Author's personal copy The Lie-group shooting method for solving the Bratu equation S. Abbasbandy a,⇑ , M.S. Hashemi a,b , Chein-Shan Liu c a Department of Mathematics, Imam Khomeini International University, Ghazvin 34149, Iran b Bonab University, Bonab 55517, Iran c Department of Civil Engineering, National Taiwan University, Taipei, Taiwan article info Article history: Received 21 November 2010 Accepted 27 March 2011 Available online 2 April 2011 Keywords: Lie-group shooting method Group preserving scheme Bratu problem abstract For the Bratu problem, we transform it into a non-linear second order boundary value problem, and then solve it by the Lie-group shooting method (LGSM). LGSM allows us to search a missing initial slope and moreover, the initial slope can be expressed as a function of r 2 [0, 1], where the best r is determined by matching the right-end boundary condition. The calculated results as compared with those calculated by other methods, illuminate the efficiency and precision of Lie-group shooting method (LGSM) for this problem. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Simplification of the solid fuel ignition model in thermal combustion theory yields an elliptic nonlinear partial differential equation, namely the Bratu problem. Also it is used in a large variety of applications such as the model of thermal reaction process, the Chandrasekhar model of the expansion of the Universe, chemical reaction theory, nano technology and radiative heat transfer [1–7]. The Bratu problem is a nonlinear boundary-value problem (BVP) that is used as a benchmark problem to test the accuracy of many numerical methods. Consider the Bratu problem given by the following boundary-value problem: u 00 ðxÞþ ke uðxÞ ¼ 0; uð0Þ¼ uð1Þ¼ 0; 0 6 x 6 1; ð1:1Þ where k > 0. The Bratu problem has an analytical solution given in the following form: uðxÞ¼2 ln cosh ðx  1 2 Þ h 2   coshð h 4 Þ " # ; ð1:2Þ where h is the solution of h ¼ ffiffiffiffiffi 2k p coshð h 4 Þ. The Bratu problem has zero, one and two solutions when k > k c , k = k c and k < k c respectively, where the critical value k c satisfies the equation 1 ¼ 1 4 ffiffiffiffiffiffiffi 2k c p sinh h c 4  : It was evaluated in [8,9] that the critical value k c is given by k c = 3.513830719. Many analytical and numerical methods have been applied to solve Eq. (1.1). For example Wazwaz [1] has been applied the Adomian decomposition method to determine exact solutions of Bratu-type equations. Aregbesola used the method of 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.03.033 ⇑ Corresponding author. Tel.: +98 912 1305326; fax: +98 281 3780040. E-mail address: abbasbandy@yahoo.com (S. Abbasbandy). Commun Nonlinear Sci Numer Simulat 16 (2011) 4238–4249 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns