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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

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 [512]. 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 [1315]. The method has been applied by a number of researchers for solution of ordinary and partial differential equations [1621]. 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.

tab1
Table 1: Comparison of absolute errors of OHAM and ADM [5] for , , , and .
tab2
Table 2: Comparison of absolute errors obtained by OHAM and ADM [5] for , , , and .
tab3
Table 3: Comparison of absolute errors obtained by OHAM and ADM [5] for , , , and .
tab4
Table 4: Absolute errors of OHAM for , , , and .

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.

tab5
Table 5: Comparison of absolute errors obtained by OHAM and ADM [5] for , , and .
tab6
Table 6: Comparison of absolute errors obtained by OHAM and ADM [5] for , , and .
tab7
Table 7: Absolute errors of OHAM for and .

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.

References

  1. C. A. J. Fletcher, “Burgers' equation: a model for all reasons,” in Numerical Solutions of Partial Differential Equations (Parkville, 1981), J. Noye, Ed., pp. 139–225, North-Holland, Amsterdam, The Netherlands, 1982. View at Zentralblatt MATH · View at MathSciNet
  2. D. L. Young, C. M. Fan, S. P. Hu, and S. N. Atluri, “The Eulerian-Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers' equations,” Engineering Analysis with Boundary Elements, vol. 32, no. 5, pp. 395–412, 2008. View at Publisher · View at Google Scholar · View at Scopus
  3. J. M. Burgers, “Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion,” Transactions of the Royal Dutch Academy of Sciences in Amsterdam, vol. 17, no. 2, pp. 1–53, 1939. View at Zentralblatt MATH
  4. J. Satsuma, “Topics in soliton theory and exactly solvable nonlinear equations,” in Proceedings of the Conference on Nonlinear Evolution Equations, Solitons and the Inverse Scattering Transform Held at the Mathematical Research Institute, Oberwolfach, July-August, 1986, M. Ablowitz, B. Fuchssteiner, and M. Kruskal, Eds., p. 342, World Scientific, Singapore, 1987. View at MathSciNet
  5. H. N. A. Ismail, K. Raslan, and A. A. Abd Rabboh, “Adomian decomposition method for Burger's-Huxley and Burger's-Fisher equations,” Applied Mathematics and Computation, vol. 159, no. 1, pp. 291–301, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. X. Y. Wang and Y. K. Lu, “Exact solutions of the extended Burgers-Fisher equation,” Chinese Physics Letters, vol. 7, no. 4, pp. 145–147, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  7. M. Javidi, “Modified pseudospectral method for generalized Burger's-Fisher equation,” International Mathematical Forum, vol. 1, no. 29-32, pp. 1555–1564, 2006. View at Zentralblatt MATH · View at MathSciNet
  8. D. Kaya and S. M. El-Sayed, “A numerical simulation and explicit solutions of the generalized Burgers-Fisher equation,” Applied Mathematics and Computation, vol. 152, no. 2, pp. 403–413, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. P. Chandrasekaran and E. K. Ramasami, “Painleve analysis of a class of nonlinear diffusion equations,” Journal of Applied Mathematics and Stochastic Analysis, vol. 9, no. 1, pp. 77–86, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. H. Chen and H. Zhang, “New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation,” Chaos, Solitons & Fractals, vol. 19, no. 1, pp. 71–76, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. E. S. Fahmy, “Travelling wave solutions for some time-delayed equations through factorizations,” Chaos, Solitons & Fractals, vol. 38, no. 4, pp. 1209–1216, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. L. Jiang, Y.-C. Guo, and S.-J. Xu, “Some new exact solutions to the Burgers-Fisher equation and generalized Burgers-Fisher equation,” Chinese Physics, vol. 16, no. 9, pp. 2514–2522, 2007. View at Publisher · View at Google Scholar · View at Scopus
  13. V. Marinca and N. Herişanu, “Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer,” International Communications in Heat and Mass Transfer, vol. 35, no. 6, pp. 710–715, 2008. View at Publisher · View at Google Scholar · View at Scopus
  14. V. Marinca, N. Herişanu, C. Bota, and B. Marinca, “An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate,” Applied Mathematics Letters, vol. 22, no. 2, pp. 245–251, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. V. Marinca, N. Herişanu, and I. Nemeş, “Optimal homotopy asymptotic method with application to thin film flow,” Central European Journal of Physics, vol. 6, no. 3, pp. 648–653, 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. R. Nawaz, M. N. Khalid, S. Islam, and S. Yasin, “Solution of tenth order boundary value problems using optimal homotopy asymptotic method (OHAM),” Canadian Journal on Computing in Mathematics, Natural Sciences, Engineering & Medicin, vol. 1, no. 2, pp. 37–54, 2010.
  17. S. Iqbal, M. Idrees, A. M. Siddiqui, and A. R. Ansari, “Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 2898–2909, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. S. Haq, M. Idrees, and S. Isalam, “Application of optimal homotopy asymptotic Method to eighth order boundary value problems,” Journal of Applied Mathematics and Computing, vol. 2, no. 4, pp. 38–47, 2008.
  19. M. Idrees, S. Haq, and S. Islam, “Application of optimal homotopy asymptotic method to fourth order boundary value problems,” World Applied Sciences Journal, vol. 9, no. 2, pp. 131–137, 2010.
  20. M. Idrees, S. Islam, S. Haq, and S. Islam, “Application of the optimal homotopy asymptotic Method to squeezing flow,” Computers and Mathematics with Applications, vol. 59, no. 12, pp. 3858–3866, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. M. Idrees, S. Haq, and S. Islam, “Application of optimal homotopy asymptotic method to special sixth order boundary value problems,” World Applied Sciences Journal, vol. 9, no. 2, pp. 138–143, 2010.
  22. X. Y. Wang, Z. S. Zhu, and Y. K. Lu, “Solitary wave solutions of the generalised Burgers-Huxley equation,” Journal of Physics A, vol. 23, no. 3, pp. 271–274, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus