 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 KdVBurgers equation with timespacefractional 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 DrinfeldSokolovWilson system by residual power series method. In [15â€“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 Phi4 equation in [24], the fractional population diffusion model [25], the generalized BurgerHuxley equation [26], and the time fractional twocomponent evolutionary system of order 2 [27].
In this paper, an analytical solution of the time fractional BenjaminBonaMahonyBurger equation (called BBMBurger equation) is proposed by residual power series method. The BBMBurger equation describes the mathematical model of propagation of smallamplitude long waves in nonlinear dispersive media. It is well known that the BBM equation is a refinement of the KdV equation. The BBMBurger 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 BBMBurger was proposed; it described unidirectional propagation of long waves in a certain nonlinear dispersive system [28â€“30]. The integer order of the BBMBurger equation can be written as where and are positive constants and is a domain partition.
In order to discuss the dynamic physical system, the time fractional BBMBurger equation was proposed. The BBMBurger equation can be written in time fractional operator form as [31] where 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 , 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 BBMBurger equation specifically. Numerical results and discussions are presented by graphics and charts in Section 4. At last, the conclusion was drawn in Section 5.
2. 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 be a function and be the upper positive integer of . 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 ElAjou 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 and , is called fractional power series about , where is a variable and are constants called the coefficients of the series.
Theorem 2 (see [11]). Suppose that has a fractional power series representation at of the form If , then coefficients of (8) are given by the formula where and is the radius of convergence.
Definition 3 (see [12]). A power series of the form for and , is called multiple fractional power series about , where is a variable and are functions of called the coefficients of the series.
Theorem 3 (see [11, 12]). Suppose that has a multiple power series representation at of the form
If are continuous on then coefficients of (11) are given as where and , in which is the radius of convergence of the fractional power series .
Now, the RPSM can be proposed by
In order to obtain the approximate value of (13), the form of the ith series of is proposed. Then the truncated series is defined by
If , . We define the ith residual function as follows:
In order to get , we look for the solution of where .
3. Solution of the Time Fractional BBMBurger Equation by Residual Power Series Method
The purpose of this paper is to use modified residual power series method to solve the time fractional BBMBurger equation. The initial condition of the time fractional BBMBurger equation is (3), and the exact solution of the time fractional BBMBurger equation is (4). In this section, we use residual power series method to solve the time fractional BBMBurger equation specifically.
is the ith residual function of (2), which is defined as
Step 1. For , the residual function of the time fractional BBMBurger equation can be written as where can be written by (13) as Then we get For , we have In addition, since and then according to , we have
Step 2. For , the residual function of the time fractional BBMBurger equation can be written as with the condition Therefore, we can attain Then we solve ; thus,
Step 3. For , the residual function of the time fractional BBMBurger equation can be written by with the condition Then we can get So, from , we get
Step 4. For the residual function of the time fractional BBMBurger equation can be written as with the condition Using the same method, through the equation of , we can get as follows: Thus, the approximate solution of the time fractional BBMBurger equation is where is given in the initial condition at (3) and , , and are given at (24)â€“(35).
4. Results and Discussion
In this section, the approximate analytical solution of the time fractional BBMBurger equation by using residual power series method is calculated. We can compare the exact solution of the BBMBurger equation with the analytical approximate solution by graphics and charts.
In Figure 1, the approximate solutions and the exact solutions are presented by drawing threedimensional graphics. Figure 1(a) presents the approximate solution, where , and Figure 1(b) presents the exact solution at .
(a) when
(b) when
In Figure 1, when approaches 1, the approximate solution is close to the exact solution. So, we can conclude that when approaches 1, the threedimensional graphic is accurate and when approaches 0, the threedimensional graphic is inaccurate. In such phenomena, one can say that when alpha approaches 0, the solution bifurcates or admits chaotic behavior.
In Figure 2, the threedimensional graphics show the influence of different on analytical solutions. Figure 2(a) presents approximate solutions when , and Figure 2(b) presents approximate solutions when . In Figure 3, the threedimensional graphics show approximate solutions when and (Figures 3(a) and 3(b), respectively).
(a)
(b)
(a)
(b)
In Figures 2 and 3, we find that the larger the value of is, the smoother is the plane. As parameter increases, the graphics get closer and closer to the exact solution of the graphic.
For any , the exact value of is 0. The difference between the 4th approximate solutions and the exact solutions can be shown by the value of . In Figure 4, the twodimensional graphics show the influence of different and on the value of . We can find the impact of different and on at and . In Figure 4, the different colors present different curves. In Figure 4(a), the parameter is 0.01; in Figure 4(b), the parameter is 0.001.
(a)
(b)
As shown in Figure 4, if the values of and are fixed, when , the values of are close to 0. When the values of are not in this interval, the values of are not close to 0. For , if the values of and are fixed, the value of decreases with an increase in , and if the values of and are fixed, the value of increases with a decrease in .
In Figure 5, the twodimensional graphics show the impact of different and on the value of . If the value of is fixed, the relationship of and is presented in the each panel of Figure 5. Different colors present different .
(a)
(b)
(c)
(d)
In Figure 5, we can see that if the values of and are fixed, the value of decreases with an increase in constant . If the value of and are fixed, the value of decreases with an increase in . If the value of and are fixed, the value of decreases with an increase in . As shown in Figures 5(a)â€“5(d), the value of is a constant and . However, if , the value of , which is because the power series approximate solution of is a generalized Taylor expansion at . If , the precision of the power series approximate solution of is higher.
Then we compare the exact solution with the approximate solution for the time fractional BBMBurger equation. The absolute error is
In Table 1, we present the exact solutions, the 4thterm approximate solutions by residual power series method, and the absolute errors. We can find the relationship between and and the solutions of and .

In Table 1, we can conclude that the greater the absolute value of is, the smaller the and values are. At the same condition of , we can find that the smaller the value of is, the smaller the absolute errors are. There are two conditions at the same value of â€”one is that the absolute errors are smallest when the value of is zero and the other is that the smaller the values of absolute are, the larger are the values of absolute errors. In general, we can find that the absolute errors with different and between the 4thterm analytical approximate solutions and the exact solutions are within the acceptable range. The range of magnitude of absolute errors is from to .
As shown in Table 2, we compare the 4thterm approximate solutions by residual power series method (RPSM) with the 5thterm approximate solutions by fractional homotopy analysis transform method (FHATM) in [31].

By comparing 4thterm approximate solutions by residual power series method (RPSM) with the 5thterm approximate solutions by fractional homotopy analysis transform 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 BBMBurger equation. In addition, in Table 2, we can conclude that when the value of parameter is smaller, the absolute error is smaller and when the value of parameter is larger, the absolute error is smaller; in contrast, when the value of parameter approaches 1, the absolute error is smaller.
We compare the value of the 4th residual function for different at in Table 3.