On the Exact Solution of Burgers-Huxley Equation Using the Homotopy Perturbation Method

The Homotopy Perturbation Method (HPM) is used to solve the Burgers-Huxley non-linear differential equations. Three case study problems of Burgers-Huxley are solved using the HPM and the exact solutions are obtained. The rapid convergence towards the exact solutions of HPM is numerically shown. Results show that the HPM is efficient method with acceptable accuracy to solve the Burgers-Huxley equation. Also, the results prove that the method is an efficient and powerful algorithm to construct the exact solution of non-linear differential equations.


Introduction
Most of the nonlinear differential equations do not have an analytical solution. Recently, semi-analytical solutions of real-life mathematical modeling are considered as a key tool to solve nonlinear differential equations.
The idea of the Homotopy Perturbation Method (HPM) which is a semi-analytical method was first pioneered by He [1]. Later, the method is applied by He [2] to solve the non-linear non-homogeneous partial differential equations. Nourazar et al. [3] used the homotopy perturbation method to find exact solution of Newell-Whitehead-Segel equation. Krisnangkura et al. [4] obtained exact traveling wave solutions of the generalized Burgers- 1 , where , 0 α β ≥ are real constants and n is a positive integer and [ ] 0,1 γ ∈ . Equation (1.1) models the interaction between reaction mechanisms, convection effects and diffusion transports [9]. When 0 α = and 1 n = , Equation (1.1) is reduced to the Huxley equation which describes nerve pulse propagationin nerve fibers and wall motion in liquid crystals [10]. When 0 β = and 1 n = , Equation (1.1) is reduced to the Burgers equation describing the far field of wave propagation in nonlinear dissipative systems [11]. The idea of homotopy perturbation method is presented in Section 2. Application of the homotopy perturbation method to the exact solution of Burgers-Huxley equation is presented in Section 3.

The Idea of Homotopy Perturbation Method
The Homotopy Perturbation Method (HPM) is originally initiated by He [1]. This is a combination of the classical perturbation technique and homotopy technique. The basic idea of the HPM for solving nonlinear differential equations is as follow; consider the following differential equation: where E is any differential operator. We construct a homotopy as follow: It is worth noting that as the embedding parameter p increases monotonically from zero to unity the zero order solution 0 v continuously deforms into the original problem ( ) 0 E u = . The embedding parameter, , is considered as an expanding parameter [2]. In the homotopy perturbation method the embedding parameter p is used to get series expansion for solution as:  [2]. It is also assumed that Equation (2.2) has a unique solution and by comparing the like powers of p the solution of various orders is obtained. These solutions are obtained using the Maple package.

The Burgers-Huxley Equation
To illustrate the capability and reliability of the method, three cases of nonlinear diffusion equations are presented.
Case І: in this case we will examine the Burgers-Huxley equation for 0, 1, 1, 1 n α γ β = = = = , so, the equation is written as: Subject to initial condition: ( ) We construct a homotopy for Equation (3.1) in the following form: The solution of Equation (3.1) can be written as a power series in p as: Using the Maple package to solve recursive sequences, Equation (3.5), we obtain the followings: S. S. Nourazar et al.

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The Taylor  e e (3.7), we get as follow: This is the exact solution of the problem, Equation (3.1). Table 1 shows the trend of rapid convergence of the results of ( ) ( ) using the HPM. The rapid convergence of the solution toward the exact solution, the maximum relative error of less than 0.0000058% is achieved as shown in Table 1   To solve Equation (3.10) we construct a homotopy in the following form: The solution of Equation (3.10) can be written as a power series in p as: Using the Maple package to solve recursive sequences, Equation (3.14), we obtain the followings: The Taylor series expansion for e e x t x t x t x t is written as: By substituting Equation ( x t x t x t x t This is the exact solution of the problem, Equation (3.10). Table 2 shows the trend of rapid convergence of the results of ( ) ( ) using the HPM solution toward the exact solution. The maximum relative error of less than 0.00014% is achieved in comparison to the exact solution as shown in Table 2 Subject to initial condition: Using the Maple package to solve recursive sequences, Equation (3.23), we obtain the followings:   3 1  3 3 1   4  4  3 3 1   4  2  2  3  3 3 1  3 3 1  3 3 1  3 3 1  3 3 1  3 3 1  4  4  4   3 .
The Taylor series expansion for