The Lie-group shooting method for solving the Bratu equation

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  ∈ [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.

[1]  Chein-Shan Liu,et al.  One-step GPS for the estimation of temperature-dependent thermal conductivity , 2006 .

[2]  Chein-Shan Liu.  A Two-Stage LGSM for Three-Point BVPs of Second-Order ODEs , 2008 .

[3]  Muhammed I. Syam,et al.  An efficient method for solving Bratu equations , 2006, Appl. Math. Comput..

[4]  Shijun Liao,et al.  An analytic approach to solve multiple solutions of a strongly nonlinear problem , 2005, Appl. Math. Comput..

[5]  Aregbesola Y.A.S.,et al.  NUMERICAL SOLUTION OF BRATU PROBLEM USING THE METHOD OF WEIGHTED RESIDUAL , 2003 .

[6]  Chein-Shan Liu,et al.  An Efficient Backward Group Preserving Scheme for the Backward in Time Burgers Equation , 2006 .

[7]  D. Frank-Kamenetskii,et al.  Diffusion and heat exchange in chemical kinetics , 1955 .

[8]  I. H. Abdel-Halim Hassan,et al.  APPLYING DIFFERENTIAL TRANSFORMATION METHOD TO THE ONE-DIMENSIONAL PLANAR BRATU PROBLEM , 2007 .

[9]  Mehmet Pakdemirli,et al.  New perturbation-iteration solutions for Bratu-type equations , 2010, Comput. Math. Appl..

[10]  Abdul-Majid Wazwaz,et al.  Adomian decomposition method for a reliable treatment of the Bratu-type equations , 2005, Appl. Math. Comput..

[11]  Chein-Shan Liu,et al.  An LGEM to Identify Time-Dependent Heat Conductivity Function by an Extra Measurement of Temperature Gradient , 2008 .

[12]  Chein-Shan Liu.  An LGSM to identify nonhomogeneous heat conductivity functions by an extra measurement of temperature , 2008 .

[13]  A. Serghini Mounim,et al.  From the fitting techniques to accurate schemes for the Liouville‐Bratu‐Gelfand problem , 2006 .

[14]  John P. Boyd,et al.  Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation , 2003, Appl. Math. Comput..

[15]  Chein-Shan Liu,et al.  Efficient Shooting Methods for the Second-Order Ordinary Differential Equations , 2006 .

[16]  Chih-Wen Chang,et al.  A new shooting method for quasi-boundary regularization of backward heat conduction problems , 2007 .

[17]  S. Khuri,et al.  An Algorithm for Solving Boundary Value Problems , 2000 .

[18]  B. Batiha,et al.  Numerical Solution of Bratu-type Equations by the Variational Iteration Method , 2010 .

[19]  Ron Buckmire,et al.  Application of a Mickens finite‐difference scheme to the cylindrical Bratu‐Gelfand problem , 2004 .

[20]  Chein-Shan Liu,et al.  The Lie-Group Shooting Method for Quasi-Boundary Regularization of Backward Heat Conduction Problems , 2007 .

[21]  Y. Tan,et al.  A General Approach to Obtain Series Solutions of Nonlinear Differential Equations , 2007 .

[22]  Chih-Wen Chang,et al.  A new algorithm for direct and backward problems of heat conduction equation , 2010 .

[23]  Chein-Shan Liu.  A two-stage Lie-group shooting method (TSLGSM) to identify time-dependent thermal diffusivity , 2010 .

[24]  Chein-Shan Liu.  Cone of non-linear dynamical system and group preserving schemes , 2001 .

[25]  Hikmet Caglar,et al.  B-spline method for solving Bratu's problem , 2010, Int. J. Comput. Math..

[26]  Jeff S. Mcgough,et al.  Numerical continuation and the Gelfand problem , 1998 .

[27]  Suheil A. Khuri,et al.  A new approach to Bratu's problem , 2004, Appl. Math. Comput..

[28]  Chein-Shan Liu,et al.  The Lie-Group Shooting Method for Nonlinear Two-Point Boundary Value Problems Exhibiting Multiple Solutions , 2006 .

[29]  Chein-Shan Liu,et al.  The Lie-Group Shooting Method for Singularly Perturbed Two-Point Boundary Value Problems , 2006 .

[30]  Chein-Shan Liu,et al.  Solving an Inverse Sturm-Liouville Problem by a Lie-Group Method , 2008 .

[31]  Hung-Chang Lee,et al.  A modified group-preserving scheme for solving the initial value problems of stiff ordinary differential equations , 2002, Appl. Math. Comput..

[32]  Hikmet Caglar,et al.  Dynamics of the solution of Bratu’s equation , 2009 .

[33]  Weichung Yeih,et al.  A Three-Point BVP of Time-Dependent Inverse Heat Source Problems and Solving by a TSLGSM , 2009 .

[34]  Chein-Shan Liu.  A two-stage LGSM to identify time-dependent heat source through an internal measurement of temperature , 2009 .

[35]  R. Jalilian,et al.  Non-polynomial spline method for solving Bratu's problem , 2010, Comput. Phys. Commun..