A residual power series technique for solving Boussinesq-Burgers equations

In this paper, a residual power series method (RPSM) is combining Taylor’s formula series with residual error function, and is investigated to find a novel analytical solution of the coupled strong system nonlinear Boussinesq-Burgers equations according to the time. Analytical solution was purposed to find approximate solutions by RPSM and compared with the exact solutions and approximate solutions obtained by the homotopy perturbation method and optimal homotopy asymptotic method at different time and concluded that the present results are more accurate and efficient than analytical methods studied. Then, analytical simulations of the results are studied graphically through representations for action of time and accuracy of method. Subjects: Science; Mathematics & Statistics; Advanced Mathematics; Analysis Mathematics; Mathematical Numerical Analysis


PUBLIC INTEREST STATEMENT
Many phenomena in the world was described by nonlinear partial differential equations that can be solved numerically. Boussinesq-Burgers equation is one of the famous nonlinear equations. In this work, author solved Boussinesq-Burgers equations numerically by novel and new method called residual power series method (RPSM). Also, the author compared the present work via two another published method and concluded that the present results are more accurate and efficient than analytical methods studied. Action of time and accuracy of the method was studied graphically at last.

Introduction
First, we consider the generalized Boussinesq-Burgers equations given by where a, b, c, and d are real nonzero constants.
The systems of nonlinear equations are known to describe a wide variety of phenomena in physics, engineering, applied mathematics, chemistry, and biology.
The Boussinesq-Burgers equations arise in the study of fluid flow and describe the propagation of shallow water waves. Here, x and t represent the normalized space and time, respectively, u(x, t) represents the horizontal velocity and at the leading order, it is the depth averaged horizontal field, while w(x, t) denotes the height of the water surface above the horizontal level at the bottom (Wang, Tian, Liu, Lü, & Jiang, 2011).
Since many systems of nonlinear equations do not have exact analytical solutions, various analytic and numerical methods for finding approximate solutions to Boussinesq-Burgers equations have been proposed. In recent years some works have been done in order to find the numerical solution of this equation (for example Abdel Rady, Osman, & Khalfallah, 2010;Chen & Li, 2006;Gao, Xu, Tang, & Meng, 2007;Hardik & Meher, 2015;Khalfallah, 2009;Li & Chen, 2005). In this work, the residual power series method (RPSM) is applied to find the analytic solution for the Boussinesq-Burgers equations.
The RPSM is based on constructing power series expansion solution for different nonlinear equations without linearization, perturbation, or discretization (Arqub, Abo-Hammour, 2013;Arqub, El-Ajou, et al., 2013;El-Ajou et al., 2013Wang & Chen, 2015). Recently, Deng et al. discussed global existence and asymptotic behavior of the Boussinesq-Burgers equations (Ding & Wang, 2015). Also, many authors studied existence and uniqueness of Boussinesq-Burgers equations (see Changjiang & Renjun, 2003;Liu, 2016). With the help of residual error concepts, this method computes the coefficient of the power series by a chain of algebraic equations of one or more variables and finally we get a series solution, in practice a truncated series solution. The main advantage of this method over the other method is it can be applied directly to the given problem by choosing an appropriate initial guess approximation and you can discuss the nonlinear problems at large time.
It has been proven that the RPSM is a convenient and effective method in its application. The aim of this paper is solving Boussinesq-Burgers equations analytically using RPSM and compared homotopy perturbation method and optimal homotopy asymptotic method at different time. Also, the accuracy of the present method at different time was discussed.

Analysis of RPSM
To illustrate the fundamental scheme of RPSM, we set a = c = 2, and b = d = −1/2 in the system of Equations (1.1) and (1.2), the system reduces to the following system: Assume that the solutions takes the expansion where m = 0, 1, 2,… Next, we define u k (x, t) and w k (x, t) to denote the kth truncated series of u(x, t) and w(x, t), respectively, that is, Obviously u(x, t) and w(x, t) satisfy the initial conditions (2.3) and (2.4), so the 0th RPS approximate solutions of u(x, t) and w(x, t) are On the other hand, from Equations (2.7) and (2.8) the initial guess approximation the first RPS approximate solutions of u(x, t) and w(x, t) should be Consequently, one can reformulate the expansion of Equations (2.7) and (2.8) as follows: (2.10) w(x, 0) = g 0 (x) = g(x).
(2.11) To obtain the coefficients f m (x), and g m (x) m = 1, 2, 3, 4, … , k, we apply the following subroutine; substitute mth truncated series of u(x, t) and w(x, t) into Equations (2.17) and (2.18), apply the derivative formula s t s Res(x, t) = 0 on Res m (x, t) m = 1, 2, 3, 4, … , k, substitute t = 0, in the following formula, equate it to zero, and then lastly solve the obtained algebraic equation to obtain the form of the other coefficients. Any how we need to solve the following algebraic equation: In this way, we can find all the required coefficients of the multiple power series of Equations (2.1-2.4) are obtained.

Application of RPSM to Boussinesq-Burgers equation
The general Boussinesq-Burgers equation considering as follows: Subjected to initial conditions
Then, the solutions are

Numerical results and discussion
This section describes the proposed methods to obtain the approximate analytical solutions by numerical simulations of. From Table 1 compared RPSM with the exact solution and the solutions obtained by the homotopy perturbation method and optimal homotopy asymptotic method at different time, and it is clear that, the present method is accurate and provides efficient results. Figures 1-4 showed surface graphic of u(x, t) and w(x, t) with exact and RPSM solution at small time, and concluded that the present work is the same exact solution. From Figures 5 and 6 sketched the RPSM solution a wide large time, and it is the advantage of RPSM. Figures 7 and 8 show the effect of space and time described at counter-plot. Figures 9 and 10 show the curve steadiness of problem at t = 1 between exact and RPSM. Finally, the action and influence of time are examined in substantive Figures 11 and 12. (3.4) w(x, 0) = −k 2 8 sech 2 kx + ln(b) 2 ,