New Improved Variational Homotopy Perturbation Method for Bratu-Type Problems

Abstract

This research paper deals with the boundary and initial value problems for the Bratu-type model by using the New Improved Variational Homotopy Perturbation Method. The New Method does not require discritization, linearization or any restrictive assumption of any form in providing analytical or approximate solutions to linear and nonlinear equation without the integral related with nonlinear term. Theses virtues make it to be reliable and its efficiency is demonstrated with numerical examples.

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O. Abolarin, "New Improved Variational Homotopy Perturbation Method for Bratu-Type Problems," American Journal of Computational Mathematics, Vol. 3 No. 2, 2013, pp. 110-113. doi: 10.4236/ajcm.2013.32018.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] J. P. Boyd, “An Analytical and Numerical Study of the Two-Dimentional Bratu Equation,” Journal of Scientific Computing, Vol. 1, No. 2, 1986, pp. 183-206. doi:10.1007/BF01061392 [2] J. P. Boyd, “Chebyshev Polynomial Expansions for Simultaneous Approximation of Two Branches of a Function with Application to the One-Dimentional Bratu Equation,” Applied Mathematics and Computation, Vol. 142, 2003, pp. 189-200. doi:10.1016/S0096-3003(02)00345-4 [3] J. Jacobson and K. Schmitt, “The Liouville-Bratu-Gelfand Problem for Radial Operators,” Journal of Differential Equations, Vol. 184, No. 1, 2002, pp. 283-298. doi:10.1006/jdeq.2001.4151 [4] M. I. Syam and A. Hamdan, “An Efficient Method for Solving Bratu Equations,” Applied Mathematics and Computation, Vol. 176, No. 2, 2006, pp. 704-713. doi:10.1016/j.amc.2005.10.021 [5] A. M. Wazwaz, “Adomian Decomposition Method for a Reliable Treatment of the Bratu-Type Equations,” Applied Mathematics and Computation, Vol. 166, No. 3, 2005, pp. 152-633. [6] X. Feng, Y. He and J. Meng, “Application of Homotopy Perturbation Method to Bratu-Type Equations,” Topological Methods in Numerical Analysis, Vol. 32, No. 2, 2008, pp. 243-252. [7] L. Jin, “Application of Variational Iteration Method to the Fifth-Order KdV Equation,” International Journal of Engineering, Contemporary Mathematics and Sciences, Vol. 3, 2008, pp. 213-222. [8] Z. M. Odibat, “Reliable Approaches of Variational Iteration Method for Nonlinear Operators,” Mathematical and Computer Modelling, Vol. 48, No. 1-2, 2008, pp. 222-231. doi:10.1016/j.mcm.2007.09.005 [9] L. Jin, “Application of Modified Variational Iteration Method to the Bratu-Type Problems,” International Journal of Engineering, Contemporary Mathematics and Sciences, Vol. 4, 2010, pp. 153-158. [10] J. H. He, “Homotopy Perturbation Method for Bifurcation of Nonlinear Problems,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 6, No. 2, 2005, pp. 207-208. [11] J. H. He, “Some Asymptotic Methods for Strongly Nonlinear Equations,” International Journal of Modern Physics B, Vol. 20, No. 10, 2006, pp. 1141-1199. doi:10.1142/S0217979206033796 [12] J. H. He, “Variational Iteration Method for Delay Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 2, No. 4, 1997, pp. 235-236. doi:10.1016/S1007-5704(97)90008-3