Journal of Applied Mathematics

Volume 2013 (2013), Article ID 387478, 8 pages

http://dx.doi.org/10.1155/2013/387478

Research Article

Application of Optimal Homotopy Asymptotic Method to Burger Equations

R. Nawaz, H. Ullah, S. Islam, and M. Idrees

Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan

Received 8 April 2013; Accepted 10 June 2013

Academic Editor: Anjan Biswas

Copyright © 2013 R. Nawaz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We apply optimal homotopy asymptotic method (OHAM) for finding approximate solutions of the Burger's-Huxley and Burger's-Fisher equations. The results obtained by proposed method are compared to those of Adomian decomposition method (ADM) (Ismail et al., (2004)). As a result it is concluded that the method is explicit, effective, and simple to use.

1. Introduction

Nonlinear phenomena play a vital role in applied mathematics, physics, and engineering sciences. The Burger’s equation models efficiently certain problems of a fluid flow nature, in which either shocks or viscous dissipation is a significant factor. It can be used as a model for any nonlinear wave propagation problem subject to dissipation [1]. The first steady-state solutions of Burger equation were given by Young et al. [2] However, the equation gets its name from the extensive research of Burger’s [3]. The generalized Burger’s-Huxley introduced by Satsuma shows a prototype model for describing the communication among reaction mechanisms, convection effects, and diffusion transports [4]. Burger-Fisher equation has significant applications in various fields of applied mathematics and has physical applications such as gas dynamic, traffic flow, convection effect, and diffusion transport [5–12]. Marinca and Herişanu et al. introduced a new semianalytic method OHAM for approximate solution of nonlinear problems of thin film flow of a fourth-grade fluid down a vertical cylinder. In progression of papers Marinca and Herişanu et al. have applied this method for the solution of nonlinear equations arising in the steady state flow of a fourth-grade fluid past a porous plate and for the solution of nonlinear equations arising in heat transfer [13–15]. The method has been applied by a number of researchers for solution of ordinary and partial differential equations [16–21]. The motivation of this paper is to show the effectiveness of OHAM for the solution of Burger’s-Huxley and Burger’s-Fisher equations. We consider Burger’s-Huxley equation of the form

and Burger’s-Fisher equation of the form

where , , , and are parameters and , , .

The present paper is divided into three sections. In Section 2 fundamental mathematical theory of OHAM is presented. In Section 3 comparisons are made between the results of the proposed method and HAM for Burger’s-Huxley. In Section 4 solution of Burger’s-Fisher equation is presented, and absolute error of approximate solution of proposed method is compared with approximate solution of HAM. In all cases the proposed method yields better results than those of ADM.

2. Fundamental Theory of OHAM

Here we start by describing the basic idea of OHAM. Consider the partial differential equation of the form:

where is a linear operator and is nonlinear operator. is boundary operator, is an unknown function, and and denote spatial and time variables, respectively; is the problem domain and is a known function.

According to the basic idea of OHAM, one can construct the optimal homotopy which satisfies

where is an embedding parameter, is a nonzero auxiliary function for , . Equation (3) is called optimal homotopy equation. Clearly, we have

Clearly, when and , it holds that and , respectively. Thus, as varies from 0 to 1, the solution approaches from to , where is obtained from (3) for :

Next, we choose auxiliary function in the form

Here are constants to be determined later.

To get an approximate solution, we expand in Taylor’s series about in the following manner:

Substituting (10) into (4) and equating the coefficient of like powers of , we obtain Zeroth-order problem, given by (6), the first- and second-order problems are given by (11)-(12), respectively, and the general governing equations for are given by (13):

where is the coefficient of in the expansion of about the embedding parameter :

Here for are set of linear equations with the linear boundary conditions, which can be easily solved.

The convergence of the series in (10) depends upon the auxiliary constants . If it is convergent at , one has:

Substituting (15) into (1) results in the following expression for residual:

If , then will be the exact solution.

For computing the auxiliary constants, , , there are many methods like Galerkin’s Method, Ritz Method, Least Squares Method, and Collocation Method to find the optimal values of , , One can apply the Method of Least Squares as

where is the residual, , and

The constants can also be determined by another method as

at any time , where . The convergence depends upon constants , can be optimally identified and minimized by (18).

3. Application of OHAM

In this section we apply OHAM for the two problems: the first is the Burger’s-Huxley equation (1) and the second is the Burger’s-Fisher equation (2).

3.1. Application of OHAM for Burger’s-Huxley Equation

Let us consider Burger’s-Huxley equation of form (1):

Subject to constant initial condition

The exact solution of (26) with given condition is given by

where

For computational work, we have taken , , , and for various values of and .

Following the basic idea of OHAM presented in preceding section we start with

Zeroth-Order Problem

Its solution is

First-Order Problem

Its solution is

Second-Order Problem

Its solution is

Third-Order Problem

Its solution is

Adding (25), (27), (29), and (31) we obtain

For the calculations of the constants , , and using the collocation method, we have computed that

 ,  ,  .

Putting the values of these constants into (32) the third order approximate solution using OHAM is

Table 1 shows a comparison between OHAM solution and ADM solution for and . For (1) is reduced to the generalized Huxley equation which describes nerve pulse propagation in nerve fibers and wall motion in liquid crystals [22]. Tables 2 and 3 show a comparison between ADM solution and OHAM solution for and respectively. Table 4 shows absolute errors of OHAM solution for larger domain for , , and respectively.

Table 1: Comparison of absolute errors of OHAM and ADM [5] for , , , and .

ADM 

OHAM 

ADM 

OHAM 

ADM 

OHAM 

0.05

0.1

1

Table 2: Comparison of absolute errors obtained by OHAM and ADM [5] for , , , and .

ADM 

OHAM 

ADM 

OHAM 

for
ADM 

OHAM 

0.05

0.1

1

Table 3: Comparison of absolute errors obtained by OHAM and ADM [5] for , , , and .

Error ADM
Error OHAM

0.05

0.1

1

Table 4: Absolute errors of OHAM for , , , and .

, , , and
, , , and
, , , and

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

3.2. Application of OHAM for Burger’s-Fisher Equation

Consider the Burger’s-Fisher equation of form (2):

subject to constant initial condition

with exact solution given by

For computational work, we have taken , , and for various values of and .

Zeroth-Order Problem

Its solution is

First-Order Problem

Its solution is

Second-Order Problem

Its solution is

The third order approximate solution using OHAM is given by

where is obtained in same lines as for first problem.

For the calculations of the constants , , and using the collocation method we have computed that

The third order OHAM solution yields very encouraging results after being compared with Fourth order approximate solution by ADM [5].

Table 5 shows a comparison between OHAM solution and ADM solution for , , and . Table 6 compares between OHAM solution and ADM solution for , , and . Table 7 shows the reliability of OHAM for larger domain.

Table 5: Comparison of absolute errors obtained by OHAM and ADM [5] for , , and .

ADM for 
OHAM 

ADM 

OHAM 

ADM 

OHAM 

0.005

0.001

0.01

Table 6: Comparison of absolute errors obtained by OHAM and ADM [5] for , , and .

ADM for 

OHAM 

ADM

OHAM

ADM 

OHAM

0.0005

0.0001

0.001

Table 7: Absolute errors of OHAM for and .

, , and
, , and

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

4. Conclusion

We successfully applied OHAM for solution of Burger’s-Huxley and Burger’s-Fisher equations. The method is simple in applicability and is fast converging to the exact solution. The results obtained by OHAM are very consistent in comparison with ADM.

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