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This study proposes an accurate approximation to the solution of second-order nonlinear two-point boundary value problems, including the well-known Bratu problem, using an iterative technique based on Green's function. The approach relies on constructing an equivalent integral representation of the problem incorporating Green's function. The proposed methodology provides a reliable approximate solution and takes just a few iterations to achieve good accuracy. The mathematical formulation is further supported by discussing in detail the convergence analysis of this approach. Different numerical examples are used to check the robustness and effectiveness of the scheme. The numerical testing for nonlinear problems with nonlinear boundary conditions demonstrates that the proposed method outperforms other existing methods, including the finite element method, the finite volume method, the finite difference method, the B-spline method, and the Adomain's decomposition method. • A new approach is proposed for solving nonlinear two-point boundary value problems. • The proposed approach has several advantages over existing methods. • Analytical and numerical solutions can easily be computed via the proposed methodology. • Some numerical examples are considered to verify the effectiveness of the approach. [ABSTRACT FROM AUTHOR]

In this article, the one-dimensional non-linear Bratu’s boundary value problem is solved via a novel approach that combines Green’s function and fixed point iterative schemes, such as Picard’s and Krasnoselskii–Mann’s. The convergence of the introduced iterative algorithm is proved using the contraction principle. The method is supported by considering a number of numerical examples that correspond to different cases of eigenvalues. The procedure underlying the strategy reduces calculations and provides highly accurate results in comparison with the exact solution and/or numerical solutions provided in the literature. The current method overcomes the difficulty of treating the problem for eigenvalues near and at the critical value, such as λ = 3 and λ = 3.51 , and handles them reliably and very efficiently. [ABSTRACT FROM AUTHOR]

Abstract: Smooth approximate solutions of the one-dimensional Bratu''s problem are developed by using non-polynomial spline function. The inverse of some tridiagonal matrices are obtained which are required in proving the convergence analysis of the presented method. By considering the maximum absolute errors in the solution at grid points and tabulated in tables for different choices of step size we conclude that our presented method produces more accurate results in comparison with those obtained by B-spline, Laplace and decomposition methods. [Copyright &y& Elsevier]