Communications in Numerical Analysis

Volume 2013 (2013), Article ID cna-00139, 14 Pages

doi: 10.5899/2013/cna-00139


Research Article


Analytic approximate solution for the Bratu's problem by optimal homotopy analysis method


Hany N. Hassan1 *, Mourad S. Semary1


1Department of Basic Science, Faculty of Engineering at Benha, Benha University, Benha 13512, Egypt


* Corresponding author. Email address: h_nasr77@yahoo.com; Tel: +201225839389


Received: 31 March 2012; Accepted: 04 January 2013


Copyright © 2013 Hany N. Hassan and Mourad S. Semary. 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

In This paper, we present analytic approximate solutions for Bratu's problem with high accuracy for different values of \lambda. We solve this nonlinear problem without any approximations or transformation in the problem and we successfully obtain the two branches of solutions for different values \lambda using homotopy analysis method. A new efficient approach is proposed to obtain the optimal value of convergence controller parameter \hbar to guarantee the convergence of the obtained series solution.


Keywords: Optimal homotopy analysis method; Bratu's problem; Series solutions; Two boundary value problem; Multiple solutions.

References

  1. R. Buckmire, Investigations of nonstandard Mickens-type finite-difference schemes for singular boundary value problems in cylindrical or spherical coordinates, Numerical Methods for partial Differential equations, 19 (3) (2003) 380-398.


  2. M. I. Syam, A. Hamdan, An efficient method for solving Bratu equations, Applied Mathematics and Computation, 176 (2) (2006) 704-713.


  3. A. M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Applied Mathematics and Computation, 166 (3) (2005) 652-663.


  4. S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D thesis, Shanghai Jiao Tong University, (1992).


  5. S. J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman & Hall/CRC Press: Boca Raton, (2003).


  6. S. J. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Communications in Nonlinear Science and Numerical Simulation, 14 (4) (2009) 983-997.


  7. H. N. Hassan, M. A. El-Tawil, An efficient analytic approach for solving two-point nonlinear boundary value problems by homotopy analysis method, Mathematical Methods in the Applied Sciences, 34 (8) (2011) 977-989.


  8. R. A. V. Gorder, Gaussian waves in the Fitzhugh-Nagumo equation demonstrate one role of the auxiliary function H (x, t) in the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 17 (3) (2012) 1233-1240.


  9. J. Biazar, B. Ghanbari, HAM solution of some initial value problems arising in heat radiation equations, Journal of King Saud University-Science, 24 (2) (2012) 161-165.


  10. B. Raftari, K. Vajravelu, Homotopy Analysis Method for MHD Viscoelastic Fluid Flow and Heat Transfer in a Channel with a Stretching Wall, Communications in Nonlinear Science and Numerical Simulation, 17 (11) (2012) 4149-4162.


  11. H. N. Hassan, M. A. ElTawil, Solving cubic and coupled nonlinear Schrodinger equations using the homotopy analysis method, International Journal of Applied Mathematics and Mechanics, 7 (8) (2011) 41-64.


  12. H. N. Hassan, M. A. El-Tawil, Series solution for continuous population models for single and interacting species by the homotopy analysis method, Communications in Numerical and Analysis, Volume 2012 (2012) 1-21.


  13. H. N. Hassan, M. A. El-Tawil, A new technique of using homotopy analysis method for solving high-order nonlinear differential equations, Mathematical Methods in the Applied Sciences, 34 (6) (2011) 728-742.


  14. H. N. Hassan, M. A. El-Tawil, A new technique of using homotopy analysis method for second order nonlinear differential equations, Applied Mathematics and Computation, 219 (2) (2012) 708-728.


  15. S. Abbasbandy, A. Shirzadi, A new application of the homotopy analysis method: Solving the Sturm-Liouville problems, Communications in Nonlinear Science and Numerical Simulation, 16 (1) (2011) 112-126.


  16. Wu. Yongyan, K. F. Cheung, Homotopy solution for nonlinear differential equations in wave propagation problems, Wave Motion, 46 (1) (2009) 1-14.


  17. M. Inc, On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method, Physics Letters A, 365 (5-6) (2007) 412-415.


  18. Z. Wang, L. Zou, H. Zhang, Applying homotopy analysis method for solving differential-difference equation, Physics Letters A, 369 (1-2) (2007) 77-84.


  19. H. Jafari, S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 14 (5) (2009) 1962-1969.


  20. S. Abbasbandy, E. Shivanian, Prediction of multiplicity of solutions of nonlinear boundary value problems: novel application of homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 15 (12) (2010) 3830-3846.


  21. S. J. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation, 15 (8) (2010) 2003-2016.