A simple solution of the Bratu problem

A brief survey of the properties and different treatments of the one-dimensional (1D) and (2D) Bratu problems is presented. Different iterative treatments of the resulting nonlinear system of equations are discussed. The finite-difference treatment of the problem is considered. Nonstandard finite-difference methods with a simple sinusoidal starting function having an appropriate amplitude are recommended. Bounds on the amplitude for yielding both lower and upper solutions are given.

[1]  Xiaomin Wang,et al.  A wavelet method for solving a class of nonlinear boundary value problems , 2013, Commun. Nonlinear Sci. Numer. Simul..

[2]  Cornelis W. Oosterlee,et al.  KRYLOV SUBSPACE ACCELERATION FOR NONLINEAR MULTIGRID SCHEMES , 1997 .

[3]  Kang Tai,et al.  Radial point interpolation collocation method (RPICM) for the solution of nonlinear poisson problems , 2005 .

[4]  B. Gidas,et al.  Symmetry and related properties via the maximum principle , 1979 .

[5]  H. D. Mittelmann.  Multilevel continuation techniques for nonlinear boundary value problems with parameter dependence , 1986 .

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

[7]  Ruhaila Kasmani.  A comparison of numerical methods for solving the bratu and bratu-type problems , 2005 .

[8]  Gisela TIMMERMANNAbstract.  Path following for Large Nonlinear Equations by Implicit Block Elimination Based on Recursive Projections Path following for Large Nonlinear Equations by Implicit Block Elimination Based on Recursive Projections , 1996 .

[9]  J H He.  Variational approach to the Bratu's problem , 2008 .

[10]  H. Keller,et al.  Continuation-Conjugate Gradient Methods for the Least Squares Solution of Nonlinear Boundary Value Problems , 1985 .

[11]  Yousef Saad,et al.  Hybrid Krylov Methods for Nonlinear Systems of Equations , 1990, SIAM J. Sci. Comput..

[12]  Paul Houston,et al.  Adaptivity and a posteriori error control for bifurcation problems I: the Bratu problem , 2010 .

[13]  Y. S. Choi.  A singular boundary value problem arising from near-ignition analysis of flame structure , 1991 .

[14]  John P. Boyd,et al.  One-point pseudospectral collocation for the one-dimensional Bratu equation , 2011, Appl. Math. Comput..

[15]  Martin Hermann,et al.  RWPKV: A software package for continuation and bifurcation problems in two-point boundary value problems , 1992 .

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

[17]  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..

[18]  H. N. Hassan,et al.  An efficient analytic approach for solving two‐point nonlinear boundary value problems by homotopy analysis method , 2011 .

[19]  Bryan D. Davidson.  Large-Scale Continuation and Numerical Bifurcation for Partial Differential Equations , 1997 .

[20]  Peng Jiang,et al.  Performance Analysis of Block Jacobi Preconditioning Technique Based on Block Broyden Method , 2006, International Conference on Computational Science.

[21]  Beong In Yun.  Solving nonlinear equations by a new derivative free iterative method , 2011, Appl. Math. Comput..

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

[23]  J. Boyd.  An analytical and numerical study of the two-dimensional Bratu equation , 1986 .

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

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

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

[27]  M. Zarebnia,et al.  Parametric Spline Method for Solving Bratu's Problem , 2012 .

[28]  Muhammed I. Syam,et al.  The modified Broyden-variational method for solving nonlinear elliptic differential equations , 2007 .

[29]  D. Calvetti,et al.  Iterative methods for large continuation problems , 2000 .

[30]  Raja Muhammad Asif Zahoor,et al.  Numerical treatment for solving one-dimensional Bratu problem using neural networks , 2012, Neural Computing and Applications.

[31]  Jalil Rashidinia,et al.  Sinc-Galerkin method for numerical solution of the Bratu’s problems , 2012, Numerical Algorithms.

[32]  R. B. Simpson.  A Method for the Numerical Determination of Bifurcation States of Nonlinear Systems of Equations , 1975 .

[33]  Youdong Lin,et al.  Enclosing all solutions of two-point boundary value problems for ODEs , 2008, Comput. Chem. Eng..

[34]  Ahmet Yildirim,et al.  Comparison between Adomian's method and He's homotopy perturbation method , 2008, Comput. Math. Appl..

[35]  D. O’Regan.  Existence Theory for Nonlinear Ordinary Differential Equations , 1997 .

[36]  S. A. Odejide,et al.  A note on two dimensional Bratu problem , 2006 .

[37]  Eugene L. Allgower,et al.  Continuation and path following , 1993, Acta Numerica.

[38]  Wolfgang Hackbusch,et al.  Comparison of different multi-grid variants for nonlinear equations , 1992 .

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

[40]  Tony F. Chan,et al.  Arc-Length Continuation and Multigrid Techniques for Nonlinear Elliptic Eigenvalue Problems , 1982 .

[41]  Basem S. Attili.  Superconvergence and the computation of generalized turning points for B.V.P's , 2006, Appl. Math. Comput..

[42]  A. Mohsen,et al.  New smoother to enhance multigrid-based methods for Bratu problem , 2008, Appl. Math. Comput..

[43]  Robert E. Kalaba,et al.  Some numerical experiments using Newton's method for nonlinear parabolic and elliptic boundary-value problems , 1961, Commun. ACM.

[44]  Ron Buckmire,et al.  Applications of Mickens Finite Differences to Several Related Boundary Value Problems , 2005 .

[45]  Ravi P. Agarwal,et al.  Positive solutions for nonlinear singular boundary value problems , 1999 .

[46]  Basem S. Attili,et al.  Efficient finite differences for parameter-dependent singularities in two-point boundary value problems , 2001 .

[47]  Fu-Hsiang Wong,et al.  Existence of positive solutions of singular boundary value problems , 1993 .

[48]  Yimin Wei,et al.  On integral representation of the generalized inverse AT, S(2) , 2003, Appl. Math. Comput..

[49]  Michael R. Osborne,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.

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

[51]  Muhammad Aslam Noor,et al.  Variational Iteration Method for Solving Initial and Boundary Value Problems of Bratu-type , 2008 .

[52]  Lakshmanan Rajendran,et al.  Analytical Solutions of Some Two-Point Non-Linear Elliptic Boundary Value Problems , 2012 .

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

[54]  Peng Jiang,et al.  Performance analysis of preconditioners based on Broyden method , 2006, Appl. Math. Comput..

[55]  G. Bratu.  Sur les équations intégrales non linéaires , 1913 .

[56]  Panagiotis G. Asteris,et al.  Interpolation Based Numerical Procedure for Solving Two-Point Nonlinear Boundary Value Problems , 2004 .

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

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

[59]  Alexander I. Fedoseyev,et al.  Continuation for nonlinear Elliptic Partial differential equations discretized by the multiquadric Method , 2000, Int. J. Bifurc. Chaos.

[60]  Saeid Abbasbandy,et al.  The Lie-group shooting method for solving the Bratu equation , 2011 .

[61]  Ji-Huan He.  SOME ASYMPTOTIC METHODS FOR STRONGLY NONLINEAR EQUATIONS , 2006 .

[62]  Muhammed I. Syam,et al.  Weighted residual method for obtaining positive solutions of two-point nonlinear boundary value problems , 2006, Appl. Math. Comput..

[63]  G. Moore,et al.  The Calculation of Turning Points of Nonlinear Equations , 1980 .

[64]  Qian Guo,et al.  Thermo-electro-hydrodynamic model for electrospinning process , 2004 .

[65]  Zhao Zeng-qin.  Existence of Multiple Positive Solutions of Boundary Value Problems for Systems of Nonlinear Differential Equations , 2009 .

[66]  Hans D Mittlemann.  A pseudo-arclength continuation method for nonlinear eigenvalue problems , 1986 .

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

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

[69]  F. Kikuchi.  Finite element approximations to bifurcation problems of turning point type , 1979 .

[70]  K. Schmitt,et al.  The Liouville–Bratu–Gelfand Problem for Radial Operators , 2002 .

[71]  Rakesh K. Kapania.  A pseudo-spectral solution of 2-parameter Bratu's equation , 1990 .

[72]  Kazimierz Wanelik,et al.  On the iterative solutions of some nonlinear eigenvalue problems , 1989 .

[73]  Yong-Hoon Lee,et al.  Existence of multiple positive solutions of singular boundary value problems , 1997 .

[74]  T. Chan,et al.  PLTMGC: A Multigrid Continuation Program for Parameterized Nonlinear Elliptic Systems , 1986 .